Spatial Models of Travel Behavior and Land Use Restriction

Andrew G. Mueller; Stephan Weiler

doi:10.1080/19498276.2023.2174661

Spatial Models of Travel Behavior and Land Use Restriction

Mueller, Andrew G.; Weiler, Stephan 2023-12-31 00:00:00 JOURNAL OF SUSTAINABLE REAL ESTATE 2023, VOL. 15, NO. 1, 2174661 ARES https://doi.org/10.1080/19498276.2023.2174661 American Real Estate Society a b Andrew G. Mueller and Stephan Weiler a b Burns School of Real Estate and Construction Management, University of Denver, Denver, CO, USA; Department of Economics, Colorado State University, Fort Collins, CO, USA KEYWORDS ABSTRACT Land use restrictions; spatial This paper develops a spatial econometric model of transportation mode choice and tests econometrics; transportation the association between zoning and other built environment variables and the choice of mode choice; travel auto and non-auto transportation. We provide an extensive review of spatial econometrics behavior; walkability; and demonstrate the importance of using models that treat space formally when investigat- zoning ing urban transportation behavior. Using a unique combination of travel, employment, and built environment datasets from Denver, Colorado, we confirm previous results that built environment variables have a small association with choice of transportation mode and show the benefits of formal spatial modeling to the traditional probit model. Introduction logit, or multinomial logit models to estimate impacts on mode choice (Ewing & Cervero, 2010). To investi- Location of economic activity has always been an gate the importance of formal spatial modeling, we use important aspect of regional economics and real estate. the dataset from a previous study on the impacts of Spatial proximity plays a key role in many decisions zoning on mode choice (Mueller & Trujillo, 2019)and made by individuals when it comes to weighing the formally model spatial dependence and spatial hetero- benefits and costs of purchase decisions, allocation of geneity to test whether the results are significantly dif- resources, and other economic behavior. Transportation ferent from the standard econometric approach where choices are particularly affected by location, and distan- space is dealt with informally. It is particularly import- ces between origins and destinations undoubtedly influ- ant to investigate the presence of spatial dependence ence travel decisions among individuals. and heterogeneity in regard to land use restrictions The development of formal econometric techniques because each individual faces a unique set of transpor- to address location is a more recent development in tation choices based on their residential location and the field. Of particular importance to the field of spa- the proximity of this residential location to available tial econometrics is the treatment of spatial depend- goods, services, recreation, transportation, and employ- ence (spatial autocorrelation) and spatial heterogeneity ment opportunities. While built environment variables (spatial structure). Spatial dependence and spatial het- have been studied for their impact on driving behavior erogeneity are important in applied economic models (Vance & Hedel, 2008), their role in land use segmen- because the presence of these phenomena may invali- tation (Levkovich et al., 2018), development and dens- date or bias results from mainstream approaches. In ity (Butsic et al., 2011;Newburn & Ferris, 2016), and addition, these issues have been largely ignored in the sprawl (Vyn, 2012), to the author’s knowledge this is mainstream literature (Anselin, 1988). This paper focuses on consumer transportation the first study that employs formal spatial modeling to investigate the association of zoning restrictions with mode choice in a spatial context to demonstrate the travel mode choice. From a policy perspective, models importance of formal spatial modeling techniques in transportation behavior research. Previous studies of that inform the impact that a change in zoning laws transportation choice largely implement OLS, probit, might have on transportation choice can further inform CONTACT Andrew G. Mueller andrew.mueller@du.edu Burns School of Real Estate and Construction Management, University of Denver, Denver, CO 80208, USA. 2023 The Author(s). Published with license by Taylor & Francis Group, LLC This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 A. G. MUELLER AND S. WEILER alternative transportation agendas. Non-auto forms of construction, see Mueller and Trujillo (2019)and the transportation and built environments that support descriptive statistics in Table 1. their use can provide traffic congestion relief, and so In this work, we formally test for spatial autocorrel- are the primary focus of this research, although other ation and spatial heterogeneity in the data and apply agendas, such as the transition to electric vehicles may spatial econometric methods to correct for spatial com- have other benefits. ponents found in the data generating process. The Previous work has used a choice-specific multi- paper reviews the relevant spatial econometric theory, nomial logit model to estimate the effect of built and applies the Moran’s I and Geary’s C tests for spatial environment, socioeconomic, and land restriction processes at work in the data generating process. We measures on the propensity of survey