BAYSIAN INFERENCE FOR ORDERED RESPONSE DATA WITH A DYNAMIC SPATIAL-ORDERED PROBIT MODEL

Abstract

Many databases involve ordered discrete responses in a temporal and spatial context, including, for example, land development intensity levels, vehicle ownership, and pavement conditions. An appreciation of such behaviors requires rigorous statistical methods, recognizing spatial effects and dynamic processes. This study develops a dynamic spatial-ordered probit (DSOP) model in order to capture patterns of spatial and temporal autocorrelation in ordered categorical response data. This model is estimated in a Bayesian framework using Gibbs sampling and data augmentation, in order to generate all autocorrelated latent variables. It incorporates spatial effects in an ordered probit model by allowing for interregional spatial interactions and heteroskedasticity, along with random effects across regions or any clusters of observational units. The model assumes an autoregressive, AR(1), process across latent response values, thereby recognizing time-series dynamics in panel data sets. The model code and estimation approach is tested on simulated data sets, in order to reproduce known parameter values and provide insights into estimation performance, yielding much more accurate estimates than standard, nonspatial techniques. The proposed and tested DSOP model is felt to be a significant contribution to the field of spatial econometrics, where binary applications (for discrete response data) have been seen as the cutting edge. The Bayesian framework and Gibbs sampling techniques used here permit such complexity, in world of two-dimensional autocorrelation. Copyright (c) 2009, Wiley Periodicals, Inc.

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JOURNAL OF REGIONAL SCIENCE, VOL. 49, NO. 5, 2009, pp. 877–913
BAYSIAN INFERENCE FOR ORDERED RESPONSE DATA
WITH A DYNAMIC SPATIAL-ORDERED PROBIT MODEL
Xiaokun Wang
Department of Civil and Environmental Engineering, Bucknell University,
Lewisburg, PA 17837. E-mail: cara.wang@bucknell.edu
Kara M. Kockelman
Department of Civil, Architectural and Environmental Engineering, The University
of Texas at Austin, 6.9 ECJ, Austin, TX 78712-1076. E-mail: kkockelm@mail.utexas.edu
ABSTRACT.
spatial context, including, for example, land development intensity levels, vehicle own-
ership, and pavement conditions. An appreciation of such behaviors requires rigorous
statistical methods, recognizing spatial effects and dynamic processes. This study de-
velops a dynamic spatial-ordered probit (DSOP) model in order to capture patterns of
spatial and temporal autocorrelation in ordered categorical response data. This model
is estimated in a Bayesian framework using Gibbs sampling and data augmentation,
in order to generate all autocorrelated latent variables. It incorporates spatial effects
in an ordered probit model by allowing for interregional spatial interactions and het-
eroskedasticity, along with random effects across regions or any clusters of observational
units. The model assumes an autoregressive, AR(1), process across latent response val-
ues, thereby recognizing time-series dynamics in panel data sets. The model code and
estimation approach is tested on simulated data sets, in order to reproduce known pa-
rameter values and provide insights into estimation performance, yielding much more
accurate estimates than standard, nonspatial techniques. The proposed and tested DSOP
model is felt to be a significant contribution to the field of spatial econometrics, where
binary applications (for discrete response data) have been seen as the cutting edge. The
Bayesian framework and Gibbs sampling techniques used here permit such complexity,
in world of two-dimensional autocorrelation.
Many databases involve ordered discrete responses in a temporal and
1. INTRODUCTION
In the fields of regional science and transportation, variables of interest
often are discrete in nature and involve temporal and spatial relationships. For
example,landuseintensity,vehicleownership,androadwayservicelevelsoften
are measured (and/or coded) as ordered discrete responses, dependent on vari-
ous influential factors. These discrete responses share a common feature: they
all exhibit some degree of temporal and spatial dependence or autocorrelation.
For example, in two slices of a panel survey of households, the count of vehicles
owned by the same household will be highly correlated. This phenomenon is
Received: February 2008; revised: November 2008; accepted: February 2009.
C ?2009, Wiley Periodicals, Inc.DOI: 10.1111/j.1467-9787.2009.00622.x
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878JOURNAL OF REGIONAL SCIENCE, VOL. 49, NO. 5, 2009
normally defined as temporal dependency or autocorrelation. Meanwhile, even
after controlling for household attributes, auto ownership levels are expected to
exhibit positive correlations in the spatial context. To some extent, such corre-
lation patterns can be explained by uncertainty or proximity because, in reality,
there are always influential factors that cannot be controlled (e.g., pedestrian
friendliness of all neighborhoods). The sign and magnitude of such uncertain-
ties tend to vary rather gradually over space. Of course in a spatial context, in
contrast to time-series data, such dependencies are two dimensional—which
adds complexity. Like temporal relationships, correlation tends to diminish
with increases in distance between any two households/observed units.
