ESSAYS IN REGIME SWITCHING POLICY AND ADAPTIVE LEARNING IN
DYNAMIC STOCHASTIC GENERAL EQUILIBRIUM
by
NIGEL J. MCCLUNG
A DISSERTATION
Presented to the Department of Economics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2018
DISSERTATION APPROVAL PAGE
Student: Nigel J. McClung
Title: Essays in Regime Switching Policy and Adaptive Learning in Dynamic Stochastic
General Equilibrium
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Economics by:
George Evans Co-Chair
Bruce McGough Co-Chair
Jeremy Piger Core Member
Wenbo Wu Institutional Representative
and
Sara Hodges Interim Vice Provost & Dean of the Graduate
School
Original approval signatures are on file with the University of Oregon Graduate School.
Degree awarded June 2018.
ii
c© 2018 Nigel J. McClung
iii
DISSERTATION ABSTRACT
Nigel J. McClung
Doctor of Philosophy
Department of Economics
June 2018
Title: Essays in Regime Switching Policy and Adaptive Learning in Dynamic Stochastic
General Equilibrium
This dissertation studies monetary-fiscal policy interactions and adaptive
learning applications in regime-switching DSGE models. A common thread through
my research is understanding how policymakers may be affected by the interaction of
policy regime change and agents’ beliefs about past, current or future policy in general
equilibrium. The work I present in this dissertation shows that conventional and
unconventional policy outcomes, as well as the existence, uniqueness and expectational
stability of rational expectations solutions, depend heavily on the expectational effects
of time-varying policy. These findings suggest that uncertainty over future fiscal policy
may curb the effectiveness of monetary policy, or otherwise constrain the actions of
central bankers. In carrying out this research agenda, my work also examines the
relationship between determinacy and expectational stability in a general class of
Markov-switching DSGE models.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Nigel J. McClung
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
Gonzaga University, Spokane, WA
DEGREES AWARDED:
Doctor of Philosophy, Economics, 2018, University of Oregon
Master of Science, Economics, 2014, University of Oregon
Bachelor of Arts, Economics and History, 2012, Gonzaga University
AREAS OF SPECIAL INTEREST:
Macroeconomics, Monetary Economics, Learning
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow/Graduate Employee, University of Oregon, 2013-2018
GRANTS, AWARDS AND HONORS:
Kleinsorge Research Fellowship, University of Oregon, 2017
Department of Economics GTF Teaching Award, University of Oregon, 2016
Best Econometrics Performance Award, University of Oregon, 2014
v
ACKNOWLEDGEMENTS
I thank Professors George Evans, Bruce McGough and Jeremy Piger for their 
assistance in the preparation of this manuscript, and for their guidance over the years. 
I also thank Professors Shankha Chakraborty and David Evans for helpful comments 
and advice. This research was supported in part by a Kleinsorge Research Fellowship 
at the University of Oregon Department of Economics.
vi
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
II. E-STABILITY VIS-A-VIS DETERMINACY IN MS-DSGE MODELS . . . . 4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Brief Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 6
MS-DSGE Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Determinacy and E-Stability . . . . . . . . . . . . . . . . . . . . . . . 36
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
III. THE POWER OF FORWARD GUIDANCE AND THE FISCAL
THEORY OF THE PRICE LEVEL . . . . . . . . . . . . . . . . . . . . . . . 47
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Fixed Coefficient Exercises . . . . . . . . . . . . . . . . . . . . . . . . 56
Markov-Switching Forward Guidance Experiment . . . . . . . . . . . . 62
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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Chapter Page
IV. MATURITY, DETERMINACY, AND E-STABILITY . . . . . . . . . . . . . 68
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Maturity and Determinacy . . . . . . . . . . . . . . . . . . . . . . . . 70
Determinacy and E-stability . . . . . . . . . . . . . . . . . . . . . . . 80
Infinite-Horizon Learning . . . . . . . . . . . . . . . . . . . . . . . . . 82
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
V. PERFORMANCE OF SIMPLE INTEREST RATE RULES
SUBJECT TO FISCAL POLICY . . . . . . . . . . . . . . . . . . . . . . . . 85
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Model and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Short-term Debt and Rational Expectations . . . . . . . . . . . . . . . 102
Adaptive Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Long-term Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
VI. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Concluding Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Extensions and Future Work . . . . . . . . . . . . . . . . . . . . . . . 134
APPENDICES
A. DERIVATION OF FIRST E-STABILITY CONDITION . . . . . . . . . . 136
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Chapter Page
B. NEW KEYNESIAN MODEL DERIVATION . . . . . . . . . . . . . . . . . 140
C. FIXED REGIME FORWARD GUIDANCE EXPERIMENT . . . . . . . . 149
D. REGIME SWITCHING FORWARD GUIDANCE EXPERIMENT . . . . . 151
E. TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
F. LAGGED INFORMATION ONE-STEP-AHEAD E-STABILITY . . . . . . 155
G. INFINITE HORIZON STABILITY CONDITIONS DERIVATION . . . . . 161
H. AN AND SCHORFHEIDE (2007) MODEL DERIVATION . . . . . . . . . 171
I. PRIOR AND POSTERIOR DISTRIBUTIONS . . . . . . . . . . . . . . . 176
J. MISCELLANEOUS FIGURES . . . . . . . . . . . . . . . . . . . . . . . . 177
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
ix
LIST OF FIGURES
Figure Page
1. Impulse Responses to One-Unit Shock . . . . . . . . . . . . . . . . . . . . . . 58
2. The 12-quarter Forward Guidance Horizon Experiment. . . . . . . . . . . . . 62
3. The 12-quarter Forward Guidance Horizon Experiment (Regime F). . . . . . 63
4. Regime Switching Forward Guidance Experiment 1 . . . . . . . . . . . . . . 64
5. Regime Switching Forward Guidance Experiment 2 . . . . . . . . . . . . . . 65
6. Regime Switching Forward Guidance Experiment 3 . . . . . . . . . . . . . . 66
7. Determinacy and Maturity: Fisher Model . . . . . . . . . . . . . . . . . . . . 77
8. Determinacy and Maturity: New Keynesian Model . . . . . . . . . . . . . . . 79
9. Globally Passive Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10. Globally Active Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11. Globally Switching Policy (Strong) . . . . . . . . . . . . . . . . . . . . . . . . 101
12. Globally Switching Policy (Weak) . . . . . . . . . . . . . . . . . . . . . . . . 101
13. Coefficient Estimate Errors and Observed State Learning . . . . . . . . . . . 116
14. Estimating the Policy State (Lagged Information) . . . . . . . . . . . . . . . 126
15. Estimating the Policy State (Contemporaneous Information) . . . . . . . . . 127
16. Coefficient Estimate Errors in Hidden Markov Model Learning . . . . . . . . 127
17. Lengthening Maturity Expands Policy Menus . . . . . . . . . . . . . . . . . . 130
18. Shortening Maturity Expands Policy Menus . . . . . . . . . . . . . . . . . . . 130
19. Determinacy and E-Stability (Parameterization 1) . . . . . . . . . . . . . . . 177
20. Determinacy and E-Stability (Parameterization 2) . . . . . . . . . . . . . . . 178
21. Determinacy and E-Stability (Parameterization 2) . . . . . . . . . . . . . . . 179
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LIST OF TABLES
Table Page
1. Fixed Coefficent Model Determinacy Conditions . . . . . . . . . . . . . . . . 56
2. Leeper (1991) Determinacy Conditions . . . . . . . . . . . . . . . . . . . . . 95
3. Overall Stance of Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4. Optimal Inflation Reaction Coefficients under RE when ρ = 0 . . . . . . . . . 110
5. Optimal Coefficients under Adaptive Learning . . . . . . . . . . . . . . . . . 117
6. Fixed Coefficient Model Parameterization . . . . . . . . . . . . . . . . . . . . 154
7. Regime-Switching Model Parameterizations . . . . . . . . . . . . . . . . . . . 154
8. Prior and Posterior Distribution Statistics . . . . . . . . . . . . . . . . . . . . 176
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CHAPTER I
INTRODUCTION
This dissertation studies monetary-fiscal policy interactions and adaptive
learning applications in regime-switching DSGE models. A common thread through
my research is understanding how policymakers may be affected by the interaction of
policy regime change and agents’ beliefs about past, current or future policy in general
equilibrium. The work I present in this dissertation shows that conventional and
unconventional policy outcomes, as well as the existence, uniqueness and expectational
stability of rational expectations solutions, depend heavily on the expectational effects
of time-varying policy. These findings suggest that uncertainty over future fiscal policy
may curb the effectiveness of monetary policy, or otherwise constrain the actions of
central bankers. In carrying out this research agenda, my work also examines the
relationship between determinacy and expectational stability in a general class of
Markov-switching DSGE models.
Chapter 2 of my dissertation generalizes McCallum (2007) and is the first to
address the relationship between determinacy and E-stability in Markov-switching
Dynamic Stochastic General Equilibrium (MS-DSGE) models with lagged endogenous
variables. I prove that the sufficient conditions for determinacy in Cho (2016) imply
the E-stability of the forward solution in MS-DSGE models with lagged endogenous
variables when agents condition their expectations of future endogenous variables on
current endogenous and exogenous variables. The class of models studied in this paper
is very general, and nests a wide array of models that are frequently studied in modern
macroeconomics.
1
In Chapter 3, I study the impact of expansionary forward guidance in a simple
New Keynesian model with recurring or permanent active fiscal policy. This work
addresses and offers a potential solution to the simple New Keynesian model’s
prediction that expansionary forward guidance can generate an implausibly large
stimulus. I find that the introduction of permanent or recurring active fiscal policy
dampens the response of output and inflation to forward guidance in the New
Keynesian model. Moreover, the presence of regime-switching policy introduces
expectational effects that cause forward guidance to be less stimulative in our regime-
switching model’s active money, passive fiscal policy regime. Finally, the introduction
of long-term debt affects the magnitude of the stimulus resulting from forward
guidance in models with active fiscal policy.
In Chapter 4, I explore determinacy and E-stability in a New Keynesian model
with switching fiscal and monetary policy. Here I present three categories of results.
First, the maturity structure of government debt matters for determinacy and
the existence of stable equilibria in our switching model, which is not true in the
analogous fixed coefficient model. I use two numerical solution techniques to show that
maturity affects both the multiplicity of stable solutions, and the existence of sunspot
equilibria. Second, determinacy generally implies E-stability when agents do not
observe contemporaneous observable variables, but not for certain arguably unrealistic
regions of the model parameter space. Third, this chapter presents conditions for
stability under infinite-horizon learning in Markov-switching DSGE models and
compares stability under infinite horizon and one-step-ahead learning. To the best of
my knowledge, this is the first paper to derive these stability conditions in a model
with switching coefficients.
2
Finally, Chapter 5 examines the performance and robustness of simple monetary
policy rules in models with learning agents subject to: (1) permanent or occasionally
active fiscal policy; and/or (2) the presence of long-term government debt. My analysis
indicates that the “global” response of the fiscal policymaker to debt determines the
optimal monetary policy response. When fiscal policy is globally passive or globally
active the optimal monetary policy rule typically features time-invariant coefficients
with high inflation reaction coefficients in globally passive models and interest rate
pegs in globally active models. In cases where fiscal policy features balanced or strong
switching between active and fiscal policy stances, the optimal monetary policy rule
features switching coefficients. These results extend to models with adaptive learning,
including a hidden Markov model of learning never seen before in the literature.
3
CHAPTER II
E-STABILITY VIS-A-VIS DETERMINACY IN MS-DSGE MODELS
Introduction
Under rational expectations, a given macroeconomic model may have multiple
equilibria. When studying models that admit a multiplicity of rational expectations
equilibria, researchers must confront the issue of equilibrium selection: which, if any, of
the equilibria are economically reasonable? To that end, the notion of “determinacy”
is used extensively as a selection criterion. When a model is determinate only
one equilibrium exists, and this eliminates the need to choose between equilibria.
Alternative criteria advance robustness to bounded rationality as a means of selecting
plausible equilibria. In the adaptive learning literature the “learnability” or E-
stability of a specific equilibrium is viewed as a criterion for the selection of a rational
expectations equilibrium. Under adaptive learning, rational expectations are replaced
with a forecasting model that assumes the same functional form as the equilibrium
law of motion, and agents are assumed to update the parameters of their model
each period. If, in the limit, agents’ parameter estimates converge to the parameters
consistent with a given rational expectations equilibrium, then this equilibrium is said
to be “E-stable” or “stable under learning.”
Despite its popularity, determinacy has some weaknesses as an equilibrium
selection criterion. First, determinacy only guarantees the existence of a unique stable
rational expectations solution, and it is well known that determinate models may
have explosive equilibria (see Cochrane (2007)). Second, determinacy does not explain
how agents coordinate on a unique rational expectations equilibrium. The E-stability
4
criterion mitigates these two problems as follows. First, the determinate equilibrium,
but not explosive solutions in determinate models, tend to be stable under learning
as shown by McCallum (2009). Second, learning explicitly examines how and when
boundedly agents can learn to become rational forecasters and therefore coordinate on
a rational expectations equilibrium. Because an E-stable determinate equilibrium is
robust to these weaknesses of determinacy as a selection criterion, it is important to
study and characterize the relationship between determinacy and E-stability.
The relationship between determinacy and E-stability has been extensively
explored in general classes of linear models with different assumptions about agents’
information sets and the horizons over which agents form expectations.1 In models
that feature Markov-switching parameters, relatively little is known about determinacy
and E-stability, and for the following reasons: (1) tractable necessary conditions for
determinacy in Markov-switching models are not known; (2) variation in the set of
current and past Markov states that agents use when forming expectations generates
distinct classes of equilibria. As shown in Branch, Davig, and McGough (2013), this
second limitation makes it hard to establish whether an equilibrium is unique to the
model, or merely to its class of equilibria. Despite these limiting factors, some research
has successfully isolated general relationships between determinacy and E-stability in
purely-forward looking Markov-switching rational expectations (MS-DSGE) models.
In this paper, we explore the relationship between determinacy and E-stability
in a very general class of Markov-switching rational expectations models with lagged
endogenous variables. Specifically, we demonstrate that a set of tractable sufficient
conditions for determinacy from Cho (2016) imply the learnability of the unique
1The following papers, among others, have studied the relationship between determinacy and E-
stability: McCallum (2007, 2009), Cochrane (2009), Ellison and Pearlman (2011), Bullard and Eusepi
(2014)
5
mean-square stable rational expectations solution if agents know current endogenous
variables when forming one-period-ahead expectations. This result contributes to
the relevant literature in three ways. First, this result extends McCallum (2007),
which finds that determinacy implies E-stability in a general class of linear rational
expectations models. Second, this is the first study to explore the relationship between
determinacy and E-stability in MS-DSGE models where agents know contemporaneous
variables. Third, our result applies to models with lagged endogenous variables,
whereas previous research focused exclusively on classes of purely-forward looking
models.
The paper is organized as follows: first, we review the literature on determinacy
and E-stability in linear and Markov-switching models, as well as papers from the
fiscal theory of the price level literature that are relevant for applications; second, we
define the class of models, model equilibria and determinacy and E-stability conditions
under consideration; third, we provide the main analytical result; fourth, we present
applications to models of monetary-fiscal policy interactions with Markov-switching
policy parameters; finally, we conclude.
Brief Literature Review
Determinacy and E-stability in LRE and MS-DSGE Models
This section explores three strands of the literature surrounding the connection
between determinacy and E-stability in linear and Markov-switching rational
expectations models. First, a robust literature examines the relationship between
determinacy and E-stability in linear rational expectations (LRE) models. Second, a
recent research program seeks both necessary and sufficient conditions for determinacy
in rational expectations models with Markov-switching. Third, a couple of papers have
6
explicitly examined the connection between determinacy and E-stability in MS-DSGE
models. We conclude with a review of the fiscal theory of the price level literature that
emphasizes New Keynesian models with Markov-switching monetary and fiscal policy
parameters and establishes a foundation for the applications in 2.5.
McCallum (2007) offers what is perhaps the most general analytical connection
between determinacy and E-stability. The paper employs a broad class of linear
rational expectations models with lagged endogenous variables. These models are
populated with agents who observe contemporaneous endogenous and exogenous
variables when forecasting tomorrow’s endogenous variables, and who utilize a
forecasting model that shares a functional form with the minimal state variable
solution. In these settings, necessary and sufficient conditions for determinacy imply
that the determinate equilibrium is stable under learning. This result does not
condition on any other assumptions 2, but is not robust to alternative assumptions
about agents’ information sets. For example, McCallum is not able to isolate
restrictions that guarantee the abovementioned implication when agents are unable
to observe contemporaneous endogenous variables–an issue that Ellison and Pearlman
(2011) further studies.
Ellison and Pearlman (2011) provides a “completely general” link between
determinacy and E-stability by requiring agents to use a saddlepath learning rule.
Their approach departs from McCallum’s in one notable way: agents’ forecasting
model assumes the same form as the saddlepath relationship. Econometrically, this
approach imposes zeros on the coefficients on lags of non-predetermined variables in
the regression model used in McCallum (2007). When agents know time-t variables,
the same relationship between determinacy and E-stability holds with either learning
2Except for a few regularity assumptions that McCallum dismisses as innocuous
7
rule. However, the saddlepath learning rule requires agents to condition future
endogenous variables on estimates of current predetermined variables. This restriction,
they argue, ensures that the unique solution in any determinate model will also be
E-stable when agents don’t know the time-t endogenous variables (i.e. only know
the time-t exogenous variables). Moreover, Ellison and Pearlman prove that only one
minimal state variable solution will be “iteratively” E-stable in indeterminate models
when agents use a saddlepath learning rule. This notion of “iterative” E-stability is
a discretization of the E-stability conditions in Evans and Honkapohja (2001), and
without it, McCallum could not obtain general E-stability results in indeterminate
models.
Bullard and Eusepi (2014) extends Ellison and Pearlman (2011) and McCallum
(2007) in two directions: (1) they allow for richer lag structures in the information set
of agents; (2) they permit agents to form expectations over infinite-horizons. Their
results are clear: determinacy does not generally imply E-stability under infinite-
horizon (IH) learning or finite-horizon (FH) learning. In particular, the presence
of delays in the information set breaks the McCallum (2007) relationship between
determinacy and E-stability in models of finite-horizon learning. They illustrate this
using a simple New Keynesian model with Calvo pricing, a cash-in-advance constraint
and a basic Taylor-type interest rate rule. When the model under study is determinate
and agents do not know contemporaneous endogenous variables and the monetary
policy rule, the unique solution is generally E-unstable. Abstracting from the New
Keynesian setting, Bullard and Eusepi identify additional sources of E-instability in
determinate models. For instance, they find that IH and FH learning rules give us
stability conditions that differ only in magnitude, with magnitude being determined
by “discount factors” that “capture reduced-form discounting of future variables in the
8
equilibrium dynamics of the model.” Due to these differences in magnitude, a change
in the underlying decision rule may render an E-stable solution E-unstable and vice-
versa. As with the papers before, these results apply to linear DSGE models.
Attempts to study determinacy and E-stability in MS-DSGE models have been
stymied, in part, by a lack of useful necessary and sufficient conditions for determinacy.
However, numerous papers have made great strides in the direction of these conditions.
These papers are best categorized by the concept of stability they employ. Papers by
Davig and Leeper (2007) and Branch, Davig and McGough (2013) use the familiar
concept of bounded stability and identify conditions under which a purely-forward
looking model is determinate.3 In the other camp, Farmer, Waggoner, and Zha [FWZ]
(2009, 2011) and Cho (2016) use mean-square stability–a notion of stability often used
in engineering–to formulate their own conditions for determinacy.
One of the earliest papers in this literature, Davig and Leeper (2007), studies
a standard New Keynesian model with a Markov-switching Taylor-type interest rate
rule, and finds that monetary policy can occasionally switch into an indeterminate
(“passive”) regime without generating a multiplicity of stable equilibria. This occurs
provided that monetary policy satisfies the “long-run Taylor principle” (LRTP), which
allows monetary policy to be sometimes passive if the Taylor principle is satisfied
sufficiently often. While this result by itself has significant policy implications, the
methods they employ constitute a separate contribution to the MS-DSGE literature. In
particular, they demonstrate how to render the original non-linear system of equations
linear by “conditioning the structural form of the model” on the underlying Markov
state. When the model is written in this form, a straightforward application of the
3Determinate in the sense that a unique regime-dependent equilibrium (RDE) exists, where a RDE
is an equilibrium in which agents condition expectations on the current Markov state, and no lags of
the Markov state.
9
Blanchard and Khan (1980) conditions yields the LRTP. Branch, Davig and McGough
(2013) develops an analogous condition in a more general purely-forward looking
model, that they call the “conditionally linear determinacy condition” or CLDC. The
CLDC is easy to use, and generalizes conditions for determinacy in LRE models to
a class of MS-DSGE model, but it does not apply to models with lagged endogenous
variables. Additionally, Branch, Davig and McGough (2013) are able to show that
the CLDC only guarantees the existence of a unique regime-dependent equilibrium;
if agents condition expectations on past states of the economy, they may generate
additional “history-dependent equilibria” when indeterminate regimes exhibit negative
feedback and the CLDC is satisfied. Similarly, FWZ (2010) identifies the potential for
indeterminacy in a New Keynesian model that satisfies the LRTP.
Because of the shortcomings of the LRTP, FWZ (2009) abandons the concept of
bounded stability in favor of mean-square stability. This decision to work with mean-
square stability also allowed them to borrow necessary-and-sufficient conditions for
mean-square stability from a rich engineering literature on the subject. In their paper,
they show how to write any rational expectations solution to a purely forward looking
MS-DSGE model as the sum of a fundamentals and non-fundamentals component.
The task of identifying determinacy is then twofold: (1) isolate conditions under which
the fundamentals component is a mean-square stable process; (2) isolate conditions
under which all non-fundamental components are mean-square unstable processes. One
can complete these tasks by minimizing the spectral radius of a single matrix with
respect to the dimensions spanned by the non-fundamental solutions. This amounts
to identifying the full set of sunspot solutions–a task that becomes drastically more
difficult as the number of states and equations increases–and restricting these solutions
to be unstable. As a result, this approach is very difficult to use in practice. Moreover,
10
the approach only applies to a class of MS-DSGE models that do not have lagged
endogenous variables.
Whereas FWZ (2009) seeks to identify determinacy by finding all sunspot
solutions, FWZ (2011) presents a numerical algorithm that may detect indeterminacy.
Their method applies to forward-looking models with lagged endogenous variables,
and is able to detect indeterminacy if the algorithm converges to more than one
fundamental solution of the model under study. There is no proof that their method
finds all fundamental solutions to a model. As such, their method’s inability to detect
multiple fundamental solutions does not preclude indeterminacy.
Foerster et al. (2016) introduces a very general, and tractable perturbation
approach for finding higher-order solutions to MS-DSGE models. In their approach,
they only perturb time-varying parameters that affect the model’s steady state. In
doing so, they perturb the smallest permissible set of time-varying parameters, and
they show how to obtain reasonable approximations around the ergodic mean of the
time-varying parameters that impact steady state. Foerster et al. (2016) also advocates
for the use of a Gro¨bner basis approach to solving MS-DSGE models. The theory of
Gro¨bner bases is particularly useful in the MS-DSGE framework because MS-DSGE
equilibrium coefficients solve quadratic polynomials that typically cannot be solved
using standard solution techniques, and because we can obtain the full set of minimum
state variable (MSV) solutions to a MS-DSGE model using a Gro¨bner basis. As such,
Foerster et al (2016) offers a means for researchers to study issues of uniqueness and
existence in the class of MSV solutions. Importantly, their methods only study MSV
solutions to the model; they do not consider sunspot solutions in their analysis.
While the approach of FWZ (2009) and (2011) are limited in terms of tractability
and sufficiency, respectively, and Foerster et al. (2016) is limited with respect to its
11
treatment of sunspot solutions, Cho (2016) offers a set of implementable sufficient
conditions for determinacy and indeterminacy. Cho’s approach builds on prior work
by McCallum (2007) and Cho and Moreno (2011) in a manner that simplifies the task
of restricting all sunspot solutions to be unstable. Unlike FWZ (2009), which requires
one to solve a complicated minimization problem to identify the full set of sunspot
solutions, Cho’s method solves a model forward for a unique solution, and computes
the spectral radii of two matrices. Cho’s method is discussed in greater detail later in
this chapter.
A much sparser literature examines the link between determinacy and E-stability
in MS-DSGE models. The biggest contribution along these lines is Branch, Davig and
McGough (2013). They prove that satisfaction of the CLDC implies E-stability of the
minimal state variable solution in purely forward looking MS-DSGE models. They also
obtain an intriguing result about the learnability of history-dependent equilibria: if the
CLDC is satisfied, then a history-dependent equilibrium can be learned when agents
condition their expectations on a “sunspot variable which captures the self-fulfilling
serial correlation in the equilibrium.” Reed (2015) extends their results by proving
that satisfaction of the conditions for determinacy in the mean-square stability sense
implies E-stability of the minimal state variable solution when agents do not know
time t endogenous variables. This is because the conditions in Cho are stronger than
the CLDC in purely forward looking models. The results in Reed only apply to purely
forward looking models–which I attempt to expand on in this section.
Fiscal Theory of the Price Level and New Keynesian Models
MS-DSGE models commonly feature Markov-switching fiscal and monetary
policy rules. This is even true to the extent that many, if not most, MS-DSGE models
12
in the New Keynesian literature would be LRE models without Markov-switching
policy rules. In this chapter, we focus on New Keynesian MS-DSGE models that
allow fiscal policy and monetary policy rules to switch between various configurations
of “active” and “passive” fiscal and monetary policy. Models that feature these
switching rules are often associationg with the Fiscal Theory of the Price Level
(FTPL) literature. To better motivate the significance of these models, we briefly
introduce the FTPL and discuss some of the salient papers in this literature below.
The FTPL is developed in a host of papers that study how fiscal and monetary
policy jointly determine inflation. This complements the more canonical view of
inflation, which is rooted in the Quantity Theory and often abstracts away from fiscal
policy by assuming Ricardian equivalence. It’s important to emphasize that the FTPL
complements–and not contrasts–the canonical view, because even the canonical view
expresses inflation as the outcome of a joint fiscal-monetary policy configuration in
which fiscal policy stabilizes the real market value of debt and does not affect prices.
While the FTPL has faced considerable criticism, it raises interesting questions about
the role of fiscal policy in price-level determination. In particular, the FTPL argues
that inflation is a fiscal phenomenon in the absence of Ricardian fiscal policy. Given
that growing evidence challenges Ricardian equivalence, the FTPL is arguably quite
relevant.
The FTPL largely begins with Leeper (1991). This seminal paper introduces
a dichotomy in the conduct of monetary and fiscal policy: a “passive” policymaker
is “constrained to stabilize debt”, whereas an “active” policymaker does not act to
stabilize government debt. To formalize this distinction, Leeper identifies regions of
the parameter space consistent with passive and active policymaking in a stylized
endowment economy with a monetary authority that adjusts interest rates in response
13
to inflation and a fiscal authority that adjusts lump-sum taxes in response to real debt.
Since each policymaker is in one of two disjoint parameter regions, there are a total of
four disjoint regions of the parameter spaces. Of these four, one yields indeterminacy
(passive fiscal and monetary policy), one yields explosive solutions (active fiscal
and monetary policy), and two of them yield determinacy: (1) the active monetary
and passive fiscal policy configuration; (2) the active fiscal and passive monetary
configuration.
Economists typically choose parameters from the first of these determinate
parameter regions for conventional macroeconomic models of inflation and output.
In this region, the monetary authority responds aggressively to inflation, and the fiscal
policymaker executes a Ricardian policy. Because a Ricardian fiscal policy does not
affect consumption and inflation, economists can ignore the fiscal authority when
studying the determination of inflation and corresponding equilibrium dynamics of
the model. In the second region, however, the fiscal authority does not raise enough
new revenue to service the interest and pay down the principal on newly issued debt.
This allows increases in debt to have first-order wealth effects that raise household
consumption and inflation since forward-looking agents do not expect governments to
tax away the benefits of the new debt wealth in the future. If the monetary authority
responds aggressively to the run up in inflation caused by debt issuance in this
environment, it will raise real interest rates, and therefore generate higher debt service
costs that an active fiscal authority will, again, not pay for. This generates explosive
model dynamics. A stable debt path therefore requires the monetary authority to
respond weakly to inflation–that is, a monetary authority must be “accommodative.”
Henceforth we embrace the terminology from Leeper and Leith (2016) and refer to an
active fiscal, passive monetary regime configuration as a “Regime F” configuration,
14
and an active monetary and passive fiscal policy configuration as a “Regime M”
configuration.
Woodford (1995) and Sims (1994) shed further light on the relationship between
fiscal policy and inflation by studying endowment economies where monetary
authorities have direct control over the money supply. In these two papers, Woodford
and Sims build on Leeper (1991) by introducing an equilibrium condition sometimes
referred to as a “bond valuation” equation that determines prices in relation to
outstanding debt and the discounted present value of expected future real surpluses.4
Sims (1994) show that the same holds in a similar model. Sims also uncovers more
nuanced dependencies of monetary policy efficacy on fiscal policy. For example, in
cases where exogenous money growth does not keep up with agents’ discounting,
the fiscal authority must credibly back the value of the currency or else inflationary
sunspot equilibria may arise. In cases where the money supply rule targets an interest
rate peg, a larger menu of fiscal policy rules–including a constant tax rate–ensure a
unique, stable equilibrium.
Woodford (1998a, 1998b, 2001) are among the earlier papers to extend the
Leeper (1991) framework to include endogenous output and nominal rigidities. In this
framework, Woodford (1998a) clearly shows that a fiscal inflation may arise through
the private behavior of households–irrespective of monetary policy. One can illustrate
this point in a variety of ways, including: when nominal bond holding changes, real
wealth changes against sluggish prices, and these wealth effects vary aggregate demand.
This contrasts Sargent and Wallace (1981) which argues that fiscal policy causes
inflation only insofar as the monetary authority monetizes runaway debt. Under
4In contrast, Leeper (1991) studies bounded solutions to systems of linearized equilibrium
conditions. These conditions, when combined with a transversality condition, imply the bond
valuation equation, but Leeper (1991) does not explicitly derive this equation
15
Woodford’s interpretation, however, fiscal inflation is not simply the result of monetary
policy, and it is this point that helps separate the fiscal theory from previous work
on the relationship between inflation and fiscal policy. These papers also study fiscal-
monetary policy interactions in the presence of debt ceilings, the zero lower bound,
and bounded rationality. For instance, Woodford (1998a) finds that a debt ceiling of
the kind proposed in the Maastrict Treaty constrains fiscal policy to be Ricardian.
Woodford (2001) suggests that the post-war bond-price support regime is an example
of Regime F behavior in U.S. data, as policymakers defended an interest rate peg. In
this analysis he further examines the zero lower bound as a subcase of interest rate
pegs.
Though Woodford (2001) describes the role of maturity structure in the fiscal
theory, Cochrane (2001) thoroughly characterizes how and when variation in the
maturity of debt affects the relationship between fiscal policy and inflation. Cochrane
studies a simple frictionless model of consumer optimization combined with a flow
constraint on fiscal policy, and a present value condition that requires the real value
of debt to equal the discounted present value of expected primary fiscal surpluses. He
calls this last condition the “equilibrium valuation equation” because it follows from
market-clearing and no-arbitrage conditions, and is therefore true in equilibrium.5 This
condition explicitly relates fiscal policy to inflation when debt is nominal: if surpluses
change exogenously, either nominal debt or price must change to restore the valuation
equation. If all debt is short-term debt, then debt is predetermined, so that a variation
in surpluses necessitates a price adjustment. To illustrate the complexity of this
relationship in the presence of long-term debt, Cochrane engages in two comparative
statics exercises. The first exercise studies the effects of debt on price. With a richer
5Conchrane’s valuation equation generalizes the aforementioned bond valuation equation from
Woodford (1995) and Sims (1994) to environments with rich maturity structures of government debt
16
maturity structure, debt variations cause outstanding debt to be revalued, which
creates a second channel for restoring the present value condition. This implies that
policymakers can trade current inflation for future inflation by lenthening the maturity
structure of debt. The second exercise looks at the effects of surplus on price, and finds
that price is more responsive to current surpluses as the maturity lengthens. This is
because outstanding long-term debt is not a claim to the current surplus; variation
in the current surplus cannot revalue outstanding debt. Similarly, variation in future
surpluses revalues outstanding long-term debt, thereby reducing the need for price
changes today. The paper concludes with an optimal policy exercise that minimizes
the variance of inflation with respect to the scale and composition debt. Cochrane
finds that longer maturities are optimal when the present value of surpluses is more
volatile than current surpluses. The intuition follows from the second comparative
static exercise.
The preceding works assume that policy is fixed, but a growing body of empirical
research suggests that regime switches in policy are the norm. In this vein, Clarida
et al (2000) identifies a regime change in monetary policy under Volcker. They
estimate a forward-looking Taylor-type policy reaction function6 combined with the
assumption that the Federal Reserve partially adjusts the Funds rate to its target.
They then estimate the reaction function using GMM and find that pre-Volcker policy
is passive and post-Volcker policy is active. This suggests, as they demonstrate in
a New Keynesian model, that the Great Moderation occurred because post-Volcker
policymakers pursued stabilizing policies, while pre-Volcker regimes permitted sunspot
instability. Lubik and Schorfheide (2004) helps to confirm these results.
6the authors point out that Taylor (1993) proposes a rule where policy is backward-looking, but
if lagged inflation or a linear combination of lagged inflation and output“is a sufficient statistic for
forecasting future inflation” then their rule nests the Taylor rule
17
While Clarida et al. (2000) offers evidence of a one-time regime change, Cogley
and Sargent (2002, 2005) find evidence of drift in monetary policy rules using a
random coefficients model. Evidence of recurring structural change appears in Sims
and Zha (2006). They estimate a backward-looking Markov-switching model that
allows for switching in monetary policy parameters and shock volatilities, and conclude
that switches in shock variances explain the Great Moderation.
