Directional Heterogeneity in Distance Profiles
in Hedonic Property Value Models
by
Trudy Ann Cameron, University of Oregon
JEL Classifications: Q25, R21, H23
Correspondence: Department of Economics 435 PLC, 1285 University of Oregon, Eugene, OR 97403-1285
cameron@uoregon.edu
Acknowledgments: Research support accompanying the R.F. Mikesell Chair in Environmental and Re-
source Economics at the University of Oregon is gratefully acknowledged. Helpful suggestions and comments
have been provided by J.R. DeShazo, Ryan Bosworth, Graham Crawford, Joe Stone, and participants at the
AERE Annual Workshop in Madison, Wisconsin in June, 2003.
1
Directional Heterogeneity in Distance Profiles
in Hedonic Property Value Models
ABSTRACT
Failure to allow for directional heterogeneity can obscure otherwise statistically significant distance eff ects
in hedonic property value models. If ambient pollution data are unavailable, researchers often rely upon
distance from a point source of pollution as a proxy for ambient environmental quality. However, damages
from all types of point-source disamenities may exhibit directional heterogeneity. We generalize conventional
distance models to allow for directional eff ects and show that commonly used linear and quadratic spatial
trend variables capture directional heterogeneity in a manner that has not previously been recognized.
Appropriate spatial models can also inform the social planner’s problem of optimal allocation of source
reduction across polluters. When independently calibrated tranport functions are not available, individual
properties can be viewed as ambient receptor sites. Hedonic models can yield estimates of the product of
marginal social damages from ambient concentrations and the change in ambient concentration per unit of
emissions from each source. Optimal emissions depend upon the spatial distribution of all aff ected properties
relative to each source, the parameters of the hedonic model, and marginal abatement costs.
2
1Introduction
In some hedonic property value (HPV) models, researchers are able to quantify and use objective measures
of localized environmental quality as specific attributes of each property in their sample. In other cases,
however, it is more diffi cult to obtain objective measures of the level of an environmental attribute. When
objective measures are unavailable or incomplete, researchers often resort to using distance from the source of
pollution as a measurable characteristic. In this paper, the importance of testing for directional heterogeneity
in distance eff ectsisexplored.Weidentifyanumberofdiff erent modeling strategies that can accommodate
this heterogeneity, should it be present.
There have been a number of very competent general reviews of hedonic property value (HPV) models
(for example, see Bartik and Smith (1987), Palmquist (1991), Freeman (1993), and most recently, Palmquist
(2003)). Farber (1998) provides a useful inventory of empirical studies, including many that estimate distance
gradients in housing prices. In particular, Michaels and Smith (1990) and Kohlhase (1991) find that distances
from Superfund sites in Boston and Houston have a positive eff ect on house prices. The suite of papers by
Kiel and her coauthors all control for distance to Superfund sites or hazardous waste incinerators and focus
on a number diff erent sites in Massachusetts (Kiel (1995), Kiel and McClain (1995), Kiel and Zabel (2001)).
Dale, et al. (1999) emphasize housing prices over time as a function of distance from a lead smelter in Dallas,
and McMillen and Thorsnes (2003) study the property value eff ects of the cleanup of a copper smelter site
in Tacoma.
Hedonic models that do not allow for directional heterogeneity in distance eff ects assume that a two-
dimensional price-distance profile in a single quadrant is adequate to capture distance eff ects on property
values. We can extend this distance profile to three dimensions by rotating it around the vertial axis. This
leads to a price-distance surface that has circular level curves defining the depression in property values
nearest the source of pollution. But distance eff ects may not be the same in all directions. These level curves
may be asymmetric around the pollution source. For example, many point sources of pollution produce
3
either noticeable odors or airborne "fallout" so that prevailing winds can create directional asymmetries
in distance eff ects. Industrial hog farming is an important regional example that has been addressed in
the HPV literature by Palmquist, et al. (1997). However, the level curves for distance eff ects may also
be asymmetric for other types of pollutants. The movement of surface or groundwater can propagate the
damages from water pollution farther in some directions than others, and visual and noise pollution can be
aff ected by topography.
We demonstrate in this paper that if substantial directional eff ects are present, but are ignored, one would
expect a model with simple homogeneous distance eff ects to exhibit larger than necessary standard errors
and apparent heteroscedasticity. Parameters defining the shape of the distance profile can also be biased if
the directional distribution of distances for observed property transactions is non-uniform. We explore a
number of models that can be used to test for the presence of directional heterogeneity in distance eff ects
and can accommodate a wide variety of systematic directional eff ects when they are present.
A second important theme in this paper concerns an insight for the social planner’s problem of optimal
allocation of abatement responsibility across multiple point sources of non-uniformly mixing pollutants.
Emissions transport and spatial heterogeneity in ambient environmental quality have long been recognized
as important issues in the theory of pollution control policy. However, the systems of dedicated ambient
monitors needed to calibrate the required "transport" functions are often incomplete so that the information
embodied in these these key transport functions is unavailable to policy makers. We show how hedonic
property value models with rich enough spatial diff erentiationindistanceeff ects can yield estimates of the
product of marginal damages and marginal transport eff ects. If marginal abatement costs at each source can
be quantified, the estimated hedonic property value function and the spatial distribution of all properties
aff ected by each source of emissions can be combined to suggest socially optimal emissions reductions for
each source.
Published hedonic property value models that employ distance measures pay no attention to direction.
4
Some authors use general spatial trend variables, either linear or quadratic. These researchers have implicitly
allowed for directional eff ects but have generally failed to recognize that these trend variables contain key
information about directional distance eff ects that cannot be ignored. There appear to be only two explicit
considerations of direction in any form in the published hedonic literature. Gillen, et al. (2001) relegate
directional considerations to the nature of spatial autocorrelation in the error terms in their model of isotropic
versus anisotropic error autocorrelation in house prices.1 In their study of the eff ects of industrial hog-farming
operations on house prices, Palmquist, et al. (1997) mention the problem of prevailing winds as an area for
future research. However, confidentiality of specific locational data concerning hog farms prevented them
from pursuing these issues. While prevailing winds would seem to be an important consideration in work
concerning the “aroma of Tacoma” by McMillen and Thorsnes (2003), they do not consider direction either.
