NCEE Working Paper
Hot Spots, Cold Feet, and Warm
Glow: Identifying Spatial
Heterogeneity in
Willingness to Pay
Dennis Guignet, Christopher Moore, and
Haoluan Wang
Working Paper 20-01
March, 2020
U.S. Environmental Protection Agency fgf
National Center for Environmental Economics fw
https://www.epa.aov/environmental-economics environmental\conomics
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Hot Spots, Cold Feet, and Warm Glow:
Identifying Spatial Heterogeneity in
Willingness to Pay
Dennis Guignet*1, Christopher Moore2, and Haoluan Wang3
Last Revised: March 10, 2020
Corresponding Author
Department of Economics
Appalachian State University
416 Howard Street
ASU Box 32051
Boone, NC 28608
Phone: +1 828-262-2117
gui gnetdb @app state. edu
1 Department of Economics, Appalachian State University
2National Center for Environmental Economics, U.S. EPA
3 Department of Agricultural and Resource Economics, University of Maryland
The views expressed in this paper are those of the authors and do not necessarily reflect the views
or policies of the U.S. EPA. Although the research described in this paper may have been funded
entirely or in part by the U.S. EPA, it has not been subjected to the Agency's required peer and
policy review. No official Agency endorsement should be inferred. We thank participants at the
Northeastern Agricultural and Resource Economics Association's 2018 meeting, the Agricultural
and Applied Economics Association's 2019 meeting, the 1st D.C. Area Student/Professor
Environmental and Energy Economics Workshop, and faculty of Appalachian State University
and Salisbury University for helpful comments. We especially thank Robert Johnston, Adan
Martinez-Cruz, and Anna Alberini for providing valuable suggestions on earlier versions of this
paper.
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Hot Spots, Cold Feet, and Warm Glow: Identifying Spatial Heterogeneity in
Willingness to Pay
Abstract:
We propose a novel extension of existing semi-parametric approaches to examine spatial patterns
of willingness to pay (WTP) and status quo effects, including tests for global spatial
autocorrelation, spatial interpolation techniques, and local hotspot analysis. We are the first to
formally account for the fact that observed WTP values are estimates, and to incorporate the
statistical precision of those estimates into our spatial analyses. We demonstrate our two-step
methodology using data from a stated preference survey that elicited values for improvements in
water quality in the Chesapeake Bay and lakes in the surrounding watershed. Our methodology
offers a flexible way to identify potential spatial patterns of welfare impacts, with the ultimate goal
of facilitating more accurate benefit-cost and distributional analyses, both in terms of defining the
appropriate extent of the market and in interpolating values within that market.
JEL Classification: Cll (Bayesian Analysis); C14 (Semiparametric and Nonparametric
Methods); Q51 (Valuation of Environmental Effects); Q53 (Air Pollution; Water Pollution; Noise;
Hazardous Waste; Solid Waste; Recycling)
Keywords: Bayesian; hotspot analysis; semi-parametric; spatial heterogeneity; stated preference;
water quality
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1. INTRODUCTION
A better understanding of the spatial distribution of welfare impacts is necessary for conducting
accurate benefit-cost and distributional analyses, both in terms of defining the appropriate extent
of the market and in interpolating values within that market. There is a rich body of literature on
the spatial dimensions of stated preference (SP) studies, focusing on various analytical and
statistical methods (see Glenk et al., 2020 for a review). These approaches include the
incorporation of space within the survey design (Wang and Swallow, 2016; Badura et al., 2020),
the combining of spatial variables with traditional econometric methods (J0rgensen et al., 2013;
Schaafsma et al., 2013; Olsen et al., 2020), and the application of spatial econometrics and geo-
statistics (Czajkowski et al., 2017; Budzinski et al., 2017).
To examine spatial heterogeneity in willingness to pay (WTP), most SP studies have applied the
distance decay paradigm, where WTP is hypothesized to diminish with distance from the resource
(Bateman et al., 2006). Generally, estimating the effect of distance on WTP depends on the nature
of the distance measure (e.g., travel, Euclidean, or geodesic distance) and the econometric model
specifications once a particular distance measure is chosen (e.g., linear or non-linear distance
decay). While there is a growing number of SP studies using traditional econometric methods to
control for "distance decay" or other forms of spatial heterogeneity (e.g., Hanley et al., 2003; Rolfe
and Windle, 2012; Olsen et al., 2020), these parametric approaches can sometimes fail to identify
existing spatial patterns.
Disciplines in the natural sciences employ more spatially-oriented analytical tools to examine
spatial patterns. These tools include tests for global spatial autocorrelation (Getis, 2007), spatial
interpolation techniques (Anselin and Gallo, 2006), and local cluster or hotspot analyses (Wang
and Qiu, 2017). These tools have been increasingly applied in economics, and in particular, in the
nonmarket valuation literature. For example, studies have tested for and generally found positive
global spatial autocorrelation of individual-specific WTP values (Campbell et al., 2009;
Meyerhoff, 2013; Johnston and Ramachandran, 2014; Johnston et al., 2015). SP studies have also
employed local indicators of spatial association (LISAs) or hotspot analysis to identify local
clusters of systematically higher or lower WTP values (i.e., hot and cold spots, respectively)
(Meyerhoff, 2013; Johnston and Ramachandran, 2014; Johnston et al., 2015). In general, these
studies found non-continuous, local spatial patterns of WTP.
In contrast to applications of these spatial tools in the natural sciences, the measures under study
by economists are often estimates (e.g., WTP), and not observed values. Although previous studies
have qualitatively recognized this fact, and its potential importance, no study to date formally
accounts for the statistical precision of those estimates when conducting spatial analyses. We are
the first to do so by incorporating techniques borrowed from meta-analytic methods into our spatial
autocorrelation, interpolation, and hotspot analyses.
We set out to accomplish three main research objectives. First, we develop a two-stage spatial
econometric approach to account for the fact that economic analyses typically observe estimated
values of WTP and other measures of interest. We use Bayesian modeling techniques to estimate
not only household-specific WTP, but also household-level measures of the variances of those
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estimates. Doing so can be important because some households may be, for example, less
knowledgeable of the environmental commodity, have less defined preferences, or even be less
engaged when taking the SP survey. If such households are systematically distributed over space,
then not accounting for the statistical precision of the household-specific WTP estimates can
confound subsequent spatial analyses. With the household-specific empirical WTP distributions
in-hand, we are able to treat the variables of interest not as given data, but as statistically derived
estimates. The household-specific variances of the WTP estimates are directly incorporated into
the spatial weights used in the second-stage spatial analyses.
Our second objective is to demonstrate our proposed two-stage methodology using data from a SP
survey that elicited values for improvements in water quality in the Chesapeake Bay and lakes in
the surrounding watershed. Using our proposed variance-adjusted tests for global spatial
autocorrelation, spatial interpolation techniques, and hotspot analyses, we examine the spatial
distribution and clustering of marginal WTP (MWTP) for improvements in several environmental
attributes. We also examine the spatial distribution of status quo (SQ) effects, which are intended
to capture potential biases for (e.g., "warm glow") or against (e.g., "cold feet") a policy option that
are not explained by the choice attributes defining that policy option. To our knowledge, this is the
first study to examine the spatial distribution of respondents exhibiting potential biases associated
with SP methods. Such an examination provides insights to improve SP methods, welfare analysis,
and future survey designs.
The third objective is to illustrate the potential policy implications of our proposed variance-
adjusted spatial analyses. We use our two-stage methodology to estimate total WTP for projected
improvements resulting from the Chesapeake Bay Total Maximum Daily Loads (TMDLs). The
total benefit estimates are compared to spatial interpolation approaches that do not account for the
statistical precision of the WTP estimates, similar to those used in earlier studies (e.g., Campbell
et al., 2009; Johnston and Ramachandran, 2014; Johnston et al., 2015), and to conventional models
that assume homogeneity of WTP across the population or control for heterogeneity parametrically
based on observed household characteristics (Moore et al., 2018).