Davig and Leeper (2006, 2011) combine Markov-switching reduced-form
estimation of policy rules and MS-DSGE models to examine the general equilibrium
implication of switching in policy parameters. In the first of these two papers, they
estimate a standard Markov-switching Taylor-type interest rate rule and a lump-
sum transfers rule that responds to real debt, government purchases, and output.
Each rule follows a two-state Markov process and they are estimated separately to
determine when each authority exhibits active or passive behavior. These rules are
then embedded in an on otherwise standard New Keynesian model with government
purchases. We emphasize three results. First, they uncover evidence of recurring
Regime F policy regimes in U.S. economic history. Second, they find that aggregate
demand responds to lump-sum transfer shocks, which gives additional reason to
question the assumption of Ricardian equivalence in models of the U.S. economy. In
the model, for example, a $1 tax cut increases the discounted present value of output
by 76 to 102 cents. Third, the U.S. can be described as a single unique equilibrium.
This conclusion breaks with Clarida et. al (2000) and Lubik and Schorfheide (2004),
which both suppose that the economy unexpectedly jumps between regions of
determinacy, indeterminacy and explosiveness, so that agents always expect permanent
regimes. When agents expect regime change, as is assumed by Davig and Leeper, the
18
economy can temporarily pass through indeterminate and explosive regimes without
undermining the determinacy of the model.
Additional attempts to extend and complement the work of Davig and Leeper
come from Bianchi (2012, 2013), Bianchi and Ilut (2017), and Bianchi and Melosi
(2013).7 Bianchi (2012) and Bianchi and Ilut (2017) estimate a regime-switching New
Keynesian model using Bayesian techniques and techniques from FWZ (2011) and Kim
and Nelson (1999). They assume a “circular structure” for transition probabilities that
forces the economy to move from Regime F to an active-active regime to either Regime
M or Regime F.8 Unlike Davig and Leeper (2006, 2011), they jointly estimate policy
rules and model parameters. They find that the economy was in Regime F pre-Volcker,
in the explosive regime during the early 1980s, and then transitioned into Regime M.
Bianchi and Ilut (2017) juxtaposes the impulse response functions from a similar
rational expectations model and impulse response functions from a counterfactual
model in which agents maintain different assumptions about the persistence and
number of regimes. These comparisons of “actual” and “counterfactual” impulse
response functions suggest that the magnitude of fiscal inflation under an active fiscal
regime depends on the horizon over which agents expect active fiscal policy to persist.
They conclude that the Great Inflation of the 1970s would not have occurred if agents
either believed fiscal policy was passive, or felt that a return to passive fiscal policy
was imminent. Specifically, their estimated model indicates that monetary policy
switched from passive monetary to active monetary under Volcker in 1979, but fiscal
policy did not switch from active to passive until 1981. Their counterfactual exercises
7Other papers not mentioned here include Bhattarai et al (2012), Bhattarai et al (2014), Bhattarai
et al (2016), Gonzalez-Astudillo (2013), Chung et al (2007)
8Bianchi (2012) employs a similar circular structure that forces the economy to transition from
active-active regimes to Regime M
19
predicts that a coordinated monetary-fiscal regime change in 1979 would have reduced
inflation sooner. Similary, Bianchi (2013) conducts counterfactuals that suggest the
Great Inflation might not have occurred had agents believed that the economy could
have switched to a very active monetary regime (“Eagle” regime) in the 1980s and on.
These results also echo findings from Bianchi and Melosi (2013), in which policy
regimes are hidden from rational agents who experience recurring deviations from
passive fiscal policy to two active fiscal regimes that are identical apart from their
average persistence. As agents observe more deviations from passive or “virtuous”
policy regime, they grow more pessimistic about the probability of being in the
more persistent regime. In this environment, the growing pessimism causes inflation
to become more responsive to fiscal policy, and this means that once “dormant”,
persistent shocks to government debt can have creeping inflationary effects as agents
become more pessimistic.
Though Bianchi and Melosi study Bayesian learning in MS-DSGE models of
fiscal-monetary policy interactions, little research has studied adaptive learning in
this class of models. A number of papers, however, approach adaptive learning in
fixed regime models of the fiscal theory of prices. Namely, Evans and Honkapohja
(2007) examines stability of the “fiscalist” and “monetarist” solutions in McCallum
(2001), and the Regime M and Regime F equilibria in Leeper (1991). To this end,
they employ a model of an endowment economy with constant government purchases,
money in utility, and policy rules from the two aforementioned papers. They find
that the explosive fiscalist equilibrium obtained in a non-stochastic model with point
expectations is unstable under learning when agents use a one-period ahead learning
rule. Using the same rule, the fixed price, zero-debt monetarist solution is E-stable. In
the context of the Leeper (1991) model, they find that Regime M equilibria are stable
20
while only a subregion of the parameter space consistent with Regime F equilibria yield
learnable solutions. These results lend some credence to Regime F equilibria.
Evans and Honkapohja (2005) studies learning and fiscal-monetary policy
interactions in the presence of a liquidity trap. They first examine a nonlinear
frictionless endowment economy to demonstrate the existence of a desired high
inflation steady state and a low inflation liquidity trap steady state. Under passive
fiscal policy, a unique high steady state equilibrium exists, under active policy, a
unique low steady state exists. The stability of these equilibria under learning are
examined using two learning rules. First, agents engage in a process of steady state
learning under which the high state is E-stable, and the low state is E-unstable. In the
second learning model, agents use a perceived law of motion that shares a functional
form with the equilibrium law of motion. Under this second learning scheme, the high
state is E-stable given passive fiscal policy, and the low state is learnable when policy
is active. However, the basin of attraction at the lower steady state is small, and this
presents the potential for a cumulative deflation. If the economy is pushed into such
a deflation, the monetary authority must pursue an aggressive policy to regain the
desirable high steady state. In other words, active fiscal policy is not enough. This
subject is further studied in a production economy with sticky prices in Evans, Guse
and Honkapohja (2008).
Eusepi and Preston tackle a number of adaptive learning questions in New
Keynesian models of monetary-fiscal policy interactions with nominal government
debt. Their analysis in Eusepi and Preston (2012) shows that a restricted subset of the
Regime F equilibria in a simple New Keynesian model with Leeper-style policy rules
are E-stable when agents are uncertain about the prevailing monetary-fiscal policy mix.
However, the removal of this uncertainty renders some of these previously E-unstable
21
equilibria stable under learning. In Eusepi and Preston (2011), the relationship
between expectational stability and the scale and composition of debt is examined.
To capture scale effects, they vary the steady state level of debt, and to capture
composition effects, they vary the rate of decay in the government’s geometrically
decaying bond portfolio. They find that models with very short or very long average
maturities tend to be the most expectationally stable. This is because variation in
maturity has two competing effects. First, a lengthening in maturity structure subjects
debt to changes in inflation expectations through revaluation effects. This means
that a shock to expectations that revalues debt may feedback to agents’ expectations
in a destabilizing way. Second, a lengthening in maturity reduces the percentage of
outstanding debt that must be rolled over in each period. This makes it easier to
finance new debt should an unexpected exogenous shock place pressure on government
finances. These effects combine to make expectations least stable at medium-to-long
average maturities. Across all maturities, expectations are most stable when the fiscal
policy is active against an interest rate peg.
For more information on the FTPL, consult Leeper and Leith (2016). Other
relevant papers not introduced in this section are discussed in Chapters 3 and 5 of
this dissertation.
MS-DSGE Models
General Class of MS-DSGE Models
In this paper, we consider Markov-Switching Rational Expectations (MS-DSGE)
models of the following form:
Xt = M(st)Et(Xt+1) +N(st)Xt−1 +Q(st)Ut (2.1)
22
where Xt is n × 1 vector of endogenous variables, Ut is m × 1 vector of exogenous
variables that follows
Ut = ρUt−1 + t
where ρ is a diagonal matrix and t is a white noise process. By assumption, Ut is a
covariance stationary process. st is a S-state Markov Chain and pij = Pr(st+1 = j|st =
i) is the (i, j)-th element of the transition probability matrix, P . From Proposition 1
in Cho (2016), any rational expectations solution to (2.1) can be written as a linear
combination of a minimal state variable solution that depends on Xt, st, and Ut and a
non-fundamental solution component, wt, as:
Xt = Ω(st)Xt−1 + Γ(st)Ut + wt (2.2)
wt = Et(F (st)wt+1) (2.3)
where the coefficient matrices satisfy the following conditions for all realizations of the
Markov Chain:
Ω(st) = {In − Et[M(st)Ω(st+1)]}−1N(st) (2.4)
Γ(st) = {In − Et[M(st)Ω(st+1)]}−1Q(st) + Et(F (st)Γ(st+1)ρ) (2.5)
F (st) = {In − Et[M(st)Ω(st+1)]}−1M(st) (2.6)
A minimal state variable solution takes the form given in (2.2) with wt = 0n×1:9
Xt = Ω(st)Xt−1 + Γ(st)Ut (2.7)
9Non-fundamental solutions arise with wt = 0n×1 where wt satisfies (2.3)
23
Any Ω(st) and Γ(st) that satisfy (2.4) and (2.5) give us a minimal state variable
solution of the form given in (2.7). Moreover, Γ(st) is uniquely determined by Ω(st)
and can be obtained by vectorizing (2.5) for each state and stacking the vectorized
equations as follow:
vec
⎛
⎜⎜⎜⎜⎝
Γ(1)
...
Γ(S)
⎞
⎟⎟⎟⎟⎠ = vec
⎛
⎜⎜⎜⎜⎝
Ξ(1)−1Q(1)
...
Ξ(S)−1Q(S)
⎞
⎟⎟⎟⎟⎠+ ((⊕Sj=1ρ′ ⊗ F (j))(P ⊗ In×m))vec
⎛
⎜⎜⎜⎜⎝
Γ(1)
...
Γ(S)
⎞
⎟⎟⎟⎟⎠
where ⊕Sj=1ρ′ ⊗ F (j) = diag(ρ′ ⊗ F (1), . . . , ρ′ ⊗ F (S)) and
Ξ(i) = In − Et[M(st)Ω(st+1)] (2.8)
Also, P denotes the transition probability matrix
P =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11 p12 · · · p1S
p21 p22 · · · p2S
...
. . .
...
pS1 pS2 · · · pSS
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
where pij is the probability of transitioning from state i to state j.
The Forward Method for Solving MS-DSGE Models
In this section, we discuss the forward method for solving MS-DSGE models
developed in Cho (2016) and Cho and Moreno (2011). When the forward method
works, we obtain the “forward solution” to the MS-DSGE model given in (2.1). We
discuss this method and the “forward solution” for two reasons: (1) the determinacy
24
results used in this paper only apply to the forward solution; (2) the presence of
lagged endogenous variables in MS-DSGE models generates an infinite regress problem
that often precludes use of standard solution techniques such as the method of
undetermined coefficients. As stated in Cho (2016), the forward method “simply
amounts to text-book style ‘solving the model’” forwards. To that end, consider the
following proposition, which is Proposition 3 in Cho (2016):
Proposition 1
Consider (2.1). For given st, Xt and Xt−1 there exists a unique sequence of
real-valued matrices (Ωk(st),Γk(st), Fk(st)) for k = 1, 2, 3, ... such that
Xt = Et[Mk(st, st+1, ..., st+k)Xt+k] + Ωk(st)Xt−1 + Γk(st)Ut (2.9)
where Ω1(st) = N(st), Γ1(st) = Q(st), F1(st) = M1(st) = M(st),
Ωk(st) = {In − Et[M(st)Ωk−1(st+1]}−1N(st) (2.10)
Γk(st) = {In − Et[M(st)Ωk−1(st+1)]}−1Q(st) + Et(Fk(st)Γk−1(st+1)ρ)(2.11)
Fk(st) = {In − Et[M(st)Ωk−1(st+1)]}−1M(st) (2.12)
and
Mk(st, st+1, ..., st+k) = Fk(st)Mk−1(st+1, ..., st+k)
Proof: See Appendix D in Cho (2016)
Definition 1
(2.1) satisfies the forward convergence condition if the coefficients of
the state variables, (Ωk(st),Γk(st), Fk(st)) in (2.10)-(2.12) converge as k
25
goes to infinity for every st. Let Ω
∗(st) = limk→∞Ωk(st) and Γ∗(st) =
limk→∞ Γk(st).
Then the model implies:
Xt = lim
k→∞
Et(Mk(st, ..., st+k)Xt+k) + Ω
∗(st)Xt−1 + Γ∗(st)Ut (2.13)
Now, since Ω∗(st) and Γ∗(st) satisfy (2.4) and (2.5). The following equation describes a
rational expectations solution of (2.1):
Xt = Ω
∗(st)Xt−1 + Γ∗(st)Ut (2.14)
Equation (2.14) is the forward solution to (2.1). (2.13) and (2.14) jointly imply that
Et(Mk(st, ..., st+k)Xt+k) = 0n×1, and Cho (2016) shows that (2.14) is the unique
solution obtained via the forward method that satisfies the No Bubble Condition
(NBC): Et(Mk(st, ..., st+k)Xt+k) = 0n×1. Because Γ∗(st) does not exist if Ω∗(st) does
not exist, one should first check to see if Ωk(st) converges. If Ω
∗(st) exists, and the
spectral radius of (⊕Sj=1ρ′⊗F (j))(P ⊗ In×m) is less than one, then Γ∗(st) exists and can
be recovered from the following equation:
vec
⎛
⎜⎜⎜⎜⎝
Γ∗(1)
...
Γ∗(S)
⎞
⎟⎟⎟⎟⎠ = vec
⎛
⎜⎜⎜⎜⎝
Ξ∗(1)−1Q(1)
...
Ξ∗(S)−1Q(S)
⎞
⎟⎟⎟⎟⎠+ ((⊕Sj=1ρ′ ⊗ F ∗(j))(P ⊗ In×m))vec
⎛
⎜⎜⎜⎜⎝
Γ∗(1)
...
Γ∗(S)
⎞
⎟⎟⎟⎟⎠
where we have substituted Ω∗(st) for Ω(st) in (2.6) and (2.8) to form F ∗(st) and
Ξ∗(st), respectively.
26
E-Stability
In this paper, we focus on the stability of the forward solution of (2.1) under
adaptive learning. Specifically, we show that a set of sufficient conditions for
determinacy in this class of models imply that the forward solution of (2.1) is stable
under learning. This result depends on the information that agents have at their
disposal. Suppose agents observe st, P , Ut, and Xt at time t. Agents have the following
perceived law of motion:
Xt = A(st) + B(st)Xt−1 + C(st)Ut
where A(i) is n × 1, B(i) is n × n and C(i) is n × m. Consistent with the PLM, the
learning agent believes that if st = i then Xt = A(i) + B(i)Xt−1 + C(i)Ut for i =
1, 2, ..., S. The coefficient matrices of the PLM may be estimated using a recursive least
squares procedure that updates A(i), B(i), and C(i) each time st = i. Notice that we
allow agents to learn steady state values by including a constant term in the PLM. In
this section, we solve for agents’ state-contingent expectations and derive the state-
contingent T-map. If st = i then
Et(Xt+1) = E(Xt+1|st = i;Xt, Ut)
=
S∑
j=1
E(Xt+1|st+1 = j, st = i;Xt, Ut)
=
S∑
j=1
pij(A(j) + B(j)EtXt + C(j)EtUt+1))
=
S∑
j=1
pij(A(j) +B(j)Xt + C(j)ρUt))
27
Substituting Et(Xt+1) into (2.1) yields the actual data generating process:
Xt = {I −M(i)(
S∑
j=1
pijB(j))}−1M(i)(
S∑
j=1
pijA(j))
+ {I −M(i)(
S∑
j=1
pijB(j))}−1N(i)Xt−1
+ {I −M(i)(
S∑
j=1
pijB(j))}−1(M(i)(
S∑
j=1
pijC(j))ρ+Q(i))Ut
If we define B = (B(1) B(2) · · ·B(S)) and Ξ(i, B) = {I − M(i)(∑Sj=1 pijB(j))} then
the state-contingent T-map is:
A(i) → Ξ(i, B)−1M(i)
S∑
j=1
pijA(j)
B(i) → Ξ(i, B)−1N(i)
C(i) → Ξ(i, B)−1(M(i)
S∑
j=1
pijC(j)ρ+Q(i))
Notice that the forward solution is a fixed point of the T-map. The block of the T-map
associated to B = (B(1) B(2) · · ·B(S)) decouples from the other equations. This block
is given by:
TB(B) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
(Ξ(1, B)−1N(1))′
(Ξ(2, B)−1N(2))′
...
(Ξ(S,B)−1N(S))′
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
′
=
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
T 1B(B)
′
T 2B(B)
′
...
T SB(B)
′
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
′
28
To assess E-stability, we begin by examining the following differential equation:
dB
dt
= TB(B)− B
Let DTB(B¯) denote the Jacobian of TB evaluated at the forward solution, B¯ = (Ω
∗(1)
Ω∗(2) · · ·Ω∗(S)). Since TB is continuously differentiable, Proposition 5.6 in Evans and
Honkapohja (2001) tells us that B¯ is locally asymptotically stable if the eigenvalues of
DTB(B¯) have real parts less than one. As demonstrated in Appendix A. this process
yields the following Jacobian:
DTB(B¯) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11Ω
∗(1)′ ⊗ F ∗(1) p12Ω∗(1)′ ⊗ F ∗(1) · · · p1SΩ∗(1)′ ⊗ F ∗(1)
p21Ω
∗(2)′ ⊗ F ∗(2) p22Ω∗(2)′ ⊗ F ∗(2) · · · p2SΩ∗(2)′ ⊗ F ∗(2)
...
. . .
...
pS1Ω
∗(S)′ ⊗ F ∗(S) pS2Ω∗(S)′ ⊗ F ∗(S) · · · pSSΩ∗(S)′ ⊗ F ∗(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
≡ (⊕Sj=1Ω∗
′
(j)⊗ F ∗(j))(P ⊗ In2)
E-stability requires the real parts of (⊕Sj=1Ω∗′(j) ⊗ F ∗(j))(P ⊗ In2) to be less
than one. It is important to note that our derivation of the E-stability conditions
hinges on the following: Ξ(i, B¯)−1N(i) = {I − M(i)(∑Sj=1 pijΩ∗(j))}−1N(i) =
{I − M(i)(Et(Ω∗(st+1))}−1N(i) = Ω∗(i) and Ξ(i, B¯)−1M(i) = {I −
M(i)(
∑S
j=1 pijΩ
∗(j))}−1M(i) = {I − M(i)(Et(Ω∗(st+1))}−1M(i) = F ∗(i) where Et
denotes conditional expectations here. We now turn to the equation for A = (A(1)′
A(2)′ · · ·A(S)′)′:
29
TA(A) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Ξ(1, B)−1M(1)(
∑S
j=1 p1jA(j))
Ξ(2, B)−1M(2)(
∑S
j=1 p2jA(j))
...
Ξ(S,B)−1M(S)(
∑S
j=1 pSjA(j))
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
=
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11Ξ(1, B)
−1M(1) p12Ξ(1, B)−1M(1) · · · p1SΞ(1, B)−1M(1)
p21Ξ(2, B)
−1M(2) p22Ξ(2, B)−1M(2) · · · p2SΞ(2, B)−1M(1)
...
. . .
...
pS1Ξ(S,B)
−1M(S) pS2Ξ(S,B)−1M(S) · · · pSSΞ(S,B)−1M(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
A
Using the same methods as before we obtain the following Jacobian evaluated at the
REE where A¯ = 0Sn×1 :
DTA(A¯, B¯) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11F
∗(1) p12F ∗(1) . . . p1SF ∗(1)
p21F
∗(2) p22F ∗(2) . . . p2SF ∗(2)
...
. . .
...
pS1F
∗(S) pS2F ∗(S) . . . pSSF ∗(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
≡ (⊕Sj=1F ∗(j))(P ⊗ In)
E-stability requires the real parts of ((⊕Sj=1F ∗(j))(P ⊗ In)) to be less than one.
Finally, we consider the equation for C = (C(1)′ C(2)′ · · ·C(S)′)′:
30
TC(B,C) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Ξ(1, B)−1(M(1)(
∑S
j=1 p1jC(j))ρ+Q(1))
Ξ(2, B)−1(M(2)(
∑S
j=1 p2jC(j))ρ+Q(2))
...
Ξ(S,B)−1(M(S)(
∑S
j=1 pSjC(j))ρ+Q(S))
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
=
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11Ξ(1, B)
−1M(1) p12Ξ(1, B)−1M(1) · · · p1SΞ(1, B)−1M(1)
p21Ξ(2, B)
−1M(2) p22Ξ(2, B)−1M(2) · · · p2SΞ(2, B)−1M(1)
...
. . .
...
pS1Ξ(S,B)
−1M(S) pS2Ξ(S,B)−1M(S) · · · pSSΞ(S,B)−1M(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Cρ
+
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Ξ(1, B)−1Q(1)
Ξ(2, B)−1Q(2)
...
Ξ(S,B)−1Q(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Using the same methods as before we obtain the following Jacobian evaluated at the
REE where C¯ = (Γ∗(1)′ Γ∗(2)′ · · ·Γ∗(S)′)′:
DTC(B¯, C¯) = ρ⊗
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11F
∗(1) p12F ∗(1) . . . p1SF ∗(1)
p21F
∗(2) p22F ∗(2) . . . p2SF ∗(2)
...
. . .
...
pS1F
∗(S) pS2F ∗(S) . . . pSSF ∗(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
≡ ρ⊗ ((⊕Sj=1F ∗(j))(P ⊗ In))
31
The REE solution A¯, B¯, C¯ is E-stable if:
i. all the eigenvalues of (⊕Sj=1Ω∗′(j)⊗ F ∗(j))(P ⊗ In2) have real parts less than 1,
ii. all the eigenvalues of (⊕Sj=1F ∗(j))(P ⊗ In) have real parts less than 1, and,
iii. all the eigenvalues of ρ⊗ ((⊕Sj=1F ∗(j))(P ⊗ In)) have real parts less than 1
The solution is not E-stable if any of these three conditions fail with eigenvalues
strictly greater than one.
Mean-Square Stability
As noted in the literature review, the MS-DSGE literature utilizes two notions of
stability: (1) bounded stability; (2) mean-square stability. We focus on mean-square
stability in this paper, but offer definitions for both mean-square stability and bounded
stability below.
Definition 2
An n × 1 stochastic process yt is mean-square stable (MSS) if there exists
an n × 1 vector y¯ and n × n matrix Q s.t. limt→∞(E[yt]) = y¯ and
limt→∞(E[yty′t]) = Q.
Intuitively, a stochastic process is MSS if its first and second moments converge
to well-defined limits. The concept of mean-square stability is closely related to other
often used concepts of stability. For example, asymptotic covariance stationarity
implies mean-square stability in general, and is even equivalent to mean-square
stability in models with exogenous asymptotic covariance stationary shock processes.
32
Definition 3
An n-dimensional stochastic process yt is bounded if there exists a real
number N such that ||yt|| < N for all t
where || • || is any well-defined norm, including the uniform norm. In LRE models
with bounded shocks, determinacy in the mean-square stability sense and determinacy
in the bounded stability sense are equivalent equilibrium selection criteria. In
models with unbounded shocks–such as the normal or lognormal exogenous shocks
commonly used in applications–the criteria differ; a mean-square stable process can
have unbounded support while bounded processes cannot. Of course, this paper, and
other papers in the MS-DSGE literature, work with models that are linearized around
a nonstochastic steady state, and therefore function best with small shocks. This
substantially reduces the meaningfulness of embracing a concept of stability that allows
for unbounded shock processes. Even with bounded shocks, the two concepts are not
equivalent in MS-DSGE models. For example, Farmer, Whaggoner and Zha (2009)
argues that MS-DSGE models with persistent unstable regimes may admit unbounded
MSS rational expectations equilibria. This means that bounded stability may rule out
equilibria of theoretical importance in certain applications (i.e. equilibria in economies
that experience recurring hyperinflation). These reasons notwithstanding, we work
with mean-square stability because there are not tractable conditions for determinacy
in the boundedly stable sense in MS-DSGE models with lagged endogenous variables.
Though Definition 2 both formalizes the concept of mean-square stability, and
offers helpful intuition, it does not help us identify mean-square stability. To that end,
Costa et al. (2005) developed a set of tractable conditions for identifying whether or
not a given process is MSS. We state these conditions by considering a very general
33
process:
yt+1 = D(st, st+1)yt +H(st+1)ηt+1 (2.15)
where D(st, st+1) is n × n, H(st+1) is n × l and ηt+1 is some l × 1 mean-square stable
(MSS) process. D(st, st+1) can depend on st alone, st+1 alone, or both st and st+1.
Given certain conditions on Ut, the forward solution is a stochastic process of this form
10:
Xt = Ω
∗(st)Xt−1 + Γ∗(st)Ut (2.16)
We can also form a stochastic process of this form by forcing the coefficient matrices in
the forward solution to depend on last period’s state:
Xt = Ω
∗(st−1)Xt−1 + Γ∗(st−1)Ut (2.17)
Theorem 3.34 in Costa et al. (2005) allows us to focus solely on the homogeneous part
of (2.15): yt+1 = D(st, st+1)yt. Consider the following matrices:
Ψ¯D =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11D(1, 1) p21D(2, 1) · · · pS1D(S, 1)
p12D(1, 2) p22D(2, 2) · · · pS2D(S, 2)
...
. . .
...
p1SD(1, S) p2SD(2, S) · · · pSSD(S, S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
ΨD =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11D(1, 1) p12D(1, 2) · · · p1SD(1, S)
p21D(2, 1) p22D(2, 2) · · · p2SD(2, S)
...
. . .
...
pS1D(S, 1) pS2D(S, 2) · · · pSSD(S, S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
10If Ut consists of m independent covariance stationary AR(1) processes then Γ(st)Ut will be MSS
and these conditions are met
34
Ψ¯D⊗D =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11D(1, 1)⊗D(1, 1) p21D(2, 1)⊗D(2, 1) · · · pS1D(S, 1)⊗D(S, 1)
p12D(1, 2)⊗D(1, 2) p22D(2, 2)⊗D(2, 2) · · · pS2D(S, 2)⊗D(S, 2)
...
. . .
...
p1SD(1, S)⊗D(1, S) p2SD(2, S)⊗D(2, S) · · · pSSD(S, S)⊗D(S, S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
ΨD⊗D =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11D(1, 1)⊗D(1, 1) p12D(1, 2)⊗D(1, 2) · · · p1SD(1, S)⊗D(1, S)
p21D(2, 1)⊗D(2, 1) p22D(2, 2)⊗D(2, 2) · · · p2SD(2, S)⊗D(2, S)
...
. . .
...
pS1D(S, 1)⊗D(S, 1) pS2D(S, 2)⊗D(S, 2) · · · pSSD(S, S)⊗D(S, S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Definition 4
Let rσ(X) = max1≤i≤n(|λi|), where λ1, ..., λn are the eigenvalues of the n×n
matrix X.
Theorem 1
The process (2.15) is mean-square stable if and only if rσ(Ψ¯D⊗D) < 1.
Proof: See Proposition 3.9 in Costa et al. (2005).
To derive this result, Costa et al. develop first-order difference equations that
describe the evolution of the first and second moments of an arbitrary stochastic
process. They show that Ψ¯D⊗D governs the evolution of the second moment such
that the second moment equation converges in the limit if and only if the spectral
radius of Ψ¯D⊗D is less than one. Conveniently, this condition is also sufficient for
the convergence of the first moment process, whose evolution is governed by Ψ¯D (see
Theorem 2). As a result, rσ(Ψ¯D⊗D) < 1 is necessary and sufficient for the mean-square
stability of the process in (2.15). The following is also a useful result from Costa et al
(2005).
35
Theorem 2
If rσ(Ψ¯D⊗D) < 1 then rσ(Ψ¯D) < 1
Proof: See Proposition 3.6 in Costa et al. (2005)
Corollary: If rσ(ΨD⊗D) < 1 then rσ(ΨD) < 1
Proof: Since rσ(ΨD⊗D) = rσ(Ψ¯D′⊗D′) < 1, Theorem 2 implies that rσ(Ψ¯D′) =
rσ(ΨD) < 1.
The stochastic processes that we study in this paper have homogeneous component
matrices that strictly depend on st or st+1, but not both. For example, let
D(st, st+1) = Ω
∗(st+1) as in (2.16). In this case: Ψ¯D⊗D = (⊕Sj=1Ω∗(j)⊗Ω∗(j))(P ′⊗ In2);
Ψ¯D = (⊕Sj=1Ω∗(j))(P ′ ⊗ In); ΨD⊗D = (P ⊗ In2)(⊕Sj=1Ω∗(j) ⊗ Ω∗(j)); ΨD =
(P ⊗ In)(⊕Sj=1Ω∗(j)). If D(st, st+1) = Ω∗(st) as in (2.17), then: Ψ¯D⊗D = (P ′ ⊗
In2)(⊕Sj=1Ω∗(j)⊗Ω∗(j)); Ψ¯D = (P ′ ⊗ In)(⊕Sj=1Ω∗(j)); ΨD⊗D = (⊕Sj=1Ω∗(j)⊗Ω∗(j))(P ⊗
In2); ΨD = (⊕Sj=1Ω∗(j))(P ⊗ In)
Cho (2016) Sufficient Conditions for Determinacy
As shown in Cho (2016), the following two conditions are sufficient for
determinacy in the mean square stability sense:
1. rσ((⊕Sj=1Ω∗(j)⊗ Ω∗(j))(P ′ ⊗ In2)) < 1
2. rσ((⊕Sj=1F ∗(j)⊗ F ∗(j))(P ⊗ In2)) ≤ 1
where all coefficient matrices are taken from the forward solution to the model.
Determinacy and E-Stability
In this section, we show that the abovementioned sufficient conditions for
determinacy imply E-stability in the class of MS-DSGE models under consideration
36
in this paper. Simply put, determinacy implies rσ(((⊕Sj=1F ∗(j))(P ⊗ In))) < 1 and
rσ((⊕Sj=1Ω∗′(j) ⊗ F ∗(j))(P ⊗ In2)) < 1. rσ((⊕Sj=1Ω∗′(j) ⊗ F ∗(j))(P ⊗ In2)) < 1 implies
that condition (i) for E-stability holds, and rσ((⊕Sj=1F ∗(j))(P ⊗ In)) < 1 implies that
conditions (ii) and (iii) are satisfied as well.
E-Stability and Cho (2016) Conditions
Theorem 3
If the MSV solution to a model of the form (2.1) satisfies the sufficient
conditions for determinacy in Cho (2016), then the MSV solution is stable
under learning.
Proof: The proof consists of two parts. First, we demonstrate that both
determinacy conditions imply rσ((⊕Sj=1Ω∗′(j) ⊗ F ∗(j))(P ⊗ In2)) < 1, which implies
the first sufficient condition for E-stability. Then, we demonstrate that the second
determinacy condition implies rσ((⊕Sj=1F ∗(j))(P ⊗ In)) < 1, which implies the
second and third conditions for E-stability . We begin by considering the following
proposition:
Proposition 2
Consider the two arbitrary stochastic processes with the following
homogeneous components:
Xt+1 = G(st)Xt
ωt+1 = H(st)
′ωt
37
where X and ω are both n × 1. If rσ((⊕Sj=1H(j) ⊗H(j))(P ⊗ In)) < 1 and
rσ((P
′⊗In)(⊕Sj=1G(j)⊗G(j))) < 1 then rσ((⊕Sj=1G′(j)⊗H(j))(P⊗In2)) < 1.
Proof: See proof of Lemma 1 in Cho (2016). Set F ′(st−1, st) = H(st−1)′ and
G(st−1, st) = G(st−1) and Proposition 2 follows. 
Now suppose G(st) = Ω
∗(st) and H ′(st) = F ∗(st)′ so that (⊕Sj=1G′(j)⊗H(j))(P ⊗
In2) = (⊕Sj=1Ω∗′(j)⊗ F ∗(j))(P ⊗ In2). If the sufficient conditions for determinacy imply
that we can form MSS processes with the following homogeneous components:
Xt+1 = Ω
∗(st)Xt (2.18)
wt+1 = F
∗(st)′ωt (2.19)
then rσ((⊕Sj=1Ω∗′(j) ⊗ F ∗(j))(P ⊗ In2)) < 1 by Proposition 2. We now prove this
implication. From the first condition for determinacy, rσ((⊕Sj=1Ω∗(j) ⊗ Ω∗(j))(P ′ ⊗
In2)) < 1. Since (⊕Sj=1Ω∗(j) ⊗ Ω∗(j))(P ′ ⊗ In2) and (P ′ ⊗ In2)(⊕Sj=1Ω∗(j) ⊗ Ω∗(j))
have the same characteristic equation, the first condition for determinacy implies
rσ((P
′ ⊗ In2)(⊕Sj=1Ω∗(j) ⊗ Ω∗(j))) < 1. From Theorem 1, it follows that (2.18) is
a MSS homogenous component. Now consider the second condition for determinacy:
rσ((⊕Sj=1F ∗(j) ⊗ F ∗(j))(P ⊗ In)) < 1. Since ((⊕Sj=1F ∗(j) ⊗ F ∗(j))(P ⊗ In))′ =
(P ′ ⊗ In)(⊕Sj=1F ∗(j)′ ⊗ F ∗(j)′), the second condition for determinacy also implies
rσ((P
′ ⊗ In)(⊕Sj=1F ∗(j)′ ⊗ F ∗(j)′)) < 1. It follows from Theorem 1 that (2.19) is also
a MSS homogeneous component. Together, the stability of (2.18) and (2.19) give us
rσ((⊕Sj=1Ω∗′(j)⊗ F ∗(j))(P ⊗ In2)) < 1 which satisfies the first E-stability condition
Now, we show that conditions (ii) and (iii) are satisfied. As demonstrated,
determinacy implies rσ((P
′ ⊗ In)(⊕Sj=1F ∗(j)′ ⊗ F ∗(j)′)) < 1. From Theorem 2 and
its corollary, rσ((P
′ ⊗ In)(⊕Sj=1F ∗(j)′)) = rσ((⊕Sj=1F ∗(j))(P ⊗ In)) < 1 follows.