Section 2 of this paper explains why direction, as well as distance, may be an important feature of
many HPV models. Section 3 outlines how commonly used spatial trend variables implicitly contribute to
directional heterogeneity in distance profiles. Prevailing winds or other physical processes are likely to be
a common denominator in cases where there is directional heterogeneity in distance profiles. Thus, section
4 reviews how external information about the annual average or seasonal directions of prevailing winds or
other physical processes can be incorporated into the model. Section 5 outlines how the level curves for
a directionally diff erentiated distance profile can be mapped. Section 6 pulls together the intuition from
directionally heterogeneous distance profiles for housing prices and the social planner’s problem of optimal
allocation of pollution abatement responsibility among multiple sources of non-uniformly mixing pollutants.
Section 7 provides some caveats before Section 8 concludes.
1 There is no point-source environmental disamenity in their data from which distances are being measured; the only distances
in the model are the distances between individual houses in the sample.
5
2 Distance Profiles as a Function of Direction
Ignoring heterogeneity in distance eff ects with respect to direction from a localized pollution externality can
potentially obscure what might otherwise be a clear price-distance relationship. Figure 1 illustrates just two
diff erent directions, East and West, rather than the full 360o of the compass. In this case, the observations
for the dependent variable, Y i , are shown lying very close to the directionally-specific E [Y i ].Eachofthese
two directional distance profiles is depicted as a linear function of distance, d , with a common intercept, α ,
but diff erent slopes.2
Figure 1 is drawn under the assumption that prevailing winds are from the west. If the researcher
controlled for direction before estimating the parameters of the distance gradient, the data in the example
would yield very precise estimates of the common intercept, α , and two separate slopes, β E and β W .The
steeper profile to the west indicates that the prevailing winds limit the westward diff usion of the pollutant. In
contrast, the flatter profile to the east captures the fact that prevailing winds carry the pollutant considerably
farther in that direction. Ignoring direction is equivalent to superimposing the two diff erent distance profiles
in the right-hand quadrant of the diagram and fitting one common distance profile to the pooled data. We
illustrate the eff ect of ignoring direction by showing the western distance profile rotated around the vertical
axis. The more heterogeneous the directional distance profiles, the greater will be the dispersion around
the common “average” distance profile when direction is ignored. A second artifact of failure to control for
direction will be apparent heteroscedasticity, with error variances increasing with distance.
In hedonic property value studies, researchers typically consider the “extent of the market” for proximity
eff ects to consist of all housing transactions within a particular radius of an environmental hazard. Alterna-
tively, the relevant market may consist of all census tracts or zip codes within a particular absolute distance
from the site. In these cases, it may be approximately true that the distribution of distances at which prop-
erty transactions are observed is independent of direction. When distance and direction are uncorrelated
2 Contrast this form of heterogeneity with the type commonly assumed in fixed eff ects models for panel data. There, we
typically assume a common slope, but diff erent intercepts across groups.
6
in the data, omitted variables bias from failure to control for direction will be minimized. However, there
are a number of ways in which hedonic property value data may exhibit correlations between direction and
distance. We will review these in turn.
In Figure 2, we expand the right-hand quadrant of Figure 1 to include an additional set of points
(represented by open dots) for the intermediate north and south distance profiles. These north and south
profiles are assumed to be identical since the prevailing winds are coming from due west. Suppose the
researcher recognizes the potentially greater influence of pollution downwind, to the east, and therefore
collects data to a greater distance in that direction. Figure 2 shows that if she fails to control for direction in
the estimation process, it is possible that she may find no statistically significant relationship at all between
Y and distance, despite precise relationships in each direction considered separately. The empirical estimates
could suggest a negligible or statistically insignificant distance eff ect—a "false negative"—when in fact distance
eff ects are substantial and would be easy to discern when controlling for direction. The omitted variables
bias in this case stems from the presence in the data of observation at greater distances only for a subset of
all possible directions.
Figure 3 shows another potential source of omitted variables bias when direction is omitted. Even if
the researcher collects data on Y within a constant radius, regardless of prevailing winds, it is possible
that observations are not naturally identically distributed with respect to distance in all directions from the
Superfund site. This is a particular concern with data such as housing transaction price information. In
the downwind direction, in the presence of a localized environmental externality, there may simply be fewer
nearby houses. Alternately, there may be fewer nearby transactions if more-aff ected properties are more
diffi cult to sell. If the nearest distance at which observations occur is greater in the downwind direction,
slope distortions may result when direction is omitted from the model.
A third way in which correlations between distance and direction may naturally be present in hedonic
property value data is when features of the landscape surrounding a pollution source preclude housing at
7
those locations. Examples will include coastal areas, areas where there are large parks, or urban areas with
expanses of industrially or commercially zoned properties.
In sum, if both distance and direction matter to housing prices, the two variables are correlated, and
direction is omitted, the coeffi cient on the distance variable can be expected to be biased. Ignoring direction
while estimating a distance gradient can significantly compromise both the accuracy and the precision with
which distance gradients are measured. This bias and loss in precision is amplified when directional eff ects
are more prominent.