respondents to then explore several models that address spatial auto- use four mode choice alternatives; auto, transit, bike, correlation, spatial heterogeneity, and both spatial auto- and walk (Ewing & Cervero, 2010; Mueller & Trujillo, correlation and heterogeneity jointly and compare them 2019). Due to the lack of observations in the data for with the results of the canonical probit approach. walk (7.42%), bike (1.5%), and transit (7.71%) which Estimates of each model are performed with the two individually represent a small subset of the dataset, we most commonly used spatial weights matrices in the lit- condense the dependent variable choice set into a bin- erature, the row standardized binary weights matrix and ary choice between auto and non-auto transportation the row standardized inverse distance matrix. The (16.6%). This paper adds to the previous work by for- results from each model show that there is a high likeli- mally implementing econometric techniques that expli- hood of the presence of spatial processes in the data citly deal with bias and inefficiency in the estimation of generating process and that these models are preferable effects that are introduced if spatial autocorrelation to canonical approaches to estimating travel mode and/or spatial heterogeneity are present in the underly- choice behavior in this travel survey sample. We pro- ing data generating process. The dataset does not allow vide evidence that the spatial Durbin model (SDM) for testing of causality in the relationship between the with a binary spatial weights matrix is the superior built environment and transportation choice, and we model that accounts for both spatial dependence and acknowledge that there may be simultaneity and self- heterogeneity, and accounts for spatial processes that selection present in the dynamic process that deter- are not modeled in the traditional probit approach. mines the relationship between mode preference and Particularly, if spatially lagged forms of the dependent zoning variables by respondents choosing to locate in and independent variables show signs of spatial depend- areas that align with their transportation preferences, ence or heterogeneity, only the spatial Durban model and possibly also influencing the political process that addresses both spatial phenomena simultaneously determines zoning in their neighborhoods. Testing of (LeSage & Pace, 2009). such relationships would require repeat observations of households in a longitudinal survey, and such a survey Reviewing the Spatial Econometric Approach is not available currently. Using transportation survey data from the City and Spatial econometrics differentiates itself from main- County of Denver in 2010 and corresponding built stream econometric approaches by applying formal environment measures during that time, we test the spatial modeling best summarized in Luc Anselin’spio- association of zoning and other built environmental neering work on the topic: “I will consider the field of variables on survey respondents’ choice of auto and spatial econometrics to consist of those methods and non-auto transportation while controlling for demo- techniques that, based on formal representation of the graphic and preference variables collected in the survey. structure of spatial dependence and spatial heterogen- The unit of observation is a “tour” taken by individuals eity, provide the means to carry out the proper specifi- where a respondent leaves the home, makes stops for cation, estimation, hypothesis testing, and prediction various activities, and returns home. The key independ- for models in regional science” (Anselin, 1988,p.10). ent variables of interest in this study are the land use restriction variables encompassed in percentages of dif- Spatial Effects ferent zoning types (low, medium, and high density residential, low and high density business, and indus- Real estate economics and regional science inherently trial) within a 1/4, 1/4–1/2, 1/2–3/4, and 3/4–1-mile deal with issues related to human behavior across radius surrounding each survey respondent’s home. For space, cities, and regions. The term spatial economet- a more thorough explanation of the dataset and its rics and its designation as a distinct branch of JOURNAL OF SUSTAINABLE REAL ESTATE 3 Table 1. Descriptive statistics. Variable N Mean St. Dev. Min Max Household vehicles 5,123 1.895 0.901 0.000 6.000 Household bikes 5,123 1.627 1.794 0.000 30.000 Household size 5,123 2.419 1.193 1.000 7.000 Male 5,123 0.468 0.499 0.000 1.000 Age 5,123 52.314 15.689 16.000 93.000 Income 5,123 79.369 50.587 0.000 160.000 College education 5,123 0.374 0.484 0.000 1.000 Employed 5,123 0.658 0.474 0.000 1.000 Driver’s license 5,123 0.945 0.228 0.000 1.000 Transit pass 5,123 0.136 0.343 0.000 1.000 Miles bike land < 1 mile 5,123 11.667 5.534 0.000 33.048 Mile bus routes < 1/2 mile 5,123 8.596 12.314 0.000 90.313 Bus stops < 1/2 mile 5,123 29.342 17.621 0.000 138.000 Rail stops < 1 mile 5,123 0.262 0.907 0.000 7.000 Miles rail lines < 1 mile 5,123 2.184 3.180 0.000 14.346 Intersection < 1/2 mile 5,123 125.574 28.072 9.000 255.000 CBG population density 5,123 7,174.166 4,497.568 27.723 33,612.000 CBG employment density 5,123 4,862.703 20,847.640 0.000 278,470.500 Home < 2 miles of downtown 5,123 0.173 0.378 0.000 1.000 Number of tours 5,123 1.486 0.773 1.000 7.000 Travel mode ¼ non-auto 5,123 0.166 0.372 0.000 1.000 