PARAMETER UNCERTAINTY, CASHFLOW
BETAS, AND EARNINGS
ANNOUNCEMENT
PREMIA
by
CAMERON S. PFIFFER
A DISSERTATION
Presented to the Department of Finance
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctorate of Philosophy
September 2022
DISSERTATION APPROVAL PAGE
Student: Cameron S. Pfiffer
Title: Parameter Uncertainty, Cashflow Betas, and Earnings Announcement Premia
This dissertation has been accepted and approved in partial fulfillment of the requirements
for the Doctor of Philosophy degree in the Department of Finance by:
Roberto Gutierrez Chair
Robert Ready Core Member
Youchang Wu Core Member
Jeremy Piger Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of Graduate
Studies.
Degree awarded September 2022.
2
© 2022 Cameron S. Pfiffer
3
DISSERTATION ABSTRACT
Cameron S. Pfiffer
Doctor of Philosophy
Department of Finance
September 2022
Title: Parameter Uncertainty, Cashflow Betas, and Earnings Announcement Premia
I apply Bayesian methods to estimate parameters describing the relationship between
firm earnings and unobserved common earnings shocks. I estimate a firm’s Bayesian cash-
flow beta, which measures the comovement between firm earnings and a latent aggregate
earnings factor, along with estimating the uncertainty about the firm’s cash-flow beta.
Firms with high parameter uncertainty have higher expected stock returns and lower stock
price reactions to earnings, consistent with investors’ rational learning in the presence of
parameter uncertainty. A novel measure summarizes the capacity of a firm’s earnings news
to convey information about the macroeconomic state and reveals that earnings responses
and announcement risk premia increase with a firm’s informativeness. The most informa-
tive firms tend to announce earlier in earnings seasons.
4
ACKNOWLEDGEMENTS
I am extraordinarily grateful for the work of my advisor, Ro Gutierrez, who went to
great lengths to help me achieve my personal and academic goals. Ro has been a superla-
tive advisor with the marvelous ability to challenge my basic assumptions in an informa-
tive and open dialogue. I thank my committee members Robert Ready, Youchang Wu, and
Jeremy Piger for their expertise, availability, kindness, and especially for their willingness
to help me through uncommon circumstances. I thank Shoshana Vasserman for affording
me a bright new world to explore when I needed it most, and for putting on a masterclass
on intelligence, rigor, curiosity, and compassion.
I acknowledge the support of all the past and present faculty members of the Univer-
sity of Oregon’s Department of Finance who have provided assistance to me during my
time at the University. In alphabetical particular order: Ioannis Branikas, Maria Chade-
rina, John Chalmers, Diane Del Guercio, Brandon Julio, Stephen McKeon, Phil Romero,
Albert Sheen, and Jay Wang. I also thank my past and current PhD student peers for
their companionship and comments during my studies: Gretchen Gamrat, Arash Dayani,
Donghyeok Jang, Wensong Zhong, Yuwen Yuan, Sina Davoudi, Wendi Wu, and Xuanyu
Bai.
I am grateful for the assistance of the various communities I have been involved with. I
acknowledge the support and kindness of the individuals who founded the Microstructure
Exchange with me, and who treated me as an equal: Björn Hagströmer, Andriy Shkilko,
Katya Malinova, and Andreas Park. I am grateful to Hong Ge and the Turing.jl team for
inspiring me to study Bayesian methods, with particular thanks to David Widmann for his
help in making me a better engineer.
I acknowledge the emotional, social, and academic support provided by the broader
economics and statistics community. Specific and pointed suggestions were given by Brad
Ross, David Childers. I thank various individuals treating me with respect and kindness:
Toni Whited, Andrew Baker, Luke Stein, Tony Cookson, Ashvin Ghandi, Akhil Rao, Demitri
Pananos, Daniel Bergstresser, Arpit Gupta, Benn Eifert, Anne Laski, and John Horton. I
further thank uncountable individuals in the community for their small acts of kindness,
inspiration, and humor.
I acknowledge the support and resources of Stanford University’s computational cluster,
Sherlock, Café Roma for cheap, delicious coffee and baked goods, and the piano located in
the Erb Memorial Union for allowing me to practice. Lastly, I thank the members of the
Oregon Association of Rowers for providing me an outlet and teaching me to row, a skill I
now hold dear.
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DEDICATION
I dedicate this dissertation to everyone who helped me in my educational journey. Tammy
Schrader, who was so patient with me and my inability to complete any mathematics as-
signment. Maria Wickwire, who thought I could do whatever I put my mind to. My fifth
grade teacher Patrick Webb, who saw that I need to live to thrive. My undergraduate
teacher Curtis Bacon, who inspired me to continue to graduate school. I dedicate this dis-
sertation to every educator or authority figure who believed in my capabilities even when I
did not.
I dedicate this work to all the people who have made me a better person or supported
my personal development. I thank my brother, Quinlan Pfiffer, who continues to show me
how to live, how to make time for joy, and how to push yourself. I thank Patricia Levi for
providing me with a vision of a prismatic future as we spin around the loop for a little
while. I thank Elizabeth Koonce for her kindness, spirit, and bottomless support.
Above all, I dedicate this dissertation to my mother, Lynnette Pfiffer. She demon-
strated an unparalleled certainty of character during challenging years. My mother gave
my brother and I the gift of her personal values. Her only wish for us was that we be Re-
naissance men — a wish that inspired us to be skilled, artistic, and kind, while specifying
no path or objective other than that we explore our curiosity with an open heart. We were
left with the space to attend our inner and outer lives, to build what we could, and, above
all, to enjoy the act of learning and doing. The generality, openness, and singularity of
purpose afforded to us by the desire that we pursue a breadth of knowledge ensures that
we have whatever we need to thrive. I am grateful for my mother inspiring me to find joy
in infinite improvement.
So far, I think I’ve done a pretty good job as Renaissance man. How do you think I did,
ma?
6
TABLE OF CONTENTS
1 INTRODUCTION......................................................................................................... 10
2 MODEL ........................................................................................................................ 15
2.1 Model discussion .................................................................................................... 17
3 TOPICS ........................................................................................................................ 21
3.1 Cashflow beta......................................................................................................... 21
3.2 Earnings news ........................................................................................................ 25
3.3 Firm informativeness.............................................................................................. 27
3.3.1 Earnings announcement order ...................................................................... 30
4 ESTIMATION .............................................................................................................. 33
4.1 Model estimation ................................................................................................... 33
4.2 Estimation results .................................................................................................. 35
4.2.1 Latent factor results ..................................................................................... 37
4.2.2 Firm results .................................................................................................. 39
5 EMPIRICAL RESULTS ............................................................................................... 42
5.1 Cross-sectional results ............................................................................................ 44
5.2 Earnings announcements........................................................................................ 48
5.3 Firm informativeness and earnings response.......................................................... 51
5.4 Earnings announcement order................................................................................ 55
6 DISCUSSION................................................................................................................ 58
6.1 The latent factor .................................................................................................... 58
6.2 Time variation in the cashflow beta....................................................................... 61
6.3 The Normality assumption..................................................................................... 64
6.4 The role of the prior............................................................................................... 66
6.5 Additional topics .................................................................................................... 69
7 ALTERNATIVE MODEL SPECIFICATION............................................................... 72
8 CONCLUSION ............................................................................................................. 75
BIBLIOGRAPHY.............................................................................................................. 77
7
LIST OF FIGURES
4.1 Sample trace and histogram plots for estimated parameters ................................. 36
4.2 Time series posterior densities of parameter estimates .......................................... 37
4.3 Estimated latent factor Mt .................................................................................... 38
4.4 The time series of latent process parameters ......................................................... 39
4.5 Cross-sectional dispersion in firm parameter estimates, grouped by decade .......... 40
4.6 Firm variance ratios VRi,t ...................................................................................... 41
5.1 Average variance ratios VRi,t by industry, filtered to the top five largest in-
dustries in my sample ............................................................................................ 53
5.2 Average days-to-announcement from the end of firm fiscal quarters ..................... 55
6.1 Comparison of the first principal component of earnings growth, compared to
the latent process Mt ............................................................................................. 59
6.2 Density of the posterior expectation Et[Xi,t], realized earnings growth Xi,t,
and the model error Et[Xi,t]−Xi,t starting from 1990 .......................................... 65
8
LIST OF TABLES
5.1 Overview of quarterly data for firms in my sample by industry ............................ 43
5.2 Correlation matrix of regressors............................................................................. 44
5.3 Fama-MacBeth regressions of weekly returns (in percent) on covariates ............... 45
5.4 Average weekly returns in percent grouped by the firm’s quintile of Et[βi] and
by the quintile of a firm’s parameter uncertainty, Vart[βi]..................................... 47
5.5 Regression of cumulative abnormal return (in percent) during an earnings
announcement, CAR(0,1). Subsamples are by NBER recession indicators............ 49
5.6 Fama-MacBeth regressions of weekly returns (in percent) on announcement
week indicators....................................................................................................... 50
5.7 Summary statistics of firms in my sample, grouped by the quintile of their
variance ratio ......................................................................................................... 52
5.8 Regression of cumulative abnormal return (in percent) across a firm’s earn-
ings announcement, testing earnings responses as a function of firm informa-
tiveness................................................................................................................... 54
5.9 Cross-sectional regressions of the difference between a firm’s earnings date
and the firm’s fiscal quarter end in days ................................................................ 56
6.1 Sample normality statistics on the model error Et[Xi,t]−Xi,t, by variance ratio
quintile ................................................................................................................... 66
9
Chapter 1
INTRODUCTION
Firm earnings announcements are important events for market participants. Quarterly
earnings news contains information about the firm profits, business activities, and chal-
lenges. For the econometrician, a cross-section of quarterly earnings announcements pro-
vide a rich data source of a firm’s exposure to common macroeconomic factors. Earnings
accrued during the same quarter that co-move between firms is indicative of shared ex-
posure to the business cycle. Importantly, investors do not know with certainty whether
there is a latent earnings component, what form it takes, or how firm earnings interact
with the latent factor. Investors can only infer commonality in earnings growth through
the data they can observe. Investors face a learning problem where they are attempting
to understand firm and common factors simultaneously. The purpose of this paper is to
quantify how much can be learned from earnings announcements once it is acknowledged
that the underlying commonality in earnings may not be observed with certainty.
I propose a novel Bayesian model of learning through earnings growth and use my
model to address several topics of interest to the financial economist. I derive a high-
dimensional Bayesian statistical model of the economy where no parameter is known to
an investor – the true state of the economy is unobservable, and can only be inferred
through observed firm earnings growth. Investors in the real economy face a learning prob-
lem whereby they jointly estimate the parameters that govern the factor process and the
parameters that govern firm-level earnings shocks. My modelling approach is unique and
permits me to study the uncertainty of a firm’s exposure to aggregate earnings growth,
and grants me a wealth of information about the joint distribution of the economy, such
as average firm earnings growth, firm systematic risk exposure, and aggregate earnings
growth risk.
A primary output of my model is the cashflow beta, which captures the linear exposure
of a firm’s earnings growth to unobserved common earnings growth shocks. The canoni-
cal cashflow beta measures the growth in raw earnings as a function of aggregate earnings
growth. The cashflow beta is distinct from the CAPM beta which measures the sensitiv-
ity of an asset’s return to the market’s return, in that the CAPM beta includes both a
cashflow component and a discount-rate component. Estimation of the cashflow beta is a
non-trivial exercise. Methods include vector autoregressions (Campbell and Vuolteenaho
2004), fitting ARIMA models on consumption growth (Da 2009), or estimating princi-
pal components (Ball, Sadka, and Sadka 2009). In each case, researchers typically proxy
for aggregate earnings growth with one or more measures, or estimate aggregate earnings
10
growth through some model. I estimate a cashflow beta that differs from previous esti-
mates, in that it is a cashflow beta on an uncertain and unobserved latent factor. Directly
estimating a cashflow beta partially frees my cashflow beta from the specification issues in
Campbell and Vuolteenaho (2004), as highlighted by Chen and Zhao (2009).
Commonality in asset payoffs has a well-studied history and in some sense functions
as the primary causal driver of asset prices. Asset prices vary with the exposure of the as-
set’s payoffs to systematic risk. Assets that tend to provide low cashflows when aggregate
cashflows are low should have higher expected returns, as cashflows arrive precisely when
those cashflows are the least desirable. However, asset pricing research has shown that
there are many facets of systematic risk. To capture the timing, size, and risk of cashflows
and their arrival times, researchers have described an asset’s systematic risk exposure in
several forms. The original Sharpe-Lintner CAPM market beta, for example, has been de-
composed into a discount rate beta and a cashflow beta (Campbell and Vuolteenaho 2004).
Da (2009) further decomposes the cashflow beta into a covariance measure, describing the
covariance between cashflows and contemporaneous aggregate consumption, and into a
duration measure that describes the temporal relationship between current cashflows and
future cashflow growth. The cashflow beta and its associated characteristics has gener-
ally been found to drive cross-sectional asset prices. Campbell and Vuolteenaho (2004),
Da (2009), Ball, Sadka, and Tseng (2021), and Bansal, Dittmar, and Lundblad (2005) all
find positive risk premia associated with the cashflow beta1. I do not find a positive risk
price using my cashflow beta estimate. Perhaps quarterly firm earnings are insufficient for
estimating relevant cashflow risk, or perhaps there are model limitations that need to be
addressed. I discuss alternative modelling approaches to consider.
A novel contribution of my paper is examining the effects of uncertainty about the cash-
flow beta. There may be insufficient evidence at a point in time to conclude the precise
point estimate of a cashflow beta, a phenomenon known as parameter uncertainty or esti-
mation risk. Parameter uncertainty is suggested to play a role in the traditional CAPM-
style market beta (Handa and Linn 1993; Kumar et al. 2008), but there does not currently
exist a paper studying the role of the cashflow beta and its associated parameter uncer-
tainty. The lack of existing literature on parameter uncertainty as it relates to the cash-
flow beta is surprising, considering that the literature on parameter uncertainty in the
CAPM beta highlights significant effects in expected returns. Insofar as the CAPM mar-
ket beta is considered to be composed of some measure of a cashflow beta (Campbell and
Vuolteenaho 2004), it stands to reason that parameter uncertainty in the cashflow beta
should influence expected returns just as it does in the CAPM beta. I find, surprisingly,
1Positive risk prices for the cashflow beta is not always the case. As shown in Ellahie (2021), there are inconsistent
cashflow beta prices associated with different measures of aggregate earnings – some cashflow beta prices are positive, some
are negative.
11
that the cashflow beta is not a priced risk in my sample of stock returns, but that there
seems to be partial evidence that uncertainty about the cashflow beta is. I highlight sev-
eral reasons why my unintuitive finding could arise, including model misspecification, lack
of time variation in beta, or unidentifiability in the latent factor. I provide guidance for
future analysis in how to better model a cashflow beta and its associated parameter uncer-
tainty.
The cashflow beta also provides for a natural extension into research questions other
than cross-sectional asset prices. If a firm’s cashflows are in part determined by a shared
but unobserved common shock (through the cashflow beta), then firm earnings news should
function as a signal about that common shock. News from different firms should have in-
formation content that varies in quality – firms that have large betas and low idiosyncratic
cashflow volatility are more precise signals about the common shock, while firms with low
betas or high idiosyncratic volatility are noisy signals that reveal little of the aggregate
earnings shock.
I provide an additional set of tests motivated by the informational content of firm cash-
flow signals. First, I study the unconditional earnings announcement premia accrued by
firms with different characteristics. Numerous papers have highlighted the existince of ele-
vated abnormal returns across earnings announcements without conditioning on earnings
surprise. The literature highlights two general mechanisms for the earnings announcement
premia. Evidence suggests that (a) risk increases during an earnings announcement, or (b)
that risk is unchanged, but the market premium is detectable only during announcement
weeks. In case (a), Savor and Wilson (2016) and Patton and Verardo (2012) highlight that
announcing firms earn a premium over non-announcing firms because firm earnings are
noisy signals about aggregate cashflow news. Savor and Wilson (2016) provide a theoreti-
cal model of this premium, their model does not permit any form of heterogeneity in the
characteristics of announcing firms. For case (b), Chan and Marsh (2022) shows that the
return-to-beta association only appears to hold during earnings announcement periods.
In either case, the premium accrued seems to be a function of systematic risk exposure
and asset informativeness. I find evidence in support of both explanations for earnings
announcement premia, and further find evidence that heterogeneity in firm characteris-
tics plays a significant role in the earnings announcement premia. Firms that are more
informative accrue the majority of the earnings announcement premium, as do firms with
larger cashflow betas. My findings suggest that a model of announcement premia that per-
mits variation in firm characteristics would better inform financial economists about why
risk appears to be elevated across earnings announcements, and to which firm characteris-
tics that risk is attributable to.
Second, I explore the response to firm earnings news as a function of the cashflow beta
12
and its uncertainty. Earnings news constitute an information revelation to the market. In-
vestors can revise their beliefs about firm characteristics after observing an earnings report.
News from firms with higher cashflow betas (holding idiosyncratic risk constant) implies
that a larger proportion of the firm’s news is systematic, and thus there is a lower infor-
mational content about the idiosyncratic elements of a firm. When much of the news of
a firm has little to do with the firm itself, we should expect reduced earnings responses.
Similarly, if parameter uncertainty is high, the ability of an investor to attribute observed
earnings news to idiosyncratic or common shocks is diminished – the same unit of news ob-
served by an uncertain firm and a certain firm is processed differently. Schmalz and Zhuk
(2019) and Beyer and Smith (2021) both predict that higher cashflow beta uncertainty im-
plies lower responses to unexpected earnings in normal economic conditions. Consistent
with the literature’s predictions, I find that earnings responses are lower for firms with
high cashflow betas and with high parameter uncertainty.
Third, I produce a sufficient statistic that describes the signal-to-noise ratio of a firm’s
earnings news about common cashflow shocks. Informative firms (high signal-to-noise
ratios) are commonly referred to as “bellwethers” (Bonsall, Bozanic, and Fischer 2013).
News from bellwether firms conveys high-quality information about the broader industry
or macroeconomy in which it operates, and commensurately draw more attention from
analysts (Hameed et al. 2015). I derive a novel, parametric, continuous measure of firm
bellwetherness, and use the measure to demonstrate that bellwether firms have different re-
turn behavior during earnings announcements, relative to non-bellwether firms. My statis-
tic, the variance ratio, summarizes the posterior joint density of a firm’s parameters and
its ability to convey macroeconomic news. I show that my variance ratio is qualitatively
distinct from an alternative and simpler measure of cashflow informativeness like R2 of a
regression of aggregate earnings on firm earnings, a statistic commonly employed in the
bellwether literature (see Hameed et al. (2015) for one example). I show that the most in-
formative firms, as measured by the variance ratio, have higher earnings responses, higher
earnings announcement premia, and tend to announce earlier.