When direction is not part of the model, we have already noted that the spatial level curves of the E [Y i ]
are implicitly assumed to be circular. With directional diff usion of airborne pollutants, one would expect the
contours of the dispersal pattern to be non-circular. Relaxing the implicit assumption of circular contour
lines in the three-dimensional distance profile allows the researcher to estimate the direction of the main axis
of a more general set of elliptical level curves. In the simple multiple-direction case illustrated in Figures 2
and 3 with data for four directions (North, South, East and West profiles), the implied three-dimensional
distance profile will have the main axis of its elliptical level curves running from due West to due East. This
is because of the assumption that the prevailing winds from due west. However, any appropriate empirical
model needs suffi cient flexibility to allow the orientation of the main axis of the elliptical contour lines to
vary freely, based on spatial patterns in the dependent variable. Alternatively, one might constrain the main
axis to correspond to the actual historical average wind direction. We now turn to viable specifications for
such models.
8
3 Spatial trend models
3.1 Implications of a simple “planar spatial trend” model
First, we show that the simplest linear-in-distance model with directional eff ects is equivalent to a model
that includes just a simple linear distance eff ect plus a pair of incidental linear spatial trend variables. With
GIS software, one can readily identify point locations of housing units or census tract centroids in decimal
degrees (conventionally, to six decimal places). The simplest specification for a generic dependent variable
Y i involves overlaying a conical direction-independent distance profile with some tilted plane defined over
longitude and latitude.3 The combination of the lat/long eff ects on the dependent variable and symmetric
distance eff ects can readily mimic distance eff ects that are non-constant around the points of the compass:
Y i = α + βd i +(γ 1long i + γ 2lat i )+ε i (1)
Keep in mind that one degree of latitude is not the same distance as one degree of longitude. The
length of one degree of longitude depends upon the latitude at which that distance is being calculated. In
general one degree of longitude = cos(latitude)* 111.325 kilometers. In contrast, one degree of latitude is
well approximated by 110.6 kilometers. This complicates the task of determining the direction of steepest
descent when longitude and latitude are used directly as explanatory variables. It is preferable to compute
location in (x,y)-space in common units in each direction.
Fortunately, it is not necessary to use the Greenwich Meridian and the equator as the origins of mea-
surement for the absolute spatial location of each property in the sample. We recommend expressing both
longitude and latitude in kilometers and shifting the origin of measurement to coincide with the site of the
environmental disamenity. Denote the longitude and latitude of the site as (x s ,y s ). Usingthesourceof
pollution as the origin of measurement, let x i be the east-west coordinate of the property, and let y i be the
3 Any real specification will of course also control for a host of structural and neighborhood variables characteristics.
9
north-south coordinate. Then
Y i = α + βd i +(γ 1x i + γ 2y i )+ε i (2)
Here we assume that the longitude-to-kilometers conversion factor can be approximated for both the specific
property and the pollution source by the latitude correction corresponding to their average latitude, so that
x i = (111. 325)(long i − long s )cos[(lat i + lat s )/ 2] (3)
y i = (110. 6)(lat i − lat s )
In equation (2), both the x i and y i distances are measured in kilometers, as is the distance d i .4 The
parameters γ 1 and γ 2 in equation (2) will be diff erent from their counterparts in equation (1) due to the
change of location and scale.
It is convenient to convert equation (2) so that it is expressed entirely in terms of polar coordinates.
Recall that x i = d i cosθ i and y i = d i sinθ i where θ i is the direction from the site to housing unit i (measured
in radians counter-clockwise from due east). Making this substitution, equation (2) becomes:
Y i = α + βd i +(γ 1d i cosθ i + γ 2d i sinθ i )+ε i (4)
Collecting the terms in distance, we get:
Y i = α +(β + γ 1 cos θ i + γ 2 sinθ i )d i + ε i (5)
Instead of having a constant distance eff ect, the distance eff ect depends upon the direction in which it is
being calculated.
4 In implementing these transformations, it is crucial to remember that cartographers measure latitude in degrees from the
equator, rather than in radians. The map measures of latitude must first be converted into the equivalent number of radians
before using econometric software to calculate the cosine of the term in square brackets in the formulas in (3).
10
One implication of this result is that researchers who have specified models with a linear spatial trend
but no linear distance eff ect (i.e. β =0) are still estimating models with implicit direction and distance
eff ects. However, the origin of measurement of these distances may be the intersection of the equator and the
Greenwich Meridian, unless some other arbitrary point of origin is selected. More insidiously, any researcher
who adds a set of linear spatial trend variables to a model that includes a specificdistancemeasuremay
overlook the fact that the full distance eff ect is β + γ 1 cos θ i + γ 2 sinθ i , rather than simply β.
In describing fitted models involving heterogeneous parameters, it is customary to simplify the results
by reporting key varying derivatives calculated at the “means of the data.” While the average angle θ in
any sample will depend upon the spatial distribution of observations in that sample, it will usually be more
convenient to use the fact that the average values of both cosθ and sinθ would be zero if this angle was
uniformly distributed around the compass from 0 to 2π . The distance eff ect in this theoretical “average”
direction corresponds to just the β coeffi cient. If the researcher desires to know the predicted distance eff ect
in each of the four main compass directions (N,W,S,E), these distance eff ects can be calculated as (β + γ 2,
β − γ 1, β − γ 2, β + γ 1). The magnitudes of the directional distance eff ectsthusclearlydependuponthe
signs and sizes of the two directional coeffi cients, γ 1 and γ 2.
3.2 Implications of “quadratic spatial trend” specifications
Some researchers control for systematic spatial trends by including quadratic and interaction terms in x and
y . (Consider the empirical example in Dubin (1998).) We show now that the combination of a fully quadratic
spatial trend with a linear-in-distance specification leads to an implicitly quadratic-in-distance specification.