Our semi-parametric results of the spatial interpolation suggest distinct local patterns in MWTP
estimates for all attributes, and evident spatial heterogeneity across the study area. The hotspot
analysis confirms statistically significant spatial clusters of high and low MWTP values.
Comparison of the conventional spatial analyses to our variance-adjusted results reveals some
differences. In general, accounting for the variances of the MWTP estimates diminishes spatial
variation, suggesting that not accounting for the statistical precision of the first-stage MWTP
estimates could lead analysts to falsely identify patterns of global and local spatial heterogeneity
in the second stage. Our analysis of spatial variation of individual-specific SQ effects reveals
substantial differences when accounting for the statistical precision of the estimates. In particular,
our proposed variance-adjustment leads to an increased identification of clusters of individuals
exhibiting "warm glow" or other biases for a policy option. Lastly, although differences in local
patterns are revealed, our policy simulations suggest that accounting for local spatial heterogeneity
(with or without our variance-adjusted extension) may not yield substantial differences in terms of
broader welfare implications, at least not in this particular application of water quality and
ecosystem improvements in an iconic and well-known estuary.
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The remainder of the paper is organized as follows. In the next section we present our two-step
empirical methodology. We then discuss the data for the specific application demonstrating that
methodology. The empirical results and implications are then presented, followed by some
concluding remarks.
2. METHODS
2.1. Random Utility Models (RUM)
Stated choice models are often estimated in a random utility framework, where vtj denotes the
deterministic component of utility that respondent i receives from alternative j in choice occasion
t. Each respondent is given three choice questions in the application presented, but the choice
occasion subscript t is omitted for notational ease. The random component of utility is denoted as
Eij. Assuming is independently and identically distributed following a type I extreme value
distribution allows the model to be estimated as a conditional or mixed logit (Maddala, 1983;
Greene, 2003; Train, 2003).
The deterministic component of indirect utility is a function of the vector of environmental
improvements Xj, the cost of living increase incurred by the household costj, and a binary
indicator denoting the status quo option SQj. To better capture preference heterogeneity, we
interact Xj with a vector of dummy variables denoting whether individual i is a user of the
corresponding resource userWe adopt a linear model and log-transform the environmental
attributes to capture diminishing marginal utility, while also preserving more degrees of freedom
than a model with higher order effects. Marginal utility of income is assumed constant across users
and nonusers. Using this specification of v(-), the conditional probability that household i would
choose alternative j is:
„ /¦. , % exp\ln(xt)p+(ln(xi)xuserj)8+Ycostt+(pSQi} , _
Pj(] Xa,COSta,SQa,USerj) = f / . , / . r —: (1)
L\J * H 4^-4 lJ Y>q exp{ln(Xq)p+(ln{Xq)xuseri)8+YCOStq+(pSQq}
where q indexes all available alternatives in a given choice occasion. The coefficients to be
estimated are /?, S, y, and (p. The first three coefficients can then be used to derive the MWTPt
vector for respondent i:
MWTP = HHuserOS
-yx
Given the natural log specification in the model, MWTP is nonlinear and varies at different levels
of the environmental attribute. In equation (2), x denotes the environmental attribute reference
levels from which MWTPt is calculated. In the empirical analysis, we set x equal to the baseline
values shown in the survey.
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2.2. Household-level MWTP distributions
To derive the household-level MWTP distributions that inform our spatial analyses, we adopt a
random parameters framework to estimate the logit model characterized by equation (1). To
simplify notation, we stack the variables ln(x), cost, SQ, and the user interaction terms into a single
M-element vector z and label the corresponding vector of coefficients "k. The typical exposition of
mixed logit models estimates the distribution of the utility parameters over the population gfXJu),
where d are the parameters of the distribution such as mean and variance. When estimating
household-level parameters, the central concept is the distinction between two distributions: the
distribution of preferences in the population and the distribution of preferences in the sub-
population who make particular choices (Train, 2003, p. 263). To that end, let rj(k\y,z,v) be the
distribution of X,in the sub-population that chose a particular set of responses to the repeated choice
experiments contained in the ^-element vector y.
The conditional distribution rj{) can be estimated in a classical framework via maximum
likelihood, and household-level parameters for that distribution can then be found by substituting
in values ofy and z(Revelt and Train, 2000). Generally, those expressions will be integrals without
closed forms and require simulation. We adopt a Bayesian approach to estimation because the
household-level distributions are more easily simulated as part of the Metropolis-Hastings (MH)
algorithm, from which any moments of those distributions can be found. We provide a derivation
of rj{) and a detailed description of the estimation algorithm in Appendix A. There is nothing
fundamentally different about our approach from the hierarchical Bayes' iterative estimation that
has become standard in the literature (Train, 2003). What allows us to characterize household-
level parameter distributions is the omission of a step in the standard algorithm that is usually
included for computational efficiency.
Typically, when hierarchical Bayes is executed, the draws of the household-level parameters are
only used to condition draws of the population-level parameters and then discarded to preserve
computational memory and processing speed. In this application, however, we store the household-
level parameter estimates and treat them as draws from the empirical distributions of each
respondent, rj(X\y,z,v). When the simulation is complete, we have a multivariate distribution of h
for each respondent i from which we can calculate draws from the distribution of MWTP via
equation (2). Those vectors of MWTP values characterize the distribution for each individual
respondent and can be used to calculate the mean and variance of MWTP at the household level.
The latter gives us a measure of statistical precision for the MWTP estimates for each respondent.
Except for the coefficient on cost, all parameters are drawn from a multivariate normal distribution.
We follow the common practice of treating the cost coefficient as fixed rather than random to
ensure MWTP has defined moments (e.g., Revelt and Train, 1998; Layton and Brown, 2000), and
we recognize this imposes a fixed marginal utility of income over the population. There are at least
two alternatives to modelling the cost parameter that avoid this assumption. One is to choose a
distribution with a strictly positive domain for the cost coefficient, such as log-normal. This is
problematic in our application because our algorithm requires storing all draws of every
coefficient, rather than just the mean value for each iteration. As a result, there can be draws of the
cost coefficient small enough to result in very large MWTP values which create computational
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difficulties, requiring ad hoc solutions. The second alternative is to estimate the model in WTP
space (Scarpa et al., 2008). This method brings its own computational issues and can result in less
precise WTP estimates (Hole and Kolstad, 2012), so we opt for the simplest approach of fixing the
cost parameter and acknowledge the implication on our results.
2.3. Household-level status quo effect distributions
In addition to household-level MWTP, we closely examine the SQ effects at the household level.
Inclusion of the status quo indicator SQj allows us to capture a respondent's tendency to vote for
or against the SQ option, irrespective of the cost and attribute levels defining the two alternative
policy options. Such tendencies estimated by (p may capture respondents' consideration of omitted
variables. It could also reflect "warm glow" if negative or "cold feet" if positive. For these reasons
we omit (p from the MWTP calculations (see equation (2)). Nonetheless, the spatial distribution of
(p could reveal important information about the validity and reliability of SP responses.
To provide an intuitive interpretation of the magnitude of the SQ effects, we express the effects in
terms of probability differences. We compare the probability of choosing the SQ option to a
constructed alternative that is identical with respect to the attribute levels but omits (p. We start
with equation (1), which expresses the multinomial logit probabilities as a function of
environmental attributes, the SQ effect, and the cost. We find the probability that each respondent
would choose the SQ option (j=SQ) in a given choice occasion, P^SQ \ xq, costq, SQq). We then
specify a second probability function for a constructed SQ alternative that omits the SQ constant,
Pi(SQ\xq, costq} = -—exP{^(^Q)P+(in(x^)XM^r')g}— an(j estimate the SQ effect as the
iV -ci q> qj Y,qexp{ln(xq)P+(ln(xq)xuseri)S+Ycostq} ^
difference between the two probabilities:
SQ effect = P^SQ \ xq,costq,SQq) - Pi(SQ\xq, costq). (3)
We perform spatial analyses of the SQ effects and account for household-level variances in the
same manner as the MWTP calculations, generating household-level distributions for the
probability differences.