Fourth, I provide novel evidence on the realized ordering of firm earnings announce-
ments. I show that, on average, firms announce earnings in a way that appears to be rela-
tively optimal. Suppose a social planner were to assign earnings announcement order with
the goal to reveal the most information about the aggregate earnings shock as early as pos-
sible. My model predicts which firms should generally tend to announce earlier under the
social planner’s problem, and I find evidence that more informative firms tend to announce
earlier. I contribute to the literature on the effects of earnings announcement timing and
socially optimal earnings disclosure, and highlight future areas of research.
The paper proceeds as follows. Section 2 outlines the model I use and derives measures
13
for informativeness and ambiguity. Section 3 summarizes the existing literature and for-
mulates predictions that can be made on each topic I consider. Section 4 produces the
estimated quantities and details summary statistics about the MCMC estimation routine.
Section 5 applies the estimated parameters from my model to each topic. Section 6 dis-
cusses the implications of my modelling choices and outlines future research areas. Section
7 details an alternative model specification. Section 8 concludes.
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Chapter 2
MODEL
In this section, I propose a novel model of firm earnings growth that is simplistic enough
to form a basic economic intuition, yet complex enough to require a joint estimation pro-
cess. As is common with contemporary Bayesian models, I focus first on describing the
laws of motion of the economy that dictate the likelihood function. I focus next on my
choice of priors. The section concludes with a discussion of the advantages and disadvan-
tages of my model.
I assume that the structure of firm earnings growth is linear in parameters unknown to
the investor. I assume that earnings growth, denoted Xt, is partly determined by a firm’s
own loadings on a one-dimensional latent process Mt. Furthermore, I make no claims
about the interpretation on Mt, other than that it is a driver of common sources of varia-
tion in earnings growth.
Xt = µ+ βMt + νt (2.1)
Mt = δ + ΦMt−1 + ϵt (2.2)
where
• µ is an N × 1 vector of firm-specific average earnings growth.
• β is an N × 1 matrix of firm loadings on the latent process Mt. Denote βi as firm i’s
loadings on Mt.
• νt is an N × 1 vector of firm-specific mean-zero shocks with variance γ2.
• δ is the unconditional mean value for the latent process, E[Mt] = δ.
• ϕ is the AR(1) persistence parameter.
• ϵt is a white-noise shock to the latent process, with variance σ2.
All the above parameters and latent states are unknown to a rational Bayesian investor.
Denote the parameter vector of the system with θ(t) = (µ, β, σ, γ, ϕ, δ, ϵ1:t). I omit the
idiosyncratic parameter νt.
1 The parameter vector has length 3N + 2 + (N + t).
1Omitting “nuisance parameters” like νt is a common practice that speeds up inference since it reduces the number of
parameters sampled. Note that, conditional on all other parameters, the distribution of Xt is Gaussian with mean µ+ βMt
and variance γ. Notably, this allows a modeler to ignore νt as a parameter and instead use the variance of νt as a part of the
likelihood.
15
Note that the latent factor shocks ϵt are treated as parameters of the model. I estimate
all latent factor shocks to reconstruct the distribution of the latent process Mt in full. Do-
ing this has the advantage of simplifying model estimation at the cost of some extra com-
putational burden.
Each quarter t, the investor observes the realized earnings growth of all firms announc-
ing in that quarter. Conditional upon observing earnings growth, the investor updates
their beliefs about the joint density of the parameters P (θ | X1:t), where X1:t is the filtra-
tion X1:t = {X1, X2, . . . , Xt}. Bayes’ rule gives
| P (X1:t | θ)P (θ)P (θ X1:t) =
P (X1:t)
The explicit posterior given above is unlikely to be analytically tractable due to the
complexity in calculating the denominator P (X1:t). However, as is common in numerical
Bayesian inference, I can state that the posterior is proportional to the likelihood times
the prior:
P (θ | X1:t) ∝ P (X1:t | θ)P (θ)
Recasting posterior inference as a proportionality problem is what allows me to make any
statements about the empirical distribution of parameters in my model, and, more broadly,
is a simplification that permits the use of many modern Bayesian inference methods.2
The first component of the posterior, P (X1:t | θ), is referred to as the likelihood. My
likelihood function is proportional to the Gaussian PDF centered at the µ+ βMt for firms
announcing in time t, with variance γ:
∏t
P (X1:t | θ) ∝ N (X 2s | µ+ βMt, γ I)
s=1
The second component of the posterior, the prior P (θ), was chosen to predispose my
estimates towards nonzero cashflow betas.3
• Firm means: µ ∼ N (µ̂, I). I set firm mean earnings growth to the average of that
firm’s earnings growth in first five years if the firm was in existence at that time, or
to the average earnings growth across all firms between 1980-1985 if a firm enters
after 1985 (this vector is denoted µ̂). The prior variance is 1, which assumes that
about 10% of the 15.52% aggregate earnings growth variance in my sample is due to
prior uncertainty in average earnings growth.
2See Gelman et al. (2013) for a rigorous mathematical treatment of contemporary Bayesian methods, or McElreath (2018)
for a more casual text.
3Selecting priors for a high-dimensional model is nontrivial, and the computational burden of my model means that it is
infeasible to conduct prior sensitivity analyses.
16
• Cashflow betas: βi ∼ N (1, 0.1I)∀i. I center the prior mean on βi at one, with the be-
lief that firms are on average positively exposed to aggregate earnings growth shocks.
The prior variance of 0.01 reflects my relatively strong belief that the model should
not permit too many firms to have a cashflow beta of zero.4
• Firm idiosyncratic variances γ2i follow an Inverse Gamma prior, where each firm has a
different distribution. To set my priors on firm idiosyncratic risk, I estimate a firm’s
earnings growth variance between 1980 and 1985 and set the α and β hyperparame-
ters of the Inverse Gamma distribution to the pair that makes the mean of the dis-
tribution equal to half the firm’s estimated earnings growth variance. Firms that did
not exist between 1980 and 1985 begin the sample with a prior equal to the aggregate
earnings growth variance. My choice of prior is equivalent to assuming that half of a
firm’s observed earnings growth variance is idiosyncratic.
• Latent parameter mean: δ ∼ N (0, 0.5). I assume that, on average, aggregate earnings
growth shocks are close to zero in line with Kothari, Lewellen, and Warner (2006).
• Latent variance: σ2 ∼ InverseGamma(10, 143.83). I estimate the variance of all firm’s
earnings growth between 1980 and 1985, and set the α and β hyperparameters of
the Inverse Gamma distribution to the pair that sets the variance of the latent shock
equal to the empirical variance.
• Latent process persistence: ϕ ∼ N (0, 0.2) ∈ [−1, 1]. I assume that earnings growth
persistence is on average centered around zero to permit negative autocorrelation,
though empirically there is some evidence that persistence is as high as 0.7 (Kothari,
Lewellen, and Warner 2006). I truncate my distribution between -1 and 1 to rule out
AR(1) processes that are not covariance stationary.
• Latent process shocks: ϵ 2t ∼ N (0, σ )∀t. I assume latent process shocks are zero on
average, with a hierarchical prior parameter of σ.
2.1 Model discussion
Existing models of cashflow betas, such as Campbell and Vuolteenaho (2004), employ a
vector autoregression (VAR) to model cashflow betas. Campbell and Vuolteenaho (2004)
use firm responses to macroeconomic conditions measured by (1) excess market returns,
(2) the yield spread between long- and short-term bonds, (3) the market’s smoothed price-
to-earnings ratio, and (4) the small-stock value spread. In the context of my model, this
4If the prior variance on βi is too large, the posterior distribution for βi could be centered tightly around zero, and the
latent factor could be removed entirely as a source of variation. Earlier versions of the model used a variance of 1, and many
estimates of βi were tightly centered around zero. This had the unfortunate byproduct of inflating firm idiosyncratic risk γ
2
i
as well.
17
is roughly equivalent to replacing the latent shocks with data, rather than estimating the
latent shocks directly. There are several distinguishing features of my model relative to
Campbell and Vuolteenaho (2004) that I contribute to the study of asset pricing.
First, I demonstrate how to examine cashflow betas in the context of a Bayesian model
when the latent factors are unknown. A firm’s latent factor loading βi is important to un-
derstand jointly with the latent factor, as we do not generally know the latent factor and
instead use rough proxies (as in Campbell and Vuolteenaho (2004)). Unfortunately, esti-
mating the latent factor jointly with the firm’s loading on that factor can be challenging to
do analytically – the posterior product distribution of βiMt is challenging to derive analyti-
cally even with very simple priors5. My model has the advantage of permitting the statis-
tician to understand a firm’s loading on an uncertain factor in a way that is new to the
literature. When the latent factor is estimated in a way that permits the econometrician
to be uncertain in its exact form, latent factor uncertainty can appropriately propagate to
estimates of a firm’s beta.
Previous papers have modeled latent factors as principal components and used firm
loadings on principal components (Ball, Sadka, and Sadka 2009). My method is distinct
from principal component analysis (PCA) in that my posterior densities are not strictly
linear in the data, as in PCA. The posterior densities in my model are jointly determined
and thus capable of describing the higher moments of my parameters in an internally con-
sistent way, i.e. the posteriors are simply functions of the likelihood and of the prior. Ad-
ditionally, PCA does not generally permit a degree of uncertainty in either the structure
of the principal components, nor in the weightings on the principal components. Thus, the
simple form of principal components analysis6 is not able to speak to parameter uncer-
tainty in the same way as is my model.
Second, to my knowledge, I am the first to estimate a model that parameterically de-
composes firm earnings into common and idiosyncratic components. Prior studies gener-
ally proxy for commonality in earnings using ratios of R2 (Brown and Kimbrough 2011),
whereas my paper provides an explicit parametric form of the economy. Additionally, my
model permits an intuitive understanding of the uncertainty a statistician should have
in that decomposition – if the posterior joint samples of a firm’s common factor loading
βi are highly negatively correlated with a firm’s idiosyncratic average earnings growth µi,
then it is difficult for the statistician to infer whether a firm’s observed performance comes
from µ 7i or from βi.
5For example, if Mt was strictly Gaussian (which it is not, due to the AR parameter ϕ), the resulting distribution re-
quires a nontrivial evaluation of modified Bessel functions (Cui et al. 2016).
6I distinguish the “simple form” of PCA from more sophisticated methods like probabilistic PCA (Tipping and Bishop
1999).
7I find a negative posterior correlation between µi and βi, particularly for large firms that are tightly linked to aggregate
earnings shocks. The reason this occurs is that, when a firm’s observed earnings closely mimic the aggregate earnings growth
series, the model is not able to distinguish between earnings that are because of a firm’s idiosyncratic average performance µi
18
Third, my model permits partial time-variation in cashflow parameters8, in that I re-
estimate the model each quarter using new data. The model permits the statistician to
change their entire understanding of a firm’s parameters based on all the data available at
the time. I opt for “constant” parameters in the laws of motion of my economy, but the
estimates of those parameters are continuously updated each quarter.
My model has a weakness that is important to consider when interpreting my results.
In particular, firm idiosyncratic performance is homoscedastic, while the latent factors are
heteroskedastic. Recall the laws of motion
Xt = µ+ βMt + νt
Mt = δ + ΦMt−1 + ϵt
with ν 2i,t ∼ N (0, γi ) and ϵi,t ∼ N (0, σ2). Firm earnings growth is thus a function of (a)
their own characteristics (average growth µi and stochastic shock ϵt) and (b) some loading
(denoted with βi) on an autoregressive process Mt.
Astute readers will note that both Mt and Xt have constant prior variances through-
out time. Latent factor shocks ϵt have variance σ
2 and firm earnings growth has vari-
ance γ2i . In frequentist approaches to econometrics, it is extremely common to handle het-
eroscedasticity through clustering or robust standard errors, and the absence of an obvious
heteroskedasticity-robust estimator may strike the frequentist reader as flawed at best and
incorrect at worst.
In practice, however, note that σ2 is a hierarchical parameter, and that the actual
shocks ϵt have their own posterior distribution that is estimated using σ
2 as a prior. The
posterior distribution for two different shocks ϵr and ϵs need not have the same posterior
variance, even if they share the same prior variance. Few of the latent factor shocks I
estimate have identical posterior variance, which addresses any concerns about the het-
eroscedasticity problem in full for the latent process and in part for the firm processes.
Using the latest model where all 165 quarters are available, 2008q2 has a posterior stan-
dard deviation of 0.24, nearly twice that of the 2019q2’s posterior standard deviation of
0.13.
Unfortunately, the issue still remains that the model I estimate treats νi,t as a parame-
ter with a constant variance – firms are not permitted to have higher residual uncertainty
and the firm’s exposure to the aggregate earnings process βi. The “exchangeability” of µi and βi for these firms is essentially
a lack of identifiability quantified through the posterior correlation in µi and βi.
8It is not clear that time-variation in parameters is critical to my empirical tests in Section 5. Campbell and Vuolteenaho
(2004) find (notably) that firm cashflow and discount rate betas have significantly different characteristics between the first
and second half of their sample period. Fortunately, Campbell and Vuolteenaho (2004) also replicate their cross-sectional
tests using a rolling estimator, and find that constant and time varying parameters both produce comparable prices for
cashflow beta risk.
19
at different time periods, since γ2i is constant. An outcome of my modelling choice is that
firm idiosyncratic variances are biased upwards to rationalize observed earnings growth in
cases where a firm’s earnings growth history is discordant with the estimated latent factor.
While I do not use γ2i directly in my tests, an upwards-biased idiosyncratic variance can
have the effect of attenuating the expectation of βi towards the prior mean of 1, since a
high idiosyncratic variance rationalizes a large set of observed earnings growth. The model
can then reduce the joint density of a parameter space by pushing βi closer to 1. This is
not a definite outcome in my estimation and depends on the relative strength of the priors
on γ2i , βi, and on the degree of observed commonality in earnings growth.
My model is a novel approach to statistical modelling in contemporary economics. I
can reduce a complex economic system to a manageable level, while maintaining the un-
certainty and complexity of the system. Further, I demonstrate how one might apply con-
temporary Bayesian methods to financial economics. The subsequent section details the
estimation of my model.
20
Chapter 3
TOPICS
This section details several topics in financial economics that I can address. Section
3.1 details existing research on the cashflow beta, and provides several tests to determine
whether the cashflow beta I estimate measures exposure to systematic risk. Section 3.2
highlights common market responses to a firm’s earnings announcements and outlines tests
using my model outputs. Section 3.3 describes the role of firm informativeness and details
several hypotheses about observable characteristics for highly informative firms.
3.1 Cashflow beta
The primary motivation of this paper is the study of the asset pricing effects of a firm’s
exposure to systematic risk. Early works in asset pricing theorized that some measure of
an asset’s covariance (beta) with consumption should be priced (Rubinstein 1976; Lucas
1978; Breeden 1979). This style of model, commonly referred to as the conditional CAPM
or CCAPM, theorizes that differences in asset prices should be driven by differences in the
covariance between an asset’s returns and aggregate consumption (the consumption beta).
Early tests of the CCAPM did not find that the consumption beta explained variation in
asset price (Campbell 1996; Cochrane 1996).
Tests of the CCAPM varied in their approaches to attempt to explain the failure of
the unconditional CCAPM to explain variation in returns. One branch proposes that con-
sumption is simply measured poorly and better proxies for consumption growth provide
consumption betas that better explain prices (Savov 2011; Kroencke 2017). Others esti-
mate a time-varying CCAPM (Lettau and Ludvigson 2001). Some tackle the failure of the
CCAPM by applying different preferences (Campbell and Cochrane 1999).
The focus of this paper is about one such extension intended to address failures of the
CCAPM, the cashflow beta. The cashflow beta differs from the consumption beta in that
it describes the covariance of an asset’s cashflow with consumption, rather than the covari-
ance of asset returns and consumption. The seminal work of Bansal and Yaron (2004) ties
asset prices to their cashflow beta. A later work Bansal, Dittmar, and Lundblad (2005)
notes that “[Under Epstein-Zin preferences t]he cash flow betas and standard consumption
betas differ and do not provide the same information as risk premia.” The cashflow beta is
a quantity with a solid theoretical grounding and some empirical suport.
A variety of empirical tests find strong support for a positive cashflow beta premium
(Da 2009; Campbell and Vuolteenaho 2004; Botshekan, Kraeussl, and Lucas 2012; Ball,
21
Sadka, and Sadka 2009). However, a large porportion of these papers were called into ques-
tion by Chen and Zhao (2009), who note that the cashflow betas estimated by any paper
using the return decomposition approach of Campbell and Shiller (1988) could be identi-
fying severely missepcified cashflow betas. Chen and Zhao (2009) shows that the common
approach of first estimating discount rate news and then treating the residual effect as
cashflow news can produce severe misspecification errors. Discount rate news is difficult to
estimate and using proxies to capture discount rate news effect can leave large and poten-
tially biased measurement errors in cashflow news, which produces biased and nonsensical
cashflow betas. Papers with questionable results due to their use of the Campbell and
Shiller (1988) include Campbell and Vuolteenaho (2004), Botshekan, Kraeussl, and Lucas
(2012) and Bansal, Dittmar, and Lundblad (2005). Papers untouched by the Chen and
Zhao (2009) critique include Da (2009) and Ball, Sadka, and Sadka (2009), but it remains
the case that much of the evidence for a cashflow beta has been called into question for
econometric purposes.
My key contribution is an alternative approach to modelling commonality in earnings
growth. Rather than select a particular proxy for earnings growth, I allow the data, guided
by my model, to estimate a latent earnings growth factor directly. Estimating the latent
factor by parametric fit rather than by an alternative method like PCA (discussed in Sec-
tion 6.1) allows my model to better interpret past earnings shocks in addition to current
ones. For example, earnings news today that is inconsistent with a non-persistent latent
factor can revise the posterior distribution of the common shock that appeared years previ-
ous.
My cashflow beta is not dissimilar to the ones estimated by Da (2009) or Ellahie (2021),
who project observed or forecasted earnings growth on aggregate earnings growth. The
Bayesian model I employ has a less obvious intuitive interpretation than the linear projec-
tion in that the posterior distribution has a non-zero weighting towards the prior density
of my joint parameters θ. I select priors in my model to tilt the estimation towards the be-
lief that the single latent factor generally influences many firms by setting the variance of
the prior density of βi to 0.1. The tightness of the prior on βi means that the model must
be presented with large amounts of data to reduce the latent factor to meaninglessness.
The interpretation of the cashflow beta, however, is not complicated. The parameter βi
is the channel the model has available to it to determine a firm’s exposure to a common
shock that is shared by all firms in the sample. The cashflow beta should be thought of as
an uncertain parameter governing the firm’s loading on an uncertain first common com-
ponent of aggregate earnings, though the common factor I find is not a maximal variance-
reducing component, but an uncertain, evidence-weighted composite informed by firm
earnings. Ball, Sadka, and Sadka (2009) shows that firm loadings on the principal compo-
22
nents of aggregate earnings are priced.