Tour distance 5,123 10.816 12.269 0.012 135.415 Trip duration 5,123 49.936 37.055 2.000 452.000 Tour crosses highway 5,123 0.746 0.435 0.000 1.000 Arrival hour 5,123 11.206 4.260 0.000 23.000 Departure hour 5,123 6.706 6.565 0.000 23.000 Work stops 5,123 0.570 1.098 0.000 30.000 Shopping stops 5,123 0.710 0.984 0.000 10.000 Health stops 5,123 0.068 0.264 0.000 3.000 Social stops 5,123 0.252 0.504 0.000 4.000 Res. low density, 0–1/4 mile 5,123 63.435 36.588 0.000 100.000 Res. med. density, 0–1/4 mile 5,123 9.200 18.181 0.000 100.000 Res. high density, 0–1/4 mile 5,123 12.527 22.923 0.000 100.000 Industrial, 0–1/4 mile 5,123 1.793 8.243 0.000 73.680 Bus. low density, 0–1/4 mile 5,123 2.044 3.909 0.000 29.601 Bus. High Density, 0–1/4 mile 5,123 8.047 17.695 0.000 100.000 Res. low density, 1/4–1/2 mile 5,123 59.288 30.483 0.000 100.000 Res. med. density, 1/4–1/2 mile 5,123 10.837 15.047 0.000 98.131 Res. high density, 1/4–1/2 mile 5,123 11.964 15.999 0.000 76.586 Industrial, 1/4–1/2 mile 5,123 3.297 9.920 0.000 76.188 Bus. low density, 1/4–1/2 mile 5,123 1.968 2.294 0.000 14.610 Bus. high density, 1/4–1/2 mile 5,123 9.622 13.830 0.000 99.991 Res. low density, 1/2–3/4 mile 5,123 56.181 26.581 0.000 100.000 Res. med. density, 1/2–3/4 mile 5,123 11.910 14.836 0.000 98.207 Res. high density, 1/2–3/4 mile 5,123 11.836 13.089 0.000 83.916 Industrial, 1/2–3/4 mile 5,123 4.814 11.310 0.000 94.524 Bus. low density, 1/2–3/4 mile 5,123 2.032 2.107 0.000 12.704 Bus. high density, 1/2–3/4 mile 5,123 9.978 11.800 0.000 88.327 Res. low density, 3/4–1 mile 5,123 54.202 23.843 0.000 99.894 Res. med. density, 3/4–1mile 5,123 12.842 14.669 0.000 99.635 Res. high density, 3/4–1 mile 5,123 11.314 10.409 0.000 88.762 Industrial, 3/4–1 mile 5,123 6.535 11.893 0.000 97.455 Bus. low density, 3/4–1 mile 5,123 1.881 1.737 0.000 10.890 Bus. high density, 3/4–1 mile 5,123 9.825 10.309 0.000 68.460 econometrics dates back to the seminal work of spaces, (4) differentiation between ex post and ex ante Paelinck and Klaassen (1979) that collected a growing interaction, and (5) explicit modeling of space. While body of literature in the regional sciences that it is possible to measure and model spatial data using attempted to formally deal with the problems inherent standard econometric techniques by including variables in modeling spatial data in the context of regional in the model that have a spatial nature to their meas- econometric models. The primary characteristics that urement, the distinction to be made here is that spatial delineate the field according to Paelinck and Klaassen econometrics formally deals with specific spatial (1979) and summarized by Anselin (1988,p. 7) are: aspects of the data that preclude the use of traditional (1) the role of spatial interdependence in spatial mod- econometric techniques, and more particularly, address els, (2) the asymmetry in spatial relations, (3) the spatial dependence and spatial heterogeneity formally importance of explanatory factors located in other (Anselin, 1988; LeSage & Pace, 2009). 4 A. G. MUELLER AND S. WEILER Spatial dependence addresses the lack of mutual Formally Modeling Spatial Interaction independence across observations in cross-sectional At the center of spatial econometrics is defining spatial data sets and is often referred to as spatial autocorrel- association amongst observations (Anselin, 1988, 2010; ation following the path-breaking work of Cliff and Arbia, 2006). To formally address the spatial connect- Ord (1969, 1973). In essence, addressing spatial edness of observations across space, an approach has dependence is the development of formal statistical been developed which uses a decision rule that deter- specifications of economic models that address mines whether two observations are spatial neighbors Tobler’s first law of geography, that “everything is and thus close enough to exert influence on each other. related to everything else, but near things are more The typical convention is to formally define spatial related than distant things” (Tobler, 1970, p. 236). connectedness through the use of a symmetric matrix Spatial dependence is estimated by the relative loca- W of dimensions equal to the number of observations tion of one observation in the dataset to another, with n, whose strictly non-negative elements w indicate the ij an emphasis on the effect of distance between obser- spatial connectedness between units i and j 6¼ i: With vations. Spatial dependence is caused by a variety of the spatial neighbor matrix constructed, spatial model- measurement errors, by spatial spill-over effects, or ing proceeds by re-weighting each row to develop a spatial externalities (Anselin, 1988), by spatially auto- spatial weights matrix, then pre-multiplying either the correlated variables (Fingleton & Lopez-Bazo, 2006), dependent or independent variables by the spatial or any situation in which the covariance of observa- weights matrix and estimating a vector of coefficients tions across geographical space is not equal to zero that includes a spatial dependence parameter. This (Anselin, 2001). For example, spatial autocorrelation modeling approach formally connects variables of is often found in hedonic pricing models of residential neighboring observations through the spatial weights real estate, where the sale price of one residential matrix W and produces an estimate of spatial associ- property is