2.4. Spatial analyses
The main contribution of this study is the formal incorporation of the underlying statistical
precision around each individual household's MWTP estimates into our spatial analyses. To our
knowledge spatial clustering and interpolation studies examining spatial variation to date, have
treated the variables of interest as observed values, and not as statistically derived estimates
(Campbell etal., 2009; Meyerhoff, 2013; Johnston andRamachandran, 2014; Johnston etal., 2015;
Czajkowski et al., 2017). Although these studies appropriately caveat their findings, none have
formally accounted for the underlying statistical precision of the WTP estimates. We do so by
borrowing techniques from meta-analytic methods. Conventional meta-analyses synthesize
estimates from different primary studies, and in doing so often weight the primary study estimates
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according to their inverse variance (Borenstein et al., 2010). We incorporate this same idea into
our three sets of spatial analyses - spatial interpolation, global autocorrelation, and hotspot
analyses. The following discussion focuses on the MWTP estimates, but the same procedures are
applied to the estimated SQ effects.
2.4.1. Variance-adjusted spatial interpolation
Spatial interpolation entails the creation of a raster (or grid) surface that visually depicts the
distribution of household MWTP over space. The weighted household-specific MWTP values are
used to predict MWTP for all locations in the study area, which in practice are identified as the
centroid of each grid cell. The following equation is used to interpolate the MWTP value assigned
to each cell /:
MWTPi is the predicted MWTP estimate at an unsampled location I. MWTPh is household /i's
estimated MWTP value, and o)lh is the element from the spatial weights matrix that links locations
I and h. We adopt the following functional form for our weighting equation to account for both
the spatial relationships and statistical precision of the primary estimates:
where is the distance from the location of the centroid of cell / to household h, and vh is the
variance of the MWTP estimate derived for household /?, which comes from the empirical
distribution generated through the 10,000 iterations of our Bayesian modelling approach (after
burn-in). The summation in the denominator is over the "K-nearest neighbors" to location I
(denoted by the set H{). Households at greater distances than the K-nearest neighbor are given a
weight of zero.
Nelson and Boots (2008) have discussed several ways to define spatial weights matrices that
include fixed distance, K-nearest neighbor, and shared boundaries. Following Johnston and
Ramachandran (2014) and Johnston et al. (2015), we adopt the K-nearest neighbor method (K=8).1
This spatial weighting scheme is appropriate for several reasons. First, K=8 is the number at which
the permutation distribution of the test statistic used in the later hotspot analysis approaches
normality (Ord and Getis, 1995). Second, this method ensures that very far households across the
large study area do not influence the MWTP estimates, so our results are not overly influenced by
1 Johnston and Ramachandran (2014) conduct sensitivity analyses based on the assumed spatial relationships and
found similar results across alternative assumptions.
MWTP, = Zh=i(6>ih X MWTPh).
(4)
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outliers. Third, the K-nearest neighbor method naturally adapts the neighborhood definition to
account for different population densities in urban, suburban, and rural areas across our study area.
Of particular interest in equation (5) is the assumed value for the parameter p, which must satisfy
0 < p < 1. The p parameter determines how much influence spatial proximity versus statistical
precision of an estimate has on the spatially interpolated MWTP value for cell /. If p = 1, then
equation (5) simplifies to the inverse distance weighting scheme commonly used in past spatial
analyses. If p = 0, then for the K-nearest neighbors, equation (5) is analogous to the common
fixed effect size (FES) weighting scheme often utilized in meta-analyses (Borenstein et al., 2010).
The choice of p is admittedly arbitrary but given our interests in accounting for both statistical
precision and spatial patterns, we assume an equal influence of both factors and set p = 0.5. A
sensitivity analysis is then conducted for alternative values of p, and most notably for the case
where p = 1, which allows for comparison of our variance-adjusted weights to the conventional
spatial weights used in previous studies.
2.4.2. Variance-adjusted global spatial autocorrelation and hotspot analyses
To test for global and local spatial autocorrelation, it is again necessary to define the neighborhood
in which relationships across space are evaluated. In contrast to the interpolation exercise above,
where the weights matrix denotes the spatial relationships between each interpolated cell centroid
and the households in our sample, in the next set of analyses the spatial relationship defined is
between each household and the K-nearest neighboring households, including the household itself
(i.e., where dih = du = 0). This mathematically prevents us from assuming an inverse distance
relationship, as done in equation (5). Given our interests in identifying statistically significant high
or low clusters of MWTP, a simple uniform weight of 1/K among the K-nearest neighbors (and
zero otherwise) is assumed here.
Again, our novel contribution is to account for the statistical precision of the individual household
MWTP estimates. More specifically, the weight given to each neighbor is basically re-distributed
among the K neighbors, giving more weight to households where the observed MWTP value was
estimated with greater statistical precision (i.e., smaller variance). The weight used for household
h in explaining the spatial relationship with household i is:
The summation in the denominator is over the K-nearest neighbors to household i (denoted by the
set Hi). The assumed value for the parameter a must satisfy 0 < a < 1. Notice that equation (6)
is a more general form of the usual K-nearest neighbor weighting scheme. When a = 1, a>ih
simplifies to 1/K for those K-nearest neighbors. If a = 0, then similar to before, for the K-nearest
neighbors equation (6) simplifies to the common FES weighting scheme used in meta-analyses
(Borenstein et al., 2010).
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To test whether the household-specific MWTP values are a result of random spatial process, we
apply Moran's / statistic to test for global spatial autocorrelation (Getis, 2010). The underlying
expectation of Moran's / test is that the spatial process promoting the observed pattern of the
attribute being analyzed is random. A rejection of the null hypothesis suggests that spatial
autocorrelation, either spatial clustering or dispersion, exists. Moran's / statistic ranges from -1 to
+1, with scores near +1 indicating spatial clustering and scores near -1 indicating spatial
dispersion. Moran's / statistic is defined as:
_ n g=1 Zh=i C0ih(.MWTPi-MWTPXMWTPh-MWTP)
Y,f=1(MWTPi-MWTP)2 ^ '
where n is the number of households in the data sample, MWTPt and MWTPh are the household
Y™ a MWTP;
MWTP values, MWTP = is the mean of all households' MWTP values in the sample,
and (joih is the variance-adjusted spatial weight that links households i and h, as defined in equation
(6). Notice that equation (7) is essentially the standard Moran's / statistic formula (Getis, 2010),
with our adjusted spatial weights illustrated in equation (6).
While Moran's / test for global spatial autocorrelation provides a means to test for spatial patterns
across the broader study area, little is revealed about local spatial heterogeneity among households.
Local indicators of spatial association (LISAs) have been developed to measure local spatial
autocorrelation. Commonly referred to as hotspot analysis, the approach provides a statistical test
for identifying spatial clusters of high values (hot spots) or low values (cold spots) of a variable of
interest beyond what can be explained by random coincidence (Anselin, 1995). Among the most
common LISAs is the Getis-Ord G* statistic proposed by Getis and Ord (1992). Our proposed
variance-adjusted Getis-Ord G* statistic is calculated as follows:
g; =
ZLi [coihMWTPh}-MWTP
^2l_(MtyrP)2
M n-1
(8)
Equation (8) is simply the standard Getis-Ord G* statistic (Getis and Ord, 1992), but with our
variance-adjusted spatial weights illustrated in equation (6).