38
Therefore, if the forward solution satisfies the sufficient conditions for determinacy
in Cho (2016), then E-stability condition (ii) is satisfied. To see that (iii) is satisfied,
observe the following: rσ(ρ ⊗ ((⊕Sj=1F ∗(j))(P ⊗ In))) = rσ(ρ)rσ((⊕Sj=1F ∗(j))(P ⊗ In)).
Since ρ is a diagonal matrix and all of its entries are bounded below by 0 and above
by 1, rσ(ρ) is less than one. Hence, rσ(((⊕Sj=1F ∗(j))(P ⊗ In))) < 1 implies condition
(iii). We conclude that the sufficient conditions for determinacy imply E-stability of
the forward solution for any S-state Markov chain. 
McCallum (2007): A Special Case
Though we implicitly pursue this result in order to further our understanding of
nondegenerate Markov-switching models (i.e. models with S > 1), we draw attention
to the fact that linear rational expectations models are nested in the class of models we
examine. This means that our methods should agree with the main result in McCallum
(2007), which shows that determinacy implies the learnability of the minimal state
variable solution when agents use an information set that is identical to the one used
in this paper. To see what our approach says about linear models we let S = 1. When
this is true, the forward solution A¯, B¯, C¯ is E-stable if:
i. all the eigenvalues of (Ω∗)′ ⊗ F have real parts less than 1,
ii. all the eigenvalues of F ∗ have real parts less than 1, and,
iii. all the eigenvalues of ρ⊗ F ∗ have real parts less than 1
Likewise, the sufficient conditions for determinacy become:
1. rσ(Ω
∗ ⊗ Ω∗) < 1
2. rσ(F
∗ ⊗ F ∗) < 1
39
A well known result in matrix algebra says that for any matrix, A, rσ(A ⊗ A) < 1 if
and only if rσ(A) < 1. Therefore, we can use the following equivalent conditions for
determinacy:
1. rσ(Ω
∗) < 1
2. rσ(F
∗) < 1
McCallum (2007) shows that these last two determinacy conditions imply the three
E-stability conditions above. Hence, our results agree with McCallum (2007).
Numerical Example
We apply our results to a basic New Keynesian model of the kind Woodford
(1998a) uses. This model is augmented to allow for Markov-switching in fiscal and
monetary policy parameters, and features a representative household and firm,
monopolistic competition in the production of intermediate goods, and price stickiness
a la Calvo (1983) according to which 1−θ fraction of firms can change their prices each
period. The model also allows government to sell one-period nominal bonds, Bt, at a
price that equals the inverse of the monetary policy instrument, 1 + it. The government
collects lump-sum taxes in accordance with an endogenous primary surplus rule, τt
and government purchases are assumed to equal 0, so that income, Yt, equals Ct in
equilibrium.
The model is linearized around the non-stochastic steady state with zero
inflation. Let zˆt ≡ ln(zt)− ln(z¯) where z¯ is the value of z in steady state. The behavior
40
of households and firms then reduces to two-equations 11:
yˆt = Etyˆt+1 − σ−1(ˆit − Etπˆt+1) (2.20)
πˆt = βEtπˆt+1 + κyˆt (2.21)
where β is the household discount factor, σ−1 is the intertemporal elasticity of
substitution and κ is defined in Appendix B. β, κ, and σ are positive by assumption,
and β is also bounded above by 1. Monetary policy follows a standard interest rate
rule of the form:
iˆt = α(st)πˆt + zmt (2.22)
zmt = ρmzm,t−1 + mt (2.23)
where st follows a 2-state Markov chain, zmt is an AR(1) exogenous monetary
policy shock, and m is an exogenous i.i.d. mean-zero innovation. Fiscal policy is
characterized by the following linearized rule for primary surpluses:
τˆt = γ(st)(bˆt−1 − iˆt−1) + zft (2.24)
zft = ρF zf,t−1 + ft (2.25)
where bˆt is the percentage deviation of real bonds from steady state, zft is an
exogenous fiscal policy shock, and f is an exogenous mean-zero i.i.d innovation. γ is
the fiscal authority’s policy parameter and it follows the same Markov process as α.
11See Appendix B. for full-derivation of (2.20), (2.21), and policy equations
41
Fiscal policy must also satisfy the following budget constraint:
bˆt − iˆt + (β−1 − 1)sˆt = β−1(bˆt−1 − πˆt) (2.26)
To reduce the number of equations in the system, we substitute (2.24) into (2.26),
which yields:
bˆt − iˆt + β−1πt = (β−1 − γ(st)(β−1 − 1))bˆt−1 + γ(st)(β−1 − 1)ˆit−1 + (1− β−1)zft (2.27)
To characterize the time-varying behavior of policymakers, let st = M if the economy
is in Regime M, and let st = F if the economy is in Regime F. In Regime M, α(M) > 1
and γ(M) > 1; in regime F, 0 ≤ α(F ) < 1 and 0 ≤ γ(F ) < 1. As discussed,
a LRE model that features either a Regime F or Regime M policy configuration is
determinate. In Regime F (M), fiscal policy is active (passive) and monetary policy
is passive (active). If the economy is not in Regime F or Regime M in a LRE model,
then monetary and fiscal policy are both passive (active), in which case the model
is indeterminate (explosive). Though the assumption that policy strictly switches
between Regime M and Regime F configurations is highly restrictive, we assume this
for expositional purposes, and will study a more general model of switching in the
future.
The system given by (2.20)-(2.23), (2.25), and (2.27) yields a solution for xt =
(yˆt, πˆt, iˆt, bˆt)
′. To utilize the forward method for solving this model, we must write the
model in the form of (2.1). Before doing this, however, we add the following constraint
42
to (2.21):
λEt(bˆt+1 − iˆt+1 + β−1πt+1 − (1− β−1)ρF zft
− (β−1 −
2∑
j=1
pijγ(j)(β
−1 − 1))bˆt −
2∑
j=1
pijγ(j)(β
−1 − 1)ˆit)) = 0
where λ is any nonzero scalar, and st = i. To derive this constraint, we forward the
government budget constraint one-period and take expectations. The constraint forces
expected inflation to be consistent with agents’ expectations of future fiscal policy. Cho
(2015) demonstrates the need for this constraint when using the forward method to
solve “block-recursive” models that have an autonomous block and a dependent block.
An autonomous block is subsystem of equations that have no behavioral dependence
on the dependent block. For example, the Phillips Curve, IS curve, and interest
rate rule constitute the autonomous block in our New Keynesian model because
these equations do not explicitly depend on the government budget constraint and
fiscal surplus rule, which constitute the dependent block. When we solve the model
forward, expectations of inflation and output will not depend on, or be consistent with,
current and expected future fiscal policy. In other words, agents will condition their
expectations on information set, {πˆt, yˆt, iˆt, zmt, st}, when we are interested in a class of
equilibria corresponding to the following information set: {πˆt, yˆt, iˆt, bˆt, zmt, zft, st}. If
fiscal policy is always passive, then current and expected future output and inflation
do not depend on fiscal policy, and the information adjustment is unnecessary. If,
however, fiscal policy is active sufficiently often, then inflation and output–which are
determined in the autonomous block–will not respond to stabilize government debt in
the absence of the informational adjustment, and the conditions for determinacy will
fail regardless of whether or not the model is determinate. Because we allow for active
43
fiscal policy, we add this informational adjustment to the system of equations. After
making this adjustment the system is written as:
A(i)xt = BEtxt+1 + C(i)xt−1 +Qut
where ut = (zmt, zft)
′, st = iand
A(i) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
1 0 σ−1 0
−κ 1 λ(−(1− β−1)∑2j=1 pijγ(j)) λ(β−1 − (1− β−1)∑2j=1 pijγ(j))
0 −α(i) 1 0
0 β−1 −1 1
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
B =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
1 σ−1 0 0
0 β + λ(β−1) −λ λ
0 0 0 0
0 0 0 0
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
C(i) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0 0 0 0
0 0 0 0
0 0 0 0
0 0 γ(i)(β−1 − 1) (β−1 − γ(i)(β−1 − 1))
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Q =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0 0
0 0
1 0
0 1− β−1
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Provided A(st) is invertible for all st it is straightforward to show that M(st) =
A(st)
−1B, N(st) = A(st)−1C(st) and Q(st) = A(st)−1. We can now apply the
44
forward method and assess the uniqueness, mean-square stability and E-stability of the
resulting forward solution. Unsurprisingly, our numerical analysis supports the main
analytical result of this paper. Specifically, consider the set of parameter values
̂α(M) ∈ [1, 5] ̂α(F ) ∈ [0, 1] ̂γ(M) ∈ [1, 5]
̂γ(F ) ∈ [0, 1] κ ∈ [0, 1] β ∈ [.975, 1]
σ ∈ [0, 5]
The forward solution is found to be E-stable for all indicated combinations of
parameter values whenever the sufficient conditions for determinacy are also satisfied.
This result depends crucially on the agents information set, I1t = {xt, ut, st}. When
agents only observe I2t = {xt−1, ut, st} at time t, there are cases in which determinacy
does not imply E-stability. This appears to hinge on the fact that fiscal policy is
periodically active; a determinate purely forward-looking New Keynesian model (i.e.
a model with a passive fiscal authority) will always have E-stable forward solutions
when agents only observe I2t . We leave this topic to future research.
Conclusion
We proved that the sufficient conditions for determinacy in Cho (2016) imply
the E-stability of the forward solution in MS-DSGE models with lagged endogenous
variables when agents condition their expectations of future endogenous variables
on all current endogenous and exogenous variables. This result extends a well-
known result in McCallum (2007), and is the first study to address the relationship
between determinacy and E-stability in MS-DSGE models with lagged endogenous
variables. Through applications to models of fiscal-monetary policy interactions, we
also demonstrate that our result is not robust to changes in agents’ information set.
45
In the future, we hope to extend the model of Eusepi and Preston (2013) to
allow for switching in fiscal and monetary policy parameters. This extension allows
us to explore new issues. First, we can analyze how the size and average maturity
of government debt affects economies that face recurring changes in fiscal and
monetary policy. In the presence of a permanent Regime F configuration, medium
to long average maturities induce expectational instability. This is a significant result
inasmuch as global data reveals substantial cross-country variation in the scale and
composition of debt. The present study does not, however, account for switches in
policy that may have occurred in the U.S. and abroad since the 1960s. We therefore
see the addition of Markov-switching interest rate rule and surplus rule parameters
as a natural extension. Second, this research allows us to better understand and
characterize regions of the New Keynesian model’s parameter space consistent with
determinacy and E-stability in the presence of switching policy rules. Third, our
treatment of this model may allow us to identify cases where the forward solution
appears stable but violates the conditions for determinacy and indeterminacy in Cho
(2016). This may help us better understand the extent to which Cho’s conditions
are not necessary. Finally, we want to study how switching in fiscal policy appears
to impart expectational instability on an otherwise standard New Keynesian model
when agents do not observe current endogenous varaiables. This is a subject of
interest to us because the methods in Branch et al. (2013) and Reed (2015) show
that determinacy implies E-stability when agents do not know the contemporaneous
endogenous variables and fiscal policy is permanently passive. We take this topic up in
Chapter 4.
46
CHAPTER III
THE POWER OF FORWARD GUIDANCE AND THE FISCAL THEORY OF THE
PRICE LEVEL
Introduction
A growing literature offers strong theoretical support for the use of expansionary
forward guidance on interest rates, particularly when interest rates are constrained by
the zero lower bound (see, for example, Eggertsson and Woodford, (2003)). Despite
the effectiveness of forward guidance in theory, the predictions of workhorse New
Keynesian models do not accord well with empirical studies of the effects of forward
guidance in the U.S. (e.g. Del Negro et al (2015), D’Amico and King (2015)). That
is, while New Keynesian models predict large responses of inflation and output to
forward guidance on short-term rates, the empirical evidence points to responses that
are positive but modest. This shortcoming of the New Keynesian model is dubbed
“The Forward Guidance Puzzle” (Del Negro et al. 2015), and it calls into question
the ability of the standard New Keynesian model to predict the effects of anticipated
monetary policy.
According to Del Negro et al. (2015), McKay et al. (2015), Carlstrom et al.
(2012), Chung et al.(2014), Kiley (2014), the implausible responsiveness of output
and inflation to forward guidance stems from three signature features of the New
Keynesian model. First, consumption is excessively responsive to changes in interest
rates. Second, the lack of a discount factor in the household’s log-linearized Euler
equation implies a strong response of consumption to long-run interest rates. Because
forward guidance is designed to influence long-run rates, forward guidance naturally
47
generates a large response in consumption through the Euler equation. Third, “front-
loading” in the New Keynesian Phillips Curve renders inflation particularly sensitive
to changes in current and future output. Together, the lack of discounting in the
Euler equation and front-loading in the Phillips Curve generate a feedback loop that
exacerbates the rise in inflation and output implied by forward guidance.
Several papers have addressed this puzzle by limiting the importance of these
three features of the New Keynesian model. For instance, McKay et al (2015) mute
the response of agents to forward guidance by introducing borrowing constraints that
prevent agents from drawing down their savings over the forward guidance horizon.
Gabaix (2016) introduces an explicit discount factor into the Euler equation and an
additional discount factor into the Phillips Curve to model myopic agents. Del Negro
et al (2015) show that a positive probability of death generates effective discounting
in the Euler equation when they introduce a perpetual youth structure into the New
Keynesian model. Chung et al. (2014) and Kiley (2014) introduce “sticky information”
in the spirit of Mankiw and Reis (2002) to mitigate feedback effects from the Phillips
Curve. Cole (2015) replaces rational expectations with a model of adaptive learning
to demonstrate that bounded rationality lessens the effectiveness of forward guidance
in specific policy experiments. Cochrane (2017) argues that equilibrium selection rules
may help to explain the Forward Guidance Puzzle.
In contrast to previous attempts to explain the exaggerated response of inflation
and output to forward guidance – which focus primarily on the specification of private
sector behavior – this paper examines how the joint conduct of monetary and fiscal
policy influences the effects of expansionary forward guidance in the New Keynesian
model. Specifically, we show that the above mentioned exaggerated response of output
and inflation to forward guidance may hinge on two assumptions (in addition to the
48
three model features highlighted above): (1) the monetary authority employs an
interest rate rule that satisfies the “Taylor Principle”; (2) fiscal policy is conducted in
such a way that variation in fiscal surpluses acts to stabilize government debt, thereby
rendering fiscal policy Ricardian. Our approach is most closely related to Cochrane
(2017), which suggests that fiscal considerations may help select equilibria with smaller
initial price jumps in response to anticipated policy announcements.1 In contrast to
Cochrane (2017), we explicitly characterize fiscal policy regimes and study how wealth
effects arising in these regimes reduce the responses of inflation and output to forward
guidance. Moreover, we model recurring fiscal regimes to capture how uncertainty
about future fiscal policy impacts the effectiveness of forward guidance.
Our contribution borrows heavily from the Fiscal Theory of the Price Level
literature, which models inflation as the outcome of both monetary and fiscal
policy (see Leeper and Leith (2016) for a review of the Fiscal Theory of the Price
Level). Work in this literature distinguishes between “passive” policymakers who are
constrained to stabilize the government debt, and “active” policymakers who determine
inflation. In the standard New Keynesian model we study here, monetary policy is
active (passive) when interest rate responds respond strongly (weakly) to inflation,
and fiscal policy is passive (active) when Ricardian equivalence is satisfied (violated).
Careful consideration of the passive-active dichotomy reveals a number of channels
through which the fiscal policy stance impacts the response of inflation and output to
both fundamental and policy shocks.
This paper focuses on a specific channel through which active fiscal policy affects
agents’ perception of bond wealth. To illustrate this channel, we restrict attention to
1Cochrane (2017) uses a bond valuation equation, which we introduce in (2), to point out the idea
that the large price adjustments predicted in the standard “forward-stable” model equilibrium must
be supported by changes in the present value of expected future surpluses.
49
our model’s intertemporal household budget constraint (assuming that all government
debt is single-period debt):
Et
( ∞∑
T=t
Rt,TPTCT
)
= Et
( ∞∑
T=t
{
Rt,T [PTYT − PT τT ]
})
+Bt−1 (3.1)
where Rt,T is the stochastic discount factor from time t to T , C is consumption,
τ is the government’s real primary surplus, PT is the price level at T , Bt−1 is the
government debt stock that matures at t, and Y is income. Under the assumption that
YT = CT ∀T , and after substituting for Rt,T = βT−tu′(YT )Pt/u′(Yt)PT , this equation
reduces to
Bt−1
Pt
= Et
( ∞∑
T=t
βT−t
u′(YT )
u′(Yt)
τT
)
(3.2)
Equation 3.2 is the sticky price version of bond valuation equation in Cochrane
(2001), and it asserts that today’s price level is determined by the real present value
of expected future surpluses, Et
(∑∞
T=t β
T−t(u′(YT )/u′(Yt))τT
)
, and the predetermined
debt stock, Bt−1. From (3.1) it follows that any variation in Bt−1 affects the
household’s consumption path, all else constant. When fiscal policy is passive, however,
all else is not constant – any change in Bt−1 induces an offsetting response in {τT} that
leaves the households choice set intact. In other words, fiscal policy satisfies Ricardian
equivalence. In an active fiscal policy regime, variations in bond wealth are not totally
offset by changes in the stream of expected surpluses, and this implies that the change
in bonds has wealth effects.
One source of the aforementioned variation in bond wealth is monetary policy.
For example, a reduction in interest rates might force bonds to a lower equilibrium
path as lower rates alleviate the burden of rolling over existing debt. In a model with
active fiscal policy, households are not compensated for their lower bond holdings with
50
tax cuts, which causes the household to feel nominally constrained through (3.1).
Moreover, the fall in bonds places downward pressure on prices through (3.2). The
resulting effect of this monetary expansion is an eventual fall in output and prices.
Much of what follows in this paper depends on the fact that forward guidance on
short-term interest rates appears in the model as a series of anticipated interest rate
cuts.
We illustrate the role played by monetary-fiscal policy interactions in determining
the effects of forward guidance by allowing fiscal (monetary) policy to be permanently
or recurrently active (passive). Our results are threefold. First, we find that the
presence of active fiscal policy allows for forward guidance to have wealth effects
that dampen the response of output and inflation to forward guidance (potentially
at the cost of implausible deflation). This result depends on the fact that agents
view government debt as net wealth in a regime with active fiscal policy. Hence, an
anticipated reduction in interest rates which places downward pressure on agents’
nominal bond returns causes agents to feel more constrained today. This mutes agents’
responses to lower long-run real interest rates and induces firms to lower prices.
Second, the presence of switching in fiscal and monetary policy has expectational
spillover effects that may cause forward guidance to be less stimulative in the switching
model’s passive fiscal, active monetary policy regime. In such a setting, the possibility
that fiscal policy may become active during the forward guidance horizon causes agents
to become less optimistic about the effects of forward guidance in an economy where
monetary and fiscal policy are currently active and passive, respectively. Interestingly,
these spillover effects always attenuate the short-term effects of forward guidance,
but can lead to more persistent responses of output and inflation, as we demonstrate
51
in one specific case. Our Markov-switching approach helps to highlight the role that
expectations play in generating a response of inflation and output to forward guidance.
Third, the presence of long-term government debt in a model with active fiscal,
passive monetary policy introduces “revaluation effects” that mitigate the deflationary
effects observed in the corresponding model with only short-term debt. We observe
these effects because an anticipated reduction in short-term interest rates raises the
market value of outstanding debt. Thus, while a reduction in interest rates lowers
aggregate demand due to lower interest rate receipts, it can also raise aggregate
demand by raising the price of the debt that households own. Such an effect cannot
be observed in a model without long-term debt.
The paper is organized as follows: first, we develop the model; second, we
explore the effects of forward in active fiscal, passive monetary policy regimes without
switching; third, we extend these results to economies that experience switching in
fiscal and monetary policy parameters; finally, we conclude.
Model
We use a basic New Keynesian model of the kind Woodford (1998) uses, and
augment this model to allow for (1) a richer maturity structure of debt as in Woodford
(2001), Eusepi and Preston (2013), and Leeper and Leith (2016); (2) Markov-
switching in policy parameters as in Davig and Leeper (2011). This model features
a representative household and firm, monopolistic competition in the production of
intermediate goods, and price stickiness a la Calvo (1983) according to which 1 − θ
fraction of firms can change their prices each period. The model also allows the
government to issue both bond portfolios, Bmt , that have a geometrically decaying
maturity structure, and short-term debt, Bs, which is held in net-zero supply. The
52
government collects lump-sum taxes in accordance with an endogenous primary surplus
rule, τt, and government purchases are assumed to equal 0, so that income, Yt, equals
Ct in equilibrium.
The model is linearized around the non-stochastic steady state with zero
inflation. Let zˆt ≡ ln(zt)− ln(z¯) where z¯ is the value of z in steady state. The behavior
of households and firms then reduces to two-equations:2
yˆt = Etyˆt+1 − σ−1(ˆit − Etπˆt+1) + rnt (3.3)
πˆt = βEtπˆt+1 + κyˆt + μt (3.4)
where y is the output gap, π is inflation, β is the household discount factor, σ−1 is the
intertemporal elasticity of substitution and κ is defined in Appendix B. β, κ, and σ are
positive by assumption, and β is also bounded above by 1. Moreover, rnt and μt evolve
according to
rnt = ρnr
n
t−1 + 
n
t (3.5)
μt = ρμμt−1 + 
μ
t (3.6)
Monetary policy is given by:
iˆt = φy(st)yˆt + φπ(st)πˆt + 
MP
t + v1,t−1 (3.7)
where MP is an i.i.d monetary policy shock, and v1,t is a linear combination of L
forward guidance shocks that obeys
2See Appendix B. for full-derivation of model equations
53
v1,t = v2,t−1 + R1,t (3.8)
v2,t = v3,t−1 + R2,t (3.9)
...
vL,t = 
R
L,t (3.10)
such that v1,t−1 =
∑L
l=1 
R
l,t−l, where [
R
1,t, 
R
2,t, ..., 
R
L,t] are the L forward guidance shocks
announced at time t. This model of short-term interest rate guidance is borrowed from
Laseen and Svensson (2011) and is widely used in the forward guidance literature.
Intuitively, Rl,t is a shock announce at time t that affects interest rates at time t + l.
The general structure of forward guidance shocks given by (3.8)-(3.10) ensure that
shocks announced at t are actually realized as intended at t + l. As shown in Appendix
C and D, policymakers can use (MPt , 
R
1,t, . . . , 
R
L,t) to announce an interest rate peg
between time t and t + L. To model recent instances of forward guidance we will peg
i at or near the zero lower bound on i. Our specification allows for switching in policy
parameters: st follows a S-state Markov chain, and the value of st determines φπ and
φy. Fiscal policy is characterized by the following linearized rule for primary surpluses:
τˆt = γ(st)(bˆ
m
t−1 + βρPˆ
m
t ) + zft (3.11)
zft = ρF zf,t−1 + ft (3.12)
54
where bˆmt is the percentage deviation of real bonds from steady state, zft is an
exogenous fiscal policy shock, and f is an exogenous mean-zero i.i.d innovation. γ is
the fiscal authority’s policy parameter and it follows the same Markov process as φy
and φπ. Fiscal policy must also satisfy the following budget constraint:
bˆmt−1 = β(1− ρ)Pˆmt + βbˆmt + (1− β)τˆt + πˆt (3.13)
where Pˆmt is the price of the bond portfolio at time t and ρ ∈ [0, 1] captures the
maturity structure of the government debt. While we relegate the derivation of this
equation to Appendix B., the intuition behind the bond portfolio is fairly simple: the
government issues bˆmt units of a nominal portfolio debt at time t that pays 1 unit of
nominal income at time t+ 1, ρ units at time t+ 2, ρ2 units at t+ 3 and so forth. This
is the sense in which the maturity of debt is geometrically decaying. This structure
allows us to introduce long-term debt into our model by using a single state variable
that captures the average maturity of debt, ρ. The limiting cases of ρ illuminate how
larger values of ρ correspond to longer average maturities: when ρ = 0, all debt is short
term, and when ρ = 1, all debt is in the form of consols. As demonstrated in Appendix
B, Pˆmt satisfies
Pˆmt = −iˆt + ρβEtPˆmt+1 (3.14)
The system given by (3.3)-(3.14) yields a solution for xt = (yˆt, πˆt, iˆt, bˆt, τˆt, Pˆ
m
t )
′.
We use Sims’ (2002) method to solve the fixed regime model, and the forward method
in Cho (2016) and Cho (2017) to solve the switching model.3 A rational expectations
3The “block-recursive” structure of the model requires us to add a constraint to the switching
model that renders agents’ inflation expectations consistent with the government budget constraint.
55
equilibrium assumes the form:
xt = Ω(st)xt−1 + Γ(st)ut
Parameters are selected so that the model under study is determinate. While there
are no simple analytical conditions for determinacy in our switching model, Woodford
(1998) gives simple conditions for determinacy in the case of non-switching (see Table
1):4
TABLE 1. Fixed Coefficent Model Determinacy Conditions
φπ > 1− 1−βκ φy φπ < 1− 1−βκ φy
γ ∈ (1, β−1+1
β−1−1) determinate indeterminate
γ /∈ (1, β−1+1
β−1−1) no stable solution determinate
We say they the economy is in Regime M when φπ > 1 − (1 − β)φy/κ and γ ∈
(1, (β−1+1)/(β−1−1)). and that the economy is in Regime F when φπ < 1−(1−β)φy/κ
and γ /∈ (1, (β−1 + 1)/(β−1 − 1)). In Regime M, fiscal policy is passive while monetary
policy is active. This is the standard assumption in most New Keynesian research. In
Regime F, fiscal policy is active while monetary policy is passive.
Fixed Coefficient Exercises
We now examine the effectiveness of forward guidance in the presence of fixed
policy regimes (i.e. we constrain all policy parameters to be permanent). Our analysis
involves three different model parameterizations: (1) a Regime M parameterization;
(2) a Regime F parameterization with short-term debt (ρ = 0): (3) a Regime F
parameterization with long-term debt. Table 6 in Appendix E contains the parameter
4We assume that φπ(st) ≥ 0 for all st
56
values used in each of the three configurations, though our results are robust to
different parameterizations.5 Our analysis in this section also involves two distinct
policy exercises that are commonly used in the literature. First, we examine the
impulse responses of output and inflation to a single one unit k-period ahead forward
guidance shock to the nominal interest rate. This exercise gives us useful intuition for
the second policy experiment, which is the main result in this section. In that exercise,
we examine the impulse responses of output and inflation to an announced 12-quarter
interest rate peg that mimics aspects of the Federal Reserve’s calendar-based forward
guidance announcements in August 2011, January 2012, and September 2012 (see Del
Negro et al (2015) for more details).
Exercise 1: Inspecting the Mechanism
In order to better understand the mechanism driving our main results in the
forward guidance experiments, we examine the effects of a one-time expansionary
forward guidance shock to the short-term nominal interest rate under all three
parameterizations. The exercise takes place as follows: at time t the central bank
announces a negative one unit shock to it+k where k ≥ 0. Agents respond at time t
by adjusting hours worked, consumption, prices and bond holdings, and this generates
paths for inflation and output that are plotted in Figure 1 for the cases where k = 8.6
Overall, output and inflation respond less favorably to forward guidance shocks
in a Regime F economy. In a Regime M economy, the negative shock at the k-
5There is one exception: for small σ and as φπ approaches 1 in Regime F, the Regime F impulse
responses of output and inflation are strictly above the Regime M impulse responses before the
realization of the shock in exercise 1 (after the shock, the Regime M impulse responses are above
the Regime F responses). This applies only to our fixed regime results and we regard this as an
unrealistic parameterization of the model. Our Markov-switching results are robust to reasonable
parameterizations.
6Qualitatively similar results obtain for different choices of k
57
FIGURE 1. Impulse Responses to One-Unit Shock
the impulse responses of output and inflation to a one-unit anticipated shock to it+8
at t. The solid line shows impulse responses in the Regime M model; the dashed line
shows impulse responses in the Regime F model with long-term debt; the dashed-
dotted line shows impulse responses in Regime F with only short-term debt.
horizon causes long run real rates to drop, which induces positive responses in output
and inflation. These responses are magnified by the lack of a discount factor in the
linearized Euler equation, which causes consumption and therefore output to be highly
responsive to changes in long-run real rates. The response of inflation to the shock is
driven, in part, by the front-loading in the Phillips curve and the presence of nominal
rigidities: firms understand that demand will continue to rise until the shock is realized
58
so they raise their prices the moment they have an opportunity to do so. These effects
combine to cause a large stimulus.
In Regime F, however, output and inflation respond to the forward guidance
shock through an additional channel: the decline in anticipated short-term interest
rates reduces the return on bond holdings. Since agents in a Regime F economy treat
their bond holdings as net wealth, expansionary forward guidance has negative wealth
effects that counteract the stimulating effects of lower long-run real interest rates. In
this framework, a k-period ahead shock initially lowers long-run real interest rates and
raises consumption. At the time of the shock’s realization, nominal wealth declines
and puts downward pressure on prices through the bond valuation equation. After the
shock is realized, agents reduce consumption to replenish bond holdings and this puts
downward pressure on consumption and output. As with the Regime M case, firms
respond by changing prices well in advance of the anticipated deflationary pressure.
This might also be driven by the presence of price stickiness: firms recognize the
probability that they won’t be able to adjust prices at the time of the shock, so they
adjust their prices as soon as they have an opportunity to do so. The overall effect of
this price setting behavior is a large and persistent deflation. Hence, while the presence
of these wealth effects generate less favorable and maybe more plausible responses of
output to forward guidance, it comes at the cost of a deflation that is not obviously
reconcilable with the data.
Figure 1 also reveals that the presence of long-term debt (i.e. ρ > 0) in Regime
F leads to higher paths of output and inflation than in a Regime F economy without
short-term debt (i.e. ρ = 0). This is because the presence of long-term debt introduces
yet another channel through which forward guidance impacts output and inflation: the
anticipated decline in short-term interest rates raises the price of outstanding debt, and
59
therefore raises the market value of outstanding debt held by the household. This is a
debt revaluation effect, and it leans against the aforementioned negative wealth effects.
One notable feature of the impulse response functions is that output responds
more favorably to forward guidance on impact, i.e. at the time of the announcement,
in the Regime F economies. We attribute this to one feature of the Regime M
economy: monetary policy satisfies the Taylor Principle such that the increase in
inflation observed on impact corresponds with higher real interest rates on impact.
If we allow φπ in all regimes to approach 1 from both directions, we observe similar
responses in all economies on impact.
Exercise 2: The Fixed Regime Forward Guidance Experiment
Our second policy experiment in the fixed coefficient model assesses the effects
of forward guidance on a specific path for interest rates. Using methods inspired by
Del Negro et al. (2015), and Cole (2015), we study what happens when the central
bank announces an interest rate target, i¯, between time T and T + L.7 We chose L =
12 to mimic the September 2012 FOMC statement that called for low interest rates
through mid-2015. Additionally, i¯ = 0 is chosen as a target, but any interest rate
target between 0 and 25 basis points may reasonably approximate the path implied by
the September 2012 statement.8 The economy is simulated for T − 1 periods prior to
announcement, and the simulations are repeated 10000 times. Figure 2 and 3 report
the mean impulse responses of output, inflation and interest rates to the L + 1 period
anticipated interest rate peg. For simplicity’s sake, I shut down shocks after time t so
that it+l = Etit+l for 0 ≤ l ≤ L. If shocks are present, monetary policymakers use
7See appendix C. for further details
8The main qualitative results in this section are robust to any i¯ below steady state, i∗ = β−1 − 1
60
some combination of unanticipated and anticipated monetary policy shocks at t + 1 to
t+ L to maintain the peg and agents’ expectation of the peg over the forward guidance
horizon. As such, we regard this simplification as innocuous.
As with the previous exercises, Figures 2 and 3 demonstrate that output and
inflation respond less favorably to expansionary forward guidance on interest rates
under the assumption of active fiscal, passive monetary policy. In contrast to previous
exercises, the forward guidance shocks do not induce a dramatic fall in output in the
Regime F economies. This result may be driven by an important feature of the impulse
responses in Figure 1: each expansionary forward guidance shock raises output before
the shock is realized, and depresses output after the shock is realized (the latter effect
is driven, in part, by a sharp drop in long-term real interest rates). Therefore, when
the economy is hit by a sequence of such shocks, as in this section, the contractionary
effects of realized forward guidance shocks are partially offset by the expansionary
effects of unrealized shocks. In general equilibrium, this leads to a relatively flat
trajectory for output (see Figure 3). Also in contrast to results from previous exercises,
inflation responds much more positively to forward guidance on the interest rate path
in the presence of long-term debt. We attribute this result to a particular strong
revaluation effect, as forward guidance on L + 1 future short-term interest rates has
a huge impact on Pˆmt (which is simply a weighted sum of expected future short-term
interest rates) .
We emphasize that the strong responses of output and inflation in Regime M are
a reflection of the forward guidance puzzle. Also note that Figure 3 uses a different
vertical scale than Figure 2.
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FIGURE 2. The 12-quarter Forward Guidance Horizon Experiment.
The solid line shows impulse responses in the Regime M model; the dashed line shows
impulse responses in the Regime F model with long-term debt; the dashed-dotted line
shows impulse responses in Regime F with only short-term debt.
Markov-Switching Forward Guidance Experiment
We now allow the policy stances of the monetary and fiscal authorities to
periodically and recurrently change. Specifically, we assume that the economy
switches between a Regime F configuration (st = F ) and a Regime M configuration
(st = M). This assumption is restrictive, but it allows us to get at one important
mechanism: expectations of changing responses to forward guidance cause agents to
behave differently today. These expectational spillovers shock the impulse responses of
inflation and output in Regime M away from the paths implied by the corresponding
62
FIGURE 3. The 12-quarter Forward Guidance Horizon Experiment (Regime F).
The 12-quarter Forward Guidance Horizon Experiment. The dashed line shows impulse
responses in the Regime F model with long-term debt; the dashed-dotted line shows
impulse responses in Regime F with only short-term debt.
fixed coefficient models, and may therefore help the impulse responses agree with the
data.