To understand this intuition better, let us walk through a simplified version of the way
the model estimates a cashflow beta. If all firms tend to have highly volatile and persistent
earnings growth shocks that move together, the model will assign more weight to the pos-
terior distribution where firms have non-zero betas. If there is evidence of common shocks
that are differentially shared by firms, then the model should down weight the density of
βi for firms that are less exposed and up weight the density of βi for firms that are more
exposed. The posterior density of the cashflow beta P (βi | X1:t) thus measures a firm’s
exposure to common sources of earnings growth.
My second contribution is my estimation of parameter uncertainty about the cashflow
beta. Cashflow risk is thought to have several characteristics, of which the cashflow beta
is just one. The cashflow beta is a measure of the covariance of asset cashflows with aggre-
gate cashflows. I suggest that parameter uncertainty should play a role in asset prices just
as the true value of the cashflow beta.
Parameter uncertainty is defined as uncertainty an investor faces in the parameters
that govern the data generating process for an asset’s characteristics, such as cash flows
or returns. The effect of parameter uncertainty (also called estimation risk in older works)
on asset prices is fairly well studied, though the literature has focused on restricted models
of Bayesian learning. A notable early entry in the study of parameter uncertainty is Klein
and Bawa (1977), which finds that parameter uncertainty should generally produce mean-
variance portfolios that are disjoint from the traditional full-information portfolio of the
CAPM.
Clarkson, Guedes, and Thompson (1996) summarizes the literature by noting that “...in
a CAPM framework, the literature has shown that differential parameter uncertainty gen-
erally has a systematic component and should be priced to some degree”. Handa and Linn
(1993) expands on this in more detail, and notes that parameter uncertainty about id-
iosyncratic components (such as a firm’s alpha) tends to function as if the asset has an
elevated correlation with the stochastic discount factor. Investors attribute more system-
atic risk to assets for which they have less information, which reduces the prices for low-
information assets. Differentially informed assets (low- and high-information) can arise due
to different asset age, structural breaks in parameters, or a spin-off that generates a new
firm.
Investors in Handa and Linn’s model attribute more systematic risk to low-information
firms because their parameters are more uncertain. This (rational) attribution of risk re-
sults in lowered prices for low-information firms and higher prices for high-information
firms. Notably, Handa and Linn’s model does not require correlated priors – investors in
their model know an asset’s idiosyncratic risk, and the only source of deviation from a
23
full-knowledge APT portfolio arises because of common estimation risk in the underlying
factor betas.
The effects of parameter uncertainty on prices can be non-linear, and it does not ap-
pear to be the case that higher parameter uncertainty should unambiguously increase or
decrease asset prices. More recent papers reinforce the findings that parameter uncertainty
causes the näıve CAPM tangency portfolio be suboptimal (Kan and Zhou 2007). Cvitanić
et al. (2006) and Xia (2001) suggest that hedging estimation risk can be a significant pro-
portion of risk asset demand, and that investment horizons can play a large role in portfo-
lio construction. Importantly, in these papers, the degree to which parameter uncertainty
shifts asset demand is a function of asset cross-covariance and not simply absolute parame-
ter uncertainty. My paper only tests the simpler predicted form of parameter uncertainty
of Handa and Linn (1993), which predicts that higher uncertainty yields higher expected
returns.
I contribute parameter uncertainty in the systematic proportion of cashflow risk to a
growing body of work that study alternative aspects of cashflow risk. Da (2009) models
the duration of asset cashflows, a measure of whether cashflows arrive in the near or far
term. Botshekan, Kraeussl, and Lucas (2012) studies differential cashflow betas in good
and bad times, and finds that the bad-times cashflow beta has a higher risk price than
does the good-times cashflow beta. Li and Zhang (2017) proposes cashflow betas with
short- and long-term exposure to aggregate consumption growth.
To test my predictions, I propose two forms of tests. First, I estimate Fama-MacBeth
cross-sectional regressions, controlling for the Fama-French 3-factor betas. I include the
cashflow beta and its uncertainty. If the cashflow beta or my measure of parameter uncer-
tainty has a risk price associated with it, the cross-sectional regressions will show a signif-
icant coefficient in the cross-sectional regressions. Importantly, my proposed tests should
control for the Fama-French 3-factor betas of Fama and French (1993) to ensure that my
test is not simply a proxy for a redundant but known risk factor.
Second, a common test when there is a hypothesized risk factor is to construct port-
folios sorted by some characteristic and evaluate whether the ex-post returns of the port-
folios differ across characteristics. Typically, researchers calculate abnormal returns for
these characteristic portfolios and test whether the abnormal returns are statistically dif-
ferent across the high and low portfolios1. I provide univariate sorts on the cashflow beta
and uncertainty, as well as a bivariate sort on the cashflow and its beta to highlight any
conditional pricing effects.
My cashflow beta βi should have an associated risk price as long as exposure to aggre-
1An alternative approach when performing portfolio tests of risk factors is to explicitly test the monotonicity in portfolio
returns, sorted by the quartile of the characteristic. Patton and Timmermann (2010) provides tests on the assumption that
portfolio returns should be monotone in the sorting characteristic.
24
gate earnings growth is a systematic risk, which is generally agreed to be the case (Camp-
bell and Vuolteenaho 2004). Section 5.1 does not find evidence that my cashflow beta is a
priced factor, but that the parameter uncertainty is priced. I discuss my findings and their
implications in Section 6.2 and provide an alternative model specification for the cashflow
beta in Section 7.
3.2 Earnings news
The cashflow beta in my model permits an investor to infer the contemporaneous la-
tent state, although imperfectly. A firm’s observed earnings growth Xi,t is a function of
idiosyncratic factors like µi and γ
2
i , but also of the cashflow beta-adjusted latent factor
βiMt. Several papers provide theoretical or empirical evidence that the degree of a firm’s
covariance with the latent factor can induce observable effects in returns during earnings
announcements.
First, a vein of research focuses on the earnings announcement premium, which de-
scribes abnormal returns that accrue during firm earnings announcements. The earnings
premium occurs without conditioning on earnings surprise. Two general strands of re-
search hypothesize why the premium may accrue. The first strand of literature suggests
that stocks become more risky during earnings announcements due to rational learning
(Savor and Wilson 2016; Patton and Timmermann 2010). The second strand posits that
risk does not increase during earning announcement, but that the risk premium associated
with a firm’s market beta is better estimated during earnings announcement dates (Chan
and Marsh 2022). In either case, the premium exists and seems to be a robust empirical
feature.
Savor and Wilson (2016) show that a portfolio long on earnings announcers and short
non-announcers has positive alpha. Patton and Verardo (2012) produce a similar find-
ing. Savor and Wilson motivate the risk premium they find through a simple model of
cross-learning, where information about a common cashflow shock can only be conveyed
imperfectly through earnings news. Firms who announce in batches with other firms have
a higher covariance with the common factor because their earnings news when they an-
nounce represents a larger proportion of beliefs about the aggregate shock than during
other times. Savor and Wilson’s model assumes that all firms are identical in characteris-
tics, so their model cannot make any implications about how the premium they highlight
could accrue to firms with different characteristics. As providing such a model is beyond
the scope of my paper, I attempt to provide empirical evidence that the Savor and Wilson
premium may vary with firm characteristics.
The form of heterogeneity I focus on is the cashflow beta and its parameter uncertainty.
25
Firms with higher cashflow betas are ceteris paribus clearer signals of aggregate earnings
shocks, and thus it could be the case that the partial announcement effect of Savor and
Wilson is amplified for these firms. Parameter uncertainty has a less clear role in earnings
announcement premia, as it can arguably increase or decrease earnings announcement pre-
mia. On one hand, parameter uncertainty could increase earnings announcement premia
for the same reason that parameter uncertainty in Handa and Linn (1993) functions as
systematic risk – when investors do not know how much of firm earnings is systematic,
they act as if the firm is more systematic than it may truly be. On the other hand, ex-
ante parameter uncertainty may reduce earnings announcement premia relative to other
announcers by adjusting uncertain firm weights in the announcer portfolio’s signal about
common earnings. If two firms announce earnings in the same week, but one firm has a
highly uncertain cashflow beta, a rational response by market investors is to apply more
weight to the firm with a certain cashflow beta.
To test earnings announcement premia, I perform Fama-MacBeth cross-sectional regres-
sions with a dummy variable for a firm’s announcement week. The predictions in Savor
and Wilson (2016) are specifically on the premium for a portfolio of announcing firms rel-
ative to non-announcing firms, so any test I conduct should attempt to capture the same
premium. To capture any heterogeneity in premium, I interact the announcement week
dummy with a measure I believe should capture some degree of differential exposure to
risk premia, such as the cashflow beta or parameter uncertainty. Finding a positive load-
ing on any of the interaction terms implies a risk premium that accrues differentially to
firms with a given characteristic. There is reason to believe that the earnings announce-
ment tests may better describe the risk premium associated with my cashflow beta than
does the general cross-sectional tests described in Section 3.1, as there seems to be evi-
dence that premia are better estimated (and potentially only relevant) during earnings
announcements (Chan and Marsh 2022).
My second set of tests applies the cashflow beta and its parameter uncertainty to earn-
ings responses. Firm earnings announcements contain an expected component that is typi-
cally priced ahead of the earnings release. Earnings announcements also contain a surprise
component that can be used by investors to revise their beliefs about a firm’s parameters.
Prior works quantify the degree of price response using the earnings response coefficient,
which is the amount of abnormal returns that occur during an earnings announcement per
unit of earnings surprise (Hirshleifer, Lim, and Teoh 2009).
Schmalz and Zhuk (2019) study earnings response coefficients in a Bayesian model simi-
lar to my own, though their model has some known structural parameters. Their primary
finding is that earnings responses are higher in bad times, since earnings news is an un-
ambiguous signal about firm quality. Schmalz and Zhuk derive an expression for a firm’s
26
earnings response coefficient (ERC) as a function of the relative parameter uncertainty
about a firm’s cashflow beta. All else equal, their expression predicts that higher betas
and higher parameter uncertainty should imply lower earnings responses.
Testing earnings responses is relatively straightforward. First, I calculate the cumula-
tive abnormal returns (CAR) on the day of and the day following an earnings announce-
ment. Second, I regress the CAR on a firm’s surprise unexpected earnings, or SUE, and
a parameter I expect to influence the cumulative abnormal returns interacted with SUE.
The construction of SUE is described in Section 5.2. If the interaction between the param-
eter of interest (the cashflow beta or parameter uncertainty) and a firm’s surprise unex-
pected earnings is significant, it implies that market responses to the same unit of news
differs for firms with different characteristics.
In Section 5.2, I provide tests of earnings response coefficients that vary with parameter
uncertainty and with the cashflow beta, finding strong evidence in support of predictions
drawn from Schmalz and Zhuk’s model. I find that earnings responses are decreasing in (a)
the cashflow beta and (b) parameter uncertainty. Parameter uncertainty appears to play a
larger role in earnings responses than does the level of the cashflow beta.
Schmalz and Zhuk (2019) test earnings responses in good and bad economic periods,
since their claim is largely that parameter uncertainty declines in bad times and thus earn-
ings responses increase. I provide similar tests to see if decreasing parameter uncertainty
corresponds with higher earnings responses segmented by good and bad periods. I do not
find statistically significant evidence that earnings responses vary between good and bad
times.
Overall, I note that the literature generally predicts higher earnings response coeffi-
cients for firms with lower uncertainty and lower cashflow betas, but that the role of the
cashflow beta is unclear as it relates to earnings announcement premia. Sections 5.2 pro-
vides empirical tests of my predictions. I find evidence that firms with higher parameter
uncertainty accrue larger earnings announcement premia, but have lower earnings response
coefficients. I find no premium associated with the cashflow beta, but I do find that higher
cashflow betas reduce earnings response coefficients.
3.3 Firm informativeness
My model permits me to define the signal-to-noise ratio of a firm about common earn-
ings shocks. Firms with precise signals are often referred to as “bellwethers”. A bellwether
is typically defined in the financial economics literature as a firm that acts as a highly in-
formative signal about other firms. Hameed et al. (2015) defines a bellwether as a firm
whose fundamentals best predict the fundamentals of peer firms. They find that revisions
27
in analyst forecasts about bellwether firms causes large price revisions in peer firms. Bon-
sall, Bozanic, and Fischer (2013) shows that bellwethers provide timely information about
industry and market shocks. Hann, Kim, and Zheng (2019) shows that earnings announce-
ments from bellwethers reduce option implied volatility in peer firms, suggesting that bell-
wether firms reduce uncertainty in non-bellwether firms, as measured by implied volatility.
In the spirit of these papers, I explicitly derive a new statistical measure of bellweth-
erness that succinctly communicates the ability of a firm’s earnings to reveal aggregate
earnings news. I make two broad predictions about bellwether firms in my sample.
First, bellwether firms should generally have higher earnings response coefficients. Bell-
wether firms have high systematic cashflow exposure (high betas) and low idiosyncratic
volatility by their very definition. Further, bellwether firms draw more analyst attention
(Hameed et al. 2015) which should correspond to elevated earnings response coefficients
(Hirshleifer, Lim, and Teoh 2009). I expect that earnings responses for highly informative
firms should be larger, because each unit of news they produce is both more informative
and more economically meaningful.
Second, highly informative firms may accrue higher risk premia due to elevated mar-
ket risk from a learning channel. Informative firms are those with (a) high betas, (b) low
parameter uncertainty, and (c) low idiosyncratic risk. In the context of Savor and Wilson
(2016), bellwether firms are those that are the clearest signals about aggregate earnings
news. Announcing bellwether firms thus may have a larger earnings premium than other
firms because they have high systematic exposure. Alternatively, these firms may have
lower premia because they are very precise signals. As a counterfactual example, Savor
and Wilson (2016) note that no risk premium would be earned if firm earnings fully re-
vealed the common earnings news. Savor and Wilson (2016) is silent on modeling cross-
sectional variation in the premium, so again I provide empirical results to motivate future
analysis.
I derive a measure of firm informativeness to test my two predictions. Recall that my
model focuses on firms’ exposure to an uncertain latent factor. The latent factor is never
known or revealed to investors, so they face a learning problem where observed firm earn-
ings growth is used to infer the value of the parameter governing the latent factor Mt, its
parameters, and the history of the shocks ϵ. The information value of any firm’s earnings
growth to the latent shock varies with firm characteristics. For example, if a firm has a
large idiosyncratic earnings growth variance γ2i relative to the volatility of aggregate earn-
ings, that firm’s earnings are high-noise and cannot be used to precisely determine the
value of the current latent factor shock.
I define the variance ratio VRi,t, which summarizes a firm’s ability to precisely and re-
liably inform investors about aggregate growth shocks. VRi,t can be thought of as the de-
28
gree of “bellwetherness”, that is, how much observing the firm’s actual earnings can reduce
residual uncertainty about aggregate earnings growth shocks. Suppose a Bayesian investor
were attempting to form a nowcast of the unobserved latent process Mt after observing
firm earnings growth Xi,t. The Bayesian investor’s forecast is Var[Mt | Xi,t], with forecast
variance Var[Mt | Xi,t]. Lemma 1 derives this forecast and its variance.
Lemma 1. The variance of the latent process Mt conditional upon observing the earnings
growth Xi,t of an announcing firm j is
2
[ ] [ ]
Var[Mt | Xj,t] = Eθ̂ Var[Mt | Xi,t, θ̂] + Varθ̂ E[Mt | Xi,t, θ̂]
where
( )−1
σ2 β2σ2
Var[Mt | X ii,t, θ̂] = +(1 ) (3.1)1− ϕ2 γ2(1− ϕ2∑ i )t−1 −1(γ2(1− ϕ2) ∑t−1 )
E[Mt | X s i si,t, θ̂] = δ + ϕ (δ + ϵt−s) + 1 + β ϕ ϵ + ν (3.2)
β σ2
i t−s j,t
s=1 i s=1
Lemma 1 is straightforward to compute numerically, once equipped with posterior sam-
ples of θ̂k. Simply computing Equations 3.1 and 3.2 for all posterior samples θ̂k and cal-
culating the sum of the sample average (of Equation 3.1) and variance (of Equation 3.2)
yields the appropriate posterior Var[Mt | Xj,t].
Var[Mt | Xj,t] is not a scale-free measure, since the model has limited data available
towards the beginning of the sample and variances can be elevated. I construct a scale-
free measure that succinctly describes the information content a firm provides about the
latent factor. I denote the amount of variance about the latent process Mt reduced after
observing firm i’s earnings growth (VRi,t) as
Vart[Mt | Xi,t]
VRi,t = (3.3)
Vart[Mt]
The ratio VRi,t ratio is bounded between 0 and 1. The intuition of Equation 3.3 is that a
2This follows from the law of iterated expectations[: ] [ ]
Var[X] = E Var[X | Y ] + Var E[X | Y ]
29
low ratio indicates that, on average, firm i contains precise information about the current
value of Mt. The variance ratio is a sufficient statistic for the nowcasting capabilities of a
firm’s earnings, since (a) it is always possible to construct an unbiased estimate of Mt (in
Equation 3.2) and thus signals can be discriminated only by their precision Var[Mt | Xj,t].
Values of VRi,t close to 1 indicate that the firm’s earnings announcement provides little
new information about the common factor. Note that the variance ratio VRi,t summarizes
both average variance and the conditional expectation, which addresses both the accuracy
and location of an investor’s nowcast.
I highlight the variance ratio because it is a reasonably effective way of reducing a high-
dimensional parameter space into a statistic that describes a firm’s informativeness. The
variance ratio includes parameter uncertainty in addition to cross-learning from firms,
making it a uniquely informative statistic about the power of a firm’s realized earnings
to forecast aggregate earnings.
The estimate VRi,t has a shortcoming, in that it is a counterfactual evaluation of a
world where firm i is the first firm to announce in a quarter. The firm with the lowest vari-
ance ratio in a quarter is the firm that an investor would choose to announce first, since
the announcement carries the most information. A more rich measure of the informational
content of a firm’s earnings content would be to integrate across all possible announce-
ment orders, though this is computationally infeasible as it would require the computation
of 1535! variance ratios. Section 3.3.1 details what my model can and cannot say about
optimal ordering of firms.
To summarize, I predict that bellwether firms, firms with low variance ratios VRi,t,
should have higher earnings response coefficients. I am unclear on the role of bellweth-
erness and earnings announcement premia. In Section 5.3 I find evidence for elevated earn-
ings responses but lower earnings announcement premia.