3. DATA
We demonstrate our two-step methodology by examining the spatial distribution of household
MWTP for improvements in the Chesapeake Bay and freshwater lakes in the broader Chesapeake
Bay Watershed. These MWTP values are estimated from data obtained in a stated choice study by
Moore et al. (2018), which focused on reductions in nutrient and sediment pollution, and the
resulting improvements in conditions for recreation and aquatic wildlife. Three choice questions
were included in each survey. Each choice question presented respondents with a SQ alternative,
showing current conditions and zero cost, and two policy alternatives with improvements in some
or all of the attributes and a positive cost (see Table 1). The cost attribute was expressed as a
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permanent cost of living increase shown in annual terms for each household. The attributes
defining each policy alternative were improvements in water clarity, striped bass population, blue
crab population, and oyster abundance in the Bay. In addition to the Bay itself, freshwater lakes in
the watershed benefit from the management practices targeting the Bay. To capture these benefits,
an additional attribute reflecting the number of lakes in the watershed with low algae growth was
included in the alternatives presented in the survey. Through a series of ten focus groups, 72 one-
on-one cognitive interviews, and an extensive pre-test, these attributes were found to be the most
salient and important to the population of interest.
The survey was administered via mail to a geographically stratified random sample of households
who reside in the District of Columbia or one of the 17 U.S. states that contain at least part of the
Chesapeake Bay Watershed or lie within 100 miles of the Eastern coast of the U.S. The survey
was sent to 2,829 households, and after adjusting for undeliverable addresses achieved a response
rate of 31%. The resulting 671 surveys were screened for protest responses and hypothetical bias,
leaving 559 completed surveys with which to estimate our models. Moore et al. (2018) provide
more details on the study design and sample characteristics.
4. RESULTS
4.1. Bayesian model results
There are three sets of results from the Bayesian mixed logit that are relevant to our objectives.
The first are the summary statistics of the posterior distributions of our estimated model
parameters, g(A,|i)). From a classical perspective these statistics can be interpreted as the coefficient
estimates and standard errors, presented in the first two columns of results in Table 2. Given the
inclusion of user-interaction terms in our model, the coefficients on the logged attribute levels are
the marginal utilities for nonusers and the coefficients on the interaction terms are the differentials
in marginal utilities for users of the corresponding resource. On average, water clarity in the Bay
and striped bass population are not significant determinants of choice for non-users but they are
for people that recreate in the Chesapeake Bay. Crab populations are significant for non-users, on
average, while their contribution to marginal utility is less for users but not significantly so. Water
quality in watershed lakes is a significant attribute for the average nonuser and generates a
significantly greater marginal utility for lake-users. The MWTP estimates are reported in the last
column of Table 2. The MWTP values refer to a one-unit increase relative to the SQ quantities
shown in Table 1; for example, a one-inch increase in clarity, a one-million increase in striped bass
and crab populations, etc.
The second set of results describe how the household-level parameters are distributed in the
population, shown in Table 3. In this case the mean values refer to the average household value
and the standard deviation is an indication of how disperse the values in the population are, and
not an indication of estimation precision. The corresponding MWTP values and the inner 90th
percentile for each are shown in the right-hand side of Table 3. The subsequent spatial analyses
examine how households' values in these distributions vary over space, after adjusting for
statistical precision using the final set of Bayesian results.
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The third and final set of Bayesian results relevant to our objectives are the household-level
empirical distributions of MWTP that contain information on the central tendency and precision
of our estimates at the household level, rj(k\\yi,Zi,x>). Given the number of respondents in the data,
it is not practical to report means and standard deviations for each of them here, but those results
are available upon request to the authors.
4.2. Spatial interpolation analysis
Figures 1-6 present the MWTP results for Bass, Clarity, Crab, Lake, Oyster, and the SO effects.
In each figure we include two panels covering the results of the spatial interpolation (left) and
hotspot analysis (right). Figure 7 shows the distribution of Getis-Ord G? statistics across
households in the hotspot analysis. For each set of maps, we present the results from the
conventional spatial weights matrix and those from the variance-adjusted spatial weights matrix
introduced in this study.
We first demonstrate the spatial interpolation results on "heat" maps, where darker shades identify
higher estimates and lighter shades identify lower estimates. In general, visual inspection of the
interpolated MWTP surfaces suggest distinct spatial patterns in MWTP estimates for all attributes
and spatial heterogeneity across the study area. For instance, we see some of the highest values for
striped bass and water clarity among households living nearest to the Chesapeake Bay, or along
the Atlantic Coast and just south of the Bay (left panels of Figures 1 and 2, respectively). We also
observe relatively higher MWTP values for improvements in freshwater lakes in the Watershed
(Lake) among households who live within the watershed (left panel, Figure 5). One unexpected
spatial pattern, suggested by the left panel in Figure 3, is that households nearest the Chesapeake
Bay hold lower values for improvements in the population of blue crabs (Crab), an iconic shellfish
species in the Bay. Whereas households outside the watershed seem to hold higher values. As
suggested by the Bayesian model results in Table 2, this could reflect the relatively large values
among nonusers. Visually there is no clear global pattern in values for improved oyster abundance,
as seen in the left panel of Figure 4. There are some areas near and just south of the Bay where
residents hold relatively high values for increases in oyster populations, but there are other
scattered clusters of high MWTP values (e.g., in New England, the gulf-side of Florida, and
northwest Pennsylvania).
The interpolated surface of the SQ effects suggests an interesting spatial pattern. In West Virginia
and in Northern New York near the Great Lakes and Finger Lakes, we see large positive values
for the SQ option, suggesting that households in these areas generally hold a preference against
any policy options leading to improvements in the Chesapeake Bay. One theory driving this result
could be that households in these areas have their own substitute, and possibly equally as iconic,
environmental amenities they care about, and thus have a bias against policy options leading to
improvements in the Bay. Another possibility is that there is a protest or strategic response against
increased regulations that was not completely eliminated using our earlier screening criteria based
on responses to debriefing questions. In any case, the potential biases captured by the SQ effects
are controlled for and are excluded from the MWTP and welfare calculations.
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Comparison of the conventional inverse distance interpolated MWTP maps (p = 1), to our
variance-adjusted interpolations (p = 0.5) in the left panels of Figures 1-5 reveals some
differences. In general, we find that the range of the interpolated MWTP estimates across the study
area becomes smaller when we account for the variance of individual respondents' estimates. This
suggests that accounting for the statistical precision of the first-stage estimates reduces the
influence of outlying, often less precisely estimated, values. Although accounting for the variance
of the first-stage estimates seems to diminish spatial variation, and reveals some local differences,
the general spatial patterns appear similar.
One exception pertains to the interpolated SQ effect estimates. As shown in Figure 6, when
accounting for the individual variances (p = 0.5), many of the areas exhibiting biases against a
policy option remain, but there is now noticeably more evidence of respondents exhibiting a
relatively high preference for a policy option (i.e., SQ effect < 0), irrespective of the
improvements and costs defining the policy options. Such cold spots, for example, are now evident
in the area around New York City, as well as in southern Florida and western Maryland. In any
case, the interpolation exercise should be interpreted as suggestive at best. Although it visually
depicts relevant spatial patterns, whether such patterns are statistically significant remains an open
question. To answer that question, we turn to tests for global spatial autocorrelation and the hotspot
analysis in sections 4.3 and 4.4, respectively.
4.3. Tests for global spatial autocorrelation
As can be seen in Table 4, the Moran's / tests for global spatial autocorrelation reveal broader
spatial patterns for some of the MWTP estimates, but not all. MWTP for increases in striped bass
and blue crab populations are both spatially correlated over the broader study area. However, such
spatial trends are not generally revealed through parametric modelling of the distance gradients
(see Appendix B), thus highlighting the importance of considering potentially relevant spatial
patterns revealed by non-parametric methods beyond just distance decay. The Moran's / test
suggests that the SQ effects are also highly correlated over space.