influenced by housing prices in surround- ation in the data generating process through the spatial ing neighborhoods (von Graevenitz & Panduro, 2015). dependence parameter(s). To demonstrate the use of Spatial heterogeneity is the “lack of stability over the spatial weights matrix, the spatial autoregressive space of the behavioral or other relationship under model equation is illuminating. In its simplest form study. More precisely, this implies that functional with no independent regressors, the spatial autoregres- forms and parameters vary with location and are not sive model equation is: homogeneous throughout the data set” (Anselin, 1988, X y ¼ q w y þ e (1) i ij j i p. 9). This type of econometric model addresses these issues by formally modeling the variation in parame- ters across space to address the heterogeneous effect The term w y þ e gives a weighted sum of ij j i an independent variable may have in different loca- each neighboring observation j’s dependent variable tions. More importantly, when spatial dependence and y , j 6¼ i: The estimated spatial dependence parameter spatial heterogeneity are present in the data generating q gives a measure of the influence those neighboring process and not explicitly modeled, the results of observations have on each y observation. High values mainstream econometric techniques may be biased, of q indicate strong spatial autocorrelation between inefficient, or both (Anselin, 1988; LeSage & Pace, observations, while a value of 0 indicates no spatial 2009; Schnier & Felthoven, 2011). Spatial econometric autocorrelation. In addition to measuring the direct techniques address spatial processes within the data influence of neighbors j on observation i, the param- generating process and are generally preferred when eter q is sometimes referred to as the spatial decay differences due to spatial location are present in data. parameter, because it also indicates how fast the effect An example of spatial heterogeneity is the variation in of neighboring observations declines with higher order the effect income may have on travel mode preferen- neighbors, i.e. neighbors of neighbors (Anselin, 1988; ces across the urban landscape. Income may have the LeSage & Pace, 2009). For example, second order opposite effect on preferences to drive in suburban neighbors of y are first order neighbors of y ’s first i i locations than it does in central business districts order neighbors and have an influence on y equal to because higher incomes allow suburban dwellers q , their influence on y being exerted indirectly greater access to automobiles, while in urban locations through y ’s direct neighbors. Influence dissipates as higher income may allow individuals to live in areas observations become further removed from y , and k with better access to goods and services, thus increas- order neighbors have and influence equal to q . Thus, ing reliance on alternative forms of transportation. values of q closer to 1 indicate a slowly dissipating JOURNAL OF SUSTAINABLE REAL ESTATE 5 influence, while values close to 0 indicate an effect models to the choice of spatial weights matrix by esti- that quickly dissipates with higher order neighbors. mating models with both a binary row standardized The literature has yet to determine a formal spatial weights matrix and an inverse distance row approach to developing the spatial weights matrix, standardized weights matrix. although several approaches have widespread adop- The decision to standardize the spatial weights matrix is not at all clear from the literature, and deci- tion. The pioneering work of Moran (1950) and sions on how to form the spatial weights matrix are Geary (1954) developed the notion of a binary weights generally determined by a priori assumptions made by matrix W, where each element w was assigned a ij the researcher in the context of each study. Anselin value of 1 if two observational units were neighbors (1988) argues that in certain cases, such as inverse dis- and assumed to exhibit influence on each other, and 0 tance, the standardization of the spatial weights matrix otherwise. The spatial weights matrix was originally may eliminate the economic interpretation of the developed in the context of areal units and neighbors results. However, the consensus is that the standard- were defined as two observational units that shared a ization of the spatial weights matrix is the preferable common border (Cliff & Ord, 1973). When observa- approach to avoid magnitude complications amongst tional units are points in space rather than areal units variables and avoid certain spatially weighted variables (as the data in this study is), neighbors are identified dominating the results of spatial models (LeSage & based on distance. Two spatial point observations i Pace, 2009). and j are considered neighbors if 0 d D, where ij Formally, each element of a binary spatial weights d is the distance between points i and j and D is the ij matrix (spatial neighbor matrix) is calculated based on bandwidth after which interaction between observa- a decision rule. For contiguity neighbors, each element tions is considered non-existent and w is assigned a ij w ¼ 1 if the two areal units represented as polygons ij 0 weight (Anselin, 1988). Assignment of a zero weight share a common boundary, and 0 otherwise. For dis- does not preclude spatial effects occurring between tance