Let the model in conventional form be
Y i = α + βd i + ¡γ 1x i + γ 2y i + γ 3x 2i + γ 4x i y i + γ 5y 2i ¢ + ε i (6)
Converting this mixed Cartesian coordinate and distance specification into exclusively polar coordinates
11
yields the following equivalent specification:
Y i = α + βd i +(γ 1d i cosθ i + γ 2d i sinθ i ) (7)
+¡γ 3d 2i cos2 θ i + γ 4d 2i cosθ i sinθ i + γ 5d 2i sin2 θ i ¢ + ε i
Simplify by collecting terms in d i and d 2i to reveal the equivalent form:
Y i = α +(β + γ 1 cos θ i + γ 2 sinθ i )d i (8)
+¡γ 3 cos2 θ i + γ 4 cosθ i sinθ i + γ 5 sin2 θ i ¢ d 2i + ε i
However, even if the observed directions θ i were distributed uniformly around the compass, the expected
distance eff ect will not, in this model, collapse to just β , as was the case for equation (5). The expected
values of cosθ i , sinθ i and cosθ i sin θ i will all be zero in this case, but the expected values of cos2 θ i and
sin2 θ i will not. We are considering the expected value over a uniform distribution for θ over the interval
from 0 to 2π . This distribution has a constant density of 1/ 2π that can be factored out of the integral in
the expectation formulas:
E £cos2 θ currency1 = 12π
Z ¡
cos2 θ ¢ ∂θ =
·1
2θ −
1
4 sin2θ
¸2π
0
= 12 (9)
E £sin2 θ currency1 = 12π
Z ¡
sin2 θ ¢ ∂θ =
·1
2θ −
1
4 sin2θ
¸2π
0
= 12
The expectations of these two terms will be strictly positive and both equal to 0.5. Thus, for uniformly
distributed directions, the specification in equation (8) corresponds to an average distance eff ect as follows:
Y i = α + βd i +
µγ
3 + γ 5
2
¶
d 2i + ε i (10)
12
Clearly, even if we are considering the average distance eff ect over all possible directions, a model with
both a simple linear distance eff ect and a full set of quadratic spatial trend terms actually implies a model
with a quadratic distance eff ect. The average distance eff ect will be linear only if the average of the estimated
coeffi cients on the squared x and y coordinatesisnotdiff erent from zero.
3.3 Nonlinear-in-distance specifications
Except for (i) changes of location and scale, (ii) the slight local approximation involving cos [(lat i + lat s ) / 2],
and (iii) conversion exclusively polar coordinates, the model in equation (5) is identical to that involving the
simple planar spatial trend in terms of the longitude and latitude variables in equation (1). However, there
is nothing to mandate using only the functional form employed in equations (1) or (2). In fact, this form
has the unappealing characteristic that in any particular direction, the marginal eff ect of distance remains
constant as distance increases. We generally expect the eff ect of localized externalities to diminish with
distance until any incremental eff ect of distance essentially disappears.
To approximate a pattern of marginal distance eff ects that decrease with distance, researchers often
use models that are nonlinear but monotonic transformations of the distance variable, f (d i ). Candidate
functional forms include a logarithmic transformation f (d i )=ln(d i ), or a reciprocal function, f (d i )=1/d i .
However, both of these specifications can produce extreme values for the transformed distance variable as
distance goes to zero. An easy fix for this problem is to shift the origin of measurement for the distance
variable by adding one unit for all observations. Distance measurements are always non-negative, so this
change-of-origin by one unit can guarantee that the smallest observed value of f (d i )=ln(d i +1)is zero and
the largest observed value of f (d i )=1/ (d i +1) is 1. To keep the notation simple in what follows, we will refer
to the distance variable generically, as f (d i ). We adapt the model in equation (2) to allow for diminishing
marginal eff ects of distance, but still allow the marginal eff ect at any given distance to vary systematically
13
and smoothly with direction by substituting a suitable version of f (d i ) for d i in our earlier model:
Y i = α +(β + γ 1 cosθ i + γ 2 sinθ i )f (d i )+ε i (11)
To test statistically for the presence of directional heterogeneity in the distance eff ect, one would again test
the joint hypothesis that γ 1 = γ 2 =0.
Over all possible directions either for the model in (1) and (2) or for the alternative model in equa-
tion (11), the two extrema of the distance eff ect occur where the derivative with respect to θ i of the
systematically varying coeffi cient, β + γ 1 cosθ i + γ 2 sinθ i , goes to zero. Making use of the facts that
∂ sinθ/∂ θ =cosθ , ∂ cos θ/∂ θ = − sinθ ,andcosθ/ sinθ =tanθ , the maximum predicted distance eff ect
occurs at θ ∗ =arctan(γ 2/γ 1), while the minimum distance eff ect lies at θ ∗∗ = θ ∗ +π . If we are considering a
hedonic property value model using equation (11), housing prices would be predicted to increase most slowly
as one moves away from the site in the direction
θ ∗∗ =arctan(γ 2/γ 1)+π (12)
This direction depends upon the estimated parameters γ 1 and γ 2. This direction (measured in radians
counterclockwise from due east) can be converted to compass degrees (measured clockwise from due north)
by computing φ ∗∗ = − 180 (θ ∗∗ /π )+90. If the level curves of the property value distribution are non-circular
only because of localized air pollution levels, this direction would be interpreted as the apparent downwind
direction.
3.4 Other patterns of directional heterogeneity
Models with the slope coeffi cient on f (d i ) generalized to simply β + γ 1 cos θ i + γ 2 sinθ i do not capture the
universe of all possible directional or spatial patterns in distance eff ects, although we will concentrate on
14
these forms below. As section 3.2 suggests, a set of quadratic spatial trend variables can capture more general
spatial patterns in the same family. These models may be appropriate in some empirical applications. Other
functions of direction can also be explored, including variants that allow for systematic shifts in distance
eff ects for subsets of directions according to the topography surrounding the source of pollution. Dummy
variables for certain directions can be interacted with cos θ i and sinθ i to permit shifts in the directional
eff ects away from the simple pattern captured by just β + γ 1 cosθ i + γ 2 sinθ i . Whenever physical features
of the context may be important, their potential distinct eff ects should be explored.