To illustrate this idea, we conduct a forward guidance experiment in the
switching model. Specifically, we first assume that the economy is in Regime M when
the central bank announces a sequence of shocks at time T such that iT = ET iT+1 =
... = ET iT+L = i¯. We then assume that the economy remains in Regime M at
T + 1, when another sequence of shocks is announced such that iT+1 = ET+1iT+2 =
... = ET+1iT+L = i¯. This process is repeated until T + L. This experiment shows
63
how the switching economy responds to an announced L + 1 period interest rate
peg in Regime M. Figures 4-5 show the Regime M effects of this experiment when
L = 3 using a parameterization inspired by a similar model in Ascari et al. (2017)
(see Table 7 in Appendix E for the parameter values contained in Figures 4-6; see
Appendix D a derivation for the policy experiment). We emphasize that agents do
FIGURE 4. Regime Switching Forward Guidance Experiment 1
The 3-quarter Forward Guidance Horizon Experiment, Parameterization 1. The solid
line shows impulse responses in the fixed coefficient model Regime M; dashed line
shows impulse responses in the switching model Regime M with long-term debt; the
dashed-dotted line shows impulse responses in the switching model Regime M with
only short-term debt.
64
FIGURE 5. Regime Switching Forward Guidance Experiment 2
The 3-quarter Forward Guidance Horizon Experiment, Parameterization 2. The solid
line shows impulse responses in the fixed coefficient model Regime M; dashed line
shows impulse responses in the switching model Regime M with long-term debt; the
dashed-dotted line shows impulse responses in the switching model Regime M with
only short-term debt.
not expect the economy to remain in Regime M throughout the forward guidance
horizon. Agents form rational expectations using the true transition probabilities
(e.g. ET (sT+1 = M |sT = M) = pMM where pMM is the probability of remaining in
Regime M). We only hold M fixed to compare the Regime M impulse responses in the
switching model, to the Regime M impulse responses in the fixed regime model. More
generally, we could allow for regime-switching during our forward guidance experiment.
65
FIGURE 6. Regime Switching Forward Guidance Experiment 3
The 3-quarter Forward Guidance Horizon Experiment, Parameterization 3. The solid
line shows impulse responses in the fixed coefficient model Regime M; dashed line
shows impulse responses in the switching model Regime M with long-term debt; the
dashed-dotted line shows impulse responses in the switching model Regime M with
only short-term debt.
Relative to the fixed regime cases, expansionary forward guidance appears to be
less stimulative in the switching model’s Regime M. In Regime M, this is driven by the
positive probability that the economy will switch to a state where the expansionary
shock has negative wealth effects. Crucially, these spillover effects exist because policy
is anticipated;9 in an environment where all shocks are i.i.d unanticipated shocks, such
9A persistent unanticipated shock would deliver similar spillover effects. We leave this for future
research.
66
spillover effects are not observed because there is zero probability that a given shock
will affect the economy in a future regime.
While the qualitative results in Figures 4-5 are robust to different policy
coefficients, structural parameters, transition probabilities, and forward guidance
horizons, Figure 6 shows that the output and inflation impulse responses in the
switching model can barely overshoot and undershoot the fixed regime responses
after the interest rate peg is over when inflation reaction coefficients are high and the
fiscal policy parameter in Regime M is relatively low.10 We have two remarks about
this particular result. First, a regime switch quickly eliminates the persistent output
and inflation gaps. In the calibrated model, these switches occur every 20 periods on
average. Second, lower inflation and higher output reaction coefficients in the interest
rate rule help to raise i faster and close the gaps.
Conclusion
Standard New Keynesian models predict implausibly large and favorable
responses of inflation and output to forward guidance on interest rates. This paper
investigates the effects of forward guidance in a New Keynesian model with active
fiscal policy and passive monetary policy. We find that the presence of active fiscal
policy allows for forward guidance to have wealth effects that dampen the response
of output and inflation to forward guidance, potentially at the cost of implausible
deflation. In an active fiscal, passive money regime, the deflationary effects of forward
guidance are mitigated by the presence of long-term debt. Moreover, the presence of
switching in fiscal and monetary policy may have expectational spillover effects that
cause forward guidance to be less stimulative in a passive fiscal, active money regime.
10See Appendix E. for parameter values
67
CHAPTER IV
MATURITY, DETERMINACY, AND E-STABILITY
Introduction
A vast literature examines determinacy and E-stability in New Keynesian
models. This work may guide the design of monetary policy by offering tractable
conditions for ensuring the existence of unique, learnable equilibria. For example, the
Taylor Principle, by which central banks move real interest rates in the direction of
inflation, is widely known to guarantee determinacy and E-stability in New Keynesian
models when fiscal policy is “passive” in the sense described in Chapter 2 (Woodford
(2003), Bullard and Mitra (2002)). Taking this point one step further, the simple
Leeper (1991) conditions also apply to a broad class of New Keynesian models to
help characterize determinacy when fiscal policy is not passive. Typically, those
papers study linear approximations to economies that involve time-invariant fiscal
and monetary policy regimes, despite ample econometric work that suggests recurring
changes in the conduct of policy. Past work may assume time-invariant policy for at
least one reason: tractability. The advent of new solution techniques and conditions for
determinacy has made it possible to more fully examine determinacy and E-stability
in a New Keynesian model with 2-state Markov-switching fiscal and monetary policy
(see Cho (2016), Foerster et al (2016)). This paper uses those techniques to study
determinacy (in the mean-square stability sense) and E-stability in a New Keynesian
model with Markov-switching fiscal and monetary policy. We emphasize three results.
First, the maturity structure of government debt matters for determinacy and
the existence of stable equilibria. We regard this as a significant result in part because
68
the maturity structure of government debt is irrelevant for determinacy in the fixed
regime analog of the model without a real risk premium on long-term debt (Eusepi
and Preston (2011), Jin (2013)). Our results therefore suggest that time-variation in
the fiscal policy stance on debt matters not only for equilibrium dynamics, but for
the existence and uniqueness of rational expectations equilibria. More specifically,
our results suggest that the maturity structure of debt impacts both the existence
of sunspot equilibria and the number of stable fundamental (MSV) solutions. We
demonstrate the former claim using a numerical search method from Cho (2016), and
the latter claim using solution techniques from Foerster et al. (2016). We utilize these
two approaches because they complement each other’s strengths and weaknesses nicely.
This result is significant for at least one other reason: a policy parameterization that
yields determinacy in a model with only short-term debt may yield indeterminacy or
no stable solutions in a model with more realistic debt maturity. As such, misspecified
models that abstract away from rich maturity structures may offer suboptimal policy
recommendations.
Second, determinacy generally implies E-stability in the 2-state switching model
when agents do not observe contemporaneous endogenous variables and employ one-
step-ahead decision rules. This finding extends our result in Chapter 2 and indicates
that determinacy is mostly sufficient for the stability of an equilibrium under learning.
Importantly, these results do not extend to cases where regimes lack persistence and
fiscal policy is occasionally or permanently extremely active. While these regions of the
parameter space are arguably unreasonable, we document them in this paper.
Finally, we derive the necessary and sufficient conditions for stability under
infinite horizon learning, and numerically compare infinite horizon and one-step-ahead
E-stability conditions. To the best of our knowledge, this is the first work to derive
69
conditions for infinite horizon learning in a model with Markov-switching coefficients.
Our preliminary analysis suggests that determinate equilibria are generally stable
under both infinite-horizon and one-step-ahead learning.
The paper is organized as follows: first, we explore the relationship between
debt maturity structure and the existence and uniqueness of rational expectations
equilibria; second, we explore the relationship between determinacy and E-stability
under one-step-ahead learning; third, we present the infinite-horizon learning results.
Unless otherwise stated, our analysis involves the New Keynesian model developed
in Appendix B. The following change of notation is used for convenience: ˜γ(st) =
(1− β)γ(st)
Maturity and Determinacy
We use New Keynesian model developed in B. to show that the maturity
structure of debt (specifically, the average maturity of outstanding debt, ρ ) matters
for the existence and uniqueness of rational expectations equilibrium. As in Chapter 2,
we cast our model in the form:
Xt = M(st)Et(Xt+1) +N(st)Xt−1 +Q(st)Ut
where Xt is n × 1 vector of endogenous variables, Ut is m × 1 vector of exogenous
variables that follows
Ut = ρUt−1 + t
A MSV, or fundamental, solution to the model assumes the form:
Xt = Ω(st)xt−1 + Γ(st)Ut
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Our analysis proceeds in a series of steps. First, we solve the models under
study using the forward method in Cho (2016) and search for regions of the parameter
space consistent with determinacy, indeterminacy and explosive MSV solutions across
the parameter space. Since the conditions for determinacy in Cho (2016) are only
sufficient, we must prove indeterminacy or the non-existence of stable MSV solutions
whenever applicable. To prove indeterminacy, we either search for stable sunspot
equilibria using a search routine developed in Cho (2016), or we solve for all the
MSV solutions using methods from Foerster et al (2016) and examine their mean-
square stability one by one. If multiple fundamentals solutions are stable, the model is
indeterminate. To demonstrate explosiveness, we either appeal to a necessary condition
for mean-square stability to rule out the existence of stable fundamental solutions,
or we use the aforementioned routine in Foerster et al (2016) to demonstrate that all
fundamentals solutions are explosive. We rely on such a complex scheme for checking
indeterminacy and explosiveness because neither Cho (2016) nor Foerster et al (2016)
offer necessary and sufficient conditions for determinacy.
Our approach starts with the methods in Cho (2016), which are explored
extensively in Chapter 2. The results in that paper apply exclusively to the forward
solution, whose equilibrium coefficients are acquired by taking the limit of the
coefficients on lagged endogenous and exogenous variables in the forward solved
model. Using the forward solution coefficients, Cho (2016) constructs two matrices
with spectral radii that offer sufficient conditions for determinacy: when both spectral
radii are inside the unit circle, the forward solution is the unique rational expectations
equilibrium. This last condition is only sufficient for determinacy. If one or both
spectral radii are outside of the unit circle, we must prove indeterminacy or the non-
existence of stable MSV solutions.
71
When the forward solution is stable, but does not satisfy conditions for
determinacy, we first search for sunspot equilibria of the form:
xt = Ω
∗(st)xt−1 + Γ∗(st)ut + wt (4.1)
wt+1 = Λ(st, st+1)wt−1 + V (st+1)V (st+1)′ηt+1 (4.2)
where η is mean-zero i.i.d., and V (st+1) is a matrix with orthonormal columns.
Λ(st, st+1) and V (st+1) satisfy the following conditions for each i, j ∈ {1, 2, ..., S}:
Λ(i, j) = V (j)Φ(i, j)V (j)′ (4.3)
V (i) =
S∑
j=1
pijF
∗(i, j)V (j)Φ(i, j) (4.4)
rσ(Ψ¯Λ) = rσ
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11Λ(1, 1) p21Λ(2, 1) · · · pS1Λ(S, 1)
p12Λ(1, 2) p22Λ(2, 2) · · · pS2Λ(S, 2)
...
. . .
...
p1SΛ(1, S) p2SΛ(2, S) · · · pSSΛ(S, S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
< 1 (4.5)
for some conformable matrix Φ(j), where F ∗(st) = {I − Et(M(st)Ω∗(st+1))}−1M(st).
The last condition is necessary and sufficient for mean-square stability. If that
condition is satisfied, then equation (4.1) is mean-square stable as well. Moreover,
we can show that the existence of a single mean-square stable wt is sufficient for the
existence of a continuum of mean-square stable processes. Indeterminacy follows.
Cho(2016) presents a simple method for detecting indeterminacy: minimize
rσ(Ψ¯Λ) in (4.5) subject to constraints (4.2)-(4.4) with respect to V (j) and Φ(i, j) using
the fminsearch function in Matlab. In practice, this approach is inefficient for at least
two reasons. First, our approach attempts to minimize a function with a relatively
72
flat bottom. This means that it often takes hundreds of initial guesses to find a stable
sunspot equilibrium.1 Second, initial guesses for V (j) and Φ(i, j) are arbitrary. This is
justified by a lack of a priori reasons for picking particular values of those matrices as
initial guesses.
Since we are working with a model that includes lagged endogenous variables,
we know that the fundamental solution is not typically unique. Consequently, we
might check to see if multiple fundamental solutions are stable when indeterminacy
is suspected. As we demonstrate below, it is relatively straightforward to express the
equilibrium coefficients as the solutions to a system of quadratic polynomials. Without
switching, this system can be solved using the generalized Schur decomposition
(Uhlig(1999), Klein(2000)). With switching coefficients, however, we cannot appeal
to the cited results. Instead, we follow Foerster et al (2016) and use Gro¨bner bases. A
Gro¨bner basis is a transformed system of polynomials with the same set of solutions
as the original system of polynomials under study. While there are many ways to
accomplish the aforementioned transformation, the Shape Lemma suggests that one
very useful transformation generally exists. Here, we restate Foerster et al. (2016)’s
presentation of the Shape Lemma
The Shape Lemma There exists an open dense subset S of systems of n
polynomial in n unknowns, {x1, ..., xn} such that for every system in S, there exists a
system with same roots that assumes the following form
1Each attempt to minimize the spectral radius function takes several seconds at least. We find that
for some parameterizations involving inflation reaction coefficients close to unity, over 5,000 initial
guesses must be made to find a single stable sunspot equilibrium. The issues we discuss here are also
addressed in Cho (2016)
73
x1 = q1(xn)
x2 = q2(xn)
...
0 = qn(xn)
where each qi is a univariate polynomial in n.
According to the Shape Lemma, we can solve a complicated system of quadratic
polynomials by simply solving a series of univariate polynomials. We now apply these
methods to the New Keynesian model we derive in Appendix B. To help deliver some
key results concisely, we abstract from nominal rigidity and exogenous shocks.2 Also,
we will temporarily assume that there is only one policy Markov state (st = 1 ∀t)
to illustrate the effect of ρ on the existence and uniqueness of model equilibrium.
Together, these restrictions on the model yield the following system of equations that
govern the laws of motion for inflation, πt, bond portfolios, bt, and the price of bond
bortfolios, Pt:
3
φππt = Etπt+1
Pt = −φππt + βρEtPt+1
bt = β
−1(1− γ˜)bt−1 + (1− ρ+ ργ˜)Pt + β−1πt
2Since we are only interested in studying the determinacy properties of our models, we can exclude
shocks without affecting any of our results
3In the absence of nominal rigidity, the output gap, yt = 0 ∀t
74
where the first equation combines it = φππt and the Fisher relation, and the second
and third equations describe a no-arbitrage equation and the government budget
constraint, respectively. Without exogenous shocks, a MSV solution to the above
system of equations assumes the form xt = Ωxbt−1 for each x ∈ {π, b, P} where the
Ωx solves:
φπΩπ = ΩπΩb
ΩP = −φπΩπ(1) + βρΩP
β−1(1− γ˜) = Ωb + (1− ρ+ ργ˜)ΩP + β−1Ωπ
There are three solutions to this model. The stability of a solution is determined Ωb
which assumes one of three values depending on the solution at hand: φπ, β
−1(1 −
γ˜(1)), 1/ρβ. The first solution is the active fiscal, passive monetary policy solution,
in which inflation responds to debt, and the second solution is the active monetary,
passive fiscal policy solution. The Leeper (1991) conditions for determinacy ensure
that one and only one of those two solutions are inside the unit circle (which satisfies
the condition for mean-square stability). Crucially, these roots do not depend on ρ,
and the only root that does not on ρ is greater than one for all ρ ∈ [0, 1]. This is the
sense in which the maturity structure of debt is irrelevant for determinacy in the fixed
regime model. If, however, we allow for regime change (i.e. st ∈ {1, 2} ∀ t where st
follows a 2-state Markov process), our simple model becomes:
φπ(st)πt = Etπt+1
Pt = −φπ(st)πt + βρEtPt+1
bt = β
−1(1− γ˜(st))bt−1 + (1− ρ+ ργ˜(st))Pt + β−1πt
75
A MSV solution to the above system of equations now assumes the form xt =
Ωx(st)bt−1 for each x ∈ {π, b, P} where the Ωx(st) solve:
φπ(1)Ωπ(1) = (p11Ωπ(1) + p12Ωπ(2))Ωb(1)
φπ(2)Ωπ(2) = (p21Ωπ(1) + p22Ωπ(2))Ωb(2)
ΩP (1) = −φπ(1)Ωπ(1) + βρ(p11ΩP (1) + p12ΩP (2))Ωb(1)
ΩP (2) = −φπ(2)Ωπ(2) + βρ(p21ΩP (1) + p22ΩP (2))Ωb(2)
β−1(1− γ˜(1)) = Ωb(1) + (1− ρ+ ργ˜(1))ΩP (1) + β−1Ωπ(1)
β−1(1− γ˜(2)) = Ωb(2) + (1− ρ+ ργ˜(2))ΩP (2) + β−1Ωπ(2)
As mentioned before, we find model MSV solutions using either the forward method
outlined by Cho (2016) or the Gro¨bner basis method discussed in Foerster et al (2016).
Since we are unable to find analytical solutions to the model, we set p11 = p22 = .95,
γ˜(1) = .05, γ˜(2) = −.05, β = .99, though we can obtain similar results using vastly
different parameterizations. Figure 7 shows determinacy regions in (φπ(1), φπ(2)) space
for the model with short-term debt (ρ = 0) and the model with long-term debt (ρ =
.96).4
Figure 7 makes it clear that variation in ρ can have substantial effects on the
menu of policy options available to monetary policymakers. At point A, where φπ(1) =
2.5 and φπ(2) = .4, we use Cho (2016) to prove determinacy for the short-term model,
and we use the Gro¨bner basis routine in Mathematica to show that there is no stable
real-valued MSV solution when ρ = .96. At point B, where φπ(1) = 1.5 and φπ(2) =
4ρ = .96 roughly imitates the average duration of outstanding debt observed in recent U.S. data (25
quarters)
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FIGURE 7. Determinacy and Maturity: Fisher Model
Left panel ρ = 0; Right panel ρ = .96; pmm = .95; pff = .94; γ(M) = .05;
γ(F ) = −.05. The determinate region is dark gray; the indeterminate region is light
gray; explosive region is white
.4, techniques from Cho (2016) still yield a unique stable MSV solution for the case
where ρ = 0, but the Gro¨bner basis method now yields two stable MSV solutions
when ρ = .96. At point C, we use the Gro¨bner basis routine to show that no stable
MSV solutions exist when ρ = 0, but techniques from Cho (2016) reveal that we have
sunspot indeterminacy when ρ = .96. These points demonstrate that the addition
of long-term debt to the model can greatly complicate our determinacy analysis. In
some cases, such as at point A, the addition of long-term debt reduces the number
of equilibria, while at others, such as B and C, it can expand the set of stable MSV
solutions and lead to indeterminacy.
When we back away from the assumption of flexible prices in the model, we
allow for the output gap, y, to be nonzero. In this environment, equations of the form
yt = Ωy(st)bt−1 give us the equilibrium dynamics of the output gap. Because nominal
ridigity gives us non-zero output gaps, our equilibrium coefficients now must solve the
following more difficult system of equation:
77
Ωy(1) =
∑
j
(p1j(Ωy(j) + σ
−1Ωπ(j)))Ωb(1)− σ−1φπ(1)Ωπ(1)
Ωy(2) =
∑
j
(p2j(Ωy(j) + σ
−1Ωπ(j)))Ωb(2)− σ−1φπ(2)Ωπ(2)
Ωπ(1) = κΩy(1) + β(p11Ωπ(1) + p12Ωπ(2))Ωb(1)
Ωπ(2) = κΩy(2) + β(p21Ωπ(1) + p22Ωπ(2))Ωb(2)
ΩP (1) = −φπ(1)Ωπ(1) + βρ(p11ΩP (1) + p12ΩP (2))Ωb(1)
ΩP (2) = −φπ(2)Ωπ(2) + βρ(p21ΩP (1) + p22ΩP (2))Ωb(2)
β−1(1− γ˜(1)) = Ωb(1) + (1− ρ+ ργ˜(1))ΩP (1) + β−1Ωπ(1)
β−1(1− γ˜(2)) = Ωb(2) + (1− ρ+ ργ˜(2))ΩP (2) + β−1Ωπ(2)
Figure 8 uses this model to offer intuition on how longer maturity structures
of debt held by households can generate an expanded determinacy region in the
monetary policy parameter space. Consider point D, for example, which yields an
unique equilibrium when ρ = .96 and no stable MSV solutions when ρ = 0. The reason
point D generates explosive dynamics when ρ = 0 has to do with the extent to which
monetary policy is active in Regime M.5 The fiscal policy parameterization featured
in this figure is too active to yield a stable Ricardian equilibrium. As such, increases
in debt are viewed as net wealth, and if monetary policymakers respond using active
monetary policy, explosive dynamics may result. For determinacy to obtain, monetary
policymakers must implement a consistenly passive monetary policy rule.
As we lengthen the maturity structure, however, the aforementioned active
monetary policy regime also revalues the outstanding debt stock in a manner that
5Here Regime M refers to the least active fiscal regime
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FIGURE 8. Determinacy and Maturity: New Keynesian Model
Left panel ρ = 0; right panel ρ = .96; pmm = .95; pff = .94; γ(M) = .05;
γ(F ) = −.05. The determinate region is dark gray; the indeterminate region is light
gray; explosive region is white
tempers the explosive economic response. For example, if debt increases today, which
without Ricardian equivalence raises inflation and therefore real interest rates under
active monetary policy, agents will feel wealthier due to higher real rates of return
on their net wealth, but they will also feel poorer because the interest rate increase
reduced the price of oustanding debt. The former effect is destabilizing and present
in both the long-term and short-term debt models, whereas the latter stabilizing
effect is only prevalent in the long-term model. These revaluation effects at point D
are evidently great enough to generate a MSV process with well-defined asymptotic
variance.
As revealed by all of these figures, policy configurations that generate
determinacy in the short-term debt model may yield indeterminacy in the
corresponding long-term debt model, and vice versa. Consequently, policy
recommendations stemming from models with only short-term debt, such as the
models commonly employed in the New Keynesian literature, may yield suboptimal
outcomes when fiscal policy is characterized by switching. Additionally, Figure 8
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reveals the addition of long-term debt to the model may expand regions of the policy
parameter space consistent with determinacy. Interestingly, fiscal policy does not need
to switch to generate this expanded determinacy region. In other words, maturity
allows central banks to choose from a wider menu of policy options that are consistent
with determinacy and E-stability. This result is left for further study.
Determinacy and E-stability
This section builds on recent work by Ascari et al. (2016) and Cho and Moreno
(2016), which characterizes conditions for determinacy in our two-state model with
only short-term debt. Their analyses suggest an even more complicated role for
monetary-fiscal policy coordination than in the fixed regime case where determinacy is
completely described by the simple Leeper (1991) conditions. For example, their works
separately confirm that switching between policy configurations that satisfy the Leeper
(1991) determinacy conditions is not sufficient for determinacy in the switching model.
Moreover, switching into explosive regime configurations or indeterminate regime
configurations does not preclude determinacy in the switching model. Determinacy
in the switching environment depends on the balance of policy. If, for example, fiscal
policy is very active or monetary policy is very passive in one regime, then fiscal policy
must be very passive or monetary policy must be very active in the other. Similarly,
if monetary policy is predominantly active (passive) across regimes, then fiscal policy
must be predominantly passive (active) across regimes as well.
We extend Ascari et al. (2016) and Cho and Moreno (2016) by studying the
relationship between determinacy and E-stability in this model class. This section
also extends our analysis in Chapter 1 by relaxing the assumption that agents observe
contemporaneous endogenous variables when forming expectations of next period’s
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endogenous variables. Instead, we assume that agents observe all contemporaneous
exogenous variables, but only observe endogenous variables with a lag. That is, agents
use the full history of endogenous variables up to t − 1 when forming expectations
at t.6 First, we assume that agents observe all contemporaneous variables and study
determinacy and E-stability numerically. We find that the forward solution is E-stable
for all indicated combinations of parameter values whenever the sufficient conditions
for determinacy are also satisfied:
φ(M) ∈ [1, 5] φ(F ) ∈ [0, 1] γ˜(M) ∈ (.01, .2]
γ˜(F ) ∈ [−.1, 01) κ ∈ [0, 1] β ∈ [.975, 1]
σ ∈ [0, 5] pmm ∈ [.8, 1) pff ∈ [.8, 1)
ρ ∈ [0, 1]
where pff and pmm are the probabilities of remaining within regimes F and M,
respectively. We emphasize two things about this set of parameter values. First, we
present a very conservative set. It is certainly the case that parameter values outside of
this set or consistent with determinacy and E-stability. Second, and more importantly,
the works cited in the literature of Chapter 1 all use parameter values that fall in this
set. This analysis therefore supports the conclusion that determinacy is sufficient for
E-stability for reasonable parameterizations of the model.
To help illustrate our results, we reproduce figures from Cho and Moreno
(2016) with the addition of E-stability regions (see Appendix J). As suggested earlier,
determinacy implies E-stability for the parameter values considered therein. We
also want to point out that for indeterminate regions of the parameter space, the
forward solution is generally unstable when the conditions for determinacy in the
6Stability conditions are derived in E.
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mean-stable sense fail. However, determinacy in the mean-stability sense is neither
necessary nor sufficient for E-stability. This notwithstanding, the close relationship
between mean-stability and learnability echo results in Branch, Davig and McGough
(2013). That paper, which we describe in Chapter 2, studies a class of models without
lagged endogenous variables. They find that a condition for determinacy, known as
the Conditionally Linear Determinacy Condition (CLDC) implies E-stability. It is
straightforward to show that the CLDC is equivalent to determinacy in the mean-
stable sense.
Our results also suggest that determinacy is not sufficient for E-stability when
agents do not observe contemporaneous endogenous variables. When the model
features a persistent active fiscal policy regime with very large negative values of γ˜,
as well a very short-lived regime (i.e. p11 to zero where Regime 1 is the short-lived
regime), the relationship between determinacy and E-stability breaks down. This is
true regardless of the fiscal policy stance in the short-lived regime, and is particularly
relevant when inflation reaction coefficients are relatively high in both regimes. Our
preliminary analysis suggests that most determinate equilibria become E-unstable as
γ˜(2) approaches negative infinity–this even applies to cases where all other parameters
are insider the set offered above.
Infinite-Horizon Learning
We now present results in our model when agents employ infinite horizon decision
rules. In order to accomplish this, we need to replace equations (3.3) and (3.4) in
82
Chapter 3 with the following equations:
yt = Et
∑
T≥t
βT−t[(1− β)yT+1 − σ−1(iT − πT+1) + rnT ]
πt = κπt + Et
∑
T≥t
(αβ)T−t[καβyT+1 + (1− α)βπT+1 + μT ]
All other equations, including agents’ perceived law of motion, are the same as before.
If we assume that agents understand the aggregate probability laws associated to the
evolution of inflation and output, as is assumed under rational expectations, these two
equations reduce to (3.3) and (3.4) in Chapter 3. Under adaptive learning, however, it
is not assumed that agents understand how other agents form expectations. As such,
they may lack sufficient information to deduce the fact that their expectations are the
aggregate expectations. The infinite horizon learning approach therefore avoids the
strong assumption that learning agents know the relevant aggregate probability laws.
The actual laws of motion and stability conditions are derived in Appendix G.7
To be the best of our knowledge, this is the first work to produce infinite horizon
stability conditions in a macroeconomic model with Markov-switching coefficients.
Generally speaking, we find that stability under one-step-ahead learning generally
coincides with stability under infinite horizon learning for determinate models. This
relationship between infinite horizon E-stability and one-step-ahead stability breaks
down for intermediate average debt maturities, and when the central bank employs the
following interest rate rule: it = Et−1φππt. These results echo results in Eusepi and
Preston (2011, 2013), which show that intermediate average durations of outstanding
debt, and the abovementioned policy rule are generally inconsistent with stability
7For brevity’s sake, we present the actual laws of motion for an economy without persistent shock
processes. We can relax this assumption and derive stability conditions for models with persistent
processes as well.
83
under infinite horizon learning in an analogous model with fixed regimes. This suggests
that we can dispense with algebraically burdensome infinite horizon learning models
when studying models of this form.
Conclusion
To conclude, we explore determinacy and E-stability in a New Keynesian model
with switching fiscal and monetary policy. Here we present three categories of results.
First, the maturity structure of government debt matters for determinacy and the
existence of stable equilibria in our switching model, which is not true in the analogous
fixed coefficient model. We use two numerical solution techniques to show that
maturity affects both the multiplicity of stable solutions, and the existence of sunspot
equilibria. Second, determinacy generally implies E-stability when agents do not
observe contemporaneous observable variables, but not for certain arguably unrealistic
regions of the model parameter space. Third, we present conditions for stability under
infinite-horizon learning in Markov-switching DSGE models and compare stability
under infinite horizon and one-step-ahead learning. To the best of our knowledge,
this is the first paper to derive these stability conditions in a model with switching
coefficients.
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CHAPTER V
PERFORMANCE OF SIMPLE INTEREST RATE RULES SUBJECT TO FISCAL
POLICY
Introduction
This paper examines the performance and robustness of simple monetary policy
rules in New Keynesian models with: (1) permanent or occasionally non-Ricardian
fiscal policy; (2) long-term government debt. In these models, time-variation in the
fiscal policy stance on debt is captured by Markov-switching in fiscal policy rule
coefficients. Policy performance is measured in terms of a loss function that equals
some weighted average of the variance of inflation and the variance of the output gap,
and central banks are tasked with selecting implementable interest rate rules that
minimize loss taking fiscal policy as given. While a primary goal of this project is to
identify optimal interest rate rules in models with rational expectations, we endeavor
to construct policies that are robust to parameter and model uncertainty, and that also
perform well in models with constant gain learning.
Our contributions provide answers to three questions. First, should monetary
policymakers respond to time-varying fiscal policy stances on the debt by implementing
time-varying monetary policy? A growing body of work argues that fiscal policy
stances on the debt are time-varying.1 By offering a potential answer to this question,
we may better understand the importance of precisely identifying the timing and
magnitude of fiscal policy regime shifts.
1See Bianchi (2012), Bianchi and Ilut (2014), Davig and Leeper (2006, 2011), Gonzalez-Astudillo
(2013), Kleim, Kriwoluzky, and Sarferaz (2015, 2016)
85
We find that the long-run or “global” responsiveness of fiscal policy to
government debt determines whether the optimal interest rate rule is time-varying.
When fiscal policy is “globally active” or “globally passive”, central banks typically
lack reason to track the fiscal policy stance, and should instead implement time-
invariant interest rate rules. In the case of globally active policy, we find that there
are strikingly large regions of the parameter space for which time-invariant interest rate
pegs are optimal. For “globally switching” policies that feature more balanced fiscal
regimes, the optimal policy is both time-varying and parameter-dependent. Hence,
we identify certain cases where monetary policymakers should track the timing and
magnitude of fiscal policy regime changes.
Second, do the optimal policy rules under rational expectations perform
well in models with adaptive learning? By answering this question, we hope to
identify conditions under which the optimal rational expectations policy is robust to
misspecifications about the true model of expectations formation. We find that interest
rate pegs are also optimal in globally active models with least squares learning agents.
Moreover, our learning agents in globally switching and passive models tend to prefer
inflation reaction coefficients in passive fiscal regimes that are higher than the optimal
rational expectations coefficients.
These last points are demonstrated in a constant gain learning model with
observed states, and in a novel hidden Markov model of learning. To the best of our
knowledge, this is the first paper to study a model with least squares learning agents
who jointly estimate the equilibrium law of motion and state probabilities in a self-
referential framework. In this environment, it is unclear whether optimal policies
are robust to the exclusion of contemporaneous variables from agents’ information
sets. This is because the non-linear structure of Markov-switching DSGE models
86
prevents agents with lagged information sets from learning the MSV solution.2 This
calls into question the relevant class of equilibria in our policy analysis. Fortunately,
in this environment, agents generally succeed in identifying policy states and learning
equilibrium coefficients if they receive some contemporaneous signal of current policy.
We emphasize that certain implications of the nowcasting problem just described, as
well as the broader analysis of stability under learning in hidden Markov models, are
left for future work.
Third, we ask: how does optimal monetary policy vary with the average maturity
of debt? McClung (2012b) shows that the maturity structure of debt matters for
determinacy in models with switching fiscal policy, which is not the case in analogous
models with fixed fiscal regimes. We expand on this by showing how the menu of
potential optimal policies available to central banks is particularly susceptible to
the effects of maturity when fiscal policy is globally switching. Moreover, we show
that uncertainty over fiscal policy variables can generate the kind of tradeoff between
minimizing loss and maximizing probability of determinacy and E-stability that Evans
and McGough (2007) discusses. A companion project to this paper explores whether
the monetary authority ought to make balance sheet decisions that target the optimal
debt maturity structure.
This paper most directly builds on the works of three optimal policy papers
in the New Keynesian literature. First, this paper extends Schmitt-Grohe and
Uribe (2007) which studies optimal simple monetary policy rules in fixed regime
New Keynesian models with active and passive fiscal policy. We extend this paper
by allowing for time-variation in fiscal policy stances. Second, this paper borrows
2Here, a MSV solution only depends on lagged endogenous variables, and contemporaneous
exogenous shocks including the Markov state. We show that agents with lagged information sets
can only learn equilibria that also depend on past Markov states.
87
heavily from Orphanides and Williams (2007), which studies the performance and
robustness of simple monetary rules in simple monetary models with price-stickiness
and learning agents. We build on their work by studying the robustness of optimal
rational expectations monetary policy rules to misspecifications about private sector
expectations in models with fiscal policy. Finally, we extend Chen et al (2015), which
studies joint optimal monetary and fiscal policy in a model with switching policies.
Their paper derives fully optimal joint policy rules through a Stackelberg game. In
contrast, we look for optimal simple, implementable policy rules that are robust to
misspecifications about private sector expectations. We also take the complementary
view that fiscal policymakers do not engage in a sophisticated optimization routine to
determine fiscal surpluses.