These findings are robust as we vary the value of a. The strongest evidence of global spatial
autocorrelation occurs when using the conventional weights (a = 1) that do not take into account
the statistical precision of the first-stage estimates. These global patterns remain robust but become
less significant as we move towards the variance-adjusted weights (a = 0). The Moran's / tests
suggest no significant global spatial patterns in terms of MWTP for improvements in clarity, oyster
abundance, and freshwater lakes. We next examine the nature of these global patterns and whether
more local spatial patterns may exist that cannot be identified via the Moran's / statistic. In specific,
we conduct local spatial associations analyses to test for the presence of statistically significant
hot or cold spots using the Getis-Ord G? statistic (Getis and Ord, 1992) and our variance-adjusted
variant of the G¦ statistic. We perform this analysis separately for each attribute, where the sampled
households are the spatial units.
12
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4.4. Spatial clustering analysis
The hotspot analysis results for the MWTP of each attribute are illustrated by the set of maps in
the right panels of Figures 1-5. The maps show the status of each household (i.e., whether it is in
a hot spot, cold spot, or demonstrates no statistically significant higher or lower MWTP values
relative to other nearby households). These designations are based on the estimated G,* statistic for
each household. The corresponding distributions of the G ¦ estimates across households are shown
in Figure 7 for different values of a. This same information is shown in Table 5, which displays
the number of households identified as being in a hot or cold spot.
Gj* is assumed to be normally distributed under the null hypothesis (Getis and Ord, 1992), and so
we fail to reject the null hypothesis when —1.645 < G- < 1.645 (i.e., a statistically insignificant
result). Such instances on the maps would suggest that there is no clustering of high or low values
around the corresponding household. Hot spots (black points on the figures) represent clusters of
atypically high MWTP estimates, indicating a MWTP hot spot significant at the 90%, 95%, or
99% level depending on the size of the dot. Cold spots (white points) represent clusters of
atypically low MWTP estimates, those with parallel negative G ¦ statistics indicating a MWTP cold
spot at the same levels of significance, again varying by size of the white dot.
The hotspot analysis for Bass and Clarity reveals clusters of high MWTP values among some
households in close proximity to the Bay (Figures 1 and 2, respectively). And for water clarity, we
find clusters of systematically lower MWTP values among households near notable substitute
waterbodies, like the Great Lakes and Finger Lakes in New York. Comparison to the variance-
adjusted hotspot analyses when a = 0 reveals similar results, but accounting for statistical
precision in the underlying estimates reduces the number of households that belong to a
statistically significant local cluster, especially for identified hot spots, as shown in Table 5.
In Figure 3, we find scattered hot spots of MWTP for crabs, mainly outside the watershed. There
is also a concentration of cold spots within the watershed, mainly in central Virginia and Maryland.
The finding that households in closest proximity to the Bay have the lowest values for
improvements in crab populations, and those farthest have the highest values, is again surprising,
but is in line with the interpolation exercise. This unexpected spatial pattern could be driven, at
least partially, by relatively large nonuse values held by nonusers for this iconic resource.
Comparing the conventional hotspot analysis (a = 1) to our variance-adjusted hotspot analysis
(ia = 0) reveals little difference, but substantially reduces the number of cold spots.
Up until this point we have found little evidence of discernible spatial patterns in oyster
populations. The hotspot analysis in Figure 4, however, does suggest statistically significant
clusters of high MWTP values for increases in oyster abundance, namely among those living
closest to the Bay. There is a noticeable pattern of clustered low MWTP values, particularly around
New York City and going North along the Hudson River (near the east-most border of New York
state). Again, the variance-adjusted hotspot analysis (a = 0) seems to reduce the number of
hotspots, but we find slightly increased evidence of cold spots, as reported in Table 5.
The hotspot analyses for improvements in freshwater lakes in the broader Chesapeake Bay
Watershed are displayed in the right panel of Figure 5. As one might expect, the conventional
13
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hotspot analysis reveals evidence of a concentration of statistically higher MWTP estimates among
households living in the Watershed for improvements in freshwater lakes within the Watershed.
There are also a few scattered cold spots, and most notably a concentration of lower MWTP values
just outside the southwest corner of the Watershed; perhaps reflecting that there are several
substitute lakes in western Virginia that are outside of the watershed. As suggested by the previous
variance-adjusted hotspot analyses, we again see similar patterns in hot and cold spots, but the
number of statistically significant local clusters of high MWTP estimates are substantially reduced
when accounting for the statistical precision of the first-stage estimates (a = 0).
The broader finding that the number of identified clusters are reduced after accounting for the
statistical precision of the first-stage estimates is better demonstrated by the distributions of the
household-specific G,* statistics shown in Figure 7. In general, we see that accounting for the
statistical precision of the MWTP estimates makes one less likely to identify statistically
significant clusters (i.e., a larger portion of the distribution of G? is located towards zero). This
finding is consistent with the hotspot maps in Figures 1-6, and makes intuitive sense. Extreme
MWTP estimates are often less precise, and so when these estimates are appropriately discounted
due to this lack of precision one is less likely to falsely identify a statistically significant cluster of
high or low WTP values. When performing scoping exercises like this to try and identify spatial
patterns, this application demonstrates that it may be important to account for the fact that these
MWTP values are estimates, and not observed values. Not taking into account the precision of the
household MWTP estimates may lead researchers to falsely identify patterns of spatial
heterogeneity.
The bottom panel in Figure 7 reveals a finding that is unique to the estimated SQ effects.
Incorporating household-specific variances into the spatial weights shifts the mass of the G-
distributions for the MWTP estimates towards zero. However, for the G? statistics corresponding
to the SQ effects, we see the distribution shift more negative. In some cases, as we previously saw,
this reduces the number of identified hot spots. For example, the conventional hotspot results
(ia = 1) in Figure 7 suggest that respondents near the Finger Lakes, a notable substitute, are more
susceptible to exhibiting potentially biasing behaviors against a policy option that improves water
quality in the more distant Chesapeake Bay (e.g., "cold feet). But this identified cluster of
significantly high SQ effect estimates disappears once the statistical precision of those underlying
estimates is accounted for.
The more unique finding is that accounting for the statistical precision behind the estimated SQ
effects identifies more statistically significant cold spots. In other words, we identify more clusters
of households exhibiting "warm glow" or other potentially biasing behaviors in favor of a policy
option. For example, in Table 5 we see a 230% increase in the number of households that belong
to a SQ effect cold spot when going from a = 1 to a = 0. This is also evident in the maps in
Figure 6. The rightmost hotspot analysis in Figure 6 where a = 0 reveals noticeably more cold
spots, especially in western Maryland, and the area around New York City, the Long Island Sound,
and Narragansett Bay and Cape Cod. Perhaps respondents near these other iconic estuaries have
an implicit bias or strategic response that pushes them towards an option that leads to
improvements in the Chesapeake Bay. Alternatively, this may reflect preference heterogeneity in
14
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favor of estuary quality that is not otherwise captured by or correlated with other attributes in the
experimental design. In any case, these effects are excluded from the policy illustration we discuss
next, but the location and clustering of respondents exhibiting such behaviors are important to keep
in mind when designing future stated preference studies and could be important to identify prior
to specifying more parametric models to estimate welfare changes. For example, the results of our
spatial analyses allude to the potential importance of substitute waterbodies. Insights like this are
an example of what can be gained from semi-parametric scoping exercises to examine spatial
patterns.