Essentially,then, many phenomena involving ordered categorical data also
have temporal and spatial relationships; yet the development of rigorous meth-
ods for analyzing these phenomena is still in its infancy. This paper presents a
model that is appropriate for describing the temporal and spatial relationships
that exist in ordered categorical data. Related issues also are explored, indi-
cating model estimation techniques, model validation and model comparisons
(with simplified, less behaviorally reasonable models). Such model specifica-
tions and estimation techniques may be viewed as breakthroughs in the area
of spatial econometrics, and the results extendable to a wide range of topics,
where dependent variables are ordered discrete values and may involve tem-
poral and spatial dependencies across observations.
The following sections review existing studies, explain the intuition behind
model specification and illustrate how to estimate the unknown parameters
using Bayesian methods. Model performance is quantified using simulated
data sets.
2. LITERATURE REVIEW
As mentioned above, temporally and spatially correlated behaviors are
observed in many contexts, and several studies have directly or indirectly ex-
plored such phenomena. For example, various models control for and predict
spatial and temporal attributes of urban systems, such as land use, trade and
travel patterns. (These include Waddell’s (2002) UrbanSim, de la Barra’s (1989)
TRANUS, and Hunt and Abraham’s (2003) PECAS model.) By controlling for
a variety of local and regional characteristics, unobserved spatial effects are
lessened (see, e.g., Nelson and Hellerstein’s (1997) land cover variables in a
study of deforestation, as related to roadway networks). Recognizing the im-
portance of temporal dependencies, Wear and Bolstad (1998) controlled for
neighborhoods’ prior land uses in their study of Appalachian landscapes. Such
research is also popular in the area of biomedicine: Kelley et al. (2007) used a
mixed-effects beta regression with inverse distance weighting to study a virus’s
genetic variations, and Madden et al. (1988) calculated autocorrelation param-
eters for up to three lags in time and space, in their study of the tobacco virus.
While space and time are present in a variety of disciplines and data sets, only
spatial econometric methods allow investigators to tackle their complex data
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WANG AND KOCKELMAN: BAYSIAN INFERENCE FOR ORDERED RESPONSE DATA 879
structures directly. The following discussion provides several such examples,
in the context of discrete responses.
As in standard spatial econometrics, methods for dealing with spatial ef-
fects in discrete choice models can be categorized into three basic types. Geo-
graphically weighted regression (GWR) is most applicable when spatial varia-
tion in behavioral parameters is of strong interest. In an analysis on suburban
subcenters and employment density, McMillen and McDonald (1998) propose
the idea of applying standard logit or probit methods to distance-weighted sub-
samples of the data in place of least squares, essentially using GWR to deal
with discrete responses. LeSage (1999) provided code for producing binary logit
and probit GWR estimates, using crime data. Atkinson et al. (2003) also used
a GWR binary logit model to explore relationships between the presence (or
absence) of riverbank erosion and geomorphological controls. Vanasse et al.
(2005) incorporated GWR in a binary logit model to study spatial variation in
the management and outcomes of acute coronary syndrome.
The second method, spatial filtering, has been applied more broadly. It
saves much specification and estimation effort. In addition to several land
use/land cover models, many other works rely on this method. For example,
an early study by Boots and Kanaroglou (1988) introduced a measure of spa-
tial structure and used it as an explanatory variable when considering spatial
effects in Toronto’s intrametropolitan migration. Dugundji and Walker (2005)
controlled for spatial network independencies in their mixed logit model when
studying mode choice behavior. Coughlin, Garrett, and Hern´ andez-Murillo
(2003) incorporated global and regional spatial effects into an analysis of state
lotteries.
The third method incorporates spatial effects directly in a discrete choice
model setting and is the focus of this study. This method can be distinguished
by two approaches. The first considers spatial autocorrelation across choices
or alternatives, as often discussed for location choice models. This approach
extends the commonly used multivariate extreme value model by allowing
correlated alternative-specific error terms in a mixed logit framework. For
example, Miyamoto et al. (2004) assumed that location choice follows a spa-
tial auto regression (SAR) process, and used the weight matrix as a multi-
plier on dependent variables. Bhat and Guo (2004) used a contiguity matrix
on their latent dependent variables to represent alternative-zone correlation
patterns.
The second approach considers spatial autocorrelation across observa-
tional units (or individuals), and is the focus of this work. Currently, the studies
recognizing such spatial autocorrelation are limited to binary choice settings.
To some extent, Wang and Kockelman (2006)’s work on estimating urban land
cover evolution seems an exception, because multiple choices are studied in
a mixed logit framework. However, rather than permitting a more flexible
SAR process, Wang and Kockelman used a direct representation method and
assumed a specific distance-decay function for interobservational correlations,
making the spatial correlation pattern across observations rather arbitrary. All
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880 JOURNAL OF REGIONAL SCIENCE, VOL. 49, NO. 5, 2009
other existing spatial probit and logit work is binary in nature. Anselin (2001)
reviewed such spatial probit models and notes that McMillen (1995) first used
the EM algorithm to estimate a probit model with a SAR process. Beron and
Vijverberg (2004) specified probit models with both spatial errors and spatial
lags, and then estimated these models by using recursive importance sam-
pling to approximate the n-dimensional log-likelihood. LeSage (2000) specified
a model with a spatially correlated error term and used Gibbs sampling for
estimation. Smith and LeSage (2004) extended this study by incorporating a
regional effect and used Bayesian techniques to analyze the 1996 presidential
election results. Similar studies include Kakamu and Wago’s (2007) Bayesian
estimation of a spatial probit model for panel data to analyze the business cycle
in Japan.