This paper also contributes to a growing literature on regime-switching policy in
the context of the Fiscal Theory of the Price Level. For instance, Davig and Leeper
(2011), Bianchi (2012, 2013), and Bianchi and Melosi (2014) estimate switching New
Keynesian models such as the models in this paper and find evidence of fiscal and
monetary policy switches in the U.S. Ascari et al. (2017) and Cho and Moreno (2016)
attempt to generalize the determinacy conditions from Leeper (1991) to environments
with switching coefficients. We extend their work by showing how maturity impacts
determinacy in the presence of fiscal policy switching, and by more exhaustively
studying how the across-regimes behavior of fiscal policymakers constrains the menu
of monetary policies consistent with determinacy. Additionally this paper borrows
heavily from, and attempts to contribute to, a learning literature involving models with
monetary-fiscal policy interactions that includes Eusepi and Preston (2011, 2012, 2013)
and Bianchi (2013) and Bianchi and Melosi (2014).
88
Finally, this paper contributes a hidden Markov model of least squares
learning to the learning literature. While many hidden Markov models of learning
are introduced by papers such as Bullard and Singh (2007) and Davig (2004),
relatively few of them study agents who jointly estimate model parameters and state
probabilities. Exceptions that do exist, such as Hansen and Sargent (2010), Hansen,
Polson and Sargent (2010), and Johannes et al (2013) invariably involve Bayesian
learners, and, almost invariably, 3 Bayesian learners in models without self-referential
feedbacks.
Our approach differs from these other approaches in many ways. First, we study
conditional least squares learners in a hidden Markov model, whereas other papers
study Bayesian learning. Second, we concern ourselves with stability of rational
expectations equilibrium in a model with hidden states. Because we can appeal to
stochastic approximation papers that study convergence properties of our conditional
least squares algorithm, future research may develop stability conditions that help
extend the intuition of Evans and Honkapohja (2001) to our regime-switching models.
Third, our agents estimate a Markov-switching VAR law of motion for endogenous
variables.
The paper is organized as follows: first, we briefly introduce the model and
estimation routine; second, we derive the optimal interest rate rules under rational
expectations in models with short-term debt; third, we discuss the optimal rules under
adaptive learning; fourth, we discuss optimal policy in the presence of long-term debt;
finally, we conclude.
3Hansen and Sargent (2010) is an exception, but their approach applies only to models that assume
a very specific structure
89
Model and Method
We consider a class of log-linearized New Keynesian models that is augmented to
include time-varying fiscal policy as in Davig and Leeper (2011), long-term maturity
structure of debt as in Woodford (2001) and Eusepi and Preston (2013). In this class
of models, private sector behavior is given by two equations of the form:
y˜t = Ety˜t+1 − σ−1(ˆit − Etπˆt+1) +
∑
d
udt
πˆt = βEtπˆt+1 + κy˜t +
∑
s
ust
where all variables are expressed as percentage deviations from steady state, y˜ is
the output gap, πˆ is inflation, and iˆ is the deviation of nominal interest rates from
the nominal interest rate target.
∑
d u
d
t and
∑
s u
s
t are demand and supply shocks,
respectively, that may include any number of exogenous processes acting on technology,
preferences, market power, etc. To introduce fiscal policy into the model, we consider
the log-linearized versions of the following equations:
Pmt bt + τt =
bt−1
πt
(1 + ρPmt ) +Gt
Pmt =
1
1 + it
(1 + ρEtP
m
t+1)
where b is real debt, G is real government spending, and τ is a surplus rule. Pˆmt is the
price of the bond portfolio at time t and ρ ∈ [0, 1] captures the maturity structure
of the government debt. While we relegate the derivation of these equations to the
appendix, the intuition behind the bond portfolio is fairly simple: the government
issues bt units of a nominal bond portfolio at time t that pays 1 unit of nominal income
at time t + 1, ρ units at time t + 2, ρ2 units at t + 3 and so forth. In other words,
90
government debt exhibits a geometrically decaying maturity structure. This structure
allows us to introduce long-term debt into our model by using a single state variable
that captures the average maturity of debt, ρ. The limiting cases of ρ illuminate how
larger values of ρ correspond to longer average maturities: when ρ = 0, all debt is short
term, and when ρ = 1, all debt is in the form of consols.
In this paper, we are primarily interested in how optimal monetary policy
depends on τ , which is characterized by a rule of the form:
τt = τ¯ (bt(1 + ρP
m
t ))
γ(st) ft
ft = f
ρf
t−1e
ft
where ft is some mean-zero i.i.d shock. τ adjusts some lump-sum component of the
government’s structural surplus in response to government liabilities, bt(1 + ρP
m
t ).
The responsiveness of this fiscal rule is determined by γ(st), which is assumed to
follow a two-state Markov process given by st. As we will discuss shortly, the value
of γ determines which model variables need to stabilize government debt. Because
the parameterization of this rule has general equilibrium implications for inflation and
output, we allow the monetary policymaker to employ a log-linearized switching rule of
the form:
iˆt = ρiiˆt−1 + (1− ρi)(φπ(st)πˆt + φy(st)y˜t) + Rt
where st is the same process that drives variation in γ, iˆ is the deviation of the
nominal interest rate from its target. To impose structure on Gt and the private sector
shocks in the model, we derive a model that is similar in spirit to the simple New
Keynesian model in An and Schorfheide (2007). Specifically, government spending is
91
given by
Gt = ξtYt
gt =
1
1− ξt
ln(gt) = ρgln(gt−1) + 
g
t
According to this specification of government spending, a time varying fraction of
output is consumed by the government. If we substitute this into the government
budget constraint, it is straightforward to see that government debt depends directly
on output. Therefore, government spending in our model introduces an output channel
that may have implications for our results. In simple cases where we want to abstract
away from this output channel, we simply set G¯ = 0 and gt = 0 and model fiscal
disturbances through ft. To bring demand and supply shocks into the model, we
assume that there are both markup shocks and shocks to household preferences. The
model derivation is left for the Appendix H, but we can write our model in the form
y˜t = Ety˜t+1 − σ−1(it − Etπˆt+1) + σ−1ρzˆ zˆt
πˆt = βEtπˆt+1 + κy˜t + μˆt
iˆt = ρiiˆt−1 + (1− ρi)(φπ(st)πˆt + φy(st)yˆt) + Rt
bˆt = β
−1(bˆt−1 − πt)− (1− ρ)Pˆm,t + β−1 G¯
b¯
yˆt
−β−1((1− β) + (1− βρ)G¯
B¯
)τˆt + β
−1gˆt
τˆt = γ(st)(bˆt−1 + βρPˆm,t) + fˆt
Pˆm,t = −iˆt + βρEtPˆm,t+1
fˆt = ρfˆ fˆt−1 + 
f
t
92
gˆt = ρggˆt−1 + 
g
t
zˆt = ρzˆ zˆt−1 + zt
μˆt = ρμμˆt−1 + 
μ
t
where z is the technology shock, μ is the cost-push shock, P is the transition
probability matrix and pij = Pr(st = j|st−1 = i). All other variables are defined
as before. In the baseline analysis, we calibrate our model so that the steady state
government liabilities, b¯(1 + ρP¯ ), equals the steady state level of output. We also
set G¯/b¯ so that G¯/Y¯ = .2 conditional of ρ. My main results do not seem to depend
on these assumptions, except in rare special cases we discuss below. Having written
the model, we are now in a position to define the fiscal policy stance on debt, and the
monetary policy rule. To that end, we use the following two definitions.
Definition 5
A fiscal policy is defined by the following parameters: {p11, p22, γ(1), γ(2)}.
Stated more thoroughly, a switching fiscal policy is fully characterized by
within-regime responses to outstanding debt, given by {γ(1), γ(2)} and by
the transition probabilities {p11, p22}.
Definition 6
A monetary policy is defined by the parameters of the interest rate rule:
{φπ(1), φπ(2), φy(1), φy(2), ρi}. In late sections, we may allow a monetary
policy to be indexed by the average maturity of debt, ρ.
We subject our policy rule to a monetary policy shock to help account for fluctuations
of i around its target value, or to capture any short-lived deviation of policy from the
rule that might be caused by dissension between policymakers. In our simple model
93
without debt, R is isomorphic to a demand shock in the IS curve. As such, we do not
need monetary policy to explore optimal monetary responses to demand shocks in a
model with Ricardian dynamics. In our model, however, R shows up in both the IS
curve and the government budget constraint and this will have implications for optimal
policy.
To help distinguish between policy regimes, we follow Leeper (1991) and describe
an “active” policymaker as one who determines inflation without concern for the
stability of debt, and a “passive” policymaker as one who directly acts to stabilize
the evolution of debt. With respect to fiscal policy, γ > 1 characterizes a “passive”
policy regime. When γ > 1, bonds evolve according to a stable autoregressive process
so that changes in i and π are not needed to keep debt from exploding. Intuitively,
γ > 1 means that surpluses adjust endogenously by an amount that is sufficient to pay
down interest and principal on new debt issuance over an infinite horizon. In such an
environment, forward looking agents recognize that any wealth effects stemming from
debt issuance will be offset by future taxes and this renders policy Ricardian. Because
fiscal policy stabilizes debt, the central bank is free to contain inflation as it pleases –
ideally by employing an active monetary policy that satisfies the Taylor Principle.
When fiscal policy is active (i.e. γ < 1), surpluses do not rise by enough to offset
any wealth effects coming from any new debt issuance and this causes consumption
and inflation to rise in response to higher debt. Any rise in inflation that results
from these wealth effects must be accommodated by central banks; if central banks
raise interest rates by more than one-for-one in response to higher inflation, they will
raise real debt service costs, leading to higher debt and therefore higher inflation in
the future, and so on. Central banks therefore must respond weakly or passively to
inflation so that inflation may erode the outstanding debt stock without generating
94
additional debt service costs. Such monetary policy is said to be “passive” and is
characterized by a violation of the Taylor Principle (e.g. φπ < 1).
Parameter values are chosen to so that the resulting model is determinate. While
there are no simple analytical conditions for determinacy in our switching model,
Leeper (1991)4 gives simple conditions for determinacy in the case of non-switching
(assuming G¯ = 0):5
TABLE 2. Leeper (1991) Determinacy Conditions
φπ > 1 φπ < 1
γ ∈ (1, β−1+1
β−1−1) determinate indeterminate
γ /∈ (1, β−1+1
β−1−1) no stable solution determinate
We say they the economy is in Regime M when φπ > 1 and γ ∈ (1, β−1+1β−1−1). and
that the economy is in Regime F when φπ < 1 and γ /∈ (1, β−1+1β−1−1). In Regime M,
fiscal policy is passive while monetary policy is active. This is the standard assumption
in most New Keynesian research. In Regime F, fiscal policy is active while monetary
policy is passive. Our model features switching between Regime F and Regime M
policy configurations. That is, our model features 2 states (i.e. S = 2) where each
state is consistent with determinacy in the analogous fixed regime model6. We solve
the model and check for determinacy in the mean-square-stable sense using techniques
from Cho (2016).
4See Table 1 for Woodford (1998a) generalization of conditions
5We assume that φπ(st) ≥ 0 for all st
6Despite the fact that each regime induces determinacy in a fixed regime model, the model with
switching between determinate regimes is often explosive or yields indeterminacy
95
We now specify the optimization problem. The central bank chooses φ = (φπ(1),
φy(1), φπ(2), φy(2), ρi, ρ)∀st to minimize:
l(φ) = var(π) + λvar(y)
The choice of φ that minimizes l is referred to as the optimal policy or “optimized”
simple interest rate rule as in Orphanides and Williams (2007). The optimal policy is
said to be time-invariant if φπ(1) = φπ(2) and φy(1) = φy(2), and is said to be time-
varying otherwise. Though the average maturity of debt appears to be a fiscal policy
tool, central banks are able to engage in large-scale asset purchase (LSAP) programs
that “twist” the maturity structure of the debt held by the public. Consequently,
we include this parameter in the central’s bank’s choice set in specific exercises. In
addition to minimizing loss, the optimal policy should satisfy two criteria: (1) the
optimal policy should implement a unique mean-square stable rational expectations
equilibrium; (2) optimal inflation reaction coefficients must be non-negative.
As implied by our discussion of the Leeper (1991) conditions, the value of the
fiscal policy parameter, γ, impacts the menu of policy options that central banks must
choose from to contain inflation. When γ < 1, fiscal policy is active and monetary
responses to inflation must be dovish; when γ is high and policy is passive, monetary
responses must be aggressive. To help characterize how fiscal policy constrains central
bankers in our model with time-varying policy stances, we employ a generalization
of these conditions similar in spirit to conditions developed by Ascari et al (2017).
Our taxonomy considers three types of fiscal policy stances: (1) “globally passive”
policies that support a stable Ricardian equivalent equilibrium; (2) “globally active”
policies that are more active than passive across regimes; (3) “globally switching” or
“balanced” policies that are neither more active nor more passive across regimes. We
96
note that both globally active and globally switching policies feature non-Ricardian
dynamics; only globally passive policies are Ricardian. The following definitions help
us characterize our three categories of switching fiscal policy and provide valuable
intuition.
Definition 7
A fiscal policy is globally passive if φπ(1) = φπ(2) = α
P for all αP > 1
yields a determinate equilibrium.
A globally passive policy can be paired with any time-invariant interest rate rule
that satisfies the Taylor Principle. In order for this to be true, fiscal policy must
be Ricardian. Otherwise, we could choose a time-invariant active monetary policy
that places debt on an explosive path. Because globally passive implies Ricardian
equivalence and vice versa we can determine if a policy is globally passive using the
following conditions passive (assuming p11 + p22 > 1):
7
(p11 + p22 − 1)h21h22 < 1 (5.1)
p11h
2
1(1− h22) + p22h22(1− h21) + h21h22 < 1 (5.2)
where hi = β
−1(1 − (1 − β)γ(i)) for i = 1, 2. These conditions, which are also
presented in Ascari et al (2017), tell us when the budget constraint implies a mean-
square stable autoregressive process for debt. If a fiscal policy satisfies these conditions,
then debt evolves according to a mean-square stable autoregressive process without
accommodation from the monetary authority and this allows monetary policymakers
to determine inflation and output in the non-policy block of the New Keynesian
model. Determinacy then requires that interest rates respond aggressively to inflation.
7We are interested in highly persistent regimes, which makes this a harmless assumption
97
Figure 9 shows regions of determinacy, indeterminacy and non-existence of stable
solutions for a model with globally passive policy. As argued in Ascari et al. (2017),
the determinacy region in Figure 9 presents something akin to the Long-Run Taylor
Principle in Davig and Leeper (2007): fiscal policy can be very active for short amount
of times, or modestly active with persistence, and the resulting equilibrium may still be
Ricardian and determinate if policy is mostly passive overall. Note that a fixed passive
fiscal policy regime is merely a special case of a globally passive policy.
FIGURE 9. Globally Passive Policy
The determinacy region is dark gray; indeterminacy is light gray; no stable solutions is
white
Definition 8
A fiscal policy is globally active if φπ(1) = φπ(2) = α
A for all α < 1 yields
a determinate equilibrium.
98
Stated equivalently, policy is globally active if a unique equilibrium exists when the
monetary authority employs a time-invariant passive monetary policy. For a permanent
passive monetary policy to be consistent with determinacy in our model, fiscal policy
must be more active than passive overall. Figure 10 shows regions of determinacy,
indeterminacy and explosiveness region for globally active fiscal policy. Note that a
fixed active fiscal policy is merely a special case of a globally active policy.
FIGURE 10. Globally Active Policy
The determinacy region is dark gray; indeterminacy is light gray; no stable solutions is
white
Definition 9
A fiscal policy is globally switching if there exists αA < 1 and αP > 1
such that neither φπ(1) = φπ(2) = α
A nor φπ(1) = φπ(2) = α
P yield a
determinate equilibrium.
99
The set of globally switching fiscal policies is the complement of the set of globally
active and passive policies. Intuitively, a globally switching policy is neither active
enough in the long-run to support all passive monetary policies nor passive enough
in the long-run to support all active monetary policies. These fiscal policies are
balanced in the sense that they are not obviously more active or passive overall. For
example, a globally switching policy may feature slow-changing, strongly active and
strongly passive regimes, or fast-changing weakly active and weakly passive fiscal
policy regimes. Table 3 offers very rough qualitative examples of how fiscal policies
may be assigned to certain categories. Figure 11 shows determinacy regions for policies
that feature highly persistent and/or strongly active and passive regimes. This figure
suggests that determinacy requires monetary authorities to be hawkish during passive
fiscal regimes and dovish during active fiscal regimes. Crucially, central banks cannot
implement time-invariant policies such as permanent interest rate pegs because the
overall fiscal policy stance is no longer mostly active or mostly passive. Figure 12
shows determinacy regions for policies that feature fast-changing and/or weakly active
and passive regimes. In these scenarios, central bankers face a meager menu of policy
options. Typically, determinacy regions for globally switching policies will resemble
either Figure 11 or 12 depending on the strength of switching regime fiscal policy
responses to debt and the persistence of regimes. Table 1 offers very rough qualitative
examples of how fiscal policies may be assigned to certain categories.
Before we present results we offer some final intuition about the fiscal policy
taxonomy. Globally active and globally passive policies can be coupled with a wide
range of time-invariant policies to deliver a determinate model, while globally switching
policies must be paired with time-varying monetary policies for determinacy. One
practical benefit of using time-invariant policies is that their implementation does not
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FIGURE 11. Globally Switching Policy (Strong)
The determinacy region is dark gray; indeterminacy is light gray; no stable solutions is
white
FIGURE 12. Globally Switching Policy (Weak)
The determinacy region is dark gray; indeterminacy is light gray; no stable solutions is
white
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TABLE 3. Overall Stance of Fiscal Policy
low pers.;
weak active
high pers.;
weak active
low pers.;
strong active
high pers.;
strong active
low pers.;
weak passive
GS GA GA GA
high pers.;
weak passive
GP GS GS GA
low pers.;
strong passive
GP GS GS GA
high pers.;
strong passive
GP GP GP GS
Strength and persistence of fiscal regime and the overall stance of fiscal policy. “pers.”
= persistence; GA = “globally active”; GP = “globally passive”; GS = “globally
switching”
require policymakers to actively track any changes in responsiveness of fiscal policy
to debt. As we show next, time-invariant policies are going to perform well in models
with globally active or passive policy.
Short-term Debt and Rational Expectations
In this section, I abstract away from long-term debt by assuming ρ = 0. We
also assume that λ = 0 so that the central bank loss function equals that variance of
inflation, but the optimal policies discussed in this section appear to perform well in
applications with small λ such as the λ weights typically found in microfounded loss
functions. Because we prioritize inflation-targeting, φy(1) = φy(2) = 0 reduces loss in
our numerical search. Accordingly, we restrict our attention the interest rate rules of
the form:
it = ρiit−1 + (1− ρi)φπ(st)πt
We present our numerical results through a series of claims contained in this section.
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Claim 1
If fiscal policy is globally passive then for all parameterizations the optimal
monetary policy response is to employ an interest rate rule with coefficients
φπ(st) = φ¯π ∀st and ρi = ρ¯i
Since determinacy requires that inflation and output be determined in the non-policy
block, the optimal simple policy rule is identical to the optimal rule used in small-
scale 3-equation models that consist of an IS curve, Phillips Curve and interest rate
rule (see Woodford (2003)). Intuitively, a globally passive policy supports a mean-
square stable autoregressive process for debt. Consequently, central banks do not need
to accommodate fiscal policy and this allows monetary policymakers to determine
inflation through the non-policy block.
When λ = 0, φ¯π → ∞ (see Woodford (2003)). Two features of this result should
be emphasized. First, the optimal policy is time-invariant despite switching in the
fiscal policy stance. Second, the monetarist equilibrium can be stable in models with
persistent active fiscal policy. For example, the monetarist equilibrium is stable when
p11 = p22 = .95, γ(1) = 5, γ(2) = 0, and G¯ = 0 despite the fact that fiscal surpluses
are entirely exogenous half of the time. See Orphanides and Williams (2007) for a
treatment of the optimal ρi when interest rates are determined in this environment
Claim 2
If fiscal policy is globally active then for all reasonable parameterizations
the monetary authorities should employ a permanent interest rate peg (i.e.
φπ(1) = φπ(2) = 0) in order to minimize the variance of inflation.
While we cannot prove Claim 3 formally, our claim relies on the following numerical
support: for all globally active policies in p11 ∈ [.9, 1], p22 ∈ [.9, 1], γ(1) ∈ [−10, 10], and
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γ(2) ∈ [−10, 10], the interest rate peg is optimal. For the posterior mean calibration
with added cost-push shocks and fiscal variables, we search over approximately 64,000
globally active policies and find that the interest rate peg is optimal for each one of
them. We repeat this analysis for alternative reasonable calibrations (alternative shock
covariance-variance structure, alternative persistence parameters for structural shocks,
σ, κ) and find that this result is robust.
We add the word “reasonable” because non-Ricardian fiscal policy presents
a tradeoff between stabilizing inflation in response to private sector shocks (i.e.
demand and supply shocks), and stabilizing inflation in response to policy shocks. In
conjunction with active fiscal policy, private sector shocks call for very high ρi (e.g.
ρi = .995) and time-varying inflation reaction coefficients, while pegs perform best
in response to monetary and fiscal policy shocks. As a result, the optimal monetary
policy in a model with globally active fiscal policy depends on the net effect of these
competing influences on inflation.
As it turns out, the private sector shock variances need to be very large relative
to policy shock variances, or the private sector shocks need to be very persistent
relative to policy shocks for the interest rate peg to be suboptimal. In particular,
monetary policy shocks need to be very small relative to other shock variances. To
illustrate this last point, we calibrate the model at the posterior mean, shut down each
shock except for one private sector shock and ask: how large does the variance of the
monetary policy shock need to be for the interest rate peg to be optimal?
When we set G¯ = 0, so that output no longer impacts debt through the budget
constraint and set ρu = .99 where ρu is the supply shock persistence term, we need for
the variance of the monetary policy shock to be greater than .014% of the variance of
the i.i.d innovation to the supply shock to get an optimal interest rate peg. For ρu =
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.9, the monetary policy shock needs to be greater than .0017% of the variance of the
same innovation to the supply shock. When we increase G¯ to .2, the interest rate peg
is optimal even when cost-push is the only shock in the model.
When we set G¯ = 0, so that output no longer impacts debt through the budget
constraint and set ρz = .99 where ρz is the demand shock persistence term, the
variance of the i.i.d. innovation to the demand shock must be less than 5 times the
variance of the monetary policy shock for the peg to be optimal. This suggests that
demand shocks are a bigger threat to the optimal interest rate peg. However, when
ρz = .9, the monetary policy shock only needs to be greater than .025% of the
variance of the i.i.d innovation to the demand shock for the peg to be optimal. Of
course, these exercises exclude fiscal policy shocks, and those shocks help to select
the peg. For example, if we shut down monetary policy shocks and set all remaining
shock parameters to their posterior mean values, we can set ρz = .99 and still have an
optimal peg.
Because intuition supports the inclusion of policy shocks in our model, and
because it is highly unlikely that an estimated model will reject the inclusion of
policy shocks, we regard cases where the peg is suboptimal as special cases involving
potentially unreasonable parameterizations of the model. We also note that pegs are
quite often nearly optimal in that loss is often close to 0% higher under the peg when
compared to the optimum. However, we have found cases where loss is as much as 3%
higher under the peg.
Non-Ricardian equilibria in our model allows us to generate some intuition
concerning the optimality of interest rate pegs. Suppose some shock (e.g. a fiscal
shock) raises the outstanding debt stock today. Since agents perceive government debt
as net wealth, this will raise consumption and inflation. This is one sense in which debt
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determines inflation under a globally active policy. The amount of inflation generated
in general equilibrium depends on monetary policy, however. As such, monetary
policy determines how inflation feeds back to stabilize debt. If an interest rate peg
is in place, a large inflation will occur today, which pushes debt in the direction of
its steady state value. On the other hand, if the central bank allows interest rates to
respond positively, then debt service costs will increase today, which creates higher
debt tomorrow and so on. The higher expected path of debt raises time t inflation
expectations, so that inflation is both higher today and propagated into the future. In
a similar thought experiment, Leeper and Leith (2016) show that the present value
of inflation will be higher under the responsive interest rate than under the peg in
their small-scale New Keynesian model. Once they solve for the equilibrium path of
inflation, it’s straightforward to show that the sharp, sudden responses of inflation
under the peg are consistent with less volatility in inflation. The complexity of our
non-linear model makes it very difficult to repeat a similar experiment in this paper.
However, Claim 3 strongly suggests that their results generalize to models with time-
varying fiscal stances – even models with recurring passive fiscal policy regimes.
To understand why pegs perform so well in globally active models, it’s important
to recall the fact that debt both determines inflation and is stabilized by inflation
in any non-Ricardian equilibrium. This means that any shock to government debt
(i.e. any shock appearing in the budget constraint) will have an affect on inflation
and output. To fix things, consider a shock which raises debt. Since agents perceive
government debt as net wealth, this will raise consumption and inflation. This is one
sense in which debt determines inflation under a globally active policy. The amount
of inflation generated in general equilibrium depends on monetary policy, however.
As such, monetary policy determines how inflation feeds back to stabilize debt. If an
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interest rate peg is in place, a large inflation will occur today, which pushes debt in
the direction of its steady state value. On the other hand, if the central bank allows
interest rates to respond positively, then debt service costs will increase today, which
creates higher debt tomorrow and so on. The higher expected path of debt raises time
t inflation expectations, so that inflation is both higher today and propagated into the
future. In a similar thought experiment, Leeper and Leith (2016) show that the present
value of inflation will be higher under the responsive interest rate than under the peg
in their small-scale New Keynesian model. Once they solve for the equilibrium path
of inflation, it’s straightforward to show that the sharp, sudden responses of inflation
under the peg are consistent with less volatility in inflation. The complexity of our
non-linear model makes it very difficult to repeat a similar experiment in this paper.
However, Claim 3 strongly suggests that their results generalize to models with time-
varying fiscal stances – even models with recurring passive fiscal policy regimes.
While pegs are broadly consistent with stable inflation in our non-Ricardian
model, φπ(1) = φπ(2) = 0 does not guarantee determinacy for all fiscal policies
that violate the abovementioned conditions (i.e. mean-square stable common-factor
sunspots may exist). In particular, indeterminacy obtains if fiscal policy is too passive
in one regime. For example, if p11 = p22 = .95, γ(1) = 2, γ(2) = −5 then fiscal policy is
sufficiently active for the interest rate peg to deliver determinacy. If, however, γ(1) = 2
is replaced by γ(1) = 5, then policy is too passive in regime 1 for the interest rate
peg to deliver determinacy. These are globally switching equilibria and determinacy
requires that they be paired with special optimal equilibria.
Claim 3
Optimal GS monetary policies are time-varying and parameter dependent
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For example, the optimized inflation reaction coefficients for the policy given by p11 =
p22 = .95, γ(1) = 5, γ(2) = −5 and for the policy given by p11 = p22 = .95, γ(1) = 2,
γ(2) = 0, are (φπ(1), φπ(2), ρi) = (3.3, 0, .99) and (φπ(1), φπ(2), ρi) = (2.97, .73, .99),
respectively. These particular optimized coefficients come from the expected posterior
loss exercise we introduce in the next paragraph. Since the optimized policy favors
large swings in inflation responses, policy inertia is undesired (i.e. ρi = 0 is optimal).
While results in the globally active and globally passive settings hinge only on
fiscal policy parameters (for reasonable parameterizations of shock processes), the
optimal policy in globally switching models depends on any model parameter that
impacts determinacy conditions. This means that we need to choose parameter values
in order to draw conclusions about optimal policy in the globally switching models.
To help inform our selection of model parameter values, we estimate the following
truncated model using Bayesian techniques:8
yˆt = Etyˆt+1 − σ−1(it − Etπˆt+1) + (1− ρg)gˆt + σ−1ρzˆ zˆt
πˆt = βEtπˆt+1 + κ(yˆt − gˆt) + μˆt
iˆt = ρiiˆt−1 + (1− ρi)(φππˆt + φyyˆt) + Rt
gˆt = ρggˆt−1 + 
g
t
zˆt = ρzˆ zˆt−1 + zt
μˆt = ρμμˆt−1 + 
μ
t
y˜t = yˆt − gˆt
where the final equations reflects the fact the natural rate of output equals gˆt in our
model, thus allowing us to glean information about government shocks from estimates
8See Appendix I. for tables containing information about our prior and posterior distributions
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of the IS and Phillips Curves. Using the separated partial means test to test the
convergence of our estimates, we believe that the best results emerge when we place
dogmatic priors over ρu and σμ and estimate only the non-policy block with the
interest rate. Since this exercise intends to consider counterfactural policies, estimates
of the underlying fiscal policy stance are unnecessary. However, estimates pertaining to
the shock processes and other private sector coefficients help to discipline our analysis
towards parameter regions that agree better with the data. In extensions of the present
work, we intend to estimate a fuller DSGE model.
After sampling from the posterior distribution, we follow Cogley et al. (2011) and
compute the expected posterior loss associated with each policy parameterization. A
Monte Carlo average of the following expected posterior loss function is computed:
∫
l(φ)P (θ˜|Y )dθ˜
where θ˜ = (κ, σ, ρg, ρz, ρμ, σg, σz, σμ, σr). For the previously mentioned case where
γ(1) = 5 and γ(2) = −5, p11 = p22 = .95, the optimal policy is given by φπ(1) = 3.33,
φπ(2) = 0, ρi = 0.99. Intuitively, monetary policy should be active in the passive fiscal
regime, and very passive in the active fiscal regime.
To sum up, the optimal policy response depends on whether fiscal policy is
globally active, globally passive or globally switching. If fiscal policy is globally passive,
then optimal policy is time-invariant and calls for large inflation reaction coefficients. If
fiscal policy is not globally passive, then interest rate pegs deliver the fundamental
solutions that minimize loss. If, however, fiscal policy is globally switching then
interest rate pegs lead to indeterminacy. In those settings, and those settings alone,
the optimal policy is time-varying.
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TABLE 4. Optimal Inflation Reaction Coefficients under RE when ρ = 0
Type (γ(1), γ(2) optimal (φπ(1), φπ(2))
GP (5, 0) (∞, ∞)
GS (5, −5) (3.33, 0)
GS (2, 0) (2.73, .72)
GA (2, −5) (0, 0)
Adaptive Learning
In this section we relax the assumption that agents form rational expectations,
and study policy performance in a model where agents attempt to learn the
equilibrium law of motion for the model’s endogenous variables, and form forecasts of
future variables according to an estimated perceived law of motion. Relative to rational
expectations models, models with learning agents feature instabilities that arise from
agents’ forecast errors. Specifically, agents’ forecast errors affect the model’s data
generating process, thereby changing future data points and future estimates of the
model’s coefficients. This self-referential feature of our model fundamentally changes
the way in which policy interacts with expectations to contain inflation and output.
As such, the inclusion of adaptive learning in our analysis provides an important
robustness check. Our main conclusion is that the optimized simple policy rules
studied under rational expectations are robust to misspecifications of the underlying
model of expectations employed by agents. That is, the optimized policy rules under
rational expectations are optimal or nearly optimal in models with adaptive learning
agents, with exceptions in the case of globally switching policy.
We present our results in two sections.First, westudy a learning model in which
agents observe the model’s endogenous variables (with a reasonable lag), exogenous
driving processes and the underlying Markov state that drives variation in fiscal
and monetary policy rules. When agents observe the underlying Markov state, they
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can easily update parameter estimates using a within-state recursive least squares
algorithm that resembles the least squares algorithm developed and discussed in Evans
and Honkapohja (2001). While this learning specification provides a natural first step
away from the rather strong assumption that agents form rational expectations, it still
assumes that agents easily observe something an applied econometrician would not:
the underlying state of policy. We therefore develop a model of learning that backs
away from this assumption.
In our hidden Markov model of adaptive learning, agents estimate the same
perceived law of motion, but do not observe the underlying Markov state (i.e. agents
find themselves in a hidden Markov model). Because agents do not observe the stance
of fiscal and monetary policy, they cannot use the recursive least squares algorithm
employed in the learning model with observed states. Instead, we allow agents to use
the recursive MLE algorithm and the recursive conditional least squares algorithm
developed in Krishnamurthy and Yin (2002) and LeGland and Mevel (1997) to update
parameter estimates after observing the model’s endogenous variables and exogenous
driving processes. We emphasize two main results from this section. First, this is,
to the best of our knowledge, the first paper to study least squares learning agents
who estimate a Markov-switching autoregressive equilibrium law of motion in a self-
referential model with hidden Markov states.9 That is, previous research does not
jointly estimate perceived laws of motion and the Markov state probabilities. We
therefore regard this section as a springboard for future research on the use of hidden
markov models of learning. Second, the exogeneity10 of policy rule coefficients makes it
possible for agents to infer the underlying state with some reasonable accuracy. Hence,
9See literature review for details on related papers
10By “exogeneity” we mean that the coefficients in the equilibrium law of motion for policy
variables τ and i do not depend on agents’ beliefs
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model dynamics in a model with learning agents and hidden states are very similar
to model dynamics in the learning model with observed states. We conclude that it is
potentially reasonable to assume that agents observe policy switches, but it remains to
be seen whether with assumption is strong in models where Markov-switching affects
non-policy variables such as trend growth.