Sometimes directional heterogeneity in distance eff ects will not vary around the compass relative to the
source of the pollution as in the case of prevailing winds. There will be other cases when relative elevation,
rather than simply compass direction, is the physically relevant determinant of distance eff ects. It may not
be the angle relative to due east, but rather the angle relative to the downhill direction that matters most.
If steepness in the downhill direction also matters, the trigonometric terms involving this angle will need to
be interacted with the degree of slope in that downhill direction.
4 Estimated versus actual “downwind” direction
In this paper, we will use the term "downwind" to imply that prevailing winds constitute the physical process
that leads to more-distant eff ects of pollution in one direction. However, we emphasize that the downwind
terminology can also be generalized to imply all physical processes that contribute to spatially asymmetric
transport of pollutants from a source, including the movement of surface or groundwater, and topographical
eff ects on visual and noise polluation.
One potential problem with the unrestricted specification in equation (11) is that the estimated downwind
direction, θ ∗∗ =arctan(γ 2/γ 1)+π , may not correspond to the known downwind direction. The estimated
downwind direction may be biased by other factors unrecognized by the researcher. To avoid such omitted
variables bias, it will be important to control for other factors that can be expected to contribute to a general
15
spatial price gradient.
Once the model in equation (11) has been estimated, therefore, it will be important to test whether the
estimated “downwind” direction, φ ∗∗ measured in compass degrees, coincides with the meteorological facts.5
This statistical test of the estimated wind direction would involve constructing a point estimate and standard
error for the estimated direction θ ∗∗ from the point estimates of parameters γ 1 and γ 2 and testing whether
this direction could be equal to the actual downwind direction, θ 0. Historical prevailing wind directions for
major cities in the US are provided by NOAA (1998).
4.1 Imposing the annual average prevailing wind direction
In some cases, the downwind direction should not be estimated, but should be determined from meteorological
data and imposed upon the model. Assume initially that the direction of prevailing winds is constant over
the seasons. Let the actual downwind direction from the environmental disamenity be θ 0 radians. If we wish
to impose this downwind direction as a constraint on our estimation, it will translate into a restriction on
the admissible values of γ 1 and γ 2. Solve equation (12) for the admissible relationship between γ 1 and γ 2
when θ ∗∗ = θ 0:
γ 2 = γ 1 tan(θ 0 + π ) (13)
Substitute this restriction into equation (11) to yield
Y i = α + ¡β + γ 1 cosθ i + £γ 1 tan(θ 0 + π )currency1 sinθ i ¢ f (d i )+ε i (14)
= α + βf (d i )+γ 1 £cosθ i +tan(θ 0 + π )sinθ i currency1 f (d i )+ε i
This model can be estimated using conventional linear-in-parameters specifications since all terms inside the
square brackets are observed data or known constants. This specification allows for directional asymmetry
5 Recall that meteorologists report wind direction based on the direction from which the wind is coming, rather than the
vector describing the direction in which it is blowing. Thus, a NW wind would be blowing "out of the NW, in a SE direction."
16
in the distance eff ect, but admits for distortions only in a direction known to be consistent with prevailing
winds. To test whether there is evidence of a directional eff ect in the dependent variable that coincides with
wind direction (or other natural flows that may aff ect waterborne contaminants, for example), one would
simply test whether γ 1 =0can be rejected.
We might desire this directional restriction on the downwind eff ect because without it, there is no re-
quirement that the direction in which housing prices increase most slowly with distance actually coincides
with the direction in which pollution travels the farthest. Allowing the distance profile implied by the model
to “tilt” in any arbitrary direction will court omitted variables bias. There may be other underlying factors,
unknown to the researcher, which account for an overarching spatial trend in housing prices. These factors
could include a temperature gradient, distance from a nearby city center or coastline, or any other amenity
or disamenity that is not controlled for by its explicit inclusion in the specification.
In some applications, however, we may wish to isolate the extent of the downwind eff ect separately from
any additional nonspecific directional eff ectthatmaybesuperimposeduponthedownwindeff ect. We can
introduce parameters γ ∗1 and γ ∗2 and allow for two distinct types of directional eff ects:
Y i = α + βf (d i )+γ 1 £cosθ i +tan(θ 0 + π )sinθ i currency1 f (d i ) (15)
+(γ ∗1 cosθ i + γ ∗2 sinθ i )f (d i )+ε i
The constructed variable £cos θ i +tan(θ 0 + π )sinθ i currency1 f (d i ) will vary independently from cosθ i f (d i ) and
sin θ i f (d i ), so that distinct estimates can be obtained for the downwind eff ect parameter, γ 1, and the generic
directional eff ect parameters, γ ∗1 and γ ∗2. The researcher will need to speculate upon logical explanations for
statistical significance in these two additional parameters.6
6 Joe Stone suggested this potentially useful generalization.
17
4.2 Imposing seasonally varying prevailing winds
A more intricate model may be appropriate when there are regular seasonal diff erences in the direction of
prevailing winds. The term in equation (14) that carries the γ 1 coeffi cient captures the direction, θ i ,from
the localized environmental disamenity which will be constant over time but will vary across observations.
This term also includes the direction of prevailing winds, θ 0. This is consistent with the assumption that the
direction of the prevailing winds is fixed across observations. However, this assumption may be unrealistic.
For example, selling prices of houses are understood to vary seasonally for a variety of reasons. If they
also vary seasonally as a result of some seasonal pattern of dispersion of some point source pollutant, this
information can also be employed to enhance estimation if time-subscripted data are available. The model
could be generalized to apply to Y it and θ 0t :
Y it = α + βf (d i )+γ 1 £cos θ i +tan(θ 0t + π )sinθ i currency1 f (d i )+ε it (16)
As before, θ 0t is not a parameter to be estimated, but additional data on seasonal wind directions to be
employed in the estimation process.