Another estimation approach is the generalized method of moments, or
GMM. Pinkse and Slade (1998) first used GMM to estimate a probit model
with spatial error components. Pinkse, Slade, and Shen (2005) refined that
study by incorporating a dynamic structure for dependent variables and apply-
ing a one-step GMM. And Klier and McMillen (2007) used GMM to estimate a
spatial logit model for analyzing the clustering of auto supplier plants in the
United States. However, the use of GMM is limited because it requires orthog-
onality conditions (as discussed in works like Pinkse and Slade, 1998; Pinkse
et al., 2005; Klier and McMillen, 2007), and standard errors must be derived.
For this reason, it presently is applied only to binary response models; it has
not yet been extended to multiple-response models. All the other estimation
methods can be called simulation estimators. As Anselin (2001) concludes, all
current simulation estimators are slow, but Gibbs sampler is relatively less
slow. In other words, among all three general methods discussed above, the
most promising one for a model of multiple discrete response with spatial ef-
fects (both autocorrelation and heteroskedasticity) is Gibbs sampling within a
Bayesian framework.
In contrast to frequentist methods (i.e., classical statistical analysis), the
Bayesian approach is rather straightforward in both model estimation and re-
sults interpretation. A primary motivation is rather direct interpretation of
parameter estimates and probabilities. A Bayesian approach yields estimates
of parameter distributions (rather than relying on asymptotics for normality).
These distributions effectively define intervals that can be “regarded as having
a high probability of containing the unknown quantit(ies) of interest” (Gelman
et al., 2004). In contrast, frequentist methods focus on producing point esti-
mates and rather standard confidence intervals, and resulting probabilities
that are strictly interpreted as “long run (asymptotic) relative frequenc(ies)”
(Koop, Poirier, and Tobias, 2007).
In practice, an important advantage of a Bayesian framework is its flex-
ibility, allowing it to deal with complex estimation problems more easily. In
fact, this is the main reason for this study’s choice of Bayesian framework—in
addition to wanting to develop new methods of model estimation for regional
and transportation sciences (where frequentist methods are the norm).
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WANG AND KOCKELMAN: BAYSIAN INFERENCE FOR ORDERED RESPONSE DATA881
In general, Bayesian estimation via Markov chain Monte Carlo (MCMC)
simulation relies on a set of conditional distributions to deduce each param-
eter’s marginal distribution. In this way, models with many parameters and
complicated multiple-layered probability specifications can be decomposed into
asetofsimplersubproblems.Bycontrast,withfrequentistmethods,themodels
have to deal directly with any complicated model specification and any statis-
tical problems arising from it. Of course, another well-understood advantage
of using a Bayesian approach is that by having priors, one can make use of es-
tablished intuition and experience to balance new information found in sample
data. Thanks to its many advantages, a Bayesian approach has been used in
various areas. For example, Wallerman, Vencatasawmy, and Bondesson (2006)
relied on Bayesian estimation for remote sensing data in forested areas, and
Hamilton et al. (2005) used it to estimate expansion times and migration rates
for Swiss populations.
Albert and Chib (1993) introduced the Bayesian approach for (stationary,
nonspatial) discrete response data models. LeSage (2000) first extended Al-
bert and Chib’s approach to models involving spatial dependencies. Later work
by Smith and LeSage (2004) further extended the model, by incorporating an
error specification that allows both spatial dependencies and general spatial
heteroskedasticity. All such studies, however, deal only with binary data. As
previously discussed, many data sets offer multiple categories. No existing
studies tackle such patterns in a spatial context. While Albert and Chib (1993)
briefly mentioned possible extensions from binary data to ordered categorical
data, they did not offer any methodological details. Several years later, John-
son and Albert (1999) suggested a detailed Bayesian framework for modeling
ordinal data, and Cowles (1996) presented a method for accelerating MCMC
convergence for models like the ordered probit. Girard and Parent (2001) even
extended Albert and Chib’s study (1993) to temporally autocorrelated ordered
categorical data, but there is nothing spatial in these studies. This study is
inspired by such studies but adds sophistication while combining space and
time for ordered categorical data. It goes beyond a simple extension or combi-
nation of these works. The contribution of these prior studies will discussed in
more detail in the next section, through illustrations of model specification and
estimation techniques.
3. MODEL SPECIFICATION
Standard Ordered Probit (OP) Model
A standard-ordered probit model has been used widely for estimating dis-
crete responses of an ordinal nature (Greene, 2002). The model is built upon a
latent regression that is expressed as follows:
Ui= X?
i? + ?i,
(1)
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