Observed Markov States
We now develop a model of learning in which agents observe the underlying
Markov state (i.e. they observe the underlying policy stance). The model dynamics are
still given by an actual law of motion, which can be constructed from the log-linearized
equilibrium conditions given previously:
xt = A(st)Etxt+1 +B(st)xt−1 + C(st)zt (5.3)
where x = (π y i b τ P )′ and z = (g zˆ R μ f ). Under rational expectations, agents
know the full structure given by (5.3) and can solve for the rational expectations
equilibrium. Under adaptive learning, however, agents do not know (5.3) and are
therefore incapable of computing the true mathematical expectations of tomorrow’s
variables. Despite the fact that agents are not fully rational, we still endow agents
with sophisticated beliefs about the law of motion governing inflation, output, etc., in
equilibrium. Specifically, we give agents the following perceived law of motion (PLM):
xt = a(st) + b(st)xt−1 + c(st)zt (5.4)
Notice that this perceived law of motion has the same functional form as the rational
expectations equilibrium law of motion, which implies that agents may conceivably
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learn the rational expectations equilibrium law of motion if their estimates of a(st),
b(st), and c(st) converge to their rational expectations values (i.e. if a(st) → 0n×1,
b(st) → Ω(st) and c(st) → Γ(st) for all st). If a rational expectations equilibrium
can be learned, it is said to be “stable under learning” or “expectationally stable”
(“E-stable”) (see McClung (2016, 2017b) for more about E-stability in this class of
models). E-stable rational expectations equilibria are easier to rationalize in the sense
that adaptive learning supports a coordination story for their realization, and in the
sense that they are robust to the unreasonably strong assumptions that undergird
rational expectations. Our task in this section is to study the volatility of inflation and
output when agents beliefs about the structure of the economy are close to the unique
rational expectations equilibrium implemented by the monetary policy rule.
To make our model of learning fully operational, we must specify agents’
information set, their estimation strategy, and the full process through which
expectations interact with predetermined variables to pin down the endogenous
variable values. We begin by specifying agents’ time t information set, It, which
includes all past observations of x, and all past and current observations of z and s.
Formally: It = {yt−1, yt−2, . . . , y0; zt, zt−1, . . . , z0; st, st−1, . . . , s0}. We could exclude zt
from the information set (i.e. only include past values of z) and obtain similar results.
Using observations in It, agents will update their estimates of the coefficients in (5.4)
using the following within-regime learning algorithm:
Φ(st)st = Φ(st)st−1 + ψstR(st)
−1
st ut(xt − Φ(st)′st−1ut) (5.5)
R(st)st = R(st)st−1 + ψst(utu
′
t −R(st)st−1) (5.6)
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where Φ(st)st = (at(st), bt(st), ct(st))
′ are the time-t estimates of regime st coefficients,
ut = (1, x
′
t−1, z
′
t)
′, and st is the number of realizations of state st up until and including
time t. Alternatively, we might use a learning algorithm that estimates a dummy
variable regression where elements in u are interacted with dummy variables that
take on values of 1 or 0 depending on the underlying Markov state. The last feature
of the algorithm we need to define is the gain parameter, ψst . Intuitively, ψst attaches
a weight to each new observation and therefore determines the extent to which new
information impacts parameter estimates. If we give each observation equal weight by
setting ψ = 1/tst , where tst is the number of realizations of st up until time t, then
our learning algorithm becomes the conditional recursive least squares estimator of Φ.
Clearly, as t → ∞ the estimates converge to some value, which may be the rational
expectations equilibrium coefficients depending on initial beliefs and the E-stability
of the equilibrium under study. Alternatively, we might allow agents to give more
weight to recent observations by using a constant gain parameter,ψ = ψ¯, where ψ¯ is
some scalar. In constant gain learning algorithms, beliefs will never converge, but may
converge to some distribution centered on the rational expectations equilibrium. These
algorithms are considered appropriate in settings where agents may expected structural
changes in the model, or in settings where agents simply value recent data more than
older data.
Having specified the learning algorithm, we now outline the sequence of events
that lead to an equilibrium at time t:
1. Agents observe zt and st and add those to their information sets.
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2. Using It and time t−1 estimates at−1(st), bt−1(st), ct−1(st). Agents form forecasts,
Eˆtxt+1:
Eˆtxt+1 = (pst1a(1)t−1 + pst2a(2)t−1) +
(pst1b(1)t−1 + pst2b(2)t−1)(a(st)t−1 + b(st)t−1xt−1 + c(st)t−1zt) +
(pst1c(1)t−1 + pst2c(2)t−1)ρzt
3. xt is generated from the actual law of motion, (5.3), which gives us time t
endogenous variables as a function of beliefs and predetermined variables
4. Agents observe xt and add it to their information sets
5. Agents use (5.5)-(5.6) to update their estimates
6. Forward t to t+ 1 and repeat steps 1-5.
Before studying policy performance in this environment, we first use a decreasing
gain parameter see whether agents can learn the rational expectations equilibrium
corresponding to each of the parameterizations we consider. Initial beliefs about a(st),
c(st) for st = 1, 2 are set to zero, while initial beliefs about b(st) are perturbed around
Ω(st).
11 For all parameterizations we consider here, beliefs eventually converge to their
rational expectations equilibrium values. Figure 13 illustrates the convergence of beliefs
for the posterior mean calibration with γ(1) = 5, γ(2) = −5, φπ(1) = 3, φπ(2) = 0.
In this figure, as well Figure 16 we plot the difference of actual beliefs and rational
11We set initial beliefs about the VAR coefficients away from zero (but still far from their REE
values) to help improve the rate of convergence of beliefs to the REE. We also want to mention that
beliefs may not converge to the rational expectations equilibrium for all initial values; E-stability is
a local stability concept that only applies to beliefs that are in some neighborhood of their potential
convergence points.
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expectations equilibrium beliefs over time (i.e. a value of 0 means that beliefs equal the
rational expectations equilibrium).
FIGURE 13. Coefficient Estimate Errors and Observed State Learning
The left-hand column features the VAR-coefficients on independent variable lagged
debt in regime 1; right-hand column features the VAR-coefficients on independent
variable lagged debt in regime 2. Notice that beliefs are held fixed when they
correspond to an inactive state (e.g. notice the flat, “mesas” in the state 2 coefficients
between t=50 and t=150).
To help better understand the impact that learning has on model dynamics, we
study policy performance in a model with constant gain learning and a gain parameter
equal to .01.12 In such a model, we cannot compute the unconditional variance of
12The learning algorithm is also augmented with a ridge correction mechanism as in Slobydan
and Wouters (2012), and projection facility that prevents estimates from updating if the updated
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TABLE 5. Optimal Coefficients under Adaptive Learning
Type (γ(1), γ(2))
optimal RE coeff.
(φπ(1), φπ(2), ρi)
optimal AL coeff.
φπ(1), φπ(2), ρi)
Projection
Facility
(per 100,000)
GS (5, −5) (3.3, 0, .99) (3.68, 0, .99) 110
GA (2, −5) (0, 0, 0) (0, 0, 0) 86
The larger inflation reaction coefficients under learning echoes a result from
Orphanides and Williams (2007). 4 is the largest inflation reaction coefficient used
in this particular numerical search. Despite the small gain parameter and infrequent
use of the projection facility, the model is frequently unstable for ψ > .02
inflation and output. We therefore approximate the variance of inflation and output by
simulating the model for 100,000 periods and computing sample variances.13 Because
these simulations are more computationally intensive, we do not compute expected
posterior losses. Instead, we set non-policy parameters equal to their posterior mean
(with added cost-push shock), and make inferences based on this model calibration.
Otherwise, the procedure for measuring performance is the same as the procedure
used in the RE model: we search over monetary policy parameters and find the set
of interest rate rule coefficients that minimizes the variance of inflation and output.
Table 5 presents our main findings.
Unobserved Markov States (Hidden Markov Model)
The learning model with observed states provides valuable evidence that the
optimized simple rules under rational expectations are robust to misspecifications of
private sector expectations. However, that model makes one potentially unreasonable
parameters imply a Markov-switching VAR that is not mean-square-stable. Intuitively, the projection
facility formalizes the notion that agents reject unstable models. We invoke the projection facility and
ridge correction mechanism in far less than 1% of simulated periods
13Each simulation uses the same 100,000 realizations of shocks. We do this to help mitigate the
potential for large outlier shocks to bias our sample variances.
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assumption: agents observe the state of fiscal and monetary policy. In practice, applied
econometricians do not observe the stance of fiscal and monetary policy. Instead,
econometricians use techniques developed in papers such as Hamilton (1989) to
identify the probable state of the economy at any point in time. Because a lot of
adaptive learning research begins with the premise that our models’ agents should
be no more informed and rational than the econometricians among us, we endeavor
in this section to remove st from the information set, It, and study the model-implied
dynamics of inflation and output. We refer to this new model as the hidden Markov
model of learning. Before deriving the hidden Markov model of learning, we emphasize
that self-referential feedback in this model not only poses the risk of destabilizing
agents’ beliefs about model coefficients; forecast errors act on both future coefficient
estimates and agents’ inferences about the underlying state. One may therefore expect
additional expectations-induced volatility in this model.
As it turns out, the structure of our model makes it possible for agents to
infer the underlying state with reasonable accuracy so that the removal of Markov
states from agents’ information set only raises the volatility of inflation and output
slightly. This last point is partly explained by an argument made in Bianchi (2013)
which states that fully rational agents can perfectly infer today’s state if they observe
contemporaneous and past x, z. Their argument relies on the fact that rational agents
know all of the S within-regime systems of equations (i.e. xt = Ω(st)xt−1+Γ(st)zt) that
may determine xt. All agents in their model need to do to perfectly infer the state is
compute each of the S equations until they find the correct system of equations. Their
argument does not apply to our framework; if agents hold incorrect beliefs about the
economy – as they always do in a model of constant gain learning, or before beliefs
converge – they may make horrible inferences about the state of the economy. Despite
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this limitation, the equilibrium coefficients of the policy rules are exogenous to beliefs,
which makes it easy for agents to learn the rational expectations equilibrium law
of motion for fiscal surpluses and infer from it the underlying state of the economy
with reasonable but far from perfect accuracy. We emphasize that other equilibrium
coefficients do depend on agents’ beliefs, so that our model is still self-referential.
As before, agents beliefs about the law of motion for endogenous variables is
given by the PLM in (5.4). In what follows, we consider two information structures.
First, we assume that It = {yt−1, yt−2, . . . , y0; zt−1, . . . , z0}. After examining the
potential convergence points of beliefs, and pointing out the exogeneity of the surplus
law of motion, we then add surpluses, τt, to It and demonstrate that agents’ beliefs
can converge to the rational expectations equilibrium. Under both information
structures, agents do not observe st, which implies that they cannot use (5.5)-(5.6)
to update their beliefs. To get around the difficulty presented by the hidden Markov
process, we rely on techniques from Krishnamurthy and Yin (2002) and LeGland
and Mevel (1997), which present “online” or recursive algorithms for learning the
coefficients of an exogenous Markov-switching autoregression. Specifically, we use the
recursive maximum likelihood estimator (RMLE) from both papers, and the recursive
conditional least squares estimator (RCLS) from LeGland and Mevel (1997). While
newer alternatives to these algorithms exist outside of the stochastic approximation
literature, we rely on these papers because they present convergence results that may
prove useful in extensions of the current analysis.
The algorithms described in both papers make inferences about the coefficients,
Φ(st), and the Markov process, st, using two related recursive processes. First, agents
make inferences about st using a prediction filter of the form introduced by Hamilton
(1989). To develop this filter we first define within-regime conditional densities for x,
119
fst = f(xt|xt−1, xt−2, ...., zt, zt−1, ...., st; Φ(st)t−1). In a model with normally distributed
i.i.d innovations to our exogenous driving process, fst assumes the following form:
fst = (2π)
−t/2|Σ|−.5exp{−.5(xt − μ(st)t−1)′Σ−1(xt − μ(st)t−1)}
where μ(st)t−1 = at−1(st) + bt−1(st)xt−1 + ct−1(st)zt and Σ is the covariance-
variance matrix for the i.i.d innovations to z. To make future calculations easier, we
define the following matrices:
ft = (f1t, f2t . . . , fSt)
′
Ft = diag(f1t, f2t . . . , fSt)
Let pˆi,t|t−1 = Pr(st = i|It), and pˆt|t−1 = (pˆ1t|t−1, pˆ2t|t−1, . . . , pˆSt|t−1)′. pˆt follows the
recursion:
pˆt+1|t =
P ′Ftpˆt|t−1
f ′t pˆt|t−1
(5.7)
where it is assumed that agents know the true transition probabilities in P . The
prediction filter in the last equation completely describes how agents recursively
compute their predictions for today’s state. Because inferences about st are made
prior to time t, agents can, at best, infer st−1 perfectly. As we show below, this
feature of our model makes it impossible for agents’ beliefs to converge to the rational
expectations equilibrium studied in previous sections, and is the primary reason why
we argue for the addition of τt to It. The second recursive process in the algorithms
presented by Krishnamurthy and Yin (2002) and LeGland and Mevel (1997) updates
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the parameter estimates, Φ(st), according to:
Φt = Φt−1 + γS(xt, It; Φt−1) + tMt
where Φt is a k × 1 vector14 that contains the elements of Φ(st) for all st, γ is the
gain parameter and Mt is a correction term (i.e. we use a projection facility in our
implementation of their algorithms). Let Φlt denote the l-th element of Φt. The
function S(xt, It; Φt−1) is the only thing that varies across the two algorithms we use in
the paper. For the RMLE algorithm, S(xt, It,Φt−1) is given by the following equations:
S(xt, It,Φt−1) = (S1(xt, It,Φt−1), . . . , Sk(xt, It,Φt−1))′
where
Sl(xt, It,Φt−1) =
f ′tω
l
t
f ′t pˆt|t−1
+
(∂f ′t/∂Φ
l
t)pˆt|t−1
f ′t pˆt|t−1
(5.8)
for all l ∈ {1, ..., k} and ωlt = ∂pˆt|t−1∂Φlt . We update ω
l
t recursively as follows:
ωlt+1 = R1tω
l
t +R2t (5.9)
where
R1t = P
′(I − Ftpˆt|t−11
′
s
f ′t pˆt|t−1
)
Ft
f ′t pˆt|t−1
R2t = P
′(I − Ftpˆt|t−11
′
s
f ′t pˆt|t−1
)
(∂Ft)/(∂Φ
l
t)pˆt|t−1
f ′t pˆt|t−1
14In our model with S = 2, n endogenous variables and m endogenous variables, k = 2(n(n + 1) +
nm)
121
Equation (5.7), (5.8) and (5.9), plus initial conditions, give us the RMLE algorithm.
To derive the RCLS we only need to change our definition of Sl(xt, It−1,Φt−1) as
follows:
Sl(xt, It,Φt−1) = (φΦt−1(xt − φ′Φt−1 pˆt|t−1))′ωlt + (
∂φΦt−1
∂Φlt−1
(xt − φ′Φt−1 pˆt|t−1))′pˆt|t−1 (5.10)
where φΦt−1 is a matrix that collects the conditional mean for each state (i.e. μ(s)t−1
for each s ∈ {1, . . . , S}). Before outlining the events leading to a temporary
equilibrium, we emphasize that this algorithm is very similar to the algorithm
presented in (5.5)-(5.6). Specifically, if agents observe the state so that ω becomes a
vector of zeros (since (I − Ftpˆt|t−11′s
f ′t pˆt|t−1
→ 0S), and they replace pˆt|t−1 with pˆt|t = (1 0)′
or pˆt|t = (0 1)′ to reflect this knowledge, then this algorithm becomes the recursive
estimator used in the model with observed states with R(st)st = I. We can now outline
the sequence of events that lead to an equilibrium at time t:
1. Agents update information sets.
2. Using It and time t−1 estimates at−1(st), bt−1(st), ct−1(st). Agents form forecasts,
Eˆtxt+1:
Eˆtxt+1 = (pˆ1t|t−1p11 + pˆ2t|t−1p21)a(1) + (pˆ1t|t−1p12 + pˆ2t|t−1p22)a(2) +
pˆ1t|t−1p11b(1)(a(1) + b(1)xt−1 + c(1)zt) +
pˆ1t|t−1p12b(2)(a(1) + b(1)xt−1 + c(1)zt) +
pˆ2t|t−1p11b(1)(a(2) + b(2)xt−1 + c(2)zt) +
pˆ2t|t−1p11b(1)(a(2) + b(2)xt−1 + c(2)zt) +
((pˆ1t|t−1p11 + pˆ2t|t−1p21)c(1) + (pˆ1t|t−1p12 + pˆ2t|t−1p22)c(2))ρzt
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3. xt is generated from the actual law of motion, (5.3), which gives us time t
endogenous variables as a function of beliefs and predetermined variables
4. Agents observe xt and add it to their information sets
5. Agents use (5.7), (5.8), (5.9), or (5.7), (5.9), and (5.10) to update their coefficient
estimates and prediction filter
6. Forward t to t+ 1 and repeat steps 1-5.
Before presenting results, it is important to note that our hidden states information
structure prevents agents from learning the rational expectations equilibrium studied
in all previous sections. This is because agents only form pˆt using t − 1 information.
Hence, if agents perfectly infer st−1 – which is the best they can do – they still hold the
following beliefs about st: pˆt|t−1 = (pst−11, pst−12)
′ < (1, 1)′. In this best case scenario,
agents’ beliefs about the VAR coefficients, b(st), will not converge to a solution of
(5.3). If, instead, agents allow their beliefs about PLM coefficients to depend on both
st and st−1 then this information structure may allow agents to learn solutions to the
following fixed point condition:
b(st, st−1) = A(st)
2∑
j=1
2∑
h=1
pst−1jpjhb(h, j)b(j, st−1) +B(st) (5.11)
These solutions, which we refer to as history-dependent equilibria, do solve (5.3).
However, they do not satisfy the following fixed point condition:
b(st) = A(st)(pst1b(1) + pst2b(2))b(st) + B(st) (5.12)
which is a necessary condition for solutions of the form, b(st). While beliefs are no
longer consistent with the rational expectations equilibria we examined up until now,
123
we nonetheless find that beliefs can converge15. Hence, while beliefs never converge to
the rational expectations equilibrium, they may nonetheless be stable over time and
converge to values that may be relatively close to the original rational expectations
equilibrium.
To identify potential convergence points consistent with (5.11), we use the
Groebner basis approach from Foerster et al (2016). We then explore issues of
uniqueness and E-stability pertaining to this class of equilibria. Initial evidence
suggests that policy parameters widely associated with determinacy in the preceding
analysis may admit multiple mean-square stable history dependent equilibria that
satisfy the fixed point condition in (5.11). Moreover, these equilibria do not appear
to be stable under learning. Since this class of equilibria is arguably relevant in settings
where agents cannot observe contemporaneous variables, we intend to further explore
these issues of uniqueness and expectational stability in future work.
Figure 14 plots pˆ1 over time. In our calibration p11 = .95 so that oscillation in
their beliefs between .05 and .95 implies that they’re inferring st−1 almost perfectly. To
better understand how agents so successfully infer the underlying state of the economy,
despite initial incorrect beliefs about the structure of the economy, we redefine x =
(x˜, τ)′ where x˜ = (y, π, i, b, P )′ and point out that the actual law of motion for x (after
beliefs are substituted in) may be written as:
⎛
⎜⎝x˜t
τt
⎞
⎟⎠ =
⎛
⎜⎝Ω˜(st; Φt−1)
Ωτ (st)
⎞
⎟⎠
⎛
⎜⎝x˜t−1
τt−1
⎞
⎟⎠+
⎛
⎜⎝Γ˜(st; Φt−1)
e′6
⎞
⎟⎠ zt (5.13)
where Ωτ (st) = (0 0 0 γ(st) 0 0), and e
′
6 = (0 0 0 0 0 1). Clearly, the evolution of τt
is only endogenous to beliefs through bt−1; the coefficients governing the evolution of
15Even for constant gain parameters beliefs appear to converge to a distribution around a fixed
point
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τ are exogenous, which suggests that agents will quickly learn the law of motion for τ
and then make accurate inferences for pˆt that rely on the marginal density:
f τst = f(τt|xt−1,Φt−1) (5.14)
The marginal density in (5.14) is so essential for correct inference of st−1 that we
can redefine our prediction filter using only the marginal densities for surpluses and
get results that are nearly identical to the results displayed in Figure 14). The fact
that surpluses are determined at the beginning of t (i.e. all shocks and bt−1 have
been realized by beginning of t, so that τt is fixed before agents form expectations),
begs an important question about timing: should agents be able to observe τt at
the beginning of t? That is, should It include τt? If agents observe τt at t, they may
be able to perfectly infer st. This allows for agents to learn solutions of the fixed
point problem given by (5.12) to coincide, i.e. so that agents may actually learn the
rational expectations equilibrium under study. To support this idea numerically, we
first redefine the prediction filter:
f τt = (f
τ
1t, f
τ
2t . . . , f
τ
St)
′
F τt = diag(f
τ
1t, f
τ
2t . . . , f
τ
St)
pˆτt|t =
F τt pˆ
τ
t|t−1
f ′τt pˆτt|t−1
pˆτt+1|t = P
′pˆτt|t
Now agents use pˆτt|t instead of pˆt|t−1 when forming expectations at time t. As shown
in Figure 15 agents can now infer the current state very effectively, which allows them
to learn the rational expectations equilibrium under study in the previous section, as
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demonstrated by Figure 16. In Figure 16, we initialize beliefs away from the rational
expectations equilibrium16, set ψ = t−2/3 (as in LeGland and Mevel (1997)) and
estimate the model using the RCLS algorithm. We also use a projection facility that
prevents agents from accepting a mean-square-unstable PLM, but this facility is
invoked in far less than .1% of periods simulated. Compared to Figure 13, the rate of
convergence is slow under RCLS, but this may be driven the errors in the prediction
filter (Figure 15) and the large decreasing gain parameter t−2/3. We find that the
optimal policy results in the observed states learning section generalize to the hidden
Markov model of learning.
FIGURE 14. Estimating the Policy State (Lagged Information)
Blue line is pˆ1,t|t−1; black line equals 1 if st = 1 and 0 otherwise
16As seen in the third subplot in the second column of Figure 16, initial beliefs about the
dependence of i on b in regime F are unintentionally close to 0. Our results do not depend on this
initial belief.
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FIGURE 15. Estimating the Policy State (Contemporaneous Information)
Blue line is pˆ1,t|t; black line equals 1 if st = 1 and 0 otherwise
FIGURE 16. Coefficient Estimate Errors in Hidden Markov Model Learning
The left-hand column features the VAR-coefficients on independent variable lagged
debt in regime 1; right-hand column features the VAR-coefficients on independent
variable lagged debt in regime 2.
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Long-term Debt
In this section, we relax the assumption that ρ = 0 and introduce long-term debt
into our model. While this innovation helps to bring our model closer to reality, it also
creates a debt revaluation channel through which monetary and fiscal policy interact
to affect agents’ perceptions of bond wealth in non-Ricardian economies. This debt
revaluation works as follows: if interest rates are reduced (increased), then the price of
outstanding debt, given by Pˆm, increases (decreases) and this positively (negatively)
affects agents’ perception of their own net wealth. This revaluation channel can often
lean against the wealth effects created by movements in debt service costs, a tendency
demonstrated in a host of papers including McClung (2017a).
In addition to the creation of a revaluation channel, the introduction of long-term
debt can alter the menu of monetary policies that induce a determinate equilibrium
when fiscal policy switches and is non-Ricardian (see McClung (2017b)). The fact that
maturity matters for determinacy in our simple switching DSGE model is a novel
result insofar as the average maturity of debt does not matter for determinacy in
the corresponding fixed regime model (see Jin (2013), for example). To illustrate the
impact that maturity has on determinacy consider figures 17 and 18.
The fact that maturity matters for determinacy complicates the policymaker’s
problem in at least two ways. First, the policymaker now has an incentive to identify
the steady state average maturity given by ρ when fiscal policy is non-Ricardian.
Without knowledge of ρ, the policymaker cannot properly identify the menu of
policies that induce a unique equilibrium, which may prevent them from finding the
optimized policy. Second, the policymaker now has an incentive to consider balance
sheet decisions that affect the value of ρ when solving their optimization problem. In
our model, the relevant measure of government debt is government debt held by the
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household, not purely the debt issued by the fiscal authority itself. As such, central
banks can impact the maturity structure of debt held by households by engaging
in Operation Twist-style policies in which households and the monetary authority
exchange short-term debt for long-term debt. Figure 17 illustrates a case where
monetary policymakers may realize an incentive to lengthen the maturity debt held
by households, while Figure 18 illustrates the opposite case. We hope to use ρ as a
proxy for these debt operations by adding ρ to the central bank’s choice set.
The search for an optimal ρ is further complicated by the fact that we face
uncertainty over the true value of ρ.17 We might address this uncertainty by assigning
a prior distribution to ρ, adding ρ to θ then estimating the model using Bayesian
techniques. Using simple priors over ρ we can generate a tradeoff between expected
posterior loss and the probability that a given policy implements a unique mean-square
stable and E-stable equilibrium. For one simple prior over ρ, the policy that maximizes
the probability of determinacy and E-stability (at .985) when p11 = p22 = .95,
γ(1) = 5, γ(2) = −1, involves φπ(1) = 1.2, φπ(2) = .9 and an expected posterior
loss of 4.15. If we replace φπ(1) = 1.2, φπ(2) = .9 with φπ(1) = 1.3, φπ(2) = .8,
we reduce the probability of determinacy and E-stability to .917, but we also reduce
expected posterior loss to 2.57. The addition of uncertainty over ρ therefore introduces
a tradeoff between minimizing loss and maximizing the probability of determinacy and
E-stability, a tradeoff first recognized by Evans and McGough (2007). Uncertainty over
all other fiscal policy parameters will almost surely present a similar tradeoff.
We believe that parameter uncertainty and the addition of an extra dimension
in ρ to our policy problem generates complications that are beyond the scope of the
17The average maturity of debt, equal to (1 − βρ)−1 in our model has been estimated using U.S.
data. However, because the actual maturity structure of U.S. debt does not decay geometrically, it is
not clear whether or not such estimates should be used to select ρ
129
FIGURE 17. Lengthening Maturity Expands Policy Menus
Left panel ρ = 0, right panel ρ = .96. pmm = .98, pff = .95, γ(M) = .02,
γ(F ) = −.01. The determine region is dark gray; the indeterminate region is light gray;
explosive region is white
FIGURE 18. Shortening Maturity Expands Policy Menus
Left panel ρ = 0, right panel ρ = .96. pmm = .98, pff = .95, γ(M) = .02,
γ(F ) = −.01. The determine region is dark gray; the indeterminate region is light gray;
explosive region is white
present analysis. However, we hope to fully explore issues pertaining to long-term debt
in an estimated DSGE framework in the near future.
Conclusion
This paper examines the performance and robustness of simple monetary
policy rules in models with learning agents subject to: (1) permanent or occasionally
non-Ricardian fiscal policy; and/or (2) the presence of long-term government debt.
130
My analysis indicates that the “global” response of the fiscal policymaker to debt
determines the optimal monetary policy response. When fiscal policy is globally
passive or globally active the optimal monetary policy rule features time-invariant
coefficients with high inflation reaction coefficients in globally passive models and
interest rate pegs in globally active models. In cases where fiscal policy features
balanced or strong switching between active and fiscal policy stances, the optimal
monetary policy rule features switching coefficients. These results are robust to
adaptive learning, including a novel hidden Markov model of learning we introduce
in the paper. For this reason, we should want to better understand how the presence
of long-term debt affects the optimal monetary policy in a model with switching fiscal
policy stances.
131
CHAPTER VI
CONCLUSION
Concluding Summary
The work which comprises this dissertation demonstrates the extent to
which conventional and unconventional policy outcomes, as well as the existence,
uniqueness and expectational stability of rational expectations solutions, depend
on the expectational effects of time-varying policy. These findings suggest that
uncertainty over future fiscal policy may curb the effectiveness of monetary policy,
or otherwise constrain the actions of central bankers. Additionally, this work examines
the relationship between determinacy and expectational stability in a general class of
Markov-switching DSGE models.
Chapter 2 of my dissertation generalizes McCallum (2007) and is the first to
address the relationship between determinacy and E-stability in Markov-switching
Dynamic Stochastic General Equilibrium (MS-DSGE) models with lagged endogenous
variables. I prove that the sufficient conditions for determinacy in Cho (2016) imply
the E-stability of the forward solution in MS-DSGE models with lagged endogenous
variables when agents condition their expectations of future endogenous variables on
current endogenous and exogenous variables. The class of models studied in this paper
is very general, and nests a wide array of models that are frequently studied in modern
macroeconomics.
In Chapter 3, I study the impact of expansionary forward guidance in a simple
New Keynesian model with recurring or permanent active fiscal policy. This work
addresses and offers a potential solution to the simple New Keynesian model’s
132
prediction that expansionary forward guidance can generate an implausibly large
stimulus. I find that the introduction of permanent or recurring active fiscal policy
dampens the response of output and inflation to forward guidance in the New
Keynesian model. Moreover, the presence of regime-switching policy introduces
expectational effects that cause forward guidance to be less stimulative in our regime-
switching model’s active money, passive fiscal policy regime. Finally, the introduction
of long-term debt affects the magnitude of the stimulus resulting from forward
guidance in models with active fiscal policy.
In Chapter 4, I explore determinacy and E-stability in a New Keynesian model
with switching fiscal and monetary policy. Here I present three categories of results.
First, the maturity structure of government debt matters for determinacy and
the existence of stable equilibria in our switching model, which is not true in the
analogous fixed coefficient model. I use two numerical solution techniques to show that
maturity affects both the multiplicity of stable solutions, and the existence of sunspot
equilibria. Second, determinacy generally implies E-stability when agents do not
observe contemporaneous observable variables, but not for certain arguably unrealistic
regions of the model parameter space. Third, this chapter presents conditions for
stability under infinite-horizon learning in Markov-switching DSGE models and
compares stability under infinite horizon and one-step-ahead learning. To the best of
my knowledge, this is the first paper to derive these stability conditions in a model
with switching coefficients.
Finally, Chapter 5 examines the performance and robustness of simple monetary
policy rules in models with learning agents subject to: (1) permanent or occasionally
active fiscal policy; and/or (2) the presence of long-term government debt. My analysis
indicates that the “global” response of the fiscal policymaker to debt determines the
133
optimal monetary policy response. When fiscal policy is globally passive or globally
active the optimal monetary policy rule typically features time-invariant coefficients
with high inflation reaction coefficients in globally passive models and interest rate
pegs in globally active models. In cases where fiscal policy features balanced or strong
switching between active and fiscal policy stances, the optimal monetary policy rule
features switching coefficients. These results extend to models with adaptive learning,
including a hidden Markov model of learning never seen before in the literature.
Extensions and Future Work
Current work in progress examines the implications of debt maturity structure for
central bank balance sheet decisions and the performance of simple interest rate rules
subject to time-varying fiscal policy. Because the maturity structure of debt has major
implications for the menu of interest rate rules available to monetary policymakers,
this project may answer two questions. First, how might optimal simple interest rate
rules depend on both the maturity structure of debt and fiscal policy regimes? Second,
what central bank balance sheet decisions help to contain inflation and output in an
economy with regime switching fiscal policy?
Other work in progress explores issues of uniqueness and expectational stability
in a general class of Markov-switching DSGE models with lagged information
structures. This line of research is motivated by Chapter 5, which shows that the
beliefs of learning agents who do not observe contemporaneous variables, including
underlying Markov states, cannot converge to the class of rational expectations
equilibria considered in most Markov-switching DSGE analyses. Because learning
applications commonly exclude contemporaneous variables from information sets, this
result suggests that a separate class of equilibria that depend on past Markov states
134
may be relevant in specific policy applications. This project attempts to learn various
properties of this class of equilibria.
Additionally, I endeavor to further study the expectational stability of rational
expectations equilibria and convergence of beliefs in hidden Markov models of adaptive
learning. Papers in the stochastic approximation literature have studied the properties
of recursive algorithms that estimate parameters of Markov-switching autoregressive
processes with hidden states. I would like to apply convergence results in that
literature to my work on learning in regime-switching DSGE models.
Future work should also extend themes in the dissertation to larger, more
realistic models with more sophisticated policy rules. While simple fiscal policy rules
help us understand the link between policy interactions and general equilibrium
outcomes, the performance of monetary policy should be studied in fuller models
that include rich debt maturity structures, capital, etc., and that allow for more
sophisticated policy rules and more than two policy states. These projects could
involve additional Bayesian model estimation of regime switching DSGE models. Such
exercises help to resolve parameter and model uncertainty, therefore offering greater
insight into the robustness of optimal policy
135
APPENDIX A
DERIVATION OF FIRST E-STABILITY CONDITION
To solve for DTB(B¯), we linearize TB(B) at the forward solution and vectorize
the resulting equation. We then use the following identification rule: if vec(dTB) =
Avec(dB) then A = DTB(B), where dB = (dB(1) dB(2) · · · dB(S)) and dTB is the
linearized system of equations. Using the rule: d(F (X)−1) = −F (X)−1(dF )F (X)−1, we
obtain the following linearization of TB(B):
dTB =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
(Ξ(1, B)−1M(1)(
∑S
j=1 p1jdB(j))Ξ(1, B)
−1N(1))′
(Ξ(2, B)−1M(2)(
∑S
j=1 p2jdB(j))Ξ(2, B)
−1N(2))′
...
(Ξ(S,B)−1M(S)(
∑S
j=1 pSjdB(j))Ξ(S,B)
−1N(S))′
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
′
= Ξ(1, B)−1M(1)p11(dB)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Ξ(1, B)−1N(1) 0n · · · 0n
0n 0n · · · 0n
...
. . .
0n 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
+ Ξ(1, B)−1M(1)p12(dB)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
Ξ(1, B)−1N(1) 0n · · · 0n
...
. . .
0n 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
136
+ · · ·
+ Ξ(1, B)−1M(1)p1S(dB)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
0n 0n · · · 0n
...
. . .
Ξ(1, B)−1N(1) 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
+ Ξ(2, B)−1M(2)p21(dB)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n Ξ(2, B)
−1N(2) · · · 0n
0n 0n · · · 0n
...
. . .
0n 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
+ Ξ(2, B)−1M(2)p22(dB)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
0n Ξ(2, B)
−1N(2) · · · 0n
...
. . .
0n 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
+ · · ·
+ Ξ(2, B)−1M(2)p2S(dB)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
0n 0n · · · 0n
...
. . .
0n Ξ(2, B)
−1N(2) 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
+ · · ·
+ Ξ(S,B)−1M(S)pSS(dB)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
0n 0n · · · 0n
...
. . .
0n 0n Ξ(S,B)
−1N(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
137
Using the rule vec(ABC) = C ′ ⊗ Avec(B), and the identification rule, we obtain:
DTB(B) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
(Ξ(1, B)−1N(1))′ 0n · · · 0n
0n 0n · · · 0n
...