5 Level curves of distance profiles
For any of these directional models, it may be also be useful to derive the implied level curves for the overall
distance profile. Once the unknown parameters of the model have been estimated, one can solve for the
latitude and longitude coordinates of locations that lie along level curves for fitted Y i . The geo-coded level
curves can then be displayed using mapping software. For the basic model in equation (11), implementation
proceeds as follows. Assume ε i =0and solve the fitted model for the values of d ∗i that correspond to each
of the observed directions (θ i )representedinthesampleifY i is held constant at Y∗ . The set of polar
18
coordinates satisfying this condition will be:
[d ∗i ,θ i ]=
·
f − 1
µ Y ∗ − α
β + γ 1 cos θ i + γ 2 sinθ i
¶
,θ i
¸
(17)
Now convert these points expressed in terms of polar coordinates back into Cartesian coordinates using
x i = d ∗i cosθ i (18)
y i = d ∗i sinθ i
Then convert these simple Cartesian coordinate back into latitude and longitude by reversing the transfor-
mations in (3):
lat ∗i = lat s +(y ∗i / 110. 6) (19)
long ∗i = long s + x ∗i / [111. 325 cos [(lat ∗i + lat s )/ 2]]
If the observations are first sorted in order of θ i , the graphing routines in conventional estimation software
can be used to connect the points and draw a smooth curve. Saving the latitudes and longitudes and mapping
these pairs of points will produce an elliptical pattern of points. Alternatively, to produce an ellipse consisting
of an arbitrarily dense pattern of points, one could discard the observed directions, θ i , simulate as many
evenly spaced values as desired between 0 and 2π and perform the transformations in equations (17) through
(19) using these simulated values instead.
6 Hedonic models and optimal source reduction
In this section, we outline how hedonic property value models, with or without directional heterogeneity
in distance eff ects, can be integrated into models for optimal pollution abatement when there is spatial
19
heterogeneity in non-uniformly-mixing pollutants emitted by a number of point sources.
The usual model for non-uniformly mixing pollutants assumes that the ambient concentration at receptor
i depends upon emissions from source j , where ambient environmental quality at i is given by the transport
function T ji (e i ) (see Helfand, Berck and Maull (2003), p. 264). If we assume that damages at location
i are due to the simple aggregate of emissions reaching that place, then damages at that location are
D i (PKj =1 T ji (e i )). A simpler assumption is that the transport function is linear and additively separable in
emissions from each source (Kolstad, 2000, p. 156). Then the relationship between emissions and ambient
concentrations can be summarized by the appropriate element a ij of the “transfer matrix,” A . Assume that
the ambient concentration in parts per million of the pollutant in question at receptor i , ppm i , depends on
emissions levels e i from each source j =1, ..., K according to this approximately linear relationship:
ppm i = a 0 + a i 1e 1 + ... + a iK e K + u i ,i=1, ..., N (20)
where u i is an error term and the transfer coeffi cients a ij are measured in parts-per-million of ambient
concentration at receptor i per ton of pollution emitted at source j .Effi cient abatement at each of the
j =1, ..., K diff erent sources requires that the decision-maker adjust emission levels from all sources to
maximize net social benefits (the diff erence between overall social benefits of reduced emissions and the
overall costs to firms of abatement).
The economic theory of optimal abatement under these circumstances is relatively mature, but policy
implementation is hampered in many cases by a paucity of data on the relevant transport functions. We will
now outline how hedonic property value models can be used to quantify the product of marginal ambient
damages and transport function derivatives. In certain cases, this product of terms is all the transport
information that is needed, so separate quantification of the transport function may not be necessary. Socially
optimal allocations of abatement responsibility can be informed by hedonic property value models.
20
6.1 Housing units as implicit “ambient receptor sites”
Suppose that the type of environmental disamenity being considered is an example of a “localized external-
ity” according to the distinctions drawn in Palmquist (1992) and Palmquist (2003). This means that the
environmental disamenity aff ects only a small number of properties relative to the size of the market. We will
also assume that there are no significant transactions or moving costs. Under these conditions, first-stage
hedonic property value estimates are appropriate for welfare calculations. Hedonic rental rate diff erentials
for localized externalities can be integrated into our conventional models that articulate the equimarginal
principle for optimal abatement or emissions at a finite number of diff erent pollution sources.
6.1.1 Marginal social costs of pollution (marginal benefits of abatement)
Suppose the sum across properties of the individual decrements in implicit housing rental rates are a rea-
sonable approximation to the social benefits of eliminating pollution. For each of the housing units in the
estimating sample, the individual social benefit of reducing pollution by ∆ e j at each source is given by:
∆ rental rate i = MSC i∆ ppm
i
XK
j =1
µ∆ ppm
i
∆ e j
¶
∆ e j (21)
= MSC i∆ ppm
i
XK
j =1 a ij ∆ e j
This assumes that the rental rate diff erential for a property is the sum of the diff erentials due to the portion
of ambient pollution at receptor i attributable to each of the K diff erent sources.