based neighbors, w ¼ 1 if d D and 0 other- ij more distant neighbors, however. Instead, influence is wise, where d is the distance between observation i modeled as a higher order recursive effect through the and j, and D is a pre-determined distance threshold estimated spatial dependence parameter as discussed above which observations are said to exhibit no direct above. Thus, observations that are not direct neigh- influence on each other. The choice of the distance bors can influence each other indirectly through inter- threshold D is not well developed in the literature and mediary neighbors that connect them. Once a binary is typically based on domain knowledge or corres- spatial weights matrix is constructed which determines pondence to other distance measurements in the data- which observations are neighbors of each other, the set. Each element of a row standardized spatial spatial weight matrix is often row standardized so that weights matrix W is calculated as: each row sums to 1. Row standardization normalizes ij spatial effects across a dataset, preventing observations w ¼ P (2) ij ij that have many spatial neighbors from dominating coefficient estimates (Anselin, 1988). s with each element of W equal to 0 or 1 in a binary Additional weighting schemes have also been specification, and 1=d in an inverse distance specifica- applied to the binary weights matrix, and it is cur- s tion if the two observations are neighbors. Matrix W rently convention to row standardize the spatial is used to link neighboring observations in spatial weight matrices after applying alternative weighting regression models, which produces estimates of coeffi- schemes. If an alternative weighting scheme is applied, cients on the resulting spatially weighted variables. the construction of the spatial weights matrix becomes a two-step process, first constructing a binary spatial Measuring Spatial Dependence neighbor matrix as above, then multiplying this matrix by another measure of spatial association. Cliff Constructing a spatial weights matrix allows for for- and Ord (1973, 1981) pioneered this concept by mul- mal testing of spatial dependence in the data generat- tiplying the binary spatial neighbor matrix by the ing process. The canonical measure of spatial inverse of the distance between observations. This dependence was developed by Moran (1950) and is approach places higher weights on neighboring obser- widely used across many fields of study. Moran’s I is vations that are closer, while still placing zero weights a global test of spatial dependence. Shortly after on neighbors that are further apart than distance D. Moran, Geary (1954) developed a formal test of local- In this study, we test the sensitivity of results of all ized autocorrelation, known as Geary’s C. Moran’s I 6 A. G. MUELLER AND S. WEILER indicates the level of global spatial autocorrelation, association also being stronger across most variables while Geary’s C indicates localized spatial autocorrel- using the inverse distance spatial weights matrix. This ation and therefore the possibility that spatial hetero- result is an indication that spatial dependence may be geneity is also present in the data generating process. present both globally (spatial autocorrelation) and Moran’s I ranges between 1 and 1, with values locally (spatial heterogeneity) in the underlying data near 1 (1) indicating the positive (negative) spatial generating processes. The dataset may include cluster- autocorrelation, and values near 0 indicating weak ing that results from socioeconomic traits, political spatial dependence of the observed variables (Moran, zoning boundary determination, and transit network 1950). Moran’s I can be interpreted as a spatial ver- design among other spatial phenomena, which is not sion of a typical correlation calculation. Geary’s C surprising, considering that spatial segregation of land ranges between 0 and 2, with values <1 demonstrat- use is one of the objectives of zoning laws (Levkovich ing increasing positive spatial autocorrelation and val- et al., 2018), and socioeconomic segregation is a widely ues >1 indicating increasing negative spatial accepted phenomenon. autocorrelation (Geary, 1954). Formally, Moran’s I is The test results from Table 2 justify using spatial calculated as: econometric modeling techniques to address the spa- P P n n tial dependence and heterogeneity that is present in w ðx xÞðx xÞ n ij i j i¼1 j¼1 P P I ¼ (3) n n the data. We develop three specifications of spatial n 2 ij ðx xÞ i¼1 j¼1 i¼1 models to correct for these spatial processes; the spa- and Geary’s C is calculated as: tial autoregressive model (SAR) which addresses spa- P P n n 2 tial dependence, the spatial error model (SEM) which w ðx x Þ ðn 1Þ ij i j i¼1 j¼1 addresses spatial heterogeneity, and the spatial Durbin C ¼ P P (4) n n n 2 2 w ij ðx xÞ i¼1 j¼1 i¼1 model (SDM) which simultaneously addresses spatial dependence and spatial heterogeneity. As suggested by Using the equations above, it is possible to test LeSage and Pace (2009), we estimate each of the mod- Moran’s I and Geary’s C statistics against their theor- els using the two most common row standardized etical values under different distributional assump- spatial weights matrices and use a Lagrange multiplier tions. We test these two statistics against their test to determine which spatial weight matrix best theoretical values