. . .
0n 0n · · · 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⊗ Ξ(1, B)−1M(1)p11
+
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n (Ξ(1, B)
−1N(1))′ · · · 0n
0n 0n · · · 0n
...
. . .
0n 0n · · · 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⊗ Ξ(1, B)−1M(1)p12
+ · · ·
+
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · (Ξ(1, B)−1N(1))′
0n 0n · · · 0n
...
. . .
0n 0n · · · 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⊗ Ξ(1, B)−1M(1)p1S
+
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
(Ξ(2, B)−1N(2))′ 0n · · · 0n
...
. . .
0n 0n · · · 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⊗ Ξ(2, B)−1M(2)p21
+
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
0n (Ξ(2, B)
−1N(2))′ · · · 0n
...
. . .
0n 0n · · · 0n
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⊗ Ξ(2, B)−1M(2)p22
138
+ · · ·
+
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
0n 0n · · · 0n
0n 0n · · · 0n
...
. . .
0n 0n · · · (Ξ(S,B)−1N(S))′
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⊗ Ξ(S,B)−1M(S)pSS
At the forward solution, (B(1) B(2) · · ·B(S)) = (Ω∗(1) Ω∗(2) · · ·Ω∗(S)). Moreover, it is
straightforward to show that Ξ(i, B¯)−1N(i) = {I − M(i)(∑Sj=1 pijΩ∗(j))}−1N(i) =
{I − M(i)(Et(Ω∗(st+1))}−1N(i) = Ω∗(i) and Ξ(i, B¯)−1M(i) = {I −
M(i)(
∑S
j=1 pijΩ
∗(j))}−1M(i) = {I − M(i)(Et(Ω∗(st+1))}−1M(i) = F ∗(i) where Et
denotes rational expectations here. After substituting these equilibrium expressions
into the Jacobian, we obtain the following Jacobian evaluated at the forward solution:
DTB(B¯) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
p11Ω
∗(1)′ ⊗ F ∗(1) p12Ω∗(1)′ ⊗ F ∗(1) · · · p1SΩ∗(1)′ ⊗ F ∗(1)
p21Ω
∗(2)′ ⊗ F ∗(2) p22Ω∗(2)′ ⊗ F ∗(2) · · · p2SΩ∗(2)′ ⊗ F ∗(2)
...
. . .
...
pS1Ω
∗(S)′ ⊗ F ∗(S) pS2Ω∗(S)′ ⊗ F ∗(S) · · · pSSΩ∗(S)′ ⊗ F ∗(S)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
139
APPENDIX B
NEW KEYNESIAN MODEL DERIVATION
Throughout the dissertation we study an economy that is populated by a
large number of infinite-lived identical household-firms indexed by j ∈ [0, 1]. Each
household-firm is a monopolistically competitive producer of a unique product variety
indexed by i ∈ [0, 1], where i = j denotes the product of household-firm j. Household-
firm j engages in a decision-making process to maximize the following objective:
E0
∞∑
t=0
βt(
(Cjt )
1−σ
1− σ − ω(yt(j)))
subject to
∫ 1
0
pt(i)c
j
t(i)di+ Et(Rt,t+1B
j
t ) ≤ W jt + pt(j)yt(j) + Pt(zt − τt) (B.1)
Cjt = (
∫ 1
0
cjt(i)
1− 1
 di)

−1
where cj(i) is household-firm j’s consumption of good i; W jt denotes the nominal value
of the bond portfolio that the household holds at the beginning of t and W0 is given;
Rt,T is the stochastic discount factor between time t and T ; y(j) is the quantity of
product variety j produced by the household-firm; z is a lump-sum transfer from the
government; τ is a lump-sum tax; ω is a strictly convex function; pt(j) and Pt are
the price of product variety j and the price level, respectively. To preclude arbitrage
opportunities, we assume that all asset prices are determined by stochastic discount
factors. This implies, for example, that Qt,t+1 =
1
1+it
= Et(Rt,t+1) where Qt,t+1 is
the price of a single-period government bond at time t, and it is the nominal interest
140
rate on a riskless one-period bond. Furthermore, market completeness is assumed.
The sequence of flow constraints implied by (B.12) yields the following intertemporal
constraint:
∞∑
t=0
E0{R0,t
∫ 1
0
pt(i)c
j
t(i)di} ≤
∞∑
t=0
E0{Ro,t
(
pt(j)yt(j) + Pt(zt − τt)
)}+W jt
Since each household-firm is identical and markets are complete, we assume that
each household-firm has the same initial wealth level. This induces agents to engage
in a process of perfect risk-sharing that generates identical equilibrium paths for
household consumption and so forth. As a result, we can drop the j subscript and
treat household-firm j as the representative household and firm.
The household-firm chooses (1) how to allocate its expenditures among the
product varieties; (2) how much to consume or save in each period; (3) how much to
produce in each period. We study these three decision processes in turn. In making
these decisions, the representative household-firm acts as a “price-taker” by taking the
actions of other household-firms as given (i.e. the household-firm takes Pt and Yt as
given). Additionally, the household-firm faces a price rigidity when solving its producer
problem, and we discuss this in greater detail below.
In this environment, a rational expectations equilibrium is a collection of
stochastic processes such that each household-firm chooses sequences of consumption,
asset portfolios, and prices that maximizes its objective given {Pt, Yt, zt, τt} and a
specification for fiscal and monetary policy; such that net demand of assets by private
household-firms equals the supply of government debt. By studying the aforementioned
three decision-making processes of the representative household-firm, we uncover
conditions that characterize such an equilibrium.
141
We present the first of these problems–the problem of maximizing Ct subject to a
given level of expenditure–in the form of a Lagrangean:
L = (
∫ 1
0
ct(i)
1− 1
 di)

1− − μ(
∫ 1
0
pt(i)ct(i)di−Xt)
where Xt is the minimum level of expenditure. Differentiating with respect to ct(z)
yields the following optimality condition:

− 1(
∫ 1
0
ct(i)
1− 1
 di)

−1−1(1− 1

)ct(z)
− 1
 − μpt(z) = 0
which can be combined with the first-order condition for any other product variety
(e.g. product variety i) to obtain:
(
ct(z)
ct(i)
)−
1
 =
Pt(i)
Pt(z)
Now, we can substitute this into the expenditure function and solve for ct(i):
Xt =
∫ 1
0
pt(i)ct(i)di =
∫ 1
0
pt(i)(
pt(i)
pt(z)
)−ct(z)di
=
ct(z)
pt(z)−
∫ 1
0
pt(i)
1−di
Because Pt = (
∫ 1
0
pt(i)
1−di)1/(1−), this last equation implies:
ct(z) =
Xt
Pt
(
pt(z)
Pt
)− (B.2)
We can then substitute the analogous equation for ct(i) this into the definition for Ct:
142
Ct = (
∫ 1
0
ct(i)
1− 1
 dj)

−1 =
Xt
P 1−t
(
∫ 1
0
pt(j)
1−dj)
−
1− =
Xt
P 1−t
P−t (B.3)
∴ PtCt = Xt =
∫ 1
0
pt(i)ct(i)di (B.4)
Equations (B.2)-(B.4) therefore imply the following demand schedule for good i:
ct(i) = Ct(
pt(i)
Pt
)−
Henceforth, let government purchases equal 0 in every period. This assumption delivers
the following market-clear condition:
Yt = Ct (B.5)
Accordingly, the demand schedule for product variety i may be expressed as:
yt(i) = Yt(
pt(i)
Pt
)− (B.6)
The optimal consumption-savings plan of the representative household must satisfy:
(1) Yt = Ct for all t; (2) the intertemporal household budget constraint (with equality)
in each period; (3) a consumption Euler-equation that can be derived through a
variational argument:
βEt
{u′(Ct+1)
u′(Ct)
Pt
Pt+1
}
=
1
1 + it
(B.7)
143
Condition (B.1) along with complete risk-sharing imply 1:
∞∑
T=t
EtRt,TPtCT =
∞∑
T=t
Et
{
Rt,T [PTYT + PT (zT − τT )]
}
+Wt (B.8)
where Wt = B
m
t−1(1 + ρP
m
t ) is nominal outstanding government debt at beginning of t.
Combining with the government flow constraint,
Bmt−1(1 + ρP
m
t ) = Pt(τt − zt) + Pmt Bmt
yields the transversality condition:
lim
t→∞
Et[Rt,TWT ] = 0
We now turn to the pricing-production decision of the household-firm. Because price
determines quantity through the demand schedule, we assume that the household-
firm chooses price when solving for the optimal production schedule. The firm is
constrained by a price friction of the form developed in Calvo (1983). Each period,
1 − θ fraction of firms are randomly allowed to reset prices, while the remaining θ
fraction continue to charge last period’s price. This means that a firm expects its price
to persist for 1/(1 − θ) periods into the future each time it resets prices. As a result,
it is natural for the household-firm to treat θ as a discount factor and choose a single
price that maximizes the discounted sum of future profits:
∞∑
k=0
θk
{
ΛtEt[Rt,t+kPyt+k(P)]− βkEt[ω(yt+k((P )))]
}
1Under full insurance, identical households with identical initial wealth levels choose identical
optimal consumption paths. Moreover,
∫ 1
0
Bj(t)dj = Bt follows from the assumption that net demand
for assets by private households equals supply of government debt
144
where Λt is the marginal utility of household income at t. As in Woodford (1998a), we
treat Λt as a constant, and proceed to the first-order condition:
∞∑
k=0
θkEt
{
Λ[Rt,t+k(
P
Pt+k
)−Yt+k(1− )− βkω′(yt+k((P )))(−)P
−−1
P−t+k
Yt+k
}
= 0
Multiply both sides of the first-order condition by P
Λ(1−) to obtain
∞∑
k=0
θkEt
{
Rt,t+k(
P
Pt+k
)−PYt+k − βkΛ−1ω′(yt+k((P )))( 
− 1)
P−
P−t+k
Yt+k
}
= 0
∞∑
k=0
θkEt
{
[Rt,t+k(
P
Pt+k
)−Yt+k
(P − βk
Rt,t+kΛ
ω′(yt+k(P))( 
− 1
)} = 0
To further simplify the first-order condition, consider the following two equations:
βk
u′(Yt+k)
y′(Yt)
Pt
Pt+k
= Rt,t+k
Λt = u
′(Yt)/Pt
The first equation is a necessary and sufficient condition for household optimization,
while the second equation is an expression for the marginal utility of income. We
substitute these equations into the first-order condition to yield:
∞∑
k=0
θkEt
{
[Rt,t+k(
P
Pt+k
)−Yt+k
(P − St+k,t( 
− 1
)} = 0 (B.9)
ω′(yt+k(P))
u′(Yt+k)
Pt+k = St+k,t (B.10)
St+k,t captures the household’s expected marginal costs at time t + k. A sufficient but
not necessary condition for optimality is that P = 
−1St+k,t for all t + k. In this case,
the optimal price is always a mark-up of 
−1 over marginal costs. Since P is the same
145
for all firms who change price at t it is straightforward to show that
Pt = [θP
1−
t−1 + (1− θ)P ]
1
1− (B.11)
We are now in a position to characterize the non-policy aggregate demand (AD)
and aggregate supply (AS) blocks of the model. The non-policy AD block is given by
equations (B.5)-(B.8), and the AS block is given by equations (B.9)-(B.11). The AD
equations give us consumption demand, bond holdings and rates of return subject to
monetary and fiscal policy and a path for prices. The AS equation gives us a path for
the price index and optimal prices subject to AD. To complete the model, we discuss
simple fiscal and monetary policy arrangements. First, the monetary authority uses an
interest rate rule of the form:
it = Φ(πt, st, 
MP
t , V1,t−1)
where πt is inflation, m is an exogenous mean-one i.i.d. shock, v1,t−1 is the log of
V1,t−12, and st follows the 2-state Markov process described in Section 5. The fiscal
authority only issues a bond portfolio, Bmt , with a maturity that declines at a rate
ρ ∈ [0, 1]. Under this maturity structure, the quantity of government debt issued at
t− 1 that matures at t+ j is:
Bt−1(t+ j) = Bmt−1ρ
j
2Outside of section 3, we set V1,t−1 ∀t. The addition of V1,t−1 allows us to model forward guidance
146
The evolution of the government’s bond portfolio satisfies that following budget
constraint:
Bmt−1(1−
∑
j≥0
Qt(t+ j)ρ
j−1) = Pt(τt − zt) + Bmt
∑
j≥0
Qt(t+ j)ρ
j
where Qt(t + j) is the price of debt that matures at time t + j and is sold at t.. To
simplify the government budget constraint, we define the price of the bond portfolio,
Pmt , as:
Pmt = Et
∑
j≥0
Qt(t+ j)ρ
j
Furthermore, we can show that bond prices follow a recursive formulation:
Pmt = Qt(t+ 1)(1 + ρEtP
m
t+1) (B.12)
which allows us to rewrite the government budget constraint as
Bmt−1(1 + ρP
m
t ) = Pt(τt − zt) + Pmt Bmt
given Bm−1. The government also implements a rule that adjusts real primary surpluses
in response to the market value of real debt. If we let St = τt − zt denote the real
primary surplus, then we may characterize this rule as:
St = S
∗ + Γ(st)(
Bmt−1
Pt−1
− bm∗) + Zft
Zft =
(
ZρFf,t−1
)
f (t)
where Zft is an exogenous fiscal shock process and F is an exogenous mean-one i.i.d
fiscal policy shock.
147
We add these policy equations to the non-policy AD block to completely
characterize aggregate demand. To better analyze the equilibrium dynamics of the
model, we linearize the equations of the AD and AS blocks. The linearized AD
equations appear in equations (3.3), (3.5), (3.7)-(3.14) in Chapter 3. To arrive at
equation (3.4) in section 3 which is the linearized AS curve, we linearize equations
(B.9)-(B.11):
Pˆt = (1− θβ)
∞∑
k=0
(θβ)kEt{sˆt+k,t +
t+k∑
s=t+1
πˆs} (B.13)
sˆt+k,t = (ω
−1 + σ−1yˆt − θω−1[Pˆt −
t+k∑
s=t+1
πs]) (B.14)
πˆt =
1− θ
θ
Pˆt (B.15)
where Pˆt is the percentage deviation of optimal price over the price index from its
steady state value of 1, Sˆt+k,t is the percentage deviation of marginal costs over the
price index from its steady state value of 1 over the markup and ω = ω
′(Y ∗)
ω′′(Y ∗)Y ∗ . To
arrive at the linearized AS curve, we substitute equation (B.14) into equation (B.13)
and quasi-difference to obtain:
Pˆt = κθ
1− θ
∞∑
k=0
(θβ)kEtyˆt+k +
∞∑
k=1
Etπˆt (B.16)
where
κ ≡ (1− θ)(1− θβ)
θ
ω + σ
σ(ω + θ)
Substituting equation (B.16) into equation (B.15) yields the linearized AS equation in
(3.4).
148
APPENDIX C
FIXED REGIME FORWARD GUIDANCE EXPERIMENT
This appendix shows how to implement an anticipated interest rate peg at time
T (i.e. iT = ET iT+1 = ... = ET iT+L = i¯) using the forward guidance shocks introduced
in section 2. First, it helps to rewrite the equilibrium relationships as
Yt = GYt−1 + Ψ¯¯t + Ψ˜˜t
where Yt = (yˆt, πˆt, iˆt, r
n
t , μt, v1,t, v2,t, . . . , vL,t, τˆt, bˆ
m
t , Pˆ
m
t )
′, ¯t = (nt , 
μ
t , 
f
t )
′, and ˜t =
(MPt , 
R
1,t, . . . , 
R
L,t)
′. It follows that the equilibrium process for iˆ is given by an equation
of the form:
iˆt = GiYt−1 + Ψ¯i¯t + Ψ˜i˜t
Assume ¯T = 0, for simplicity (we can relax this). Then:
iT = GiYT−1 + Ψ˜i˜T
ET iT+1 = G
2
iYT−1 + (GΨ˜)i˜T
...
ET iT+L = G
L+1
i YT−1 + (G
LΨ˜)i˜T
where Gki and (G
kΨ˜)i denote the rows of G
k and (GkΨ˜) that correspond to the
nominal interest rate for k = 1, ..., L+1. If we set is = i¯ for all s ∈ {T, T+1, . . . , T+L},
then we have a system of L+ 1 equations in L+ 1 unknowns which are the elements of
149
˜T . The solution of this system is given by:
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Ψ˜i
(GΨ˜)i
...
(GLΨ˜)i
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
−1⎛
⎜⎜⎜⎜⎜⎜⎜⎝
i¯1L+1×1 −
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Gi
G2i
...
GL+1i
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
YT−1
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
= ˜T
where 1L+1×1 is a L + 1 × 1 vector of ones. We implement the interest rate peg by
announcing ˜T at T . If we also suppose that ¯s = 0 for all s ∈ {T, T + 1, . . . , T + L}
and ˜s = 0 for all s ∈ {T + 1, . . . , T + L} then ˜T will also successfully implement the
interest rate ex post (i.e. iT = iT+1 = ... = iT+L = i¯). If we relax this last assumption
(e.g. if ¯s = 0 for some some s ∈ {T + 1, ..., T + L}), then the central bank will have to
announce shocks after T to defend the peg. Regardless of whether shocks are present,
the central bank can always use these shocks to defend an L-horizon peg.
150
APPENDIX D
REGIME SWITCHING FORWARD GUIDANCE EXPERIMENT
First, it helps to rewrite the equilibrium relationships as
Yt = G(st)Yt−1 + Ψ¯(st)¯t + Ψ˜(st)˜t
where Yt = (yˆt, πˆt, iˆt, r
n
t , μt, v1,t, v2,t, . . . , vL,t, τt, bˆ
m
t , Pˆ
m
t )
′, ¯t = (nt , 
μ
t , 
f
t )
′, and ˜t =
(MPt , 
R
1,t, . . . , 
R
3,t)
′. It follows that the equilibrium process for iˆ is given by an equation
of the form:
iˆt = G(st)iYt−1 + Ψ¯(st)i¯t + Ψ˜(st)i˜t
We now suppose that ¯s = 0 for all s ∈ {T, T + 1, . . . , T + L} and ˜s = 0 for all
s ∈ {T + 1, . . . , T + L}. As in Appendix A.2. we can relax this assumption and allow
the central bank to defend the peg using shocks after T . Next, we define the following
matrices:
K1st = (pst1G(1)i + pst2G(2)i)
K2st = (pst1p11G(1)iG(1) + pst1p12G(2)iG(1)
+ pst2p21G(1)iG(2) + pst2p22G(2)iG(2))
K3st = (pst1p
2
11G(1)i(G(1))
2 + pst1p11p12G(2)i(G(1))
2 + pst1p12p21G(1)iG(2)G(1)
+ pst1p12p22G(2)iG(2)G(1) + pst2p21p11G(1)iG(1)G(2)
+ pst2p21p12G(2)iG(1)G(2) + pst2p22p21G(1)i(G(2))
2 + pst2p
2
22G(2)i(G(2))
2)
151
where G(st)i is the row of G(st) corresponding to the nominal interest rate, i,
for st ∈ {1, 2} and pstk is the probability of transition from state st to state k for k ∈
{1, 2}. Under these assumptions:
iT = G(sT )iYT−1 + Ψ˜(sT )i˜T
ET iT+1 = K
1
sT
(G(sT )YT−1 + Ψ˜(sT )˜T )
ET iT+2 = K
2
sT
(G(sT )YT−1 + Ψ˜(sT )˜T )
ET iT+3 = K
3
sT
(G(sT )YT−1 + Ψ˜(sT )˜T )
If we set is = i¯ for all s ∈ {T, T + 1, T + 3}, then we have a system of 4 equations
in 4 unknowns which are the elements of ˜T . The solution to this system is the set of
shocks that sets the interest rate peg. The solution is given by:
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Ψ˜(sT )i
K1sT Ψ˜(sT )
K2sT Ψ˜(sT )
K3sT Ψ˜(sT )
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
−1⎛
⎜⎜⎜⎜⎜⎜⎜⎝
i¯14×1 −
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
G(sT )i
K1sTG(sT )
K2sTG(sT )
K3sTG(sT )
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
YT−1
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
= ˜T
where 1L+1×1 is a 4× 1 vector of ones.
Since ET iT+1 = iT+1 is typically true in the presence of switching coefficients and
non-absorbing states, the central bank will typically have to announce a new sequence
of shocks at T + 1 to defend the interest rate peg (i.e. the central bank issues new
monetary shocks to ensure that iT+1 = ET+1iT+2 = ET+1iT+3 = i¯). Define ˜T+1 =
(MPT+1, 
R
1,T+1, 
R
2,T+1, 0)
′. Then at time T + 1:
152
iT+1 = G(sT+1)iYT + Ψ˜(sT+1)i˜T+1
ET+1iT+2 = K
1
sT+1
(G(sT+1)YT + Ψ˜(sT+1)˜T+1)
ET+1iT+3 = K
2
sT+1
(G(sT+1)YT + Ψ˜(sT+1)˜T+1)
If we set is = i¯ for all s ∈ {T + 1, T + 3}, then we have a system of 3 equations in
3 unknowns, which we solve for MPT+1, 
R
1,T+1, 
R
2,T+1 as before. This process repeats itself
in T + 2 where we use equations for iT+2 and iT+3 to solve for the pair (
MP
T+2, 
R
1,T+2)
that sets iT+2 = ET+2iT+3 = i¯. Then again at T +3 we use the equilibrium equation for
iT+3 to solve for the 
MP
T+3 that sets iT+3 = i¯
153
APPENDIX E
TABLES
TABLE 6. Fixed Coefficient Model Parameterization
Description Regime M
Regime F
(short-term )
Regime F
(long-term)
σ CRRA parameter 1 1 1
β Discount Factor .99 .99 .99
κ Phillips Curve Slope .1 .1 .1
φπ Feedback Inflation 1.5 0 0
φy Feedback Output 0 0 0
ρn AR(1) natural rate .5 .5 .5
ρμ AR(1) cost-push .5 .5 .5
ρ Average Debt Maturity 0 0 .93
γ Feedback Debt 2 .1 .1
TABLE 7. Regime-Switching Model Parameterizations
γ(M) γ(F ) φπ(M) φπ(F ) pMM pFF
Figure 4 20 -5 1.5 0 .95 .95
Figure 5 20 -5 1.5 .8 .95 .95
Figure 6 5 -5 1.5 .8 .95 .95
Section 4 parameterizations same as Section 3 parameterizations except for the above
values.
154
APPENDIX F
LAGGED INFORMATION ONE-STEP-AHEAD E-STABILITY
We consider the class of models developed in section 2.3.1. Suppose agents
observe the current state, st, and know the elements of P , but do not know Xt. Agents
have the following perceived law of motion:
Xt = A(st) + B(st)Xt−1 + C(st)Ut
where A(i) is n × 1, B(i) is n × n and C(i) is n × m. In this section, we solve for
agents’ state-contingent expectations and derive the state-contingent T-map. For now,
we assume S = 2, but this proof can be extended to any finite Markov-Chain. If st = i
then:
Et(Xt+1) = E(Xt+1|st = i)
= pi1A(1) + pi2A(2) + (pi1B(1) + pi2B(2))Xt + (pi1C(1) + pi2C(2))ρUt
= pi1A(1) + pi2A(2) + (pi1B(1) + pi2B(2))(A(i) + B(i)Xt−1
+C(i)Ut) + (pi1C(1) + pi2C(2))ρUt
= (pi1A(1) + pi2A(2) + (pi1B(1) + pi2B(2))(A(i)))
+((pi1B(1) + pi2B(2))B(i))Xt−1 +
((pi1C(1) + pi2C(2))ρ+ (pi1B(1) + pi2B(2))C(i))Ut
Substituting Et(Xt+1) into (2.1) in Chapter 2 yields the actual data generating process:
155
Xt = M(i)(pi1A(1) + pi2A(2) + (pi1B(1) + pi2B(2))(A(i)))
+ (M(i)((pi1B(1) + pi2B(2))B(i)) +N(i))Xt−1
+ (M(i)((pi1C(1) + pi2C(2))ρ+ (pi1B(1) + pi2B(2))C(i)) +Q(i))Ut
This delivers the state-contingent T-map:
T
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
A(1)
A(2)
B(1)
B(2)
C(1)
C(2)
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
=
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
M(1)((p11 + p11B(1) + p12B(2))A(1) + p12A(2))
M(2)((p22 + p21B(1) + p22B(2))A(2) + p21A(1))
M(1)(p11B(1)
2 + p12B(2)B(1)) +N(1)
M(2)(p22B(2)
2 + p21B(1)B(2)) +N(2)
M(1)((p11C(1)ρ+ (p11B(1) + p12B(2))C(1)) + p12C(2))ρ+Q(1)
M(2)((p22C(2)ρ+ (p21B(1) + p22B(2))C(2)) + p21C(1))ρ+Q(2)
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
The block of the T-map associated to B = (B(1)′ B(2)′)′ decouples from the other
blocks. This block is given by:
TB(B) =
⎛
⎜⎝M(1)(p11B(1)2 + p12B(2)B(1)) +N(1)
M(2)(p22B(2)
2 + p21B(1)B(2)) +N(2)
⎞
⎟⎠ =
⎛
⎜⎝T 1B(B)
T 2B(B)
⎞
⎟⎠
To assess E-stability, we begin by stabilizing the following differential equation:
dB
dt
= TB(B)− B
156
Let DTB(B¯) denote the Jacobian of Tb evaluated at the MSV solution, B¯ = (Ω
∗(1)′
Ω∗(2)′)′. Since TB is continuously differentiable, Proposition 5.6 in Evans and
Honkapohja (2001) tells us that B¯ is asymptotically stable if the eigenvalues of
DTB(B¯) have real parts less than one. Alternatively, we can analyze the stability of
(3) by analyzing the stability of the following differential equation:
dB˜
dt
= TB˜(B˜)− B˜
where B˜ = (B(1) B(2)) and
TB˜(B˜) =
(
T 1B(B) T
2
B(B)
)
Now let DTB˜(Ω¯
∗) denote the Jacobian of Tb evaluated at the MSV solution, Ω¯∗ =
(Ω∗(1) Ω∗(2)). Since TB˜ is continuously differentiable, Ω¯∗ is asymptotically stable if
the eigenvalues of DTB˜(Ω¯
∗) have real parts less than one. Finally, since DTB(B¯) is
similar to DTB˜(Ω¯
∗), the asymptotic stability of the ¯˜B in (4) implies the asymptotic
stability of B¯ in (3). To solve for DTB˜(Ω¯
∗), we linearize TB˜(B˜) at the REE and
vectorize the resulting equation. We then use the following identification rule: if
vec(dTB˜) = Avec(dB˜) then A = DTB˜(Ω¯
∗), where dB˜ = (dB(1) dB(2)) and dTB˜ is
the linearized system of equations.
157
dTB˜ =
⎛
⎜⎝(p11M(1)((dB(1))B(1) + B(1)dB(1)))′
(p22M(2)((dB(2))B(2) + B(2)dB(2)))
′
⎞
⎟⎠
+
⎛
⎜⎝(p12M(1)(B(2)(dB(1)) + (dB(2))B(1)))′
(p21M(2)(B(1)(dB(2)) + (dB(1))B(2)))
′
⎞
⎟⎠
′
= (p11M(1)B(1) + p12M(1)B(2))dB˜
⎛
⎜⎝In 0n
0n 0n
⎞
⎟⎠+ p11M(1)dB˜
⎛
⎜⎝B(1) 0n
0n 0n
⎞
⎟⎠
+ p12M(1)dB˜
⎛
⎜⎝ 0n 0n
B(1) 0n
⎞
⎟⎠+ (p22M(2)B(2) + p21M(2)B(1))dB˜
⎛
⎜⎝0n 0n
0n In
⎞
⎟⎠
+ p22M(2)dB˜
⎛
⎜⎝0n 0n
0n B(2)
⎞
⎟⎠+ p21M(2)dB˜
⎛
⎜⎝0n B(2)
0n 0n
⎞
⎟⎠
Using the rule vec(ABC) = C ′ ⊗ Avec(B), and the identification rule, we obtain:
DTB˜(B˜) =
⎛
⎜⎝In 0n
0n 0n
⎞
⎟⎠⊗ (p11M(1)B(1) + p12M(1)B(2))
+
⎛
⎜⎝B(1)′ 0n
0n 0n
⎞
⎟⎠⊗ (p11M(1)) +
⎛
⎜⎝0n B(1)′
0n 0n
⎞
⎟⎠⊗ (p12M(1))
+
⎛
⎜⎝0n 0n
0n In
⎞
⎟⎠⊗ (p22M(2)B(2) + p21M(2)B(1))
+
⎛
⎜⎝0n 0n
0n B(2)
′
⎞
⎟⎠⊗ (p22M(2)) +
⎛
⎜⎝ 0n 0n
B(2)′ 0n
⎞
⎟⎠⊗ (p21M(2))
158
At the REE, (B(1) B(2)) = (Ω∗(1) Ω∗(2)) = Ω¯∗. E-stability requires the real parts of
DTB˜(Ω¯
∗) to be less than one. We now turn to the equation for A = (A(1)′ A(2)′)′:
TA(A) =
⎛
⎜⎝M(1)(p11 + p11B(1) + p12B(2)) p12M(1)
p21M(1) M(2)(p22 + p21B(1) + p22B(2))
⎞
⎟⎠A
Using the same methods as before, we obtain the following Jacobian evaluated at the
REE where A¯ = (0′n×1 0
′
n×1)
′ :
DTA(A¯, B¯) =
⎛
⎜⎝M(1)(p11Ω∗(1) + p12Ω∗(2)) 0n
0n M(2)(p21Ω
∗(1) + p22Ω∗(2))
⎞
⎟⎠
+
⎛
⎜⎝p11M(1) p12M(1)
p21M(2) p22M(2)
⎞
⎟⎠
E-stability requires the real parts of ΨM to be less than one. Finally, we consider
the equation for C = (C(1)′ C(2)′)′:
TC(C) =
⎛
⎜⎝p11M(1)B(1) + p12M(1)B(2) 0n
0n p21M(2)B(1) + p22M(2)B(2)
⎞
⎟⎠C
+
⎛
⎜⎝p11M(1) p12M(1)
p21M(2) p22M(2)
⎞
⎟⎠Cρ
159
Using the same methods as before, we obtain the following Jacobian evaluated at the
REE where C¯ = (Γ(1)′,Γ(2)′)′:
DTC(B¯, C¯) = ρ⊗
⎛
⎜⎝p11M(1) p12M(1)
p21M(1) p22M(2)
⎞
⎟⎠
+ Im ⊗
⎛
⎜⎝M(1)(p11Ω∗(1) + p12Ω∗(2)) 0n
0n M(2)(p21Ω
∗(1) + p22Ω∗(2))
⎞
⎟⎠
The REE solution A¯, B¯, C¯ is E-stable if:
i. all the eigenvalues of DTA(A¯, B¯) have real parts less than 1,
ii. all the eigenvalues of DTB˜(Ω¯
∗) have real parts less than 1, and,
iii. all the eigenvalues of DTC(B¯, C¯) have real parts less than 1
The solution is not E-stable if any of these three conditions fail with eigenvalues
strictly greater than one.
160
APPENDIX G
INFINITE HORIZON STABILITY CONDITIONS DERIVATION
We derive the infinite horizon stability conditions in section (4). As mentioned in
a footnote in that section, we assume that all exogenous driving processes are mean-
zero i.i.d and that ρ = 0 in our derivation. We do this for expositional purposes; we
can relax these assumptions and obtain stability conditions in the more general model.
It should also be noted that all matrix infinite series of the form
∑
t≥0A
t converge to
(I − A)−1 if and only if the spectral radius of A is less than one. In the model explored
in section 4, this condition is met for nearly all of the parameterizations we consider.