Assume that the dependent variable in a typical hedonic property value model is the annualized rental
rate, Z i , that corresponds to the selling price of the property, Y i . Adapting our simple model of equation (11)
to the case of multiple sources of pollution, we can specify this rental rate as a function of the distances and
directions of the property relative to several point sources of pollution. Assume for now that all properties are
otherwise identical, in terms of size as well as structural and neighborhood characteristics, so that there is no
need to control for other sources of heterogeneity besides distance and direction from the various pollution
21
sources. A specification for the annualized rental rate function with multiple localized pollution sources,
j =1, ..., K could be:
Z i = α +
XK
j =1
¡β
j + γ j 1 cosθ ji + γ j 2 sinθ ji
¢ f
j (d ji )+ε i (22)
The compound coeffi cients ¡β j + γ j 1 cosθ ji + γ j 2 sinθ ji ¢ in equation (22) are measured in dollars per unit
of distance and the specification assumes implicitly that the quantity and type of pollutant emitted from
each source is identical. To allow for diff erent transformations of distance to apply for each pollution source,
we now index f (d ji ) by j to create f j (d ji ), although for similar pollutants, the identical transformation may
apply to distances from all sources. Suppose now that the distance eff ects are estimated perunitofpollutant
emitted, so that the overall impact of the total emissions from source j will be this marginal eff ect times the
level of emissions from that source, e j . The model in (22) can be generalized to the case of varying levels of
emissions from each source:
Z i = α +
XK
j =1
¡β
j + γ j 1 cos θ ji + γ j 2 sinθ ji
¢ f
j (d ji )e j + ε i (23)
With this modification, the change in the implicit rental rate of the i th property with respect to a change
in the emissions from the j th pollution source is a function of the distance between that source and the
property in question. Viewing this specification from the perspective of equation (21), we can identify the
correspondence
∆ rental rate i
∆ e j =
MSC i
∆ ppm i a ij =
¡β
j + γ j 1 cosθ ji + γ j 2 sinθ ji
¢ f
j (d ji ) (24)
Thus, the eff ect of a unit of emissions from the j th pollution source on the implicit rental rate of the i th
property subsumes both the marginal social cost to that property of the j th source’s contribution to ambient
22
concentrations and the relevant transfer coeffi cient.
The overall social benefit of a vector of pollution reductions at the set of K sources will be the sum
of these individual eff ects across all N housing units in the aff ected area (not just the N ∗ units in the
estimating sample from which the hedonic distance coeffi cients are derived). The overall marginal social
benefits of emissions reductions (∆ e 1, ..., ∆ e K ) will be equal to the overall marginal social costs (MSC )that
this pattern of pollution created:
MSC =
XN
i =1
µXK
j =1
¡β
j + γ j 1 cosθ ji + γ j 2 sinθ ji
¢ f
j (d ji )∆ e j
¶
(25)
6.1.2 Marginal social benefits of pollution (marginal costs of abatement)
The marginal costs of pollution abatement to the industry are the same as the marginal savings to the
industry from continuing to pollute. The aggregate marginal benefits to the K firms of the incremental
changes in emissions summarized as (∆ e 1, ..., ∆ e K ) will be:
MS =
XK
j =1 mac j ∆ e j (26)
where mac j is the marginal abatement cost (marginal cost of emissions reduction) for source j .Suppose
abatement costs are a simple quadratic function of abatement level b j with no intercept or linear term:
ac j = δ j b 2j . Then marginal abatement costs will be proportional to abatement levels: mac j =2δ j b j = c j b j .
6.1.3 Effi cient abatement
Equalizing overall marginal social costs and overall marginal social benefits can be accomplished by equalizing
the marginal social costs and marginal social benefits associated with emissions from each source:
·XN
i =1
¡β
j + γ j 1 cosθ ji + γ j 2 sinθ ji
¢ f
j (d ji )
¸
∆ e j =[c j b j ]∆ e j j =1, ..., K (27)
23
The term ∆ e j cancels from each side of this equimarginal condition and we are left with the result that the
optimal level of abatement b j for each firm will be determined by:
b j =(1/c j )
XN
i =1
¡β
j + γ j 1 cosθ ji + γ j 2 sinθ ji
¢ f
j (d ji ) (28)
There is still more information that can be brought to bear on the estimation process and the deter-
mination of the socially optimal level of emissions reduction for each source. If the pollutant is dispersed
by prevailing winds, we can impose a dispersal pattern consistent with those prevailing winds. If we adapt
equation (23) to the case with constant prevailing winds, as in equation (14), we get
Z i = α +
XK
j =1
£β
j + γ j 1
¡cosθ
i +tan(θ 0 + π )sinθ i
¢currency1 f
j (d ji )e j + ε i (29)
If the eff ect of distance and direction on the per-unit eff ects of emissions is the same across all sources, so
that β j = β and γ j 1 = γ 1 for all j , a resulting special case of this hedonic model would be:
Z i = α + β
XK
j =1 f j (d ji )e j + γ 1
XK
j =1
¡cosθ
i +tan(θ 0 + π )sinθ i
¢ f
j (d ji )e j + ε i (30)
If the hypotheses that β j = β and γ j 1 = γ 1 for all j cannot be rejected, then each unit of pollutant from
any localized source has the same eff ect on the rental rates for any housing unit at the same distance and in
thesamedirection.
With quadratic abatement costs and the estimated downwind direction constrained to match the empirical
downwind direction based on typical weather patterns, the socially optimal level of abatement for each source
24
can be determined as follows:
b j =(1/c j )
XN
i =1
£β
j + γ j 1
¡cosθ
i +tan(θ 0 + π )sinθ i
¢currency1 f
j (d ji ) (31)
= ¡β j /c j ¢
XN
i =1 f j (d ji )+
¡γ
j 1/c j
¢ XN
i =1
£cosθ
i +tan(θ 0 + π )sinθ i
currency1 f
j (d ji )
If we constrain the per-unit eff ects of pollutants from each source to be identical at the same distance and in
the same direction as in equation (30), the equation in (31) can be simplified. If all firms also face identical
cost structures, so that the slopes of their marginal cost curves are identical and c j = c for all sources j ,
then the socially optimal level of abatement for each source will be:
b j =(β/c )
XN
i =1 f j (d ji )+(γ 1/c )
XN
i =1
£cosθ
i +tan(θ 0 + π )sinθ i
currency1 f
j (d ji ) (32)
In words, even if abatement technology is identical for all firms, the spatial distribution of housing units
relative to each site will imply diff erent optimal levels of emissions b j for each firm. As expected, these
optimal abatement levels will depend on the location of houses relative to each firmandonthedirectionof
the prevailing winds.
If there is no directional heterogeneity in any of the distance eff ects for any of the individual pollution
sources, then γ 1 =0and equation (33) reduces to just
b j = β c
XN
i =1 f j (d ji )=
Nβ
c f j (d ji ) (33)
This form implies that only the number of aff ected housing units and the average of the relevant function
of distance matters, along with the common distance eff ect (β ) and the common marginal abatement cost
parameter, c .Iff j (d ji )=d ji for all sources j , then simply the average distance will enter the formula.