3.3.1 Earnings announcement order
The previous section provides a sufficient statistic for the amount of information a firm
reveals if it were to go first in each earnings season. One question is whether observed
earnings season orderings are consistent with an informationally optimal ordering. My
model is capable of providing some information about the general earnings order observed
and whether my model would predict a similar ordering.
Suppose a social planner were selecting the order in which firms should announce their
earnings to the market. Assume that the social planner’s objective is to maximize the
amount of information transmitted as early as is possible. The first firm the social plan-
ner would select is the firm with the lowest variance ratio VRi,t, since this is the firm that
most reduces uncertainty about the latent factor.
30
For the second and higher firms, the order is not simple to calculate numerically since
it requires conditioning on the set of all firms who have already announced. A brute-force
algorithm to compute optimal ordering would be to calculate a sequence of variance ratios:
1 Vart[Mt | Xi,t]VRi,t = Vart[Mt]
2 Vart[Mt | Xi,t, Xι(1),t]VRi,t = Vart[Mt | Xι(1),t]
. . .
N Vart[Mt | Xi,t, Xι(1),ι(2),...,ι(N−1),t]VRi,t = Vart[Mt | Xι(1),ι(2),...,ι(N−1),t]
where ι(j) = argmini VR
j
i,t indicates the firm with the minimum variance ratio in posi-
tion j. The firm order vector ι = [ι(1), ι(2), . . . , ι(N)] is then informationally optimal (by
construction) in that each firm’s earning news provides the most information to investors.
Unfortunately, calculating the optimal ordering is computationally infeasible as it would
require the calculating a total of 1535! variance ratios. A more reasonable first-pass at
optimal ordering would be to organize firms by their variance ratios, assuming they were
all the first to announce. Releasing the earnings news of these firms first should generally
correspond with large information events happening towards the beginning of the earnings
season.
Few of the above papers make theoretical predictions about exactly where firms should
land in the earnings announcement order. The only paper I am aware of on the topic is
Guttman, Kremer, and Skrzypacz (2014), which models a manager’s decision of when to
release information early or late. They show that the manager should release informa-
tion later rather than later, unless the manager is myopic and cares about early-period
prices. Guttman, Kremer, and Skrzypacz (2014) points to the basics of disclosure theory
and suggests that managers in a non-competitive environment would prefer to release later.
However, Guttman, Kremer, and Skrzypacz (2014) does not address the effects of informa-
tion competition in earnings disclosure. If there were two managers in the model, would
the late-announcement preference still hold? This isn’t clear, and so it remains an open
question as to what order we should expect to observe in competitive markets.
It is not clear what we should expect to see by comparing this pseudo-optimal ordering
to observed earnings announcement orderings, since there are many factors that have lit-
tle to do with a hypothetical social planner’s decision. Further, most existing papers on
earnings announcement order tend to focus on marginal deviations from scheduled earn-
ings. Kim, Pierce, and Yeung (2021) finds that firms announcing good news tend to delay
31
their earnings news, and that the delay is unrelated to earnings management. In a similar
vein, Johnson and So (2018) shows that firms who schedule earlier tend to announce worse
news. Noh, So, and Verdi (2021) shows that firms who announce earlier (through a quasi-
exogeneous shock to ordering) have higher earnings announcement premia and greater
media attention. Savor and Wilson (2016) shows that early announcers have higher earn-
ings announcement premia. Frederickson and Zolotoy (2015) finds that investors choos-
ing between the earnings news of firms who announce on the same day tend to focus on
more “visible” firms, defined in their study as firms with higher media attention, advertis-
ing spending, analyst coverage, and size.
Section 5.4 provides basic statistical tests showing that highly informative firms tend
to announce first, even when controlling for size and book-to-market. My finding provides
some validation to the model in that there is no explicit requirement in my model that
early announcing firms also be informative, but observed non-model earnings announce-
ment orders seem to coincide with a high-information equilibrium. The next section pro-
vides statistical evidence for the topics described above.
32
Chapter 4
ESTIMATION
4.1 Model estimation
My sample consists of quarterly earnings releases for 1,535 firms between January 1984
and December 2021. Firms are selected from the CRSP/Compustat dataset. I draw all
domestic firms with a stock ownership code of zero (STKO) to exclude ADRs. I use only
firms with fiscal quarters ending in March, June, September, and December to simplify my
modelling exercise. Firm-quarters without income before extraordinary items (IBQ), com-
mon shares outstanding (CSHOQ), quarterly end price (PRCCD), or total assets (ATQ)
are excluded, as are firms with a market cap below $500 million.
I calculate seasonally-adjusted earnings growth Xi,t using the method in Savor and
Wilson (2016) and Kothari, Lewellen, and Warner (2006). Firm earnings growth is calcu-
lated as the difference between a firm’s earnings (IBQ) and the firm’s earnings four quarter
prior, rescaled by the firm’s market cap in the previous quarter. Firms are thus required
to have at least four quarters of observations before they can be included in my sample. I
multiply earnings growth by 100 to reduce the risk of numerical errors that can arise with
small numbers – all parameters should thus be interpreted in terms of percentage points
(i.e. µi = 1.5 indicates an average earnings growth of 1.5%).
My model is written in the probabilistic programming language Turing.jl. I perform
Markov chain Monte Carlo (MCMC) sampling using the No U-Turn Sampler (NUTS), a
Hamiltonian Monte Carlo method. NUTS is considered to be an easy-to-use MCMC algo-
rithm that scales well to high dimensional problems (Hoffman and Gelman 2014). NUTS
obviates much of the difficulties of Bayesian inference for high-dimensional problems like
the one I have presented. As NUTS (and Markov chain Monte Carlo in general) may not
be familiar to traditional financial economists, I include a brief description of the function-
ality of MCMC.
MCMC methods vary greatly by the specific algorithm employed, but in general (most)
methods follow a common procedure:
1. Initialize model parameters θ0 to some value (typically random draws from the prior).
2. Propose a new parameterization θ̃, chosen stochastically by a MCMC kernel K(θ0).
In my case, this kernel is the NUTS.
3. Accept the proposed θ̃ with some probability given by the kernel K(θ, θ0). If accepted,
set the draw to θ1 = θ̃. If rejected, set θ1 = θ0.
33
4. Repeat steps 2-4 sequentially until a predetermined number of samples are drawn.
The kernel K(·) is replaceable with numerous MCMC algorithms. For example, a common
MCMC kernel is random-walk Metropolis-Hastings, which proposes a new parameteriza-
tion by perturbing the old parameterization with a Gaussian shock, i.e. θ̃ ∼ N (θn−1,Σ)
for a fixed proposal variance Σ. The new proposal θ̃ is accepted with probability α =
P (θ̃)/P (θn−1), where P (θ̃) is the joint density of the model (prior probability of θ̃ multi-
plied by the likelihood evaluated at θ̃). The kernel I employ, NUTS, is a significantly more
complex statistical construction that treats parameters as if they were high-energy parti-
cles with momentum, traversing the probability space of the problem. I refer the reader to
Hoffman and Gelman (2014) for detailed information about NUTS, as understanding how
it functions is both outside the scope of this paper and noncritical to understanding my
results.
For each quarter, I generate a vector of earnings growth for firms announcing in the
quarter. I then estimate a new model for each new quarter of data, where each model in-
corporates all past quarterly data. For example, the 16th model uses 16 quarters’ worth
of earnings growth data for all firms that announced. The 17th model has 17 quarters of
data available, for example. Each model contains 4,000 samples (1,000 across 4 chains set
to random initial values), with an additional 1,000 samples per chain at the beginning of
sampling for burn-in and adaptation. The 1,000 burn-in samples are discarded after sam-
pling is completed as they are generally non-representative of the true posterior.
My MCMC kernel, NUTS, is designed to have few degrees of freedom for the researcher.
The two sampling parameters I modify are the target acceptance rate and the maximum
tree depth. The target acceptance rate is the proportion of proposed samples that the sam-
pler should accept. Typically, NUTS handles higher acceptance rates by making smaller
proposals that are close to the previous sample. I set my target acceptance rate to 80%
in accordance with the suggestion in Hoffman and Gelman (2014). The other parameter,
maximum tree depth, limits the number of points evaluated used by NUTS to find a new
sample. Higher limits for tree depth permit more significant jumps from sample to sample
with higher acceptance rates, but due to computational constraints I select a slightly lower
tree depth of 5 than the default of 7. Lowering the maximum tree depth does not make
my samples less accurate or incorrect asymptotically, but it does mean that my sampler
will generally converge slower on average.
Under certain conditions, MCMC methods asymptotically guarantee1 that all parame-
ters are drawn from the posterior joint density P (θ | X1:t). In particular, the distribution
of the sampled values θn approximate the distribution P (θ | X1:t) as the number of sam-
1See Betancourt (2021) for a brief overview of finite-sample convergence in MCMC samples, or Gelman et al. (2013) for a
more comprehensive text on MCMC in general.
34
ples reach infinity. In the absence of asymptotic draws, however, statisticians must rely on
various diagnostics to indicate that a Markov chain has reached a stationary point. Var-
ious measures have been proposed, such as the Gelman-Rubin diagnostic of Gelman and
Rubin (1992). The Gelman-Rubin diagnostic, commonly referred to as R̂, tests whether
there is sufficient evidence to conclude that the first half of a Markov chain has the same
mean as the second half. In an equilibrium state, a Markov chain should have the same
mean at every point in the sample2. A second diagnostic is to conduct a visual inspection
of trace plots for samples to verify that the samples appear stationary. I provide diagnos-
tics in the next section.
4.2 Estimation results
I estimate a total of 165 probabilistic models. Each of the 165 models incorporates a
new quarter’s earnings growth for each firm that announced earnings. The 165th model
thus incorporates all earnings growth disclosures from 1984Q1-2021Q4 (165 total quar-
ters).
To verify that each of the 165 models are likely to be valid MCMC samples (in the
sense that their sampling distribution approximate the true posterior density), I visually
check trace plots for a sample of parameters. A common diagnostic technique in Bayesian
inference is to verify that the trace plot of a parameter displays no obvious autocorrela-
tion and appears to follow a stationary distribution. Figure 4.1 demonstrates a randomly
selected example of a trace and histogram plots for the parameters of IBM using 163 quar-
ters of earnings growth. The trace plots on the left are good examples of a ”caterpillar
plot”, wherein there seems to be little autocorrelation or trend in the trace. After review-
ing 100 trace plots from different chains, I can proceed with the analysis under the as-
sumption that my samples are directly drawn from a sufficiently close approximation of
P (θ|X1:t).
An additional diagnostic tool employed in Bayesian methods is R̂ (Gelman and Ru-
bin (1992)). Gelman et al. (2013) notes that an R̂ less than 1.1 is generally considered
acceptable, though there is no hard-and-fast statistical criterion for whether a chain has
converged. I calculate that the maximum R̂ across all parameters and models is 1.042, in-
dicating that all parameters are drawn from a stationary distribution3. The result of these
tests allow me to proceed with my analysis, comfortable that sampling error due to chains
that have not converged is unlikely to be driving my results.
2Vehtari et al. (2021) suggests that R̂ is an inadequate convergence diagnostic when the chain has a heavy tail or when
sample variance is different across chains. My model is a linear model with Gaussian and inverse-gamma priors, which does
not have the complex geometry of models studied in Vehtari et al. (2021).
3It is not surprising that my chains demonstrate as good a performance as they do. My model is linear, has simple priors,
and uses a relatively flat hierarchical structure.
35
(a) Trace, βi (b) Histogram, βi
(c) Trace, µi (d) Histogram, µi
(e) Trace, γi (f) Histogram, γi
Figure 4.1: Trace and histogram plots for parameters estimated for IBM using
data from 1984q1 to 1994q4.
36
(a) Idiosyncratic mean µ (b) Common factor loading β (c) Idiosyncratic variance γ
Figure 4.2: Median estimates of parameters (bounded by 10% and 90% quan-
tiles) estimates of parameters for the Walt Disney Company.
All variables in my model have distributional (at a single point in time) and time-series
interpretation (distributions across time). Figure 4.4 shows the evolution of firm-specific
characteristics for the Walt Disney company from 1984 to 2021. Firm parameters vary
meaningfully over the sample period. As an example, note that Panel B of Figure 4.2
demonstrates a steady decline in Disney’s loading on the common factor βi, though as
of 2020q2, this trend reverses and suggests a large increase in the cashflow beta. A signif-
icant proportion of Disney’s earnings comes from theatrical film releases and theme park
attendance, both of which declined substantially during the onset of the COVID-19 pan-
demic.
Next, I explore the determinants of the latent factor, and the cross-sectional distribu-
tion of firm parameters.
4.2.1 Latent factor results
Since the latent factor plays a key role in my analysis, I extract all posterior estimates
that govern the latent factor to reconstruct it. Figure 4.3 plots the latent factor using the
latest model, estimated with data from 1984-2021. I note that the latent process closely
tracks recessions, particularly the 2001, 2008, and 2020 recessions. The precision of the
latent factor is quite high, in that the posterior standard deviation is small. My model is
relatively confident in the shape and timing of the latent factor.
Figure 4.2 displays the time series posteriors for the parameters that govern the latent
factor Mt. Each point in time is the filtered estimate of the parameter in each panel, us-
ing data up to and including the quarter on the x-axis. Aggregate earnings growth shocks
have become more persistent since roughly 2008. Note that the standard deviation of la-
tent factor shocks σ appears to have declined from 3.7 on average to 1.25. While this may
appear alarming to some readers, I note that this is more a function of the posterior vari-
ance of σ contracting as more information is revealed.
The unconditional mean aggregate earnings growth δ begins the sample in 1985 with a
37
Figure 4.3: The estimated series of the latent factor Mt, using all 165 quarters
of data. The thick blue line is the mean value E[M√t | X1:165], while the shaded
ribbon represents the mean ±1 standard deviation Var[Mt | X1:165].
median posterior belief of -0.5bps and increases to 10bps at the end of the sample. The 5%
highest posterior density4 for δ as of December 31st, 2021 is [−0.125, 0.282]. A frequentist
interpretation is that δ is not significantly different from zero at the end of the sample us-
ing a 5% significance level, though I would caution against thinking this way – there is a
large mass of δ that is positive (84% of the posterior samples are greater than zero), sug-
gesting that δ is highly likely to be small but non-zero. My findings are qualitatively and
quantitatively similar to Kothari, Lewellen, and Warner (2006), who find an unconditional
mean of 0.14.
Beliefs about the latent factor persistence parameter ϕ begins near a median of 0, but
asymptotes towards a median value of 0.225 with a 5% highest posterior density of [−0.088, 0.555]
as of the end of 2021. As of 2021, the posterior probability that ϕ > 0 is 91%. Aggregate
earnings growth persistence suggest that shocks are positively autocorrelated to some ex-
tent. A non-zero value of ϕ has the benefit of an explicit time-series learning component in
the economy, since knowing the current quarter’s shock has some predictive power over the
next period’s shock.
My posterior estimates of systematic variance (σ) declined steadily since 1985, starting
near a median of 14% and declining towards a posterior median of 1.66%. The 5% highest
posterior density at the end of the sample is relatively tight around [1.363, 2.063]. A firm
with βi = 1 would thus expect to have median earnings growth with a variance of 1.66%
per quarter, assuming an idiosyncratic shock variance of zero.
To put the systematic shocks σ into context, note that the median idiosyncratic earn-
4The highest posterior density is similar to a standard confidence interval for unimodal posteriors.
38
(a) Latent factor mean, δ (b) Latent factor persistence ϕ (c) Systematic variance σ
Figure 4.4: Median estimates of parameters (bounded by 10% and 90% quan-
tiles) governing the latent process, given by the identity Mt = δ+ϕMt−1+ ϵt where
ϵt ∼ N(0, σ).
ings growth γi is 5.23% as of 2021. I estimate that the median firm with βi = 1 derives
24% of their earnings growth innovation from systematic exposure. This is roughly consis-
tent with the explanatory power of the first principal component in Ball, Sadka, and Sadka
(2009). It is half the average proportion of macroeconomic information in Bonsall, Bozanic,
and Fischer (2013), though their use of a semipartial R2 makes it difficult to compare to
my measure and to the measure in Ball, Sadka, and Sadka (2009).
4.2.2 Firm results
Summarizing the estimated firm posteriors for µi, γi, and βi is difficult due to the large
number of these parameters, so I have opted to calculate the expected posterior values for
all firms in each quarter, and display their cross-sectional dispersion by decade. Figure 4.5
displays the results.
Expected growth rates µi have declined from about 20bps per year in 1980 to about
10bps per year as of 2020. Expected cashflow betas βi have dispersed substantially by the
2010s and 2020s, and firms are much more spread out around E[βi] = 1. Lastly, firm id-
iosyncratic earnings variance γi has increased from the earlier decades. The distributions
are also not particularly smooth, in that there are several ”modes”. In the 2010s, for ex-
ample, idiosyncratic volatility has a modal point near 2% standard deviation in quarterly
earnings growth, and a much smaller mode at about 0.5%. In all cases, it is clear that the
cross-sectional distributions have moved away from the prior-dominated density of the
1980s.
The variance ratio VRi,t is calculable from the posterior draws. Figure 4.6 portrays the
results for each firm overlaid on top of each other, to convey a sense of scale and location.
I standardize the variance ratio in each quarter by subtracting the mean and dividing the
difference by the standard deviation. Firms below zero in this panel are firms that are
more informative on average about aggregate earnings growth. The distribution of vari-
39
(a) µi
(b) βi
(c) γi
Figure 4.5: Cross-sectional dispersion in firm parameter estimates, grouped by
decade.
40
Figure 4.6: Firm variance ratios VRi,t
ance ratios is fat tailed, with a larger mass of firms having low variance ratios pre-2000.
This shifts after 2000, with the bulk of firms being above the mean variance ratio. The
2008 crisis induced severe cross-sectional dispersion in variance ratios, and introduced a
regime change in the average dispersion. The COVID-19 pandemic had a similar outcome
on cross-sectional dispersion.
In this section, I outlined the mechanics of estimating my model. I then provided a
brief overview of the results and the estimated parameters. The next section summarizes
the existing literature on a set of topics that my model is capable of addressing.
41
Chapter 5
EMPIRICAL RESULTS
In this section, I perform several tests to determine whether the measures I construct
have any statistical power for (a) earnings announcement premia and (b) pricing the cross-
section of returns. All empirical tests begin at 1990 to eliminate highly volatile parameter
estimates during 1980-1989, when my Bayesian model was√relatively low-data and heavily
prior driven.1 I standardize the posterior variance of beta Var[βi] to make the results
more numerically interpretable.
Table 5.1 presents an overview of my sample, grouped by industry. The additional
columns contain the parameters I estimate, discussed below. The top three industries are
chemicals, depository institutions, and business services. These three industries collectively
compose 24% of my sample. The largest industry by average market capitalization is mis-
cellaneous retail. I calculate the average earnings growth Xi,t for each firm as well – the
fastest growing industry in my sample is stone, clay, glass, and concrete products. The
quickest contracting industry is amusement and recreation services.