When shocks are i.i.d agents employ the following perceived laws of motion for x ∈
{y, π, i} and b:
bt = a(st) + b(st)bt−1
xt = cx(st) + dx(st)bt−1
We also define the following matrix for each z ∈ {a, b, cx, dx}:
zˆ = diag(z(1), z(2), ..., z(S))
where S is the number of Markov states. In much of what follows, we let Fi denote
the ith row of the matrix F . Moreover, P is the transition probability matrix and 1s
denotes an S by 1 vector of ones. Let Et denote expectations formed using agents’
161
subjective beliefs. For T ≥ t, the following is true:
Etbt+1 = (
T−t∑
k=0
P T−t+1−kaˆ(P bˆ)k)i1s + ((P bˆ)T−t+1)i1sEtbt
Etxt+1 = P
T−t+1cˆx
+ (
T−t∑
k=0
P T−t+1−kaˆ(P bˆ)k−1P dˆx)i1s + ((P bˆ)T−t)P dˆx)i1sEtbt
To proceed, we need to find an expression for AT =
∑T−t
k=0 P
−kaˆ(P bˆ)k. We accomplish
this by first vectorizing AT :
vec(
T−t∑
k=0
P−kaˆ(P bˆ)k) =
T−t∑
k=0
vec(P−kaˆ(P bˆ)k)
=
T−t∑
k=0
(((P bˆ)k)′ ⊗ P−k)vec(aˆ)
=
T−t∑
k=0
((P bˆ)′ ⊗ P−1)kvec(aˆ)
= (I − ((P bˆ)′ ⊗ P−1)T−t+1)(I − (P bˆ)′ ⊗ P−1)−1vec(aˆ)
= vec(AT )
Returning to the original equation:
Etbt+1 = (P
T−t+1AT )i1s + ((P bˆ)T−t+1)i1sEtbt
This implies:
∑
T≥t
γT−tEtbt+1 = γ−1(
∑
T≥t
γT−t+1EtbT+1)
= γ−1(
∑
T≥t
((γP )T−t+1AT )i1S + ((γP bˆ)T−t+1)i1S)Etbt
162
We now need to calculate:
A¯ =
∑
T≥t
(γP )T−t+1AT )
=⇒ vecA¯ =
∑
T≥t
I ⊗ (γP )T−t+1vec(AT )
vec(A¯) = ((I ⊗ γP )(I − I ⊗ γP )−1
− ((P bˆ)′ ⊗ γI)(I − (P bˆ)′ ⊗ γI)−1)(I − (P bˆ)′ ⊗ P−1)−1vecaˆ
Therefore: ∑
T≥t
γT−tEtbT+1 = (γ−1A¯)i1S + γ−1((I − γP bˆ)−1P bˆ)i1SEtbt
for all reduced form discount factors in the infinite horizon model. Similarly, we can
show:
∑
T≥t
γT−tEtxT+1 = ((I − Pγ)−1P cˆx)i1s + ((γP )−1A˜P dˆx)i1S + ((I − γP bˆ)−1P dˆx)i1SEtbt
where
A˜ =
∑
T≥t
(γP )T−t+1A˜T )
A˜T =
T−t−1∑
k=0
P−kaˆ(P bˆ)k
In our model, γ ∈ {β, αβ}. We can write the model as:
Yt =
∑
γ
Mγ(st)(
∑
T≥t
γT−tEtYT+1) + Nˆ(st)Yt−1 + t
163
where Y = (y, π, i, b)′ and  is an m by 1 vector of i.i.d shocks. Next, we define A =
(cy(1) cπ(1) ci(1) a(1) cy(2) cπ(2) ci(2) a(2))′ and B = (dy(1) dy(2) dπ(1) dπ(2) di(1)
di(2) b(1) b(2))′. If Etbt = a(st) + b(st)bt−1, it follows that:
T (A) =
∑
γ
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Mγ(1)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − Pγ)−1P cˆy)11s + ((γP )−1A˜P dˆy)11S
((I − Pγ)−1P cˆπ)11s + ((γP )−1A˜P dˆπ)11S
((I − Pγ)−1P cˆi)11s + ((γP )−1A˜P dˆi)11S
(γ−1A¯)11S + ((I − γP bˆ)−1P bˆ)11Sa(1)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Mγ(2)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − Pγ)−1P cˆy)21s + ((γP )−1A˜P dˆy)21S
((I − Pγ)−1P cˆπ)21s + ((γP )−1A˜P dˆπ)21S
((I − Pγ)−1P cˆi)21s + ((γP )−1A˜P dˆi)21S
(γ−1A¯)21S + ((I − γP bˆ)−1P bˆ)21Sa(2)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
+
∑
γ
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Mγ(1)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − γP bˆ)−1P dˆy)11Sa(1)
((I − γP bˆ)−1P dˆπ)11Sa(1)
((I − γP bˆ)−1P dˆi)11Sa(1)
0
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Mγ(2)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − γP bˆ)−1P dˆy)21Sa(2)
((I − γP bˆ)−1P dˆπ)21Sa(2)
((I − γP bˆ)−1P dˆi)21Sa(2)
0
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
To simplify this expression in order to obtain E-stability conditions, consider the
following manipulation that works, for some correctly defined matrix, for any zˆ ∈
164
{aˆ, bˆ, cˆx} where x ∈ {y, π, i}:
((I − Pγ)−1P cˆy)i1s = ((I − Pγ)−1P (cy(1), cy(2))′)i
= ((I − Pγ)−1Pξ((1, 1), (5, 2)))iA
where ξ((h, j), (l,m)) is a 2 by 8 matrix with ones in the (h, j) and (l,m) entries and
zeros elsewhere. Similarly, we define ξ(h, j) as a 2 by 8 matrix with one in its (4, 1)th
entry and zeros elsewhere. Also note that vec(A¯) assumes the following form:
vec(A¯) = Qvec(aˆ)
=⇒ A¯ =
⎛
⎜⎝Q11a(1) +Q14a(2) Q31a(1) +Q34a(2)
Q21a(1) +Q24a(2) Q41a(1) +Q44a(2)
⎞
⎟⎠
= Q1ξ((4, 1), (8, 2))A
(
1 0
)
) +Q2ξ((4, 1), (8, 2))A
(
0 1
)
)
Similarly:
A˜ = Q˜1ξ((4, 1), (8, 2))A
(
1 0
)
) + Q˜2ξ((4, 1), (8, 2))A
(
0 1
)
)
165
For appropriately defined matrices Q1 and Q2, which can be recovered from our
calculations above. We can also simplify ((γP )−1A¯P dˆx)i1S for x ∈ {y, i, b} as follows:
((γP )−1A¯P dˆx)i1S = ((γP )−1)iA¯P dˆx1S
= ((γP )−1)i(Q1ξ((4, 1), (8, 2))A
(
1 0
)
+ Q2ξ((4, 1), (8, 2))A
(
0 1
)
)P dˆx1S
= (
(
1 0
)
P dˆx1S)((γP )
−1)iQ1ξ((4, 1), (8, 2))A
+ (
(
0 1
)
P dˆx1S)((γP )
−1)iQ2ξ((4, 1), (8, 2))A
Since
(
1 0
)
P dˆy1S and
(
0 1
)
P dˆy1S are scalars. These simplications allow us to
express T (A) = DT (A,B)A. Evaluated at the rational expectations equilibrium, where
ξi(k) is a 1 by 8 matrix of zeros with 1 in the (k, 1) entry:
DTA(0,Ω
∗) =
∑
γ
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Mγ(1)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − Pγ)−1P )1ξ((1, 1), (5, 2))
((I − Pγ)−1P )1ξ((2, 1), (6, 2))
((I − Pγ)−1P )1ξ((3, 1), (7, 2))+
((I − γP bˆ)−1P bˆ1s)1ξ1(4)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Mγ(2)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − Pγ)−1P )2ξ((1, 1), (5, 2))
((I − Pγ)−1P )2ξ((2, 1), (6, 2))
((I − Pγ)−1P )2ξ((3, 1), (7, 2))
((I − γP bˆ)−1P bˆ1s)2ξ1(8)
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
166
+
∑
γ
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Mγ(1)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − γP bˆ)−1P dˆy1s)1ξ1(4)
((I − γP bˆ)−1P dˆπ1s)1ξ1(4)
((I − γP bˆ)−1P dˆi1s)1ξ1(4)
0
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
Mγ(2)
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
((I − γP bˆ)−1P dˆy1s)2ξ1(8)
((I − γP bˆ)−1P dˆπ1s)2ξ1(8)
((I − γP bˆ)−1P dˆi1s)2ξ1(8)
0
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
+
∑
γ
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Mγ(1)
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
(
1 0
)
P dˆy1s((γP )
−1)1Q˜1ξ((4, 1), (8, 2))(
1 0
)
P dˆπ1s((γP )
−1)1Q˜1ξ((4, 1), (8, 2))(
1 0
)
P dˆi1s((γP )
−1)1Q˜1ξ((4, 1), (8, 2))(
1 0
)
1s(γ)
−1Q1ξ((4, 1), (8, 2))
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
Mγ(2)
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
(
1 0
)
P dˆy1s((γP )
−1)2Q˜1ξ((4, 1), (8, 2))+(
1 0
)
P dˆπ1s((γP )
−1)2Q˜1ξ((4, 1), (8, 2))(
1 0
)
P dˆi1s((γP )
−1)2Q˜1ξ((4, 1), (8, 2))(
1 0
)
1s(γ)
−1Q1ξ((4, 1), (8, 2))
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
167
+
∑
γ
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Mγ(1)
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
(
0 1
)
P dˆy1s((γP )
−1)1Q˜2ξ((4, 1), (8, 2))(
0 1
)
P dˆπ1s((γP )
−1)1Q˜2ξ((4, 1), (8, 2))(
0 1
)
P dˆi1s((γP )
−1)1Q˜2ξ((4, 1), (8, 2))(
0 1
)
1s(γ)
−1Q2ξ((4, 1), (8, 2))
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
Mγ(2)
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
(
0 1
)
P dˆy1s((γP )
−1)2Q˜2ξ((4, 1), (8, 2))(
0 1
)
P dˆπ1s((γP )
−1)2Q˜2ξ((4, 1), (8, 2))(
0 1
)
P dˆi1s((γP )
−1)2Q˜2ξ((4, 1), (8, 2))(
0 1
)
1s(γ)
−1Q2ξ((4, 1), (8, 2))
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
Again, the dˆz are evaluated at the rational equations equilibrium coefficients. Since the
ODE associated to B decouples from the system, it can be studied in isolation. Let
γ = 1 correspond to a discount factor of αβ and γ = 2 correspond to a discount factor
of β. We make use of the model structure to simplify our analysis as well:
M1(st) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Myy1 (i) M
πy
1 (i) 0 0
Myπ1 (i) M
ππ
1 (i) 0 0
Myi1 (i) M
πi
1 (i) 0 0
Myb1 (i) M
πb
1 (i) 0 0
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
M2(st) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎝
Myy2 (i) M
πy
2 (i) M
iy
2 (i) 0
Myπ2 (i) M
ππ
2 (i) M
iπ
2 (i) 0
Myi2 (i) M
πi
2 (i) M
iπ
2 (i) 0
Myb2 (i) M
πb
2 (i) M
iπ
2 (i) 0
⎞
⎟⎟⎟⎟⎟⎟⎟⎠
168
Using elements from the above matrices, we form the following:
Mγz (1) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
Mγzy(1) 0
0 0
Mγzπ(1) 0
0 0
Mγzi(1) 0
0 0
Mγzb(1) 0
0 0
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
Mγz (2) =
⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
0 0
0 Mγzy(2)
0 0
0 Mγzπ(2)
0 0
0 Mγzi(2)
0 0
0 Mγzb(2)
⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
for z ∈ {π, y, i, b}. The T-map is then given by:1
T (B) =
∑
γ
∑
i∈{1,...,S}
∑
z
Mγz (i)(I − γP bˆ)−1P dˆz1SN(i)B
1We exclude Nˆ terms because they will not affect the stability analysis and because they clutter
the proof
169
where N(1) = (0 0 0 0 0 0 1 0) and N(2) = (0 0 0 0 0 0 0 1). Define 2 by 8 matrices
N(z, i) such that N(z, 1)B = (dˆz(1), 0)
′ and N(z, 2)B = (0, dˆz(2))′ for z ∈ {y, π, i, b}.
This implies:
P dˆz = P (N(z, 1)B
(
1 0
)
+N(z, 2)B
(
0 1
)
)
Using this, we can derive the E-stability matrix associated to B:
DTB(B) = ∑
γ
∑
i∈{1,...,S}
∑
z
(((1, 0)δ(γ, i, z))′ ⊗Mγz (i)(I − γbˆ)−1PN(b, 1)
+ ((0, 1)δ(γ, i, z))′ ⊗Mγz (i)(I − γbˆ)−1PN(b, 2)
+ ((1, 0)1sN(i)B)
′ ⊗Mγz (i)(I − γbˆ)−1PN(z, 1)
+ ((0, 1)1sN(i)B)
′ ⊗Mγz (i)(I − γbˆ)−1PN(z, 2)
+ I ⊗Mγz (i)(I − γP bˆ)−1P dˆz1SN(i))
where δ(γ, i, z) = (I − γbˆ)−1P dˆz1sN(i)B. The necessary and sufficient conditions
for E-stability are that all the eigenvalues of DTB and DTA have real parts less than
one where both matrices are evaluated at the rational expectations equilibrium.
170
APPENDIX H
AN AND SCHORFHEIDE (2007) MODEL DERIVATION
We present a simple model that is inspired by An and Schorfheide (2007). As
in standard in the New Keynesian literature, the model consists of households, a
competitive final goods producing firm, monopolistically competitive intermediate
firms, a fiscal authority and a monetary authority. We briefly describe the optimization
problems facing agents in this economy, then we collect the equilibrium conditions
which are log-linearized and presented in section 2.
Households maximize a lifetime utility functions that depends positively on the
level of consumption, Ct and negatively on labor supply, Nt. Additionally, households
are subjected to a preference shock, Zt that directly impacts the contribution of time t
utility to overall lifetime utility. Formally:
max
{Ct,Nt,Wt}
Eo
∑
t≥0
(
C1−σt − 1
1− σ − χNt
)
Zt
subject to
PtCt + Et(Rt,t+1W
j
t+1) ≤ W jt + PtωtNt − Ptτt
and a tranversality condition of the form:
lim
t→∞
Et[Rt,TWT ] = 0
171
where Wt is wealth at time t, ω is the competitive real wage paid to labor, τ is a lump-
sum tax, C is consumption, and Rt,t+1 is a stochastic discount factor that equals
(Ct+1/Ct)
−σ in our model with complete markets. From the first order conditions
for Wt+1, Ct and Wt we get the familiar necessary intertemporal and intratemporal
conditions for the household optimization problem:
1 = βEt
{
Ct+1
Ct
−σZt+1
Zt
(1 + it)
πt+1
}
(H.1)
ωt = χC
σ
t
The perfectly competitive firm has technology described by:
Yt =
(∫ 1
0
Yt(j)
1−ηtdj
) 1
1−ηt
where inputs, Yt(j), are goods produced by each intermediate firm j ∈ [0, 1], and ηt is a
shock to markups. The perfectly competitive firm maximizes profits given by:
ΠFINt = PtYt −
∫ 1
0
Pt(j)Yt(j)dj
This implies the following demand schedule for each intermediate producer’s good,
Yt(j):
Yt(j) =
(
Pt(j)
Pt
)−1/ηt
Yt
Pt(j) =
(∫ 1
0
Pt(j)
ηt−1
ηt
) ηt
ηt−1
172
Intermediate firms are monopolistically competitive and utilize identical technologies
that assume the form:
Yt(j) = Nt(j)
To introduce nominal rigidities, we assume that firms face the following adjustment
costs:
ACt(j) =
φ
2
(
Pt(j)
Pt−1(j)
− π
)2
Yt(j)
Firms maximize the present value of firm profits taken real wages, ωt+s as given.
Formally, they choose labor inputs and prices to maximize the following:
ΠINT = E0
{∑
t≥0
βtR0,t
(
Pt(j)
Pt
Yt(j)− ωt(j)Nt(j)− ACt(j)
)}
Substituting the product demand schedule into the profits equation, then optimizing
with respect to Pt+s(j) and substituting for ωt+s = c
σ
t and Rt|0 = (Ct/C0)
−σ yields the
following optimality condition:
(
1
ηt
− 1
)
=
Cσt
ηt
− φ
2
(
2(πt − π)− (πt − π)
2
ηt
)
+βφ
((
Ct+1
Ct
)σ
(πt+1 − π) πt+1Yt+1
Yt
)
(H.2)
The fiscal authority only issues a bond portfolio, Bmt , with a maturity that
declines at a rate ρ ∈ [0, 1]. Under this maturity structure, the quantity of government
debt issued at t− 1 that matures at t+ j is:
Bt−1(t+ j) = Bmt−1ρ
j
173
The evolution of the government’s bond portfolio satisfies that following budget
constraint:
Bmt−1(1−
∑
j≥0
Qt(t+ j)ρ
j) + PtGt = Ptτt +B
m
t
∑
j≥0
Qt(t+ j)ρ
j−1
where Qt(t + j) is the price of debt that matures at time t + j and is sold at t. To
simplify the government budget constraint, we define the price of the bond portfolio,
Pmt , as:
Pmt = Et
∑
j≥0
Qt(t+ j)ρ
j−1
which allows us to rewrite the government budget constraint as
Bmt−1(1 + ρP
m
t ) + PtGt = Ptτt + P
m
t B
m
t (H.3)
Furthermore, we can show that bond prices follow a recursive formulation:
Pmt = Qt(t+ 1)(1 + ρEtP
m
t+1) (H.4)
given Bm−1. The government also implements a rule that adjusts real primary surpluses
in response to the market value of real debt. In equilibrium, households hold all
government debt which requires that the following condition hold ∀t:
Wt = B
m
t−1(1 + ρP
m
t )
The processes for τt and Gt are specified. Finally, monetary policy follows the following
rule:
Rt = R
ρi
t−1
(
R∗
( πt
π∗
)φπ(st)( Yt
Y ∗t
)φy(st))1−ρi
(H.5)
174
where Rt = 1 + it, R
∗ = β−1, Y ∗t is potential output defined as the level of output
that obtains without nominal rigidities and with constant markups. The log-linearized
equilibrium conditions in Chapter 5, are simply log-linearized versions of equations
(H.1), (H.2), (H.3)-(H.5). μt is a composite of ηt from (H.2), and all other shocks and
the fiscal policy rule are described in Chapter 5.
175
APPENDIX I
PRIOR AND POSTERIOR DISTRIBUTIONS
TABLE 8. Prior and Posterior Distribution Statistics
Name Prior Density Prior Param (1) Prior Param (2)
Posterior
Mean
σ Gamma 2.00 0.50 2.36
κ Uniform 0.00 1.00 .91
φπ Gamma 1.50 0.25 2.16
φy Gamma 0.50 0.25 .56
ρi Uniform 0.00 1.00 .71
ρg Uniform 0.00 1.00 .98
ρz Uniform 0.00 1.00 .93
100σm InvGamma 0.40 4.00 .2
100σg InvGamma 1.00 4.00 .75
100σz InvGamma 0.50 4.00 .2
We estimate the An and Schorfheide (2007) model using U.S. data, Q1:1983 to
Q3:2007. Param (1) and Param (2) are the lower and upper bounds for the uniform
distributions and the mean and standard deviation for the Gamma and Inverse
Gamma distributions
176
APPENDIX J
MISCELLANEOUS FIGURES
FIGURE 19. Determinacy and E-Stability (Parameterization 1)
This figures show determinacy and E-stability regions in Regime F when φπ(M) = 1.5
and γ˜(M) = γ(M)(1 − β) = .05. φπ(F ) is on the vertical axis and γ˜(F ) is on
the horizontal axis. The top right and bottom left quadrants are consistent with
determinacy in the fixed regime model. Green regions are determinate and E-stable,
yellow regions are indeterminate and E-stable, and black regions of indeterminate and
E-unstable
177
FIGURE 20. Determinacy and E-Stability (Parameterization 2)
This figures show determinacy and E-stability regions in Regime F when φπ(F ) = 0
and γ˜(F ) = γ(M)(1 − β) = −.05. φπ(M) is on the vertical axis and γ˜(M) is on
the horizontal axis. The top right and bottom left quadrants are consistent with
determinacy in the fixed regime model. Green regions are determinate and E-stable,
yellow regions are indeterminate and E-stable, and black regions of indeterminate and
E-unstable
178
FIGURE 21. Determinacy and E-Stability (Parameterization 2)
This figures show determinacy and E-stability regions in Regime F when φπ(M) = 1.5
and φπ(M) = 0. γ˜(M) = γ(M)(1 − β) is on the horizontal axis and γ˜(F ) is on the
vertical axis. The bottom right and top left quadrants are consistent with determinacy
in the fixed regime model. Green regions are determinate and E-stable, yellow regions
are indeterminate and E-stable, and black regions of indeterminate and E-unstable
179
REFERENCES CITED
An, S., & Schorfheide, F. (2007). Bayesian Analysis of DSGE Models, Econometric
Review 26 (2-4), 113-172.
Ascari, G., Florio, A., & Gobbi, A. (2016). Monetary and Fiscal Policy Interactions:
Leeper (1991) Redux. Mimeo.
Bianchi, F. (2012). Evolving Monetary/Fiscal Policy Mix in the United States.
American Economic Review Papers and Proceedings 101 (3), 167-172.
———–. (2013): “Regime Switches, Agents’ Beliefs, and Post-World War II U.S.
Macroeconomic Dynamics,” Review of Economic Studies, 80(2), 463-490.
———–. & Melosi, L. (2013). Dormant Shocks and Fiscal Virtue. NBER
Macroeconomics Annual 2013 28, 1-46.
Bianchi, F. & Ilut, C. (2017). Monetary/Fiscal Policy Mix and Agents’ Beliefs. Review
of Economic Dynamics 26, 113-139.
Bhattarai, S., Lee, J. & Park, W. (2012). Monetary-Fiscal Policy Interactions and
Indeterminacy in Postwar US Data. American Economic Review 102 (3), 173-
178.
———–. (2014). Inflation dynamics: The role of public debt and policy regimes.
Journal of Monetary Economics 67, 93-108.
———–. (2016). Policy Regimes, Policy Shifts, and U.S. Business Cycles. The Review
of Economics and Statistics 98 (5), 968-983.
Blanchard, O., & Kahn, C. (1980). Solution of linear difference models under rational
expectations. Econometrica 38, 1305-1311.
Branch, B., Davig, T., & B. McGough. (2013). Adaptive Learning in Regime-
Switching Models. Macroeconomic Dynamics 17 (5), 998-1022.
Bullard, J., & Eusepi, S. (2014). When Does Determinacy Imply Expectational
Stability. International Economic Review 55 (1), 1-22.
Bullard, J., & Mitra, K. (2002). Learning about monetary policy rules. Journal of
Monetary Economics 49 (6), 1105-1129.
180
Bullard, L., & Singh, A. (2012). Learning And The Great Moderation.International
Economic Review 53 (2), 375-397.
Calvo, G.A. (1983). Staggered Prices in a Utility Maximizing Model. Journal of
Monetary Economics 12 (3), 383-398.
Carlstrom, C., Fuerst, T., and Paustian, M. (2012). “How Inflationary Is an Extended
Period of Low Interest Rates. Manuscript.
Cho, S. (2016). Sufficient Conditions for Determinacy in a Class of Markov-Switching
Rational Expectations Models. Review of Economic Dynamics 21, 182-200.
———–. (2015). Sufficient Conditions for Determinacy in a Class of Markov-Switching
Rational Expectations Models: A Technical Guide. Manuscript.
———–. (2017). Sufficient Conditions for Determinacy in a Class of Markov-Switching
Rational Expectations Models: A Technical Guide. Manuscript.
———-, & Moreno, A. (2011). The Forward Method as a Solution Refinement in
Rational Expectations Models. Journal of Economic Dynamics and Control
35 (3), 257-272.
———-. (2016). Global Determinacy Under Monetary and Fiscal Policy Switchings.
Manuscript.
Chen, X., Leeper, E., & C. Leith. (2015). U.S. Monetary and Fiscal Policy: Conflict or
Cooperation. Manuscript.
Chung, H., Herbst, E., & Kiley, M. (2014). Effective Monetary Policy Strategies in
New Keynesian Models: A Re-examination. NBER Macroeconomics Annual 29,
289-344.
Chung, H., Davig, T., & Leeper, E. (2007). Monetary and Fiscal Policy Switching.
Journal of Money, Credit, and Banking 39 (4), 809-842.
Clarida, R., Gali, J., $ Gertler, M. (2000). Monetary Policy Rules and Macroeconomic
Stability: Evidence and Some Theory. Quarterly Journal of Economics, 115 (1),
147-180.
Cochrane, J. (2001). Long Term Debt and Optimal Policy in the Fiscal Theory of the
Price Level. Econometrica 69 (1), 69-116.
———-. (2007). Inflation Determination with Taylor Rules: A Critical Review. NBER
Working Paper 13409.
181
———-. (2009): “Can learnability save New-Keynesian models? Journal of Monetary
Economics 56, 1109-1113.
———-. (2017). The new-Keynesian liquidity trap. Journal of Monetary Economics
92, 47-63.
Cogley, T., De Paoli, B., Matthes, C., Nikolov, K., & Yates, T. (2011). A Bayesian
Approach to Optimal Monetary Policy with Parameter and Model Uncertainty.
Journal of Economic Dynamics and Control 35 (12), 2186-2212.
Cogley, T., & Sargent, T. (2002). Evolving US Post-World War II Inflation Dynamics.
NBER Macroeconomics Annual 16, 331-373.
————. (2005). Drifts and Volatilities: Monetary Policies and Outcomes in the Post
WWII US. Review of Economic Dyanmics 8, 262-302.
Cole, S. (2015). “Learning and the Effectiveness of Central Bank Forward Guidance.”
under review at Journal of Money, Credit and Banking.
Costa, O., Fragoso, M., &Marques, R. (2004). Discrete-Time Markov Jump Linear
Systems Spring, New York: Springer-Verlag London.
D’Amico, S., & King, T. (2015). Policy expectations, term premia, and macroeconomic
performance. Manuscript.
Davig, T. (2005). Regime-switching debt and taxation. Journal of Monetary
Economics 51 (4), 837-859.
Davig., T., & Leeper, E. (2006). Fluctuating Macro Policies and the Fiscal Theory.
NBER Macroeconomic Annual 21, 247-298.
————. (2007). Generalizing the Taylor Principle,” American Economic Review
97 (3), 607-635.
————. (2011). Monetary-Fiscal Policy Interactions and Fiscal Stimulus. European
Economic Review 55 (2), 211-227.
————, & Waler, T. (2010). Unfunded Liabilities and Uncertain Fiscal Financing.
Journal of Monetary Economics 57 (5), 600-619.
————. (2011). Inflation and the Fiscal Limit. European Economic Review 55 (1),
31-47.
Del Negro, M., Giannoni, M., & Patterson, C. (2015). The Forward Guidance Puzzle.
FRB of New York Staff Report 574.
182
Eggertsson, G., & Woodford, M. (2003). The Zero Bound on Interest Rates and
Optimal Monetary Policy. Brookings Papers on Economic Activity 2003 (1),
139-211.
Ellison, M., & Pearlman, J. (2011). Saddlepath Learning. Journal of Economic Theory
146 (4), 1500-1519.
Eusepi, S., & Preston, B. (2011). Learning about the Fiscal Theory of the Price Level:
Some Consequences of Debt-Management Policy. Journal of the Japanese and
International Economies 25 (4), 358-379.
————. (2012). Debt, Policy Uncertainty and Expectations Stabilization. Journal of
the European Economic Association 10 (4), 860-886/
————. (2013). Fiscal Foundations of Inflation: Imperfect Knowledge. Federal
Reserve Bank of New York 649.
Evans, G., & Honkapohja, S. (2001). Learning and Expectations in Macroeconomics.
Princeton, New Jersey: Princeton University Press.
————. (2005). Policy interaction, expectations and the liquidity trap. Review of
Economic Dyanmics 8, 303-323.
————. (2007). Policy Interaction, Learning and the Fiscal Theory of Prices.
Macroeconomic Dynamics 11, 665-690.
————, & Guse, E. (2008). Liquidity Traps, Learning and Stagnation. European
Economic Review 52, 1438-1463.
Evans, G., & McGough, B. (2005). Optimal Constrained Interest-Rate Rules. Journal
of Money, Credit and Banking 39 (6), 1335-1356.
Farmer, R., Waggoner, D., & Zha, T. (2009). Understanding Markov-Switching
Rational Expectations Models. Journal of Economic Theory 144 (5), 1849-1867.
————. (2010). Generalizing the Taylor Principle: A Comment. American Economic
Review 101 (1), 608-617.
————. (2011). Minimal State Variable Solutions to Markov-Switching Rational
Expectations Models. Journal of Economic Dynamics and Control 35 (12),
2150-2166.
Foerster, A., Rubio-Ramirez, J., Waggoner, D., & Zha, T. (2016). Perturbation
methods for Markov-switching dynamic stochastic general equilibrium models.
in Quantitative Economics 7 (2), 637-669.
183
Gabaix, X. (2016). A Behavioral New Keynesian Model. Manuscript.
Gertler, M., & Karadi, P. (2015). Monetary Policy Surprises, Credit Costs, and
Economic Activity. American Economic Journal: Macroeconomics 7 (1), 44-76.
Gonzalez-Astudillo, M. (2013). Monetary-Fiscal Policy Interaction: Interdependent
Policy Rule Coefficients. Finance and Economics Discussion Series No. 2013-
58, Federal Reserve Board.
Haberis, A., Harrison, R., & Waldron, M. Transitory interest-rate pegs under imperfect
credibility. Technical Report, 2014.
Hamilton, J. (1989). A New Approach to the Economic Analysis of Nonstationary
Time Series and the Business Cycle. Econometrica 57, 357-384.
Hansen, L., & Polson, N. (2006). Tractible Filtering.” Manuscript.
————, & Sargent, T. (2010). Nonlinear Filter and Robust Learning. Working
Paper.
Hansen, L. & Sargent, T. (2010). Fragile Beliefs and the Price of Uncertainty.
Quantitative Economics 1 (1), 129-162.
Hansen, L., & Sargent, T. (2013). Recursive Models of Dynamic Linear Economies.
Princeton University Press: Princeton, New Jersey.
Jin, H. (2013). Debt Maturity Management, Monetary and Fiscal Policy Interactions.
mimeo.
Johannes, M., Korteweg, A., & Polson, N. (2014). Sequential Learning, Predictability,
and Optimal Portfolio Returns. The Journal of Finance 69 (2): 611-644.
Kiley, M. (2014). Policy Paradoxes in the New Keynesian Model. Review of Economic
Dynamics 21, 1-15.
Kim, C., & Nelson, C. (1999). State-Space Models with Regime Switching. MIT Press:
Cambridge, Massachusetts.
Klein, P. (2000). Using the generalized Schur Form to Solve a Multivariate Linear
Rational Expectations Model. Journal of Economic Dynamics and Control 24,
1405-1423.
Krishnamurthy, V., & Yin, G. (2002). Recursive Algorithms for Estimation of Hidden
Markov Models and Autoregressive Models With Markov Regime. IEEE
Transactions of Information Theory 48 (2), 458-476.
184
LeGland, F., & Mevel, L. (1997). Recursive estimation of hidden Markov models.
Proc. 36th IEEE Conf. Decision Control, San Diego, CA, Dec. 1997.
Laseen, S., & Svensson, L. (2011). Anticipated Alternative policy Rate Paths in Plicy
Simulations. International Journal of Central Banking 7 (3), 1-35.
Leeper, E. (1991). Equilibria under ‘Active’ and ‘Passive’ Monetary and Fiscal Policies.
Journal of Monetary Economics 27, 129-147.
———–, & Leith, C. (2016). Understanding Inflation as a Joint Monetary-Fiscal
Phenomenon. Manuscript.
Levin, A., Wieland, V., Williams, J. (1999). Robustness of Simple Monetary Policy
Rules under Model Uncertainty. In J.B. Taylor (ed.), Monetary Policy Rules
(pp. 263-318). Chicago, Illinois: University of Chicago.
———–. (2003). The Performance of Forecast-Based Policy Rules under Model
Uncertainty. American Economic Review 93 (3), 622-645.
Lubik, T.A., & Schorfheide, F. (2004). Testing for Indeterminacy: An Application to
U.S. Monetary Policy. American Economic Review 94 (1), 190-219.
Mankiw, G., & Reis, R. (2002). Sticky Information versus Sticky Prices: A Proposal
to Replace the New Keynesian Phillips Curve. The Quarterly Journal of
Economics 117 (4), 1295-1328.
McCallum, B.T. (2001). Indeterminacy, Bubbles, and the Fiscal Theory of Price Level
Determination. Journal of Monetary Economics 47 (1), 19-30.
————. (2007). E-Stability Vis-A-Vis Determinacy Results For A Broad Class Of
Linear Rational Expectations Models. Journal of Economic Dynamics and
Control 31 (4), 1376-1391.
————. (2009). Inflation determination with Taylor rules: is New-Keynesian analysis
critically flawed?. Journal of Monetary Economics 56, 1101-1108.
McClung, N. (2016). E-Stability vis-a-vis Determinacy in Markov-Switching DSGE
Models. Manuscript.
McClung, N. (2017a). The Power of Forward Guidance and the Fiscal Theory of the
Price Level. submitted.
McClung, N. (2017b).Maturity, Determinacy and E-Stability in Models of Regime
Switching Fiscal and Monetary Policy. Manuscript.
185
McKay, A., Nakamura E., & Steinsson, J. (2016). The Power of Forward Guidance
Revisited. American Economic Review 106 (10), 3133-3158.
Orphanides, A., Williams, J. (2005). The Decline of Activist Stabilization Policy:
Natural Rate Misperceptions, Learning, and Expectations. Journal of Economic
Dynamics and Control 2929(11), 1927-1950.
Orphanides, A., &Williams, J. (2007). Robust Monetary Policy with Imperfect
Knowledge. Journal of Monetary Economics 54 (5), August, 1406-1435.
Reed, J.R. (2015). Mean-Square Stability and Adaptive Learning in Regime-Switching
Models. Manuscript.
Sargent, T.J., & Wallace, N. (1981). Some Unpleasant Monetarist Arithmetic. Federal
Reserve Bank of Minneapolis Quarterly Review 5 (Fall), 1-17.
Schmitt-Grohe, S., & Uribe, M. (2007). Optimal simple and implementable monetary
and fiscal rules,” Journal of Monetary Economics 54 (6): 1702-1725.
Slobodyan, S., & Wouters, R. (2012). Learning in a Medium-Scale DSGE Model with
Expectations Based on Small Forecasting Models. American Economic Journal:
Macroeconomics 4 (2), 65-101.
Sims, C.A. (1994). A Simple Model for Study of the Determination of the Price Level
and the Interaction of Monetary and Fiscal Policy. Economic Theory 4 (3), 381-
399.
———–(2002). Solving Linear Rational Expectations Models. Computational
Economics 20 (1), 1-20.
———–, & Zha, T. (2006). Were There Regime Switches in US Monetary Policy?
American Economic Review 96 (1), 54-81.
Uhlig, H. (1999). A Toolkit for Analyzing Nonlinear Dynamic Stochastic Models
Easily. In R. Marimon and A. Scott (eds.), Computational Methods for the
Study of Dynamic Economies (pp.30-61). Oxford, England: Oxford University
Press.
Woodford, M. (1995). Price-Level Determinacy Without Control of a Monetary
Aggregate. Carnegie-Rochester Conference Series on Public Policy 43, 1-46.
————. (1998a). Control of the Public Debt: A Requirement for Price Stability. G.
Calvo, & M. King (eds.), The Debt Burden and Its Consequences for Monetary
Policy (pp. 117-154). New York, New York: St. Martin’s Press.
————. (1998b). Public Debt and the Price Level.
186
————. (2001). Fiscal Requirements for Price Stability. Journal of Money, Credit,
and Banking 33 (3), 669-728.
————. (2003). Interest and Prices: Foundations of a Theory of Monetary Policy.
Princeton University Press: Princeton, New Jersey.
187