If abatement costs are approximately quadratic and the (constant) slope of the marginal abatement
25
cost function for each source is known, then equation (31), in conjunction with locational polar-coordinate
data for all aff ected housing units, (d ji, θ ji ) relative to each source j , combined with information about the
downwind direction for prevailing winds, θ 0, allows a tedious but straightforward calculation of the socially
optimal level of abatement for each source.
7 Caveats and Directions for Future Research
Researchers who explore directional heterogeneity in distance eff ects will still need to be vigilant about the
possibility of omitted variables biases in their estimated distance eff ects. Collinearity among distances to
diff erent pollution sources, and to other amenities and disamenities that can also influence housing prices,
will need to be assessed. One important class of sensitivity tests might include the demonstration that
similar functional forms for distance and direction, computed relative to some randomly selected point (or
points), exhibits no such directional distance eff ects on housing prices.7
Appropriate specifications for the form of directional heterogeneity in distance eff ects should typically
refer to the physical features of the application (including prevailing winds, direction of groundwater move-
ment or surface-water flow, and/or topography). It will be unwise to pursue any serious model without
acquiring some knowledge of these processes or spatial features.
It is worth noting that nothing about the specification in equation (11) requires that the fitted distance
eff ect be positive in all directions. If the housing price diff erentials captured by the distance eff ect are entirely
due to the localized environmental disamenity, one would expect that the distance eff ect should be strictly
positive in all directions. This hypothesis can be tested. If the distance eff ect is negative in some directions,
this will call into question the assumption that distance is exclusively a proxy for ambient environmental
quality. Negative distance eff ects in some directions may be a consequence of other omitted spatial variables.
Of course, even when the estimated distance eff ect is positive in all directions, omitted spatial variables can
7 Richard Ready contributed this suggestion.
26
still cause mischief, but the presence of some negative distance eff ects should certainly raise suspicion about
the completeness of the specification.
If the researcher is willing to resort to non-linear-in parameters specifications, the sign of the distance
eff ect could be constrained to be positive in all directions by estimating the logarithm of the distance eff ect,
rather than its level. This corresponds to using the specification:
Y i = α +[exp(β + γ 1 cosθ i + γ 2 sinθ i )] f (d i )+ε i (34)
Here, the logarithm of the distance eff ect varies systematically with direction.
If the environmental externality in question is not a localized externality, so that the hedonic price
schedule is shifted for the entire market, then the first-stage hedonic equation is insuffi cient as a measure of
the loss in social welfare due to pollution. Richer welfare models will be necessary and the simple analysis
in Section 6 pertaining to optimal levels of emissions for diff erent firms will be inappropriate. For localized
externalities, however, the method proposed here may be entirely suitable.
The data requirements for assessing the optimal levels of emissions for diff erent point sources of localized
pollution will be substantial, since it is necessary to approximate the locations of all aff ected properties, as
well as to estimate the eff ect of the localized externality on each property in a sample of housing transactions
prices. It will also be necessary to convert transactions prices into equivalent annualized rental rates, and to
quantify the annual emissions from each plant and the marginal abatement costs for each plant. However,
any formal benefit-cost analysis will typically require a lot of data.
All of the models developed in this paper pertain to just the systematic portion of a first-stage hedonic
property value model, rather than the error terms. All of these generalizations will be amenable to being
combined with state-of-the-art spatial error models, such as correlogram- or semivariogram-based models or
lattice models (see the summary of methods provided by Pace et al. (1998)).
27
8Conclusions
The significant contribution in this paper is the development of a new empirical specification for spatial data
concerning variables that may be aff ected by proximity to environmental hazards. We introduce direction
as a potentially important determinant of distance (proximity) eff ects. We include a special case of this
empirical model that can accommodate additional data about the direction of prevailing winds or other
systematic physical features or processes. This special case involves a restriction on the parameter estimates
that can still be accommodated within a conventional linear-in-parameters estimation framework. If there
are seasonal patterns in prevailing winds or other processes that may lead to seasonal diff erences in the spatial
pattern of the level of the disamenity from the environmental hazard, these data can also be exploited.
We have developed these new specifications as generally as possible. In the context of hedonic property
value models, the dependent variable will typically be data on individual house selling prices. If environmental
justice is the issue in question, however, the dependent variable could be the proportion of the population in
diff erent sociodemographic groups at diff erent distances (and directions) from the localized environmental
externality or externalities.
If direction matters but is ignored, the best-case result is that the researcher is limited to measuring the
average distance eff ect, around the compass, with lesser precision than would be possible in a directionally
heterogeneous model. A worst-case result is that the geographic distribution of observations is systematically
correlated with direction (an omitted variable) so that the estimated distance eff ects are both biased and
less precise. The implication and insight of this analysis is that prior research that has ignored substantial
directional eff ects may have failed to identify statistically significant distance eff ects or may have presented
biased results.
Finally, we have explored the possibility of interpreting the housing units in a hedonic property value
study as individual ambient receptor sites. This allows hedonic property value models to be integrated with
notions of socially optimal pollution abatement among multiple sources of localized pollution externalities.
28
The insight drawn from this model is that optimal pollution levels for locally polluting firms emitting non-
uniformly mixing pollutants can depend on more than just the marginal abatement costs of each firm. The
distribution of housing units relative to each pollution source will also figure prominently in the calcula-
tions. If independently calibrated transport functions are not available for calculations of optimal abatement
patterns, there is scope for using hedonic property value models to fill this gap.
29
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Figure 1: Bi-directional distance eff ects and imprecision in estimates
32
Figure 2: Four-directional distance eff ects and one possible source of bias
Figure 3: Four-directional distance eff ects and a second possible source of bias
33