under a normal Gaussian distribu- fits the data. The two spatial weighting schemes tion and the results are shown as significance stars in employed in the spatial weights matrix before row Table 2. As can be seen from the equations above, standardization are the two most common in the lit- both statistics I and C are measurements of the erature, binary and inverse distance. Both spatial covariance of deviation from the mean of observations weight matrices are then row standardized before esti- of a single variable x across a dataset, linked through mating each model. the spatial weights matrix W. Thus, one can think of the measures as clustering of deviations from the mean. If neighboring observations defined through W The Spatial Autoregressive Model deviate from the mean in the same direction, high The spatial autoregressive model (SAR) formally esti- spatial clustering (autocorrelation) is present. mates the presence of spatial dependence by incorpo- Table 2 shows that Moran’s I and Geary’s C for rating a spatially lagged dependent variable on the most variables in the current dataset are statistically right-hand side of the regression equation (Cliff & significant at the 1% level, providing confidence in Ord, 1973). Thus, observations of the dependent vari- their estimated values. Comparing Moran’s I using the able are influenced by other observations of the binary and inverse distance weight matrices, one can dependent variable nearby. In the context of the pre- see that the inverse distance matrix picks up higher sent study, the SAR model is a way of controlling for values of positive spatial correlation for demographic the influence of neighboring survey respondents’ variables (household size, vehicles, bikes, age, income, transportation mode choices on the observational unit and college education). This is an indication that the under study which represents a spatial clustering closest neighbors are more spatially correlated. Urban effect. In the binomial context, the choice variable form variables have Moran’s I values close to 1 using observed (transportation mode ¼ auto or non-auto) either spatial weights matrix, indicative of neighboring observations sharing a common urban form. Geary’s C depends on the underlying utility of the choice indica- statistics also show local positive auto-correlation tor observed. The underlying latent variable y ¼ under both spatial weights matrices, with the U U is assumed to follow a normal distribution 1i 0i JOURNAL OF SUSTAINABLE REAL ESTATE 7 Table 2. Moran’s I and Geary’s C statistics. Variable Moran’s I Moran’s I (1/d) Geary’s C Geary’s C (1/d) HH size 0.084 0.449 0.915 0.555 HH vehicles 0.065 0.427 0.934 0.569 HH bikes 0.072 0.477 0.935 0.527 Male 0.000 0.228 1.000 1.229 Age 0.035 0.277 0.962 0.734 Income (000s) 0.083 0.420 0.918 0.581 College education 0.070 0.175 0.931 0.829 Employed 0.011 0.088 0.988 0.902 Tour distance 0.024 0.081 0.979 0.935 Tour crosses highway 0.039 0.117 0.963 0.876 Miles bike lanes < 1 M. 0.838 0.953 0.156 0.048 Miles of bus routes < 0.5 M. 0.639 0.903 0.354 0.111 Bus stops < 0.5 M. 0.654 0.905 0.344 0.111 Rail stops < 0.5 M. 0.473 0.846 0.530 0.173 Miles of rail lines < 0.5 M. 0.687 0.899 0.301 0.101 Intersections with 0.5 M 0.475 0.836 0.496 0.150 CBG population/sq. mile 0.423 0.758 0.590 0.244 CBG jobs/sq. mile 0.179 0.629 0.854 0.361 Work stops 0.000 0.029 1.004 0.964 Shopping stops 0.006 0.083 0.996 0.907 Social stops 0.001 0.084 1.006 0.912 Res. low density, 0–1/4 mile 0.548 0.847 0.449 0.154 Res. medium density, 0–1/4 mile 0.392 0.808 0.580 0.190 Res. high density, 0–1/4 mile 0.520 0.820 0.486 0.186 Bus. low density, 0–1/4 mile 0.175 0.697 0.819 0.288 Bus. high density, 0–1/4 mile 0.397 0.815 0.579 0.187 Ind., 0–1/4 mile 0.269 0.698 0.667 0.253 Res. low density, 1/4–1/2 mile 0.681 0.899 0.315 0.103 Res. medium density, 1/4–1/2 mile 0.649 0.895 0.326 0.104 Res. high density, 1/4–1/2 mile 0.658 0.880 0.360 0.125 Bus. low density, 1/4–1/2 mile 0.355 0.778 0.667 0.224 Bus. high density, 1/4–1/2 mile 0.527 0.848 0.454 0.148 Ind., 1/4–1/2 mile 0.469 0.797 0.449 0.167 Res. low density, 1/2–3/4 mile 0.476 0.566 0.539 0.444 Res. medium density, 1/2–3/4 mile 0.356 0.477 0.611 0.503 Res. high density, 1/2–3/4 mile 0.362 0.479 0.654 0.522 Bus. low density, 1/2–3/4 mile 0.440 0.529 0.586 0.489 Bus. high density, 1/2–3/4 mile 0.282 0.388 0.737 0.591 Ind., 1/2–3/4 mile 0.363 0.475 0.599 0.476 Res. low density, 3/4–1 mile 0.515 0.599 0.500 0.406 Res. medium density, 3/4–1 mile 0.402 0.525 0.570 0.470 Res. high density, 3/4–1 mile 0.333 0.446 0.681 0.551 Bus. low density, 3/4–1 mile 0.420 0.517 0.589 0.504 Bus. high density, 3/4–1 mile 0.303 0.430 0.708 0.557 Ind., 3/4–1 mile 0.402 0.518 0.569 0.448 Note. Significance at the 1, 5, and 10% levels shown by , , and , respectively. in the probit model estimation. The general spatial Typically, the SAR model is used to adjust for autoregressive model in a binomial context can be for- dependent variables that have a direct effect on the mally stated in the system of equations as: realization of the dependent variable in close proxim- ity. The classic example is SAR hedonic pricing mod- y ¼ qWy þ Xb þ e (5) els of residential home values (e.g., Pace & Barry, e Nð0, r I Þ 2004), where the value of a house sold has a direct y ¼ 1, if y 0 impact on other residential home prices in the area and has been shown to be a valuable