Table 5.1 highlights the average estimated parameters for my model, as well as the cor-
responding parameter uncertainty. Average betas are generally near one with some at-
tenuation towards zero. Growth rates are near zero (as in Kothari, Lewellen, and Warner
(2006)) and tend to have large posterior standard deviations, suggesting most uncondi-
tional firm earnings growth is due to aggregate earnings growth. Average idiosyncratic
earnings growth suggests that between 1-3% of firm earnings growth is due to idiosyncratic
risk. √
I calculate the measures VRi,t, E[βi], and Var[βi] directly from my posterior samples
in each quarter. I assign the above measures to cumulative abnormal returns in the quar-
ter following to avoid bias, meaning all results are interpretable as the effect of a particular
posterior quantity on CAR in the follow√ing quarter. The variance ratio VRi,t, cashflow
beta Et[βi], and parameter uncertainty Var[βi] are normalized to make estimates compa-
rable across quarters. I subtract the quarterly cross-sectional mean from each observation
and divide by the cross-firm standard deviation.
Table 5.2 highlights correlations between estimated quantities in my model. Many of
the posterior variances are highly correlated. Firms with very uncertain cashflow betas
also tend to have uncertain mean growth rates. The variance ratio is highly positively cor-
related with the posterior variance of firm parameters Vart[µi], Vart[βi], and Et[γi]. There
1The exclusion of the 1980s does not change the qualitative results, but it does change the quantitative results as there
are several extreme posterior estimates I generate during the 1980s.
42
43
Industry Obs Firms Mkt cap Xi,t E[µ] E[β] E[γ] VRi,t
Chemicals and Allied Products 9748 130 20,648.4 0.06 -0.05 0.92 1.51 -0.00
Other 9718 134 13,406.9 0.23 0.05 0.96 1.68 -0.01
Business Services 9206 129 22,051.84 0.22 0.07 0.91 1.67 0.01
Depository Institutions 8910 124 13,552.6 0.20 0.15 0.97 1.59 -0.07
Insurance Carriers 6950 89 10,343.58 0.35 0.07 0.99 1.89 0.29
Electric, Gas and Sanitary Services 6910 78 9,924.22 0.05 0.05 0.90 1.48 -0.09
Electronic & Other Electrical Equipment & Components 6635 82 17,748.89 0.08 -0.04 1.03 1.72 0.02
Industrial and Commercial Machinery and Computer Equipment 6181 81 7,268.638 0.13 -0.06 1.02 1.66 -0.06
Oil and Gas Extraction 5760 85 10,147.25 -0.16 -0.09 1.02 1.95 0.52
Measuring, Photographic, Medical, & Optical Goods, & Clocks 5430 67 11,180.76 0.08 0.03 0.90 1.38 -0.31
Transportation Equipment 3844 48 14,176.23 0.45 0.03 1.06 1.95 0.14
Communications 3419 49 31,944.12 0.60 -0.03 0.95 1.89 0.11
Holding and Other Investment Offices 3123 40 8,139.993 0.29 0.06 0.93 1.67 0.08
Food and Kindred Products 2665 34 20,375.96 0.11 0.06 0.84 1.25 -0.39
Security & Commodity Brokers, Dealers, Exchanges & Services 2532 40 13,313.46 0.18 0.10 0.98 1.55 -0.18
Wholesale Trade - Durable Goods 1691 20 4,577.746 0.10 0.01 0.98 1.33 -0.58
Paper and Allied Products 1627 22 10,754.32 0.22 -0.07 1.02 1.49 -0.41
Primary Metal Industries 1611 22 4,306.214 -0.12 -0.45 1.12 2.34 0.59
Health Services 1587 20 8,334.007 0.20 0.09 0.90 1.64 0.20
Eating and Drinking Places 1522 22 11,925.99 0.06 0.07 0.86 1.30 -0.37
Metal Mining 1462 20 12,599.25 -0.07 -0.06 1.01 1.91 0.43
Petroleum Refining and Related Industries 1409 22 60,385.21 0.12 -0.10 1.05 1.65 -0.01
Fabricated Metal Products 1396 15 5,578.4 0.09 0.01 1.05 1.44 -0.34
Transportation by Air 1214 17 5,484.36 2.31 0.15 1.03 3.16 0.69
Motor Freight Transportation 1163 16 8,913.742 0.11 0.03 0.91 1.38 -0.32
Amusement and Recreation Services 1112 16 6,362.425 -0.49 -0.01 1.02 2.05 0.29
Engineering, Accounting, Research, and Management Services 1075 16 3,527.093 0.06 -0.01 0.93 1.57 0.05
Water Transportation 926 15 2,831.631 -0.21 -0.06 0.98 1.72 0.20
Wholesale Trade - Nondurable Goods 888 11 9,574.891 0.02 0.04 0.85 1.42 -0.29
Nondepository Credit Institutions 877 14 33,127.64 0.39 0.21 0.94 1.44 -0.28
Rubber and Miscellaneous Plastic Products 870 12 3,093.69 -0.05 -0.06 0.97 1.64 0.09
Apparel, Finished Products from Fabrics & Similar Materials 851 11 5,477.718 0.09 0.06 0.91 1.35 -0.42
Printing, Publishing and Allied Industries 789 10 3,287.056 0.08 -0.04 0.93 1.23 -0.54
Insurance Agents, Brokers and Service 696 9 11,165.12 0.12 0.12 0.75 1.20 -0.32
Lumber and Wood Products, Except Furniture 634 8 5,180.051 0.15 -0.07 1.09 1.43 -0.45
Railroad Transportation 616 7 19,115.12 0.15 -0.00 0.99 1.52 -0.06
Table 5.1: Overview of quarterly data for firms in my sample by industry.
E[µi] Var[µi] E[βi] Var[βi] E[γi] Var[γi] log(mcap) VRi,t
E[µi] 1.000
Var[µi] -0.027 1.000
E[βi] -0.153 0.295 1.000
Var[βi] -0.047 0.606 0.542 1.000
E[γi] 0.018 0.352 0.070 0.138 1.000
Var[γi] 0.023 0.121 0.011 0.031 0.890 1.000
log(mcap) 0.119 -0.361 -0.179 -0.371 -0.009 0.009 1.000
VRi,t -0.070 0.783 0.035 0.527 0.328 0.099 -0.194 1
Table 5.2: Correlation matrix of regressors.
is little correlation between the variance ratio and the level of the cashflow beta, and un-
derstandably so – the ability of firm earnings to convey macroeconomic news is primar-
ily a function of parameter uncertainty and fundamental uncertainty γ2. This is largely
consistent with Schmalz and Zhuk (2019) who show that information transmission about
aggregate earnings is only a function of posterior variance, not posterior expectations.
5.1 Cross-sectional results
In this section, I place my measures of systematic risk uncertainty and informativeness
into the canonical literature on cross-sectional asset pricing, and whether the factors I con-
struct represent priced sources of risk. Section 3.2 provides further context and literature
review on this topic area. I predicted that firms with higher cashflow beta parameter un-
certainty Vart[βi] should have higher cross-sectional returns due to increased systematic
risk exposure.
To test the cross-sectional implications of my parameter estimates, I perform the stan-
dard regressions√of Fama and MacBeth (1973) using several specifications. The cashflow
beta Et[βi] and Vart[βi] are cross-sectionally normalized by subtracting each variable
from the average in a week and divided by the week’s standard deviation of each variable
in order to focus on the effects of the relative position of a firm’s characteristics, since the
posterior I draw from moves substantially between the beginning and end of the sample.
Section 6.1 provides additional context on the scale and potential unidentifiability of βi.
I use Newey-West standard errors with three lags. I gather weekly returns for all firm-
weeks in my sample, and estimate Fama-French 3-factor betas2 for time t using weeks in
[t− 120, t− 2], requiring at least 60 weeks of returns. My cross-section contains 1,266,809
firm-week observations after eliminating firm-weeks where FF3 betas were not estimable.
Table 5.3 presents my results. The first specification regresses weekly returns on only a
firm’s cashflow beta E[βi], without controlling for the Fama-French 3-factor coefficients. I
2Data on the Fama-French factors is gathered from the Ken French data library.
44
Ta√ble 5.3: Fama-MacBeth regressions of weekly returns (in percent) on covari-ates. Standard errors appear in parentheses. E[βi] is a firm’s expected cashflow
beta. Var[βi] is the standard deviation of a firm’s posterior cashflow beta. Stan-
dard errors are Newey-West using three lags.
Returns
(1) (2) (3) (4) (5) (6) (7)
√E[βi] 0.006 −0.003 −0.011 −0.009(0.012) (0.007) (0.008) (0.010)
Var[βi] 0.017
∗ 0.013 0.006 0.016∗
(0.010) (0.009) (0.007) (0.009)
VRi,t √ 0.004(0.007)
E[βi]× Var[βi] 0.002
(0.002)
βHMLi 0.010 0.009 0.008 0.008 0.008
(0.033) (0.033) (0.034) (0.033) (0.034)
βSMBi 0.048
∗∗ 0.047∗∗ 0.046∗∗ 0.047∗∗ 0.040∗
(0.020) (0.021) (0.021) (0.021) (0.021)
βMKTi 0.029 0.026 0.027 0.026 0.037
(0.039) (0.039) (0.040) (0.039) (0.039)
Observations 1,266,809 1,266,809 1,266,809 1,266,809 1,266,809 1,266,809 1,242,481
R-squared 0.278 0.275 0.347 0.349 0.347 0.350 0.351
Notes: ∗∗∗p < .01; ∗∗p < .05; ∗p < .1
find no support that my cashflow beta estimate has any substantive effect on cross-sectional
asset prices. Column 1, 3, 4, and 6 all highlight no significant effect of a cashflow beta on
asset returns. In fact, the point estimates seem to suggest a negative price, though they
are insignificant – a negative cashflow beta price suggests that my model’s cashflow esti-
mate does not capture the same type of cashflow covariance risk measured in papers like
Campbell and Vuolteenaho (2004) or Da (2009), both of whom find positive and signifi-
cant cashflow betas.
The lack of a risk price associated with my cashflow beta could simply be the result of
low statistical power. Chan and Marsh (2022) shows that the expected return predictions
of the CAPM only hold during earnings announcement periods. My cashflow beta is not
the same as the market beta they estimate, but Section 5.2 highlights that the cashflow
beta does seem to be a priced risk during an earnings announcement week, consistent with
Chan and Marsh (2022).
I find (limited) support for a risk price associated with the parameter uncertainty. Col-
umn (2) regresses cross-sectional returns on only the parameter uncertainty measure, and
finds a positive risk price of 0.017. The economic magnitude is small. The median firm
should expect an extra 0.5bps of weekly returns due to their parameter uncertainty, which
annualizes to 29.7bps per annum. Firms with higher parameter uncertainty at the 80th
45
percentile of parameter uncertainty have high returns by about 0.9bps per week, or 46.9bps
per annum.
I test for any effect that omitting the cashflow beta might have on the risk price asso-
ciated with parameter uncertainty. In Column (4), I include the cashflow beta with no
interaction between the beta and its uncertainty. I find that the risk price is numerically
similar to Column (2), but that it is no longer significant. Similarly, controlling for the
Fama-French betas in Column (5) removes the effect of parameter uncertainty.
However, in Column (6), I permit an interaction between the beta and parameter un-
certainty, and the effect I detected in Column (2) becomes significant again. Et[βi] and
Vart[βi] are heavily correlated. My findings suggest that variation in parameter uncertainty
provides incremental explanatory power once the correlated component between Et[βi] and
Vart[βi] is removed by the interaction term. Importantly, the magnitude in this specifica-
tion is quantiatively similar to the one in Column (2), where parameter uncertainty was
the only regressor. Column (6) summarizes the main result of this section by providing evi-
dence that parameter uncertainty in the cashflow beta explains variation in cross-sectional
returns. My finding suggests a nonlinearity in parameter uncertainty, consistent with more
complicated hedging-based models of parameter uncertainty (Cvitanić et al. 2006).
The interpretation in Column (6) may seem unintuitive since the main effect on param-
eter uncertainty is weakly significant, while the interaction is not. Note that both Et[βi]
and Vart[βi] are normalized variables, which implies that the coefficient on Vart[βi] indi-
cates the effect of parameter uncertainty on returns is different from zero when Et[βi] is
at its cross-sectional mean. This leads to a follow-up question: why are no coefficients in
Column (4) significant, where the interaction term is omitted? There are several expla-
nations. First, it may simply be that the non-interacted specification is misspecified, and
that the effect of parameter uncertainty is only locally identified once Et[βi] is fixed. Sec-
ond, as noted by Jun and Pinkse (2009), the addition of the ”irrelevant” interaction term
Et[βi]× Vart[βi] can improve the efficiency of the estimator.
The remaining columns show no effect on weekly returns. The variance ratio VRi,t does
not suggest any difference in weekly returns. I suggest that perhaps a firm’s informative-
ness about quarterly cashflows is insufficient to describe variation in weekly returns. Sec-
tion 5.2 highlights that the informativeness measure seems to explain cross-sectional stock
return variation for a firm’s announcement week, precisely when the informativeness of
firm earnings news is most valuable.
A common test of a cross-sectional pricing story is to provide portfolio sorts, as popu-
larized by Fama and French (1993). To provide an analysis of the dual role of the cashflow
beta and its associated parameter uncertainty, I provide two sets of portfolios. I construct
univariate sorts of firms into quintiles based on their cashflow beta and on their parameter
46
Vart[βi] Et[βi]
Quintile Raw Abnormal Quintile Raw Abnormal
1 (Low) 0.220 -0.0230 1 (Low) 0.223 -0.0201
(3.79) (-1.47) (4.23) (-1.25)
2 0.230 -0.0331 2 0.244 -0.0304
(3.47) (-2.08) (3.97) (-1.90)
3 0.244 -0.0333 3 0.264 -0.0425
(3.47) (-2.00) (3.83) (-2.30)
4 0.262 -0.0348 4 0.268 -0.0314
(3.52) (-1.81) (3.60) (-1.50)
5 (High) 0.273 -0.0299 5 (High) 0.245 -0.0307
(3.41) (-1.50) (3.16) (-1.27)
5-1 0.041 -0.007 5-1 0.021 -0.011
(0.44) (-0.27) (0.23) (-0.36)
Table 5.4: Average weekly returns in percent grouped by the firm’s quintile of
Et[βi] and by the quintile of a firm’s parameter uncertainty, Vart[βi]. Abnormal
returns are the residuals of the Fama-French 3-factor model.
uncertainty. I calculate abnormal returns as the residuals from constructing the Fama-
French 3-factor betas on the portfolio’s excess returns.
Table 5.4 highlights the average returns of univariate sorts on cashflow beta and on
parameter uncertainty. Abnormal returns for portfolios sorted on the cashflow beta or
on parameter uncertainty do not seem to be statistically different between the high and
low portfolios on either axis. The limited support for my statistical results in the cross-
sectional regressions is not surprising, given that the economic magnitude of parameter
uncertainty on returns is small. Further, portfolio tests tend can be relatively low power as
compared to regression tests when the number of cross-sectional observations is small, as
in my sample (Cattaneo et al. 2020).
Note that several of the portfolios have abnormal returns that are significantly nega-
tive. This is in part the result of my sample selection, which tilts more towards firms with
low book-to-market ratios (Zhang 2013). Similar results hold for untabulated analyses on
high/low large/small portfolios provided by Ken French.
I conclude this section by noting that the cross-sectional regressions indicate that the
cashflow beta is not priced, but that the parameter uncertainty is. However, portfolio
sorts suggest that neither the cashflow beta nor its uncertainty are priced cross-sectionally,
though portfolio sorts are likely to be relatively low-power given my small cross-section. I
cannot conclude that there is no risk price associated with either attribute, but I can say
that I do not find sufficient evidence.
47
5.2 Earnings announcements
Section 3.2 summarizes the current literature on earnings announcement premia and
earnings response coefficients as it relates to parameter uncertainty. I contend that that
firms with an increase in systematic parameter uncertainty should have higher earnings
response coefficients.
I construct cumulative abnormal returns (CAR) for each firm-earnings announcement
pair. I calculate abnormal returns for each day by regressing returns from [t − 120, t −
14] (requiring at least 60 days’ worth of returns) on the Fama-French three factors (size,
value, and market). Abnormal returns are then the residual of the raw return on day t
after removing the projected FF3 effects. Cumulative abnormal returns CAR(s, k) are the
sum of all abnormal returns between [t+ s, t+ k]. This is standard practice, see Kothari and
Warner (1997) for one of many examples.
Lastly, I calculate a measure of raw earnings news, standardized unexpected earnings
(SUE). I measure SUE as the difference between actual earnings and median forecasted
earnings, scaled by the previous quarters’ ending price. I require a firm to have at 5 ana-
lysts in the 90 days preceding a firm’s scheduled earnings announcement. This is similar
to the method used by Patton and Verardo (2012), among many others. My definition of
SUE is written as
EPSi,t − E[EPSi,t]
SUEi,t =
pi,t−1
I perform two types of tests. First, I test for earnings response coefficients, which de-
scribe the abnormal returns to a stock as a function of unexpected earnings news (SUE).
Second, I test for earnings announcement premia, which are abnormal returns that accrue
to firms regardless of their unexpected earnings. The earnings announcement premium
tests are cross-sectional, while the earnings response coefficients examine within-firm varia-
tion in earnings response due to changing parameter beliefs.
In Section 3.2, I summarized the literature on parameter uncertainty, and suggested
that firms with higher total parameter uncertainty might have higher earnings announce-
ment premia due to a lower signal-to-noise ratio as per Savor and Wilson (2016). I also
suggested that firms with increased parameter uncertainty should have higher earnings
response coefficients – intuitively, if an investor is highly uncertain about a firm’s param-
eters, any new information is weighted higher than it would be than if the firm had lower
parameter uncertainty.
Table 5.5 tests these hypotheses. I find that when a firm’s parameter uncertainty in-
creases, they have lower earnings responses. Alternatively stated, this implies large earn-
ings responses when parameter uncertainty is low. Column (1) indicates that earnings
48
Table 5.5: Regression of cumulative abnormal return (in percent) during an
earnings announcement, CAR(0,1). Subsamples are by NBER recession indicators.
Abnormal returns are daily, adjusted for Fama-French 3-factor betas estimated
using returns from [t − 120, t − 14], requiring at least 60 days of returns. SUE is
the actual EPS less the median analyst estimate, scaled by the previous quarter’s
price. Standard errors are clustered by firm and by quarter. Fixed effects are by
firm and by quarter.