4.5. Policy illustration: Chesapeake Bay Total Maximum Daily Loads
To examine the potential policy implications of accounting for spatial heterogeneity in WTP
estimates, and the underlying statistical precision of those estimates, we repeat the benefit
calculations carried out by Moore et al. (2018). As reported in Table 9 of Moore et al. (2018), the
projected improvements from the Chesapeake Bay TMDLs are an average increase of 4.33 inches
in Bay water clarity, 1.03 million striped bass, 41 million blue crab, 541 tons of oysters, and 455
freshwater lakes reaching "low algae" status. Following Holmes and Adamowicz (2003), the
annual WTP for each household i is calculated as the difference of the deterministic component of
the indirect utility, divided by the marginal utility of income:
_ PiInj.*1 )+8iln(x1)useri-[f}jln(x)+Siln(x)useri] ^
where x is a vector of the baseline attribute levels (Table 1), and x1 are the projected policy levels
(baseline levels plus the improvements). The household-specific WTP estimates are derived for
all 559 households in the sample. In order to extrapolate the WTP estimates to the population of
44,353,441 households in the study area, we then create interpolated WTP surfaces using the same
procedure described in section 2.4.1. We next take the average of the interpolated cells within each
census tract and multiply that by the number of households in that tract according to the 2010 U.S.
Census. These total WTP estimates for each tract are then summed over all census tracts in the
study area. The resulting total annual WTP estimates are displayed in Table 6. We emphasize that
our policy illustration is based solely on the spatial interpolations and does not rely on the analyses
of global spatial autocorrelation and local clusters.
The first two columns in Table 6 show the total benefit estimates taken from Moore et al. (2018).
Their model 1 estimates assume homogeneity across the entire study area by applying a single
average WTP estimate to all households in the population. Their model 2 estimate is based on a
similar procedure, and although it does not explicitly account for spatial heterogeneity, it does
account for heterogeneity regarding the use of the resource and extrapolates those values based on
estimates of the proportion of the population that are users versus nonusers. The next four columns
in Table 6 show the results of our spatially-explicit extrapolation exercise, and suggest total benefit
estimates for the entire study area ranging from $6.6 to $6.9 billion per year. These total benefit
estimates are largely in line with those from Moore et al. (2018). This suggests that accounting for
spatial heterogeneity may not yield substantial differences in terms of broader policy implications,
15
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at least not in this specific context and when considering the entire study area as a whole. Such
spatially explicit details may be important, however, for more local policies. For example, we do
find significant variation in household-level WTPs across the study area, ranging from an annual
household WTP of $23 to $312.
Comparison of the total WTP estimates from the conventional spatial interpolation exercise (p =
1) to our variance-adjusted spatial interpolations (p < 1) suggests that total WTP estimates
decrease as more weight is given to the statistical precision of the first-stage estimates. This is
consistent with our broader findings that accounting for statistical precision reduces the influence
of less precisely estimated outliers that could otherwise unduly influence empirical analyses. In
this particular context, however, the differences in total WTP inferred from the spatial
interpolations may not be economically significant. Surprisingly, relatively small differences are
also revealed when examining total WTP at more local levels, such as by state or county. In fact,
even at the individual tract-level, comparing our variance-adjustment estimates when p = 0.5 to
the benefits inferred from conventional interpolation techniques (p = 1), suggests that the latter
leads to only a 10% difference in total tract-level WTP for the majority (90%) of the 27,117 census
tracts in the study area. In short, although we find that accounting for spatial heterogeneity is
important, the proposed variance-adjustment may not make much of a practical difference in this
particular setting.
5. CONCLUSION
We propose a novel extension of existing semi-parametric techniques to analyze spatial patterns
when the variables of interest are estimates and not observed values, as is the case in many
applications to nonmarket valuation. When examining spatial welfare patterns, we account for the
fact that our first-stage model will estimate some households' values less precisely than others.
The methodology in this study estimates household-specific MWTP variances using Bayesian
estimation techniques and incorporates that information into the spatial weights matrix used in
tests for global spatial autocorrelation, spatial interpolation maps, and hotspot analyses. Similar
spatial analyses have been increasingly introduced in the nonmarket valuation literature (e.g.,
Campbell et al., 2009; Meyerhoff, 2013; Johnston and Ramachandran, 2014; Johnston et al., 2015;
Czajkowski et al., 2017), but our study is the first to formally incorporate the statistical precision
of the first-stage estimates into the second-stage spatial analyses.
We demonstrate our two-step methodology using a SP study of water quality improvements in the
Chesapeake Bay. Accounting for the statistical precision of the MWTP estimates generally seems
to result in less statistically significant evidence of spatial patterns, as reflected by tests for global
spatial autocorrelation and the hotspot analysis. This tendency increases as additional weight is
given to the statistical precision of the MWTP estimates. A similar finding is found with regards
to households exhibiting "cold feet", or a tendency to disproportionally favor the status quo.
Overall, the analysis suggests that accounting for the statistical precision of the estimated
economic phenomena being analyzed reduces the chances of falsely identifying statistically
significant spatial heterogeneity. In contrast, we also find that accounting for the statistical
16
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precision of the underlying estimates can lead to increased identification of areas where
respondents disproportionally favor a policy option for reasons not explained by the choice
attributes. Identifying locations where households exhibit such "warm glow" or other potentially
biasing behaviors can aid in future survey design and help inform econometric model
specifications to estimate welfare changes.
We estimate the total benefits projected to result from the Chesapeake Bay TMDLs to examine
the importance of our extension of traditional spatial analyses from a practical standpoint. We find
that in a broader regional setting, at least with our data, the difference between the benefits inferred
from traditional spatial interpolation techniques versus those that accommodate for statistical
precision are small. More applications of the methods discussed in this study to SP data valuing
other environmental amenities are needed to see whether accounting for the statistical precision of
the first-stage WTP estimates reveals similar findings, particularly in cases where the
environmental amenities of interest are more local in nature. One might not necessarily expect as
much spatial heterogeneity in preferences for a well-known iconic resource, like the Chesapeake
Bay. Examination of more localized amenities, perhaps where familiarity with the resource is more
varied, may yield different findings in how accounting for statistical precision impacts the
identification of spatial patterns.
Although our two-step methodology provides an intuitive path for accounting for the statistical
precision of the first-stage estimates when conducting spatial analyses, and presumably allows for
more accurate identification of spatial patterns, future simulation studies are needed to formally
examine the potential improvements in accuracy. Such studies might entail analysis of simulated
data where the researcher knows the true data generating process over space. Nonetheless, given
the emphasis of these spatial analytic techniques for purposes of data diagnostics and scoping
(Johnston and Ramachandran, 2014; Johnston et al., 2015), we encourage researchers to
implement our variance-adjustment methods when attempting to identify potential spatial patterns
that may not be immediately apparent through conventional parametric models.