addition to trad- y ¼ 0, if y < 0 itional home price models (Anselin & Lozano-Gracia, Where y is the unobserved latent utility of mode 2007). Conceptually, the SAR would be the correct choice, y ¼ 1 if the binomial choice is observed, and 0 model for the underlying data generating process if a otherwise, W is the spatial weights matrix, q is an esti- survey respondent’s choice to use auto or non-auto mated spatial dependence parameter of spatial autocorrel- transportation depended upon neighboring survey ation between observations, X is a matrix of independent respondents’ transportation mode choices, i.e. a clus- variables, and b is a vector of estimated coefficients. The tering effect of mode choice. latent utility construct implies that Prðy ¼ 1Þ¼ While theoretically, the model has justifiable merit PrðU U Þ¼ Prðy 0Þ (LeSage & Pace, 2009). in controlling for spatial dependence, it is important to 1i 0i i 8 A. G. MUELLER AND S. WEILER note that this model does not distinguish the direction and the estimation of its spatial lag parameter k. of causality, only the association of built environment Unlike the SAR model, indirect and direct effects can- characteristics with transportation mode choice. It is not be estimated because there is no feedback loop of quite possible that people who enjoy non-auto forms of changes in the dependent regressors of neighboring transportation tend to live in the same locations observations on the dependent variable since there is because these locations provide employment, leisure, no autocorrelation parameter present. The parameter and shopping in close enough proximity to make non- k represents the extent to which heterogeneous inde- auto trips more convenient. However, this model does pendent coefficient estimates vary across space. This is identify if there is spatially clustered transportation the correct model to use if neighboring respondents’ behavior, and how fast this clustering effect deteriorates transportation mode choices do not affect an individ- with distance. If the spatial dependence parameter q is ual’s mode choice, but the effect of independent varia- significant, explicitly modeling spatial dependence is bles have varying effects across space, such as the justified and therefore relevant to the study of spatial variation in the effect of income described earlier. associations between zoning laws and transportation choices. While this model cannot determine the under- Spatial Durbin Model lying cause of clustering in mode choice, it does con- The spatial Durbin model (SDM) allows for the esti- trol for the spatial phenomenon, leading to unbiased mation of both spatial autocorrelation and spatial het- estimates of the association between urban form and erogeneity simultaneously by including a spatially transportation behavior. It is also important to note that in the SAR model, lagged dependent variable as well as spatially lagged the spatial dependence parameter q incorporates a independent variables in a single model. The advan- feedback loop in the effect of neighboring observa- tages of this model are the simultaneous control of tions on the dependent variable. There is a direct both spatial dependence and spatial heterogeneity, but effect of independent variables on transportation in practice can suffer from the curse of dimensional- choice, and this transportation choice then indirectly ity. One advantage of the Bayesian approach to model effects transportation mode choices of neighboring estimation employed in this study and described observations, which in turn affect the observation below is the ability to estimate such models without under study, creating a spatial feedback loop effect. running into non-convergence problems. These prob- Thus, direct, indirect, and total effects of independent lems can be a significant challenge with the maximiza- variables on the dependent variable are estimated. tion procedures employed in maximum likelihood and generalized method of moments estimation, and often lead to severe computational challenges. The binomial The Spatial Error Model probit SDM model can be formally stated as: In contrast to the spatial autoregressive model, the spa- y ¼ qWy þ Xb þ WXh þ e (7) tial error model (SEM) allows for heterogeneous effects e Nð0, r I Þ of independent regressors across space. This adaptation y ¼ 1, if y 0 of the traditional OLS or probit model allows for both i y ¼ 0, if y < 0 global coefficients (b) and local variation across the space of coefficients to be modeled in the error struc- where q is the estimated parameter of spatial autocor- ture. In the binomial choice context, the latent variable relation of the dependent variable, b is the estimated approach of unobserved utility of the resulting choice vector of parameters on the independent variables, indicator is used for the probit estimator similar to the and h is the vector of estimated parameters on the process described for the SAR model. The SEM bino- spatially lagged independent variables. The estimation mial choice model can be formally stated as: of the SDM is similar to that of the SAR model with y ¼ Xb þ u (6) the independent variables multiplied by the spatial weights matrix added as additional independent varia- u ¼ kWu þ e 2 bles, WX. The resulting model then produces a vector ðÞ e N 0, r In of global effects of the independent variables b and a y ¼ 1, if y 0 vector of local