CAR(0,1)
(1) (2) (3)
E [β ] 0.154∗∗∗ 0.147∗∗∗ 0.159∗∗∗√t i (0.048) (0.049) (0.049)
Vart[βi] 0.129
∗∗ 0.107∗ 0.127∗∗
(0.063) (0.061) (0.063)
√E[βi]× SUE −12.239
∗∗∗ −10.781∗∗
(4.566) (4.922)
Var[β ]× SUE −96.173∗∗∗ −95.072∗∗∗i
(24.117) (23.317)
SUE 102.068∗∗∗ 61.235∗∗∗ 107.900∗∗∗
(22.004) (12.333) (23.029)
Observations 79,919 79,919 79,919
R-squared 0.040 0.037 0.040
Adjusted R-squared 0.019 0.016 0.020
Notes: ∗∗∗p < .01; ∗∗p < .05; ∗p < .1
√
responses are reduced when firms have high parameter uncertainty (large Var[βi]). My
evidence is consistent with the model in Schmalz and Zhuk (2019), who show that greater
parameter uncertainty about the cashflow beta should yield lower earnings response coeffi-
cients.
Columns (2) and (3) demonstrate that higher betas also reduce a firm’s earnings re-
sponse, though the effect is smaller than the effect of parameter uncertainty. This find-
ing is not surprising. Earnings news about a firm with a high cashflow beta should yield
a weaker response, since more of the firm’s earnings news is driven by common factors,
meaning that stock returns incorporate more of the firm’s earnings news. Belief revision
about the firm’s parameters should be relatively small, since the information conveyed by
earnings news is driven more by the aggregate earnings shock.
The previous tests focus on earnings response coefficients, but do not address risk pre-
mia that accrue to firms with different parameterizations. To test risk premia, I conduct
Fama-MacBeth regressions using dummies for firm-weeks that are announcement periods.
The test of earnings announcement premia in Savor and Wilson (2016) is fundamentally
a cross-sectional test, which I attempt to replicate through a regression framework rather
than a portfolio sort.
The primary variable of interest is Announcement week interacted with the variance
ratio VRi,t, the cashflow beta Et[βi], and parameter uncertainty Vart[βi]. My hypothe-
49
Table 5.6: Fama-MacBeth regressions of weekly returns (in percent)√on an-
nouncement week indicators. E[βi] is a firm’s expected cashflow beta. Var[βi]
is the standard deviation of a firm’s posterior cashflow beta. VRi,t is firm i’s vari-
ance ratio. Standard errors appear in parentheses and are Newey-West adjusted,
using three lags.
Weekly Returns
(1) (2) (3) (4)
Announcement week 0.159∗ 0.063 0.093 0.077
(0.088) (0.117) (0.078) (0.157)
Announcement week ×VRi,t −0.221∗∗
(0.112)
Announcement week ×√E[βi] 0.626
∗
(0.379)
Announcement week × Var[βi] 0.167
(0.281)
VRi,t −0.0001
(0.010)
√E[βi] 0.004(0.010)
Var[βi] 0.004
(0.009)
βMKTi 0.074
∗ 0.080∗ 0.073∗ 0.072
(0.043) (0.045) (0.043) (0.044)
βHMLi −0.046 −0.049 −0.048 −0.048
(0.033) (0.034) (0.032) (0.032)
βSMBi 0.053
∗∗ 0.055∗∗ 0.051∗∗ 0.050∗∗
(0.025) (0.026) (0.025) (0.025)
Observations 666,573 653,789 666,573 666,573
R-squared 0.334 0.339 0.337 0.336
Notes: ∗∗∗p < .01; ∗∗p < .05; ∗p < .1
ses have concluded that, in general, it is difficult to determine the a priori expectation
whether any of these parameters should increase or decrease premia. The closest available
work that studies announcement premia as functions of my estimates is Savor and Wilson
(2016), which unfortunately does not permit heterogeneity in firms. I provide these esti-
mates to readers for further research on the effects of informativeness, cashflow beta, and
parameter uncertainty on earnings announcement premia.
To ensure that the cross-sectional results are not driven by announcement weeks with
a few announcers. I require a week to have at least thirty announcers and thirty non-
announcers. This restriction reduces my sample size by nearly half. Table 5.6 presents
the results of my cross-sectional regressions. Column (1) reinforces the Savor and Wilson
(2016) finding that announcers accrue premia relative to non-announcers – announcers
earn a premia of approximately 16bps during announcement weeks that is unexplained by
the Fama-French three-factor model. Annualized, this 8.61%, which is similar in magni-
50
tude to the 9.9% reported by Savor and Wilson (2016).
Column (2) shows that more informative firms earn higher premia during their an-
nouncement weeks relative to less informed firms. The magnitude of the elevated pre-
mium is similar to the magnitude of the unconditional premium, which provides evidence
that heterogeneity in the signal-to-noise quality of firm earnings about aggregate cash-
flow growth is an important feature of the earnings announcement premium. A 20th per-
centile informative firm has 13bps higher weekly returns for an annualized return of ap-
proximately 7.24%. The median firm has an announcement informativeness premium of
2.45bps which annualizes to 1.28%. I show that much of the premium that accrues goes
to the most informative firms, and that my estimates seem to suggest that almost all the
Savor and Wilson (2016) announcement premium is driven by firm informativeness.
Column (3) tests the effect of the cashflow beta on announcement premia. I show that
higher cashflow betas correspond to higher earnings announcement premia during a firm’s
earnings announcement. The cashflow beta effect subsumes the unconditional earnings
announcement premium just as in Column (2). The median cashflow beta firm accrues
an additional 9.1bps during their quarterly announcement week, or 4.84% in annualized
returns. Firms with relatively high cashflow betas in the 80% percentile accrue 21.62bps,
or nearly 11.88% per annum. Column (4) shows no effect of parameter uncertainty.
The results in Table 5.6 suggest that much of the results in Savor and Wilson (2016)
are driven primarily by heterogeneity in firm characteristics, and that a disproportionate
amount of earnings premia accrue to firms that are informative or have high cashflow be-
tas. As the model in Savor and Wilson (2016) does not permit any form of heterogeneity,
my results suggest that additional modelling of announcement premia with a language for
heterogeneity would be a fruitful topic for future research.
To conclude this section, I find that earnings responses for firms with high parameter
uncertainty are lower, and that earnings responses for firms with higher cashflow betas are
lower. I find that informative firms earn higher announcement premia, as do firms with
larger cashflow betas.
5.3 Firm informativeness and earnings response
The variance ratio VRi,t summarizes the ability of a firm’s earnings to precisely commu-
nicate aggregate earnings news. The previous section focuses on the effects of systematic
parameter uncertainty. This section focuses on how responsive markets are to informative
and uninformative firm news, as measured by their variance ratio. In each quarter, I di-
vide firm variance ratios into quintiles. The method for calculating a firm’s variance ratio
is described in Equation 3.3. I use the same cumulative abnormal returns, controls, fixed
51
VR Quintile Days since Std. Median Avg. Avg. Avg. √ Avg.
QE SUE market cap market cap Et[βi] Vart[βi] Et[γ2i ]
1 (informative) 26.07 0.005 7,296.2 23,559.4 0.979 0.068 1.05
2 27.72 0.006 4,137.1 13,870.5 0.928 0.086 1.30
3 29.20 0.007 3,011.3 11,917.7 0.933 0.091 1.44
4 30.17 0.011 2,607.2 11,106.7 0.958 0.093 1.65
5 (uninformative) 29.56 0.024 3,276.5 10,724.3 1.006 0.098 2.83
Table 5.7: Summary statistics of firms in my sample, grouped by the quintile
of their variance ratio.
effects, and standard error clustering as in Section 5.2.
Table 5.7 collects summary statistics sorted by their variance ratio quintile, with quin-
tile 1 being the “most informative” in the language of my model. The most informative
firms are also generally the largest, though the size effect does not have a clear direction in
the median for quintiles 2-5. The standard deviation of SUE is monotone increasing with
the variance ratio quintile. Informative firms have the smallest variance in earnings sur-
prise. This is in part an output of my model – highly predictable firms have low posterior
variance in each of their parameters because the model has frequent, repeated, and consis-
tent evidence about a firm’s parameters. The motivation for the variance ratio is that low
variance ratios indicate firms that are predictable and low-noise, which is consistent with
the correlations in Table 5.7.
Variance ratios move throughout time, and different firms and industries shift between
levels of informativeness to the market. Figure 5.1 highlights the time variation in infor-
mativeness by averaging the standardized variance ratio for each industry-quarter, filtered
to the top-ten largest industries. Recall that a lower variance ratio indicates higher in-
formativeness, so industries towards the bottom of the figure are more informative in the
language of my model.
Depository institutions were among the least informative industries at the beginning
of my sample, but towards the end are the most informative of the large industries. Cu-
riously, depository institutions trended towards more informative continuously until the
2008 banking crisis. Insurance carriers are continuously among the least informative in-
dustries. Utilities tended towards highly informative, but seem to have lost their infor-
mative role for much of the 2000s (except for a high-informativeness period during the
mid-2010s).
Next, I take my firm-level estimates of variance ratios to the data. Table 5.8 presents
the results. All specifications use the standardized variance ratio VRi,t, which is negative
for informative firms. Column (1) indicates that firms who become more informative have
higher earnings response coefficients. Since the variance ratio is standardized, the eco-
nomic conclusion one should take away is that a one-standard-deviation decrease (meaning
52
Figure 5.1: Average variance ratios VRi,t by industry, filtered to the top five
largest industries in my sample. A lower variance ratio means the firm is more
informative.
more informative) in the variance ratio means that, on average, the earnings response co-
efficient is 132.3, as compared to the median informative firm with an earnings response
coefficient of 98.99.
My finding is consistent with the model of Schmalz and Zhuk (2019). In Lemma 5, for
example, they demonstrate that when a firm’s idiosyncratic variance or parameter uncer-
tainty is high, earnings response coefficients are generally decreasing. Beyer and Smith
(2021) produce a similar result. Firms should have higher earnings response coefficients
when they are more informative.
I address the concern that the variance ratio provides little incremental information
above a simpler measure like the R2 of a regression of aggregate earnings on firm earn-
ings. The R2 would measure the capacity of a firm’s cashflow news to convey information
about the aggregate shock. I calculate the aggregate earnings growth measure of Kothari,
Lewellen, and Warner (2006), defined as the year-over-year difference in the sum of firm
earnings, rescaled by the total market capitalization of all firms in the prior quarter.
I estimate a regression for each firm and quarter t. In each regression, the data are an
expanding window using data from 1 to t. I extract the R2 statistic to compare its inter-
pretation to that of my variance ratio VRi,t. I multiply the R
2 by −1 to make its inter-
pretation comparable to the variance ratio, since more negative variance ratios are more
53
Table 5.8: Regression of cumulative abnormal return (in percent) across a
firm’s earnings announcement, testing earnings responses as a function of firm
informativeness. Abnormal returns are daily, adjusted for Fama-French 3-factor
betas estimated using returns from [t − 120, t − 14], requiring at least 30 days of
returns. SUE is the actual EPS less the median analyst estimate, scaled by the
previous quarter’s price. Standard errors are clustered by firm and by quarter.
Fixed effects are by firm and by quarter.
CAR(0,1)
(1) (2)
VR ∗i,t 0.116
(0.060)
VRi,t× SUE −34.316∗∗∗
(7.794)
SUE 97.987∗∗∗ 58.977∗∗∗
(20.701) (15.980)
−R2i,t 0.141
(0.558)
−R2i,t× SUE 46.295∗
(26.200)
Observations 79,919 51,715
R-squared 0.041 0.052
Adjusted R-squared 0.020 0.023
Notes: ∗∗∗p < .01; ∗∗p < .05; ∗p < .1
informative. Similarly, −R2i,t is more negative when firms are less informative. I find that
the correlation between −R2i,t and VRi,t is small, only 0.04.
Columns (2) perform the same tests, but this time with the R2i,t calculated from pro-
jecting aggregate earnings growth on firm earnings growth. Notably, the R2i,t measure does
not reproduce the qualitative or quantitative results as when the variance ratio is used.
The only significant coefficient related to the −R2i,t measure is on the earnings response
coefficient, which is of the opposite sign to the variance ratio tests – firms with high R2i,t
informativeness have lower earnings responses, whereas the variance ratio predicts higher
earnings responses. I cannot say which of these estimates is more “correct”, only that they
correlate with different outcomes.
I find that the most informative firms are also those with the strongest earnings re-
sponse coefficients. Markets respond more strongly to earnings news when firms are infor-
mative, consistent with rational learning about systematic risk. Further, I find evidence
that my variance ratio provides quantitatively and qualitatively different interpretations of
earnings response coefficients.
54
Figure 5.2: Average days-to-announcement from the end of firm fiscal quarters.
Informative here is defined as a firm’s quarterly variance ratio VRi,t being in the
lowest quintile.
5.4 Earnings announcement order
Section 3.3.1 highlights the existing literature on earnings announcement order, and
highlights that it remains an open question whether we should expect actual earnings or-
derings to be high-information first, low-information first, or some mix of both. To provide
some guidance for future research, I estimate whether firms I predict to be high informa-
tion announce earlier or later in my sample. I also examine whether high-information firms
tend to announce earnings by themselves or as part of a cohort, since cohort information
releases are more information-rich environments.
Using my data on earnings announcements from Section 5.2, I calculate the number
of days between a fiscal quarter end and a firm’s actual earnings announcement. Figure
5.2 graphs average days to fiscal quarter end by firm informativeness. I group all firms
in the first quintile of their variance ratio per quarter and calculate the average number
of days a firm takes to announce. The graph demonstrates a clear pattern of informative
firms announcing earlier on average. The spikes in the graph represent additional delays in
earnings announcements at year-end, when firms typically release an annual report.
I perform Fama-MacBeth cross-sectional regressions of the number of days since a
firm announced earnings on their variance ratio VRi,t, market capitalization, and book-to-
55
Table 5.9: Cross-sectional regressions of the difference between a firm’s earn-
ings date and the firm’s fiscal quarter end in days. Columns (1) and (2) use the
log number of days between a firm’s quarter-end and the firm’s earnings an-
nouncement date. VRi,t is the firm’s variance ratio, cross-sectionally standard-
ized.
log(Days since FQE) Days since FQE
(1) (2) (3) (4)
VRi,t 0.030
∗∗∗ 0.052∗∗∗ 0.829∗∗∗ 1.445∗∗∗
(0.009) (0.007) (0.248) (0.243)
BOOK TO MARKET 0.001 −0.043
(0.003) (0.084)
LOG(MCAP) −0.033∗∗∗ −1.042∗∗∗
(0.006) (0.177)
CONSTANT 0.014 0.020∗∗ 0.176 0.686∗∗∗
(0.011) (0.008) (0.308) (0.225)
Observations 79,653 115,011 79,653 115,011
R-squared 0.731 0.622 0.693 0.400
Notes: ∗∗∗p < .01; ∗∗p < .05; ∗p < .1
market ratios. I demean the number of days in each quarter to reduce the effect of differen-
tial average days-to-announce during the year-end. The specifications using market capital-
ization and book-to-market generally have a smaller sample size due to limited availability
in accounting figures for some firms in my sample. I provide specifications controlling for
market capitalization and book-to-market to demonstrate that variance ratio provides in-
cremental explanatory power above size and book-to-market, which can proxy for financial
distress.
Table 5.9 presents the regression results. As before, I use fixed effects and clustered
standard errors by firm and by quarter.I find that, on average, informative firms announce
significantly earlier than uninformative firms. All columns of Table 5.9 show positive and
significant effects of the variance ratio on the number of days between the quarter end and
the firm’s earnings announcement. Recall that informative firms have negative variance
ratios, so a variance ratio for a firm with a -1 standard deviation of informativeness of -1
would announce 0.829 days earlier (using Column 3) than firms with average informative-
ness. 0.829 days may seem a small effect, but Noh, So, and Verdi (2021) builds their work
on a sample where nearly 70% of variation in days-to-announce is ±1 day.
The results in Table 5.9 may provide some explanation of the results in Sections 5.3,
which shows that earnings responses are higher for informative firms. It is possible that
much of the result of those sections is driven by informative firms announcing earlier and
with fewer co-announcers, though a more rigorous economic model of disclosure order may
be needed.
Overall, I provide novel evidence that informative firms with highly predictable earn-
56
ings tend to announce ahead of other firms. The existing literature is too sparse to place
my findings into an appropriate context, but it is my hope that my empirical findings mo-
tivate further theoretical research to determine precisely why informative firms announce
ahead of other firms.
57
Chapter 6
DISCUSSION
In this section, I provide discussion about my model, the assumptions used, and the
results those assumptions may have on my results. Section 6.1 discusses the role of the
latent factor Mt. Section 6.2 discusses the lack of time variation in the cashflow beta βi,
and provides alternative estimation procedures for the cashflow beta. Section 6.3 discusses
the role of the Gaussian likelihood in my model. Section 6.4 analyzes my priors. Finally,
Section 6.5 suggests additional research areas to which my model could be applied.
6.1 The latent factor
This section discusses the role the latent factor plays in my analyses. A primary goal
of this paper is to quantify the uncertainty of a firm’s cashflow beta with respect to an
unknown latent factor. I propose a parametric description of the common earnings growth
factor, and allow the data to inform the parameterization. Importantly, my latent factor
is jointly estimated with firm loadings on that factor to better capture the difficulty an
investor faces in interpreting earnings news.
The first concern is why there is residual posterior uncertainty given the size of the
cross-section. One would expect that Vart[Mt | X1:t] would fall to zero as N → ∞, but in
fact there is a positive posterior standard deviation (between 0.06 and 0.3 in different time
periods and across model estimates). For example, the model I estimate for the fourth
quarter of 1984 estimates a shock of 0.11 ± 0.06, where 0.06 is the posterior standard de-
viation. As of the 2022 model, the estimate of the latent shock in 1984q4 has an estimate
of 0.35 ± 0.09, an increase of 1/3 in standard deviation. Why then does the parameter
uncertainty in the latent factors not fall to zero?
As with any estimator in a finite sample regime, my Bayesian estimate of Mt has un-
certainty just as would a frequentist estimator of Mt. My estimator likely yields a slower
convergence rate relative to a frequentist estimator since (a) the size of my dataset is small
relative to the size of the parameter space, and (b) my parameter space grows with the
cross-section – each firm added to my sample requires the estimation of βi, γi, and µi, so
simply growing the cross-section and holding the number of time periods T fixed will not
resolve posterior variance in Mt. Exact convergence rates are difficult to derive for even
simple Bayesian models, let alone models such as the one in this paper that do not have
an obvious closed-form posterior.