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TABLES AND FIGURES
Table 1. Attribute Descriptions and Levels
Attribute Description Status Quo Level Post-Policy Levels
Bay Water Clarity
Number of feet below the water
surface you can see
3 feet
3; 3.5; 4.5
Striped Bass
Population
Number of adult striped bass in
the Chesapeake Bay (millions)
24 million
24; 30; 36
Blue Crab Population
Number of adult blue crab in
the Chesapeake Bay (millions)
250 million
250; 285; 328
Oyster Abundance
Tons of oysters living in the
Chesapeake Bay
3,300 tons
3,300; 5,500;
10,000
Low Algae Lakes
Out of 4,200 freshwater lakes in
the Chesapeake Bay Watershed,
number with low algae levels
2,900 lakes
2,900; 3,300; 3,850
Annual Cost to
Household
Permanent increase in the
annual cost of living starting
the following calendar year
$0 per year
$20; $40; $60;
$180;$250; $500
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Table 2. Posterior Distributions of Coefficient Estimates
Mean
SD
Non-User MWTP
ln(clarity)a
0.505
0.548
1.47
ln(bass)
0.874
0.584
3.83*
ln(crab)
2.070***
0.638
Q gy***
ln(oyster)
0.198
0.206
0.01
ln(lake)
3.769***
0.824
q 24***
User MWTP
user x ln(clarity)
1.101***
0.720
4 g9***
user x ln(bass)
0.615**
0.863
6.52***
user x ln(crab)
-0.366
0.933
0.72
user x ln(oyster)
0.355
0.320
0.02
user x ln(lake)
1.305*
1.069
0.18**
SQC
-1.938***
0.352
cost
-0.009***
0.001
*** p<0.01, ** p<0.05, * p<0.1. All coefficient estimates modelled as random, except the
coefficient on cost is treated as fixed to ensure MWTP distributions are finite. Marginal
willingness to pay (MWTP) estimates expressed in 2014$. (a) Note that clarity is expressed
as inches in the empirical models and subsequent MWTP estimates. All other
environmental attributes are expressed in the same units originally specified in the survey,
and as reported in Table 1 (i.e., millions of bass, millions of crabs, tons of oysters, and the
number of low algae lakes).
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Table 3. Distribution of Coefficients and MWTP in the Population
Mean
SD
Mean Non-User MWTP
Inner 90th Percentile MWTP
ln(clarity)
0.4327
3.0252
1.38
-5.69
9.52
ln(bass)
0.5519
2.2868
3.84
1.63
6.92
ln(crab)
2.058
0.6847
0.87
0.74
1.03
ln(oyster)
0.2532
1.3006
0.01
-0.01
0.03
ln(lake)
3.6197
1.3837
0.14
0.08
0.24
Mean User MWTP
user x ln(clarity)
1.5235
1.6738
4.91
-4.55
14.34
user x ln(bass)
1.0359
1.2955
6.55
0.37
13.35
user x ln(crab)
-0.9166
1.4795
0.72
0.44
0.99
user x ln(oyster)
0.0865
0.6691
0.02
-0.08
0.26
user x ln(lake)
1.4167
2.1951
0.17
0.07
0.27
Marginal willingness to pay (MWTP) estimates expressed in 2014$. MWTP for an increase in Bay water clarity is expressed in inches.
MWTP for all other attributes are expressed in the same units originally specified in the survey, and as reported in Table 1 (i.e., millions
of bass, millions of crabs, tons of oysters, and the number of low algae lakes).
Table 4. Moran's / Tests for Global Spatial Autocorrelation
a =
1.0
a =
0.5
a =
0.0
Moran's I
z-score
Moran's I
z-score
Moran's I
z-score
Clarity
-0.0047
-0.1408
-0.0056
-0.1824
-0.0058
-0.1947
Striped Bass
0.0459
2.3157**
0.0419
2.1029**
0.0355
1.7525*
Blue Crab
0.0577
2.8897***
0.0533
2.6599***
0.0476
2.3411**
Oysters
0.0305
1.5682
0.0300
1.5368
0.0294
1.4872
Lakes
0.0194
1.0276
0.0163
0.8737
0.0131
0.7116
SQ Effect
0.0726
4.0285***
0.0705
3.692***
0.0708
3.2323***
*** p<0.01, ** p<0.05, * p<0.1
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Table 5. Number of Households Identified as Being in a Hot or Cold Spot
Numbers of Hot Spots Numbers of Cold Spots
a = 1.0
a = 0.5
a = 0.0
a = 1.0
a = 0.5
a = 0.0
Clarity
24
21
16
20
21
20
Striped Bass
62
42
33
1
2
1
Blue Crab
34
41
46
46
35
20
Oysters
49
38
27
23
24
31
Lakes
47
35
16
21
19
23
SQ Effect
57
37
24
52
132
172
The displayed counts show the number of households (out of the sample of 559) that are identified as being part of a spatial cluster of
statistically higher (hot spot) or lower (cold spot) values. Hot spots are those with G* > 1.645 and cold spots are those with G* <
-1.645.
Table 6. Total Annual Willingness to Pay for Improvements under Total Maximum Daily Loads (2014$, billions)
Moore et al. (2018) Spatial Interpolation in this Study
Model 1 Model 2 p = 1.00 p = 0.75 p = 0.50 p = 0.25
$6,813 $6,488 $6,870 $6,790 $6,711 $6,635
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Interpolated MWTP
<2.00
2.00 - 4.00
H 4.01 - 6.00
H 6.01 - 8.00
H > 8.00
I I Watershed Boundary
rho = 1
150 300
600 Miles
rho = 0.5
500 M es
Figure 1. MWTP for Bass
Hotspot Analysis
o Cold Spot - 99% Confidence
» Co/tf Spot • 95% Confidence
Cold Spot • 90% Confidence
Not Significant
• Hot Spot - 90% Confidence
• Hot Spot - 95% Confidence
• Hot Spot - 99% Confidence
I I Watershed Boundary
25
-------
Interpolated MWTP
<0.00
0.00 - 2.00
M 2.01 - 4.00
¦i 4.01 - 6.00
H > 6.00
I I Watershed Boundary
rho = 1
rho = 0.5
Hotspot Analysis
O Cold Spot • 99% Confidence
o Cold Spot - 95% Confidence
Cold Spot • 90% Confidence
Not Significant
• Hot Spot - 90% Confidence
• Hot Spot - 95% Confidence
• Hot Spot - 99% Confidence
I I Watershed Boundary
600 Miles X
alpha = 7
—U
r\
SJ
alpha
Figure 2. MWTP for Clarity
26
-------
Interpolated MWTP
<0.60
0.60-0.70
M 0.71 - 0.80
H 0.80 - 0.90
H >0.90
I I Watershed Boundary
rho =1
600 Miles
rho =0.5
Hotspot Analysis
O Cold Spot - 99% Confidence
» Cold Spot - 95% Confidence
Cold Spot - 90% Confidence
Not Significant
• Hot Spot - 90% Confidence
• Hot Spot - 95% Confidence
• Wor Spor - 99% Confidence
I I Watershed Boundary
alpha = 7
600 Miles
Figure 3. MWTP for Crabs
27
-------
Interpolated MWTP
<0.000
0.000 - 0.005
Hi 0.006 - 0.010
¦I 0.011 -0.015
H > 0.015
I I Watershed Boundary
Figure 4. MWTP for Oysters
Hotspot Analysis
O Cold Spot - 99% Confidence
o cold Spot - 95% Confidence
Cold Spot - 90% Confidence
Not Significant
• Hot Spot - 90% Confidence
• Hot Spot - 95% Confidence
• Hot Spot - 99% Confidence
I I Watershed Boundary
28
-------
Interpolated MWTP
<0.12
0.12-0.14
¦I 0.15-0.16
Wk 0.17-0.18
¦M>0.18
I I Watershed Boundary
Figure 5. MWTP for Lakes
Hotspot Analysis
O Cold Spot - 99% Confidence
o cold Spot - 95% Confidence
Cold Spot - 90% Confidence
Not Significant
• Hot Spot - 90% Confidence
• Hot Spot - 95% Confidence
• Hot Spot - 99% Confidence
I I Watershed Boundary
alpha =
alpha = 0
29
-------
Interpolated Status Quo Effect
< -0.30
-0.20--0.15
H -0.1 A - 0.00
¦I 0.01-0.15
H > 0.15
I I Watershed Boundary
rho = 1
600 Miles
rho = 0.5
Hotspot Analysis
O Cold Spot - 99% Confidence
» Cold Spot - 95% Confidence
Cold Spot - 90% Confidence
Not Significant
• Wor Spot - 90% Confidence
• Hot Spot - 95% Confidence
• Wor Spor - 99% Confidence
I I Watershed Boundary
alpha = 7
600 Miles
alpha
Figure 6. Status Quo Effects
30
-------
G* Statistics for Bass
-5
0
G* St3t:st-cs tor Carity
€0
CM
Ss
G* Statistics far Crab
G* Statistics for Lake
^ -
rO -
fM ¦
-5
0
G* Statistics for Oyster
G Statistics for SQ Effect
Alpha = 1
Alpha = 0,5
Alpha = 0
Figure 7. Distribution of Hotspot Analysis Getis-Ord Statistics
Note: Vertical long-dashed grey lines denote 90% confidence interval.