Inference on the latent factors can be a difficult problem in general. In a Bayesian
58
Figure 6.1: Comparison of the first principal component of earnings growth,
compared to the latent process Mt.
framework, latent parameters need not always asymptotically converge (in time periods
T ) to the Bernstein-von Mises maximum likelihood estimate (Bontemps 2011; Li, Yu, and
Zeng 2020), particularly for a fixed cross-section N . Latent parameters in my model are
“partially consistent”, as so named in Neyman and Scott (1948). Partially consistent vari-
ables have a finite amount of data attributable to them and may not even have a consis-
tent maximum likelihood estimate, and similarly may not have an asymptotically consis-
tent Bayesian estimator.
While such issues are important to consider, the primary benefit of Bayesian methods
is their application to finite samples. We do not have a set of infinite time periods nor an
infinite cross-section. Whether the Bernstein-von Mises theorem holds or not is important
but not the key consideration – the focus of this paper is on a proposed representative
learning process faced by an investor who does not live an infinite number of time peri-
ods, nor observe an infinite number of assets. The investor faces finite sample uncertainty
consistently, but may not have a consistent estimator. What they do have is an uncertain
guess about how cashflows share common cashflow risk that may not be fully resolved.
A second issue to consider of the jointly-estimated latent factor is that it may be uniden-
tified, since the variance of the latent factor σ2 and each firm’s cashflow beta βi are jointly
estimated. In a traditional frequentist paradigm, estimating these two parameters together
with maximum likelihood estimation is a fool’s errand. It is less clear to what is identified
59
under my joint Bayesian framework – the priors constrain σ2 and βi to a particular region,
but the constraint imposed by the priors could have unanticipated effects on the full joint
density P (θ(t) | X1:t). The magnitude of the latent factor shock is inversely related to the
average magnitude of the cashflow betas in my model, and so only the ratio can be identi-
fied. A common practice is to fix the variance of the latent factor Var[Mt] to 1, and allow
the betas to be identified through a normalization.
Importantly, the parameter ϕ must also be modified to identify σ2. Note that the vari-
ance of an AR(1) process is given by
σ2
Var[Mt] =
1− ϕ2
In order to fix Var[Mt] = 1, any pair (σ
2, ϕ) must satisfy the identity σ2 = 1 − ϕ2. The
identity above means that σ2 can be removed as a parameter since it is given as a direct
function of ϕ. By maintaining ϕ as a parameter sampled from the domain [−1, 1], sigma2
is a parameter with a prior density defined only through a functional transform of the
prior density on ϕ.
An alternative specification is to estimate a latent process via principal component anal-
ysis (PCA). Under this specification, one would estimate the latent factor, fix the latent
process to the estimated factors, and allow only the firm’s weightings on the latent process
to fluctuate. Doing so is akin to assuming that the econometrician knows the latent factor
with certainty but the factor loadings to be uncertain.
I provide visual evidence of the principal component alternative. Since I have an unbal-
anced panel with many missing values (as firms enter and exit at different times), I use the
imputation method of Bai and Ng (2021) to produce a principal component. I standardize
and demean all firm earnings growth Xi,t, and estimate the first principal component using
all 165 quarters of data. Since principal components are identified up to a sign, I multiply
the factor by −1 to match the peaks and troughs of Mt. Figure 6.1 demonstrates that the
first principal component is a close fit to the estimated latent process E[Mt]. Quantita-
tively, the correlation between the two measures is 0.875.
The principal component factor is much more volatile than is Mt. A key reason is that
the latent factor is a “predictability-weighted” latent process, in that common compo-
nents are shared by firms that have earnings growth well described by my model. Firms
with earnings that appear stationary and Gaussian across my sample end up molding the
shape of Mt more than do other firms. Principal components analysis does not do this –
it attempts to find a common factor that reduces the variance across all firms, without
explicitly modelling firm earnings growth.
The principal components alternative could be an acceptable alternative to the joint
60
latent factor distribution, though there are several shortcomings of the approach. First,
principal components are a non-parametric method – there are no parameters that gov-
ern a system estimated by PCA. It is an ex-post rationalization of observed data without
provision of a meaningful time-series structure. Certainly, this has a role in explaining real-
ized common variation (as in Ball, Sadka, and Sadka (2009)), but the lack of an economic
structure provided by PCA limits its use as an economic tool. For example, there is no
longer a sense in which observed earnings growth connects to forecastable quantities as
under my baseline model. Investors cannot make (noisy) predictions about future earn-
ings, and so the value of learning is reduced. Much of the results in the learning literature
are functions of explicit parametric forms for earnings or returns, and a non-parametric
method simply cannot provide an analogue to a valuable source of economic intuition.
Second, the latent factor does not have an associated level of uncertainty that can prop-
agate through to the factor coefficients in a tractable way. This may be positive or nega-
tive depending on your perspective. In some sense it is a good thing, since removing the
latent factor’s uncertainty alleviates some concern with the prior’s role in determining
the size and scale of β 2i and σ . In a less positive sense, the key analysis of this paper is to
provide a model of earnings growth that intuitively maps onto the learning process of an
investor. Investors cannot know the latent factor any more than they can know the latent
factor loadings, and one must expect uncertainty in the latent factor to drive up uncer-
tainty in the beta coefficients. Shutting down the uncertainty of the latent space does not
permit the flow of uncertainty from an uncertain space to the firm-level parameters.
To conclude, the latent factor could be replaced by a principal component estimate.
However, further study is needed to determine whether the qualitative outcomes match
those using the latent process Mt. I defer to future research further work on the estima-
tion method of the latent factor and the associated parameter uncertainty in cashflow be-
tas.
6.2 Time variation in the cashflow beta
A restriction imposed by my model is that the cashflow beta βi is time invariant. As-
suming a fixed beta is akin to assuming that a firm has the same risk level throughout its
lifetime. A motivator for my modelling choice is to capture a firm’s long-run average expo-
sure to systematic risk. Exposure to long-run risk is generally considered a key component
of the cashflow beta (Bansal and Yaron 2004).
Such a restriction may be too aggressive, particularly since researchers typically per-
mit time-varying market or discount rate betas. Ghysels (1998), for example, highlights
significant structural breaks in CAPM-style betas. Zhang (2005) micro-founds one such
61
mechanism for time-variation in betas by noting that value firms (high book-to-market
ratio) are simply riskier in bad times, due to such firms having more assets-in-place and
fewer growth options. Gomes, Kogan, and Zhang (2003) provides a similar motivation for
time-variation in betas.
Time variation is commonly detected in cashflow betas as well, though explicit mod-
elling of time-variation in the cashflow beta remains more limited than the work done for
market betas. Botshekan, Kraeussl, and Lucas (2012) performs their up-down cashflow-
discount rate beta decomposition on time-varying betas estimated on 60-month rolling
windows. Campbell, Polk, and Vuolteenaho (2010) and Campbell and Vuolteenaho (2004)
both highlight striking differences in their cashflow beta estimates after they separate their
sample into two time periods. Da (2009) provides time-varying versions of cashflow covari-
ance and duration, the paper’s two key attributes of cashflow sensitivity.
Formal economic models of time-variation in cashflow betas are sparse, though appear
capable of reconciling anomalous outcomes in asset pricing. Li and Zhang (2017) expand
the long-run consumption risk model of Bansal and Yaron (2004) to include time-varying
cashflow betas on short- and long-term cashflow risk, and find significant explanatory
power for momentum and value portfolio returns.
The fixed beta of my model is at odds with much of the prior literature, and indeed it
may be a primary cause as to why my results on the cashflow beta Et[βi] are not as clear
as might be expected. It may even play a role in the partial significance of parameter un-
certainty Vart[βi], in that parameter uncertainty may function as a catch-all variable for
firms predisposed to structural breaks in their cashflow betas. In this case, my measure-
ment of parameter uncertainty may not necessarily describe the quantity it is intended to
capture.
A potential solution to time-variation in betas is to estimate posterior densities in se-
quences on earnings growth, rather than on expanding windows. Using an iterative pro-
cedure would remove the restriction that the beta be fixed in the likelihood, but would
maintain its Bayesian interpretation. Cashflow betas would then be more flexible and reac-
tive to current data. However, there are important considerations in how the model would
be estimated in sequence.
The priors of the model must be updated in sequence. In the current version of the
model estimation, priors are fixed at the beginning of the sample, but the expanding win-
dow means that the role of the prior deteriorates as more data are used. In a sequential
estimation procedure, using the same prior constructed at the beginning of the sample is
not consistent with a Bayesian methodology – the posterior at time t should act as the
posterior in time t+ 1.
Rather than the current posterior P (θ(t) | X1:t), consider the alternative time-varying
62
joint density
( ) ( ) ( )
(P θ(t) | Xt ∝) P Xt | θ(t) P θ(t) | θ(t− 1) (6.1)
Note that the prior P θ(t) | θ(t− 1) is the density of the parameterization θ(t) evaluated
at the posterior density using the set of earnings growth Xt−1. The likelihood of the vector
Xt is then
( )
P Xt | θ(t) ∝ N (Xt | µ+ βMt, γ2I) (6.2)
Note that this is a unit-observation model, in that only a single quarter is exposed to
the posterior at a time. I do not allow for a rolling window of earnings growth. A rolling
window is inappropriate in my model, since overlapping observations between quarters will
result in excess “certainty” of the posterior. For example, consider the time t posterior
estimated using earnings growth between t − k and t. The prior, estimated at t − 1, uses
earnings growth between t− k − 1 and t− 1, meaning that the overlapping data available to
the time t posterior includes k samples that are already included in the posterior at time
t − 1. If the k observations are included in the posterior estimation for t, the resulting
density will be overfitted on replicated data.
Estimating the sequential density is simple given an analytic posterior, but unfortu-
nately my model does not permit a set of tractable closed-form posterior densities. I sam-
ple from the posterior density using Markov chain Monte Carlo instead. Markov chain
Monte Carlo is not a sequential method, in that the posterior samples cannot easily or ac-
curately be used as inputs in a subsequent prior. I propose three alternative estimation
ro(utines tha)t must be performed to permit the sequential updating of the posterior density
P θ(t) | Xt .
First, rather than use NUTS as the inference algorithm, one could instead use a sequen-
tial Monte Carlo (SMC) method. SMC is an algorithm that performs inference on a series
of distributions much like the ones in Equation 6.1 (Chopin 2002; Chopin 2004). Such an
approach is technically uncomplicated with probabilistic programming, and may be the
simplest solution in general.
Second, one could maintain NUTS as the sampler and approximate each posterior by
fitting a distribution to the marginal posterior density. For example, each βi would have
a Gaussian prior P (βi | Xt−1), with the prior mean Et−1[βi] and prior variance Vart−1[βi].
The key downside of this posterior-fitting approach is that the co-movement structure
between parameters is lost – for example, if βi and βj are highly positively correlated, fit-
63
ting priors on the marginal densities only will lose this information and require the model
to learn the co-movement from the new observation Xt. The priors in this case are not
incorrect in any sense, but they are less informative in that a large amount of valuable
information about the relationships between parameters is lost.
Third, the model could be re-written to accommodate time-variation in beta directly. I
defer this conversation to Section 7, which provides a detailed analysis of alternative model
specifications that can address time-variation in the cashflow beta.
I suggest that future research in the probabilistic estimation of cashflow betas utilize
a sequential estimation method, whether that is sequential Monte Carlo, priors set by fit-
ting known distributions to posterior estimates, or an alternative model specification. The
method applied in this paper may not be capturing an appropriate cashflow beta, since as
the model fixes the cashflow to be constant throughout the 40-year sample period.
6.3 The Normality assumption
The use of the Gaussian likelihood may play a large role in my analyses. This section
describes the role of the likelihood in my model and the influence it may have on my pos-
terior density. The likelihood is Gaussian in my model due to the assumption that idiosyn-
cratic shocks νt are Gaussian. Recall the likelihood is specified
∏t
P (X1:t | θ) ∝ N (Xs | µ+ βMs, γ2I)
s=1
for mean µ+ βMs and variance-covariance matrix γ
2I.
A key reason that I selected the Gaussian distribution for the likelihood is that it has a
simple conditional expectation. I use the conditional expectation of a multivariate Gaus-
sian in my construction of the variance ratio to derive the amount of variance reduction
attributable to a given firm’s earnings. This is the primary reason for selecting a Gaussian
shock. However, it is important to consider the role that the Gaussian likelihood plays in
my posterior estimates, and to attempt to understand in which ways it may distort my
empirical tests.
The impact of using Gaussian errors ϵt can have unintuitive results on the posterior
density. Müller (2013) highlights that misspecified likelihoods can cause certain obvserva-
tions to be weighted more strongly than others, though the degree and direction of this
weight adjustment is difficult to determine in general. My model is high-dimensionsal and
minor misspecifications could cause my posterior estimates to be pushed into regions that
may not necessarily provide sufficient coverage of the “true” parameterization of my model.
Unfortunately, the size of my model makes it difficult to determine ex-ante what role the
64
Figure 6.2: Density of the posterior expectation Et[Xi,t], realized earnings
growth Xi,t, and the model error Et[Xi,t]−Xi,t starting from 1990.
Gaussian likelihood may play.
What I can highlight is that my model seems to capture an element of non-normality in
its estimation, which may address some concerns. Figure 6.2 highlights the difference be-
tween the model-predicted earnings growth Et[Xi,t] and the observed earnings growth Xi,t.
Note that the posterior expectation has much thinner tails and muted skewness, while the
observed earnings growth in blue demonstrates significant fat tails and moderate negative
skewness. The forecast error in green displays similar fat tails but is mean zero. The form
of non-normality captured in the model prediction, Et[Xi,t], is non-normal through excess
kurtosis, but is only minorly skewed. One could say that the thin tails in Et[Xi,t] are not
correctly describing earnings growth, and that the posterior density is biased towards zero
to accomdate the skewness and fat tails of observed earnings growth Xi,t.
However, it seems that the variance ratio VRi,t describes firms that are the most prone
to this non-normality. To test this, I group all firms in my sample by their variance ratio
quintile, pool their forecast errors, and calculate Jarque-Bera normality test statistics on
the pooled model error Et[Xi,t] − Xi,t. Table 6.1 presents the results. Notably, the most
“informative” firms in my sample, those in quintile 1, have the highest deviations from nor-
mality in my sample. The variance of model errors is similarly much lower, though they
are significantly more skewed and have a higher kurtosis than do firms in other samples.
The informative firms have the lowest parameter uncertainty as well. The model detects,
65
VRi,t Quintile Jarque-Bera Mean Sd. dev. Skew Kurtosis Var[βi] VRi,t
1 39099.981 -0.015 0.962 0.290 9.402 -1.259 -1.031
2 30881.801 -0.037 0.986 0.179 8.711 -0.507 -0.033
3 16047.247 -0.044 1.111 0.138 7.138 -0.113 0.201
4 5621.551 -0.118 1.360 0.112 5.495 0.355 0.312
5 444.687 -0.156 1.738 0.081 3.749 1.485 0.525
Table 6.1: Sample normality statistics on the model error Et[Xi,t] − Xi,t, by
variance ratio quintile. The first column is the Jarque-Bera normality test statis-
tic. Other columns describe the mean, standard deviation, skewness, and kurtosis
for model errors. The last two columns detail average variance ratios and parame-
ter uncertainty.
perhaps correctly, that the firms with the highest skewness are those that are most infor-
mative about the macroeconomic state, and at the same time is able to precisely estimate
their cashflow betas.
This could be consistent with the model allocating too much weight in the likelihood
to informative firms with non-Gaussian earnings. Such an overweighting would also signif-
icantly move the posterior density of the latent factor Mt, which could cause unintuitive
outcomes on the posterior density of other firms in my sample. The firms I find to be the
more informative about the latent economy are those with the most non-Gaussian earn-
ings growth. Informative firms have highly skewed model errors, and significantly fatter
tails, suggesting that the likelihood weighting overweights firms with non-Gaussian earn-
ings.
The answer as to whether the Normal likelihood is an issue for my estimates is difficult
to determine without a formal Monte Carlo analysis or more detailed econometric analy-
sis of my model. Further, the answer depends on the object of study. If the objective of
the analysis is to construct a variance ratio or other measure of informativeness, then it
is not obvious that the Gaussian likelihood is an issue: it is tractable, easy to use, and
yields a variance ratio that captures skewness in earnings growth, and appears to produce
estimates that are consistent with theory and existing literature.
6.4 The role of the prior
In this section, I discuss the role my selection of priors may play in determining my
results. Priors play an important role in Bayesian analysis. Selecting appropriate priors
in even simple models can have outsized effects on the posterior densities. In more com-
plex models like the one applied in this paper, priors and their relative strength can influ-
ence the posterior in unintuitive ways. For example, consider a piece of evidence Xt that
is less consistent with a common source of variation in earnings growth. My selection a
low-variance prior on the cashflow beta βi means that the posterior may adjust the latent
66
factor variance downwards instead of reducing the betas if the prior variance on σ2 is rela-
tively low. Further, the adjustment of σ2 in response to the tightness in the cashflow beta
prior can influence the location and scale of the densities for the latent factor mean δ, and
even idiosyncratic means µ.
In short, a detailed prior analysis is a nontrivial exercise. The size and computational
burden of my model make it difficult to conduct the formal prior sensitivity tests common
in Bayesian analyses, so instead I provide some context from a modeler’s perspective by
commenting on the priors of relevant parameters that I believe play a significant role in
the results I obtain. The priors I believe play a significant role are those on cashflow betas
βi, idiosyncratic volatility σ
2, firm idiosyncratic volatility γ2i , and the latent persistence
parameter ϕ.
The prior on the cashflow beta, βi, is homogeneous across firms βi ∼ N (1, 0.1I) ∀ i.
The cashflow beta prior is a simple one intended to predispose my posterior towards a
common structure in firm earnings growth. Early posterior estimates of the model in this
paper with a larger standard deviation of 1 resulted in a model that was too permissive –
the posterior estimates concluded that there was very little evidence of a common variance
structure in earnings growth. Such a finding strongly disagreed with my personal priors
and with the existing literature (Ball, Sadka, and Sadka 2009).
The cashflow beta prior I applied is also overly simplistic in that it is homogeneous
for all firms. There is little reason to expect that a large and old firm that sells consumer
goods should have the cashflow risk exposure as a software firm. Indeed, one of the pri-
mary contributions of Li and Zhang (2017) is the evidence that differential exposure to
firm short- and long-term cashflow risk appears differently in firms sorted on size, value,
and momentum.
A more well-reasoned prior would be to assign a relatively lower prior for beta to firms
with lower book-to-market ratios following the evidence in Campbell and Vuolteenaho
(2004). To implement the differential priors in my model, could calculate the rational dif-
ference between the firm’s estimated beta as a proportion of the sample average cashflow
beta. Using Campbell and Vuolteenaho, this would imply an average beta of 0.55 for small
growth assets, 1.18 for small value assets, 0.27 for large growth, and 1 for large value. Per-
mitting heterogeneous priors in cashflow betas should allow for a more informed posterior
that better incorporates the existing body of work on cashflow beta estimation.