31
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APPENDIX
Appendix A: Description of Bayesian Estimation Routine
We can appeal to Bayes' Rule to define the relationship between the conditional distribution
rj(X\y,z,v) and the distribution over the population g(X|i)). First consider the probability for the
mixed logit:
p(y, \z^) = \p(y, \ a)g(M (Al)
which gives the probability of household V s set of responses, given the data and the parameters of
the population level parameter distribution. Using Bayes Rule, we express r/(k\y,z,v) as
p(yt
777—; : and since the denominator is constant with respect to X, n(k\y,z,v) is
P(yi \Zi,»)
proportional to the numerator which provides a useful interpretation of rj{). The density of X, in the
subpopulation that chose y, when faced with z, is proportional to the density of X in the entire
population, given by g(.), multiplied by the probability that someone would choose y, given the
data and the set of parameters X.
Estimating the mixed logit model via hierarchical Bayes' requires iteratively drawing from the
distributions for the mean and variance of X and the household-level parameters while always
conditioning on the most recent draw of the other parameters. We refer the reader to Train (2003,
pages 302-308) for a complete description of the algorithm but for our purposes, consider a
simulation that begins with starting values for the mean vector of X, a covariance matrix W, and a
vector of household-level parameters for each respondent h. The first step of the algorithm draws
a realization of the mean vector A. conditional on X, and W which is distributed /y' (V A \y \. The
liN' J
second step draws the covariance matrix Wfrom an inverse Wishart distribution with M+N degrees
MI+NS
of freedom and scale matrix , where / is a M-dimensional identity matrix and
M+N ' J
£(4 -X)(/L-X)'
S = _i ' ! The third and final step of the algorithm draws household-level parameter
N
ne^Ziyitt , _ ^
—r/J(A I A,W) which requires a MH algorithm.
i
After a burn-in period, the draws will converge to the joint posterior distribution of the model
parameters.
32
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Appendix B: Parametric Estimation of Distance Decay
As a preliminary step to examine whether preferences for environmental improvements vary with
distance to the resource, researchers often include a measure of the distance of respondent i to the
resource (e.g., Pate and Loomis, 1997; Hanley et al., 2003; Bateman et al., 2006). We denote this
distance measure as /(dj), and examine various functional forms for such a global distance
gradient. The model in equation (1) of section 2.1 in the main text is thus augmented as follows:
p(: ,v rn-t cn J exp\ln(xj)fi+\ln(xj)xf(dj)\Q +Ycostj+
-------
Table Bl. Conditional Logit Models with Parametric Distance Gradient
VARIABLES
Linear
ln(distance)
Inverse
Quadratic distance
Stepwise:
Stepwise:
Stepwise:
distance
distance
50km
In watershed
Geographic Strata
(1)
(2)
(3)
(4)
(5)
(6)
(7)
ln(clarity)
2.336e-01
1.973e-01
2.589e-01
1.746e-01
3.183e-01
3.367e-01
7.411e-03
(0.353)
(0.934)
(0.259)
(0.466)
(0.287)
(0.336)
(0.430)
ln(bass)
-5.943e-01
-1.388e+00
-2.395e-01
-8.126e-01*
-2.248e-02
-1.869e-02
-6.930e-01
(0.371)
(0.982)
(0.276)
(0.486)
(0.310)
(0.363)
(0.447)
ln(crab)
-1.859e-01
-5.876e-01
3.363e-01
-2.027e-01
4.288e-01
7.597e-01
-5.320e-01
(0.519)
(1.386)
(0.387)
(0.691)
(0.428)
(0.480)
(0.636)
ln(oyster)
-1.13 le-01
-7.380e-02
-5.586e-02
-7.431e-02
-6.705e-02
-1.439e-03
-1.870e-02
(0.127)
(0.335)
(0.095)
(0.166)
(0.107)
(0.123)
(0.150)
ln(lake)
1.160e+00**
4.660e-01
1.573e+00***
1.359e+00**
1.883e+00***
1.788e+00***
8.586e-01
(0.492)
(1.423)
(0.358)
(0.662)
(0.386)
(0.451)
(0.613)
SQC
-8.971e-01***
-1.293e+00***
-7.081e-01***
-1.051e+00***
-5.598e-01***
-5.090e-01***
-1.063e+00***
(0.178)
(0.432)
(0.137)
(0.227)
(0.151)
(0.172)
(0.215)
cost
-5.008e-03***
-5.009e-03***
-5.045e-03***
-5.005e-03***
-5.035e-03***
-5.004e-03***
-5.064e-03***
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
(0.000)
watershed other east
distance distanceA2
states coast states
ln(clarity) x
f(dist)
1.591e-05
1.022e-02
-1.376e-02
4.576e-04 -4.662e-07
-3.273e-01
-2.313e-01
6.53 le-01 1.747e-02
(0.001)
(0.178)
(0.474)
(0.003) (0.000)
(0.668)
(0.529)
(0.628) (0.639)
ln(bass)x f(dist)
1.257e-03
2.352e-01
3.654e-01
3.060e-03 -1.689e-06
-1.010e+00
-5.013e-01
3.860e-01 1.242e+00*
(0.001)
(0.191)
(0.577)
(0.003) (0.000)
(0.662)
(0.553)
(0.626) (0.727)
ln(crab) x f(dist)
1.862e-03
1.920e-01
7.844e-01
1.956e-03 3.062e-08
-2.114e-01
-9.659e-01
6.992e-01 2.407e+00**
(0.001)
(0.269)
(0.589)
(0.004) (0.000)
(0.950)
(0.794)
(0.875) (1.005)
ln(oyster) x f(dist)
2.262e-04
4.815e-03
2.075e-01
-1.094e-04 3.101e-07
9.369e-02
-1.228e-01
-2.197e-01 2.024e-01
(0.000)
(0.065)
(0.161)
(0.001) (0.000)
(0.224)
(0.190)
(0.223) (0.230)
ln(lake) x f(dist)
1.551e-03
2.283e-01
3.689e-01
-1.544e-04 1.592e-06
-1.455e+00
-5.109e-01
6.058e-01 2.075e+00**
(0.001)
(0.273)
(1.147)
(0.004) (0.000)
(0.977)
(0.739)
(0.841) (0.938)
Joint significance
X2(4) = 4.64
X2(4) = 1.94
X2(4) = 6.30
X2(8) = 5.51
X2(4) = 2.56
X2(4) = 2.78
X2(8) = 12.40
p = 0.3267
p = 0.7464
p = 0.1779
p = 0.7014
p = 0.6334
p = 0.5958
p = 0.1341
34
-------
SQC x f(dist)
6.697e-04
(0.000)
1.189e-01
(0.084)
6.904e-02
(0.100)
Observations 4,719 4,719 4,719
LL -1567.4058 -1569.1278 -1568.1169
Robust standard errors in parentheses; *** p<0.01, ** p<0.05, * p<0.1.
1.896e-03
(0.001)
-1.130e-06
(0.000)
-7.387e-01**
(0.304)
-4.840e-01*
(0.260)
s.^e-oi"
(0.296)
6.248e-01:|
(0.330)
4,719
-1564.6207
4,719
-1566.0111
4,719
-1568.7936
4,719
-1555.4703
35
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