The second prior to consider is the prior on common risk volatility. I choose a prior
that yields an average variance equal to the observed earnings growth variance across all
firms between 1980 and 1985. The prior thus assumes all cross-sectional variance is due
only to systematic innovations, though this is not a particularly strong assumption for
a large cross-section. Indeed, the principal component analysis conducted in Section 6.1
67
suggest the variance of the first principal component has a standard deviation of roughly
14%, which is close in magnitude to the empirical variance in earnings growth in 1980-1985
of 15.98%.
The joint selection of priors on σ2 and βi could be problematic, in particular since the
magnitude of the latent factor is not formally identified without an additional source of
data1. The choice of E[βi] = 1 and E[σ
2] = 15.98 is equivalent to an alternative prior(in
terms of expected common risk exposure) where E[βi] = 3 and E[σ
2] = 15.98/3 = 5.33.
The choice of prior location where beta is centered at one is roughly equivalent to one
where it is centered at 3, as long as the other variables are scaled accordingly. I discuss in
Section 7 an alternative specification that removes this issue by fixing σ2 = 1 and allow
the cashflow betas to dictate the magnitude of the latent factor shocks.
My choice of prior on σ2, βi, and γ
2
i is intended to suggest that the common factor pro-
vides nearly half of the observed variance for firm i, since the priors on γ2i is the result of
fitting an inverse Gamma distribution to half of a firm’s 1980-1985 variance if the firm ex-
isted. If the firm did not exist and entered much later in the sample, it will enter with the
prior assumption that it share the same variance structure as a firm from 1980-1985.
Just as with the priors on firm betas, there is reason to expect significant heterogeneity
in idiosyncratic risk γ2i . Firms in financial distress, small firms and firms in new or novel
industries should have significantly less predictable returns. Firms with low posterior vari-
ance ratios are firms with significantly lower idiosyncratic earnings volatility, and similarly
low earnings surprise volatility.
Fortunately, my priors handle this heterogeneity in part, as long as a firm’s earnings
variance in 1980-1985 is illustrative of the firm’s idiosyncratic volatility. For firms that
enter after 1985, there is zero heterogeneity in priors on idiosyncratic variance. Such a non-
informative prior is perhaps too simplistic – a better prior would be to permit the average
idiosyncratic variance to vary by industry, which is typically known quite early on in a
firm’s lifetime.
The final prior I consider is the persistence of the latent factor, ϕ. The persistence pa-
rameter ends up having a significant influence on the size of the systematic risk a firm is
exposed to. The (conditional) variance of a firm’s earnings growth attributable to the la-
tent factor is β2 2i σ /(1− ϕ2). For posterior samples of ϕ that are zero (for example), this is
simply β2 2i σ . But as ϕ approaches 1 in absolute value, the firm’s exposure to common risk
increases. The posterior mean estimate E165[ϕ] ≈ 0.44 suggests that the magnitude of a
firm’s total variance is nearly 25% higher than would be expected by just β 2i and σ .
The role of the persistence parameter in determining the total amount of common risk
1One method of rectifying the unidentifiability of the latent factor is to include a component in the likelihood for the
latent factor. For example, one could assume that the aggregate earnings measure of Kothari, Lewellen, and Warner (2006) is
simply Mt measured with some unit of noise, which would permit the latent variance σ2 to be matched to observed data.
68
faced by a firm is important, and the prior may not appropriately reflect this risk. Recall
that my prior was specified as ϕ ∼ N (0, 0.2) ∈ [−1, 1], which has a standard deviation
of roughly 0.45. This prior is quite wide relative to its domain, and may have resulted in
a much larger posterior variance for all parameters in my model. A tighter but positive
prior, perhaps ϕ ∼ N (2, 0.05) ∈ [−1, 1], would allow the model to focus more on the
parameter of interest, βi.
The remaining parameters are less critical to the discussion in that they are intended
to capture residual variation in earnings growth, such as firm- or common-level earnings
trends. I conclude this section by noting that my priors are reasonable, but could utilize
more a priori information about the fundamental processes that govern earnings growth.
6.5 Additional topics
In this section, I consider further analyses that could be studied with my model. The
model I study provides a rich description of firm earnings growth, though only parts of
the model output are used. I generate estimates of numerous quantities that are not focal
points of my analysis. For example, I estimate average firm earnings growth µi and id-
iosyncratic risk γ2i , though those quantities are generally marginalized in my computation
of the cashflow beta Et[βi], parameter uncertainty Vart[βi], and the variance ratio VRi,t.
Further study is needed on the informational effects of earnings announcement order.
It is striking that my findings that firms with low variance ratios tend to announce early,
even using the crude approximation to the true conditional informativeness of a firm’s
earnings news given by the variance ratio. Recall that the variance ratio is computed
through an implicit counterfactual assuming that a firm were to go first, without account-
ing for preceding earnings news from other firms. The value of the variance ratio as com-
puted becomes less useful for firms that empirically announce later, since announcements
from late announcers have a significant amount of conditioning information available.
Explicitly studying the full earnings announcement order would be a novel application
of the model I propose. Guttman, Kremer, and Skrzypacz (2014) studies the role of disclo-
sure timing, but in a noncompetitive environment. Suppose firms were characterized only
by the signal-to-noise ratio their earnings news provides about common earnings growth.
Firms are allowed to partially determine their announcement order. Should we expect
firms with high signal-to-noise ratios (the variance ratio in my model) to go first? There
is limited theoretical support for such a claim, but providing some guidance clarify the
extent to which my findings on the variance ratio are expected under a rigorous analytic
framework.
Using the model in this paper, a researcher could model observed and counterfactual
69
orders and compare the informational efficiencies from the perspective of a representative
investor. How efficient is the current earnings announcement order? Does the existing or-
der resolve uncertainty early or late, and how quantitatively efficient is that order relative
to the first-best optimal order? Questions like these cannot be resolved without a measure
of the informational efficiency of a firm’s earnings news and a well-grounded theoretical
model of earnings announcement ordering.
An additional expanded study built upon my model would focus more closely on the
learning process throughout the earnings season. Estimating a new model in each earn-
ings week (rather than in each quarter) to better track the evolution of beliefs about firm
parameters throughout the earnings season. One question that could be addressed is the
effects of belief revision. For example, a firm that announces bad news early during a quar-
ter may have their poor performance attributed to idiosyncratic factors. If it becomes ap-
parent that the common earnings shock is negative during the earnings season, one might
expect beliefs about the firms’ parameters to revise as more firms announce. Firms that
experience belief revision should have some form of reversal in returns following their an-
nouncement as the proportion of their earnings news believed to come from idiosyncratic
factors decreases.
Analyzing belief revision throughout the earnings season is straightforward with my
model, in that calculating belief revision can be done by calculating a prior-posterior dis-
tance measure like KL divergence or the Wasserstein metric. Large belief revisions for firm
i that occur after another firm j announces should cause proportional revisions in returns.
Zhang (2006) finds that increased uncertainty during an information release corresponds to
greater under-reaction to firm news. Zhang’s findings could be augmented by explicitly cal-
culated measures of uncertainty and of signal-to-noise ratios. Providing explicit posterior
distributions for parameters and determining if prices respond to directly estimated belief
revision can provide evidence about the persistent discussion on the relative influence of
rational learning and behavioral biases (Brav and Heaton 2002).
Lastly, my model as implications for the predictive capacity of earnings news. Much of
the analysis in this paper focuses on the ability of a firm’s earnings to reveal the contem-
poraneous common earnings shock. However, there is an as-yet unused component of the
model that can be applied to the forecasting of future earnings shocks. The parametric
form of Mt (when ϕ ̸= 0) permits some measure of forecastability, in that understanding
the current level of Mt provides a more accurate prediction of the level in Mt+1. The same
firms who provide accurate nowcasts (low variance ratios) may also be the same firms who
provide accurate forecasts, and it is worth evaluating to what degree these firms’ earn-
ings news predicts future common shocks and future firm earnings growth. Focusing on
forecasting would allow a researcher to test alternative specifications in the specification
70
for the latent factor. Comparing various time series models (ARIMA, ARMA, etc.) could
provide valuable insight into which latent factor permits firms news to convey the most
earnings news.
To conclude this section, I note that my model has many promising applications to the
study of information revelation and learning. Extending my model to a more rich analysis
of earnings announcement order would be a fruitful topic for future study due to the lim-
ited existing theoretical and empirical literature. My model produces estimates of earnings
growth that can be well-applied to the rational learning literature. Lastly, I suggest a di-
rection of study for understanding the ability of firm earnings news to forecast aggregate
earnings.
71
Chapter 7
ALTERNATIVE MODEL SPECIFICATION
This section considers an alternative model specification after considering the discussion
in Section 6. The two key issues are (a) time variation in the cashflow beta βi, and (b)
fixing the latent factor variance to provide identification of βi. Fixing the variance of the
latent factor is simple, as doing so amounts to setting σ2 to 1.
The more challenging modelling problem is to permit time-variation in firm cashflow
betas. Estimating the model in a one-step filtering approach as in Section 6.2 can address
time-variation in betas without modifying the likelihood of the model. However, one could
improve the economic interpretability of the model through the inclusion of an explicit
form for a time-varying beta.
What form would such a beta take? The literature supports time-variation in cashflow
betas, but at the same time also suggests that firms may have different exposure to short-
and long-term cashflow risk. The seminal work of Bansal and Yaron (2004) focuses on ex-
posure to long-term cashflow risks. Li and Zhang (2017) expands Bansal and Yaron (2004)
to include firm exposure to short-term shocks in consumption growth. Da (2009) moti-
vates his study through heterogeneous asset exposure to cashflows that arrive at different
times. My model should attempt to capture the same exposure, and to permit the analysis
of parameter uncertainty in addition to known effects due to the level of the cashflow beta.
An improved model would permit variation in firm betas within the model rather than
modifying the estimation procedure, as described in Section 6.2. Any model extension
should permit the ease of estimation afforded by Hamiltonian Monte Carlo methods due
to its desirable high-dimensional convergence properties. Finally, the model should permit
some notion of short- and long-term exposure to common cashflow risk.
Consider the modified laws of motion
Xt = µ+ βtMt + νt (7.1)
βi,t = ρiβi,t−1 + (1− ρi)β̄i + ξi,t (7.2)
Mt = δ + ϕMt−1 + ϵt (7.3)
where Var[ϵt] = 1 − ϕ2, Var[νt] = γ2I, and Var[ξt] = ς2I. Note that σ2i is no longer a pa-
rameter to be inferred but a function of ϕ – this addresses a major criticism in the model I
estimate, and permits better identification of firm-level cashflow betas βi,t.
The new component of the model is the time-series process for firm i’s beta in Equation
72
7.2. Under this process, a firm’s beta varies with time, and has a short- and long-term
component. The short-term cashflow beta is given by its current level βi,t. The long-term
cashflow beta β̄i, which measures the unconditional average cashflow beta, captures the
firm’s exposure to long-term systematic risk. The autocorrelation coefficient ρi ∈ [−1, 1]
measures the degree of persistence in deviations from average β̄i. The variance of beta
shocks ξi,t is ς
2
i .
My model is similar to a simplified version of the one in Li and Zhang (2017), which
incorporates an AR(1) process in the cashflow beta. Li and Zhang (2017) estimate their
model using the simulated method of moments, and find small exposure to short-run risk
but large and persistent exposure to long-run risk. In Equation 7.2, this would be equiva-
lent to a small ρi, the parameter that governs short-term risk persistence.
To estimate this model, one could estimate βi,t using a conditional filter (if ξi,t were
assumed Gaussian, this could be the Kalman filter) during each MCMC step. Doing so
would mean that the only new parameters added to the model would be ρi, β̄i, and the
variance term ς2i . For a cross-section of N firms, this yields 2N new parameters, since the
term βi in my model is replaced by β̄i.
Regarding the priors of this model, I would incorporate the priors discussed in Section
6.4, with several additions to accommodate the alternative specification for the latent fac-
tor. The prior for firm means µi would use the same mean from the pre-sample, given by
that firm’s average earnings growth between 1980 and 1985. However, for firms that en-
tered later in the sample, I would use the average earnings growth for the firm’s industry
using data from 1980 to the quarter prior to the firm’s entrance. For all firms, I would set
the prior variance to the standard deviation of the earnings growth in the years prior to
the firms entrance.
The prior on idiosyncratic risk γi would be similar to its current form. I would estimate
the variance of a firm’s earnings growth between 1980-1985, or from 1980-t − 1 for entry
date t, and fit an inverse Gamma distribution to yield an average variance equal to half of
the firm’s observed earnings variance. Maintaining the prior that half of a firm’s observed
risk comes from idiosyncratic risk is a semi-informed prior that permits the data to sway
the posterior estimates towards the evidence.
Regarding the new law of motion for βt, I would set ρi to a truncated Gaussian with
mean of 0.5 and prior standard deviation 0.4. The bounds would be [−1, 1], though I find
it unlikely that any persistence in beta would be negative. The prior distribution 0.5± 0.4
would allow the prior to cover relatively high persistence as estimated by Li and Zhang
(2017), but also allow my model to find low persistence if there is sufficient evidence to do
so.
The long-term beta β̄i would be set to predispose the model towards non-zero cashflow
73
exposure as in the current prior specification. However, the prior mean and variance must
be adjusted to match the scale of observed aggregate earnings growth, since the latent
factor variance σ2 is now normalized to one. I suggest an average cross-sectional beta of
15.98, with the assumption that β̄i,t explains the total cross-sectional aggregate variance
in my sample from 1980-1985. To adjust for heterogeneity in firms, I suggest multiplying
15.98 by the firm’s corresponding characteristic (noted in Section 6.4) when the firm first
enters the sample. For a large growth firm, this would yield an average β̄i = 15.98× 0.27 =
5.395. A small value firm would have an average β̄i = 15.98× 1.18 = 23.576. Firms not in
the extreme quintiles would have the corresponding scalar calculated from Campbell and
Vuolteenaho (2004). The variance of the cashflow beta should be correspondingly widened
– I suggest that, since my model is relatively new, the priors should be widened to allow
betas to fall ±50% within one standard-deviation. For a large growth firm, the prior would
be N (5.395, 2.6972).
The prior on the variance of beta, ς2i , should be relatively small compared to aggregate
earnings growth. I believe it highly unlikely that “true” cashflow betas should increase
or decrease between quarters by more than 5 units with a one-standard-deviation shock.
For a firm with a prior beta of 10, for example, a ςi = 10 implies that shocks to the beta
process double a firm’s exposure to systematic risk. If ρi is highly persistent (close to 1),
this would result in large and unstable processes for firm betas. I recommend the prior
InverseGamma(5, 100) as an example of a fairly wide prior with an average standard devia-
tion of 5. I do not have more informed priors about whether firms of different sizes, value,
or industry should have differential priors, in part because Li and Zhang (2017) does not
provide estimates of their shock volatility sorted by characteristics. For this parameter, I
would prefer to permit the data speak for itself.
Lastly, I would not change the priors on the latent factor priors for δ, ϕ, and latent fac-
tor shocks ϵt. I would maintain the prior δ ∼ N (0, 0.5) for the mean. The prior on latent
factor persistence would remain a truncated Gaussian ϕ ∼ N (0, 0.2) ∈ [−1, 1]. Lastly, the
latent factor innovations would only differ in that they are no longer hierarchical, since
their standard deviation is normalized to one, i.e. ϵt ∼ N (0, 1)∀t.
The model specified above can address shortcomings in the currently estimated model
by providing greater identifiability and allowing for time variation in firm cashflow betas.
Further, the subdivision of firm cashflow betas into a time-varying form βi,t and the long-
run mean β̄i,t permits the additional study of the relative parameter uncertainty investors
face in the two forms of exposure to cashflow risk.
74
Chapter 8
CONCLUSION
I propose a model of Bayesian learning through earnings growth and use my model to
address several topics of interest to the financial economist. I directly estimate the joint
density of an economy where no parameter is known and study how the levels, uncertainty,
and time variation in beliefs influences cross-sectional variation in returns as well as firm
price responses during earnings seasons.
I find evidence that investors respond to earnings news consistent with rational learn-
ing in the presence of parameter uncertainty. I find that firms with higher parameter un-
certainty have lower earnings responses. Similarly, I show that firms with earnings that
provide precise information about aggregate earnings growth accrue greater earnings an-
nouncement premia than do other firms, and have larger stock price reactions to unex-
pected earnings news. My findings regarding earnings announcements are consistent with
existing theoretical and empirical work in rational learning and parameter uncertainty,
and suggest that my methodology may have a promising application in directly estimating
parameter of interest to researchers studying the effects of learning, uncertainty, and the
behavior of rational Bayesian investors.
However, I do not find robust support that my cashflow beta or its parameter uncer-
tainty explains variation in stock returns in non-announcement weeks. I do find a posi-
tive risk price associated with the cashflow beta, but only during a firm’s announcement
week, consistent with a literature that indicates that risk prices are more salient during
announcement weeks. My results on parameter uncertainty about the cashflow beta seems
to suggest a positive risk price associated with parameter uncertainty, though I highlight a
potential nonlinearity in the pricing role of parameter uncertainty.
I provide a discussion on the shortcomings of my model and why it may be the case
that I find a cross-sectional risk price associated with my cashflow beta only during an-
nouncement weeks. I note that there is a potential identifiability issue in my cashflow
betas, since they are not pinned down in levels due to the joint estimation of the latent
factor variance. I highlight solutions and alternative model specifications that may allow
my cashflow beta to better capture asset cashflow risk, including sequential Monte Carlo
estimation, as well as a model that explicitly permits time-variation in cashflow betas.
Lastly, I contribute to the discussion and measurement of bellwether firms by deriving
a measure that summarizes a firm’s ability to communicate common earnings shocks to
the market. I find that these bellwether firms are larger on average and have larger earn-
ings response coefficients. Further, bellwether firms accrue larger earnings announcement
75
premia than uninformative firms. I show that bellwether firms tend to announce earlier
than other firms. Curiously, my model makes no requirements that informative firms an-
nounce first, so the finding that informative firms usually go earlier is novel. I highlight
that additional economic modelling is required to understand the competitive dynamics of
firm earnings announcement ordering. My results on informative firms is consistent with
rational learning and information choice. My findings imply that the information content
of firm payoff signals provides an important source of variation in stock returns during
information events such as earnings announcements.
My paper contributes more generally to the Bayesian modelling of fundamental asset
risk. I estimate a model and provide guidance for best practices in future models by an-
alyzing the role played by my prior selection and the form of the likelihood. I provide al-
ternative estimation methods and model definitions that permit time variation in cashflow
betas to guide future analyses on the effect of learning under parameter uncertainty.
76
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