effects of the independent variables h. y ¼ 0, if y < 0 LeSage and Pace (2009) detail the advantages of where W is the spatial weights matrix. The SEM each spatial modeling approach, and determine that model allows for spatial variance of the error term when the correct model is unknown and not dictated JOURNAL OF SUSTAINABLE REAL ESTATE 9 by theory, only the SDM gives unbiased results even if (1992) because the posterior distributions are available the true model is SAR or SEM. More particularly, to calculate valid inference measures of the parameter when the true data generating process is the SEM estimates, thus escaping the bias inherent in model, SAR and SDM will produce unbiased but inef- McMillen’s algorithm and the necessity to specify the ficient estimates. When the true data generating pro- functional form of model variance over space a priori. cess is the SAR model, the SEM model produces The likelihood function for the SAR, SDM, and SEM biased estimates, while the SDM does not. If the true models is: data generating process is the SDM model, the other 1 1 ðÞ Ly, Wq, b, r ¼ jI qWjexp e e ðÞ models will have omitted variable bias. The SAR, n ðÞ 2 2r 2pr SEM, and SDM versions of the travel behavior—built (8) environment models are estimated in Table 3 using both a binary and inverse distance weighted row e ¼ðI qWÞ y Xb for the SAR model, standardized spatial weights matrix. e ¼ðI qWÞðy XhÞ Xb for the SDM model, e ¼ðI kWÞðy XbÞ for the SEM model (LeSage, 2000,p. 23). Estimation Techniques It is important to note that the Bayesian approach McMillen (1992) was the first to propose techniques for to modeling is fundamentally different from that of the estimating the SAR and SEM probit models. Due to the frequentist approach employed in OLS, probit, and complicated error structure of the SAR and SEM probit other canonical statistical models. The results of models, direct maximum-likelihood estimation is not Bayesian estimation produce full distributions of par- possible; however, in McMillen’s procedure, the discrete ameter estimates, and convention is to report the mean variable is replaced by the expected value of the under- of each parameter distribution. Significance tests are lying latent variable, and the expectation is calculated then the probability of the parameter estimate contain- iteratively until convergence. McMillen (1992), among ing zero calculated directly from the parameter distri- others, deems this procedure impractical for large data- bution. This approach is fundamentally different from sets. LeSage (2000) outlines several other drawbacks to the frequentist approach, which calculates the probabil- the procedure. First, the estimation procedure requires ity of the parameter estimate being zero from the the estimation of the likelihood function, which prohib- standard errors of each estimate and the underlying its the use of the information matrix for calculating the distributional assumption (often Gaussian) of the errors precision of the parameter estimates. Attempts to cir- (Albert, 2007; Albert & Chib, 1993; LeSage, 2000). cumvent this problem produces biased estimates of the covariance matrix. Second, McMillen’s approach Econometric Models and Results requires the researcher to specify a functional form of the heteroskedastic spatial variance and leads to varying Econometric Model inferences across alternative specifications. Alternatively, Three econometric models are specified following the Bayesian estimation techniques do not require these theoretical specifications for the SAR, SEM, and SDM assumptions about the functional form of the error pro- above. The binary choice indicator variable y is set to cess. We, therefore, implement a spatial Bayesian tech- 1 if the survey respondent used non-auto transporta- nique to estimate the spatial probit models in this study. tion for an observed tour, and 0 otherwise. The spatial Following the work of Albert and Chib (1993) and probit model for the SAR, SEM, and SDM is com- Chib (1992), which detail the estimation of probit and prised of the travel choice indicator variable and the logit models for discrete choices using Markov Chain independent regressors which are the same as in Monte Carlo estimation in a Bayesian context, LeSage Mueller and Trujillo (2019). The formal equation to (2000) proposes a Bayesian estimation technique be estimated for the SDM is then: based on the Gibbs sampling approach (Albert & y ¼ qWy þ Xb þ WXh þ e (9) Chib, 1993). The estimation technique specifies a i complete set of prior distributions for all parameters e Nð0, r InÞ in the model and then samples from these distribu- y ¼ 1, if y 0 tions until a large number of parameter draws are y ¼ 0, if y < 0 obtained that converge to the true joint posterior dis- tribution of the parameters. This approach overcomes where X ¼½IS BE where I is an n 1vector of the drawbacks of the approach proposed by McMillen ones, S is a matrix of sociodemographic characteristics, 10 A. G. MUELLER AND S. WEILER Table 3. Model coefficient comparison. Dependent variable: non-auto transportation mode ¼ 1 Variable Probit SAR SAR (1/d) SEM SEM (1/d) SDM SDM (1/d) Intercept 0.029 0.419 0.279 0.433 246.400 0.329 1.346 Household size 0.050 0.052 0.052 0.108 75.260 0.067 0.061 Household vehicles 0.363 0.368 0.368 0.777 531.200 0.362 0.377 Household bikes 0.057 0.052 0.053 0.116 79.000 0.054 0.060 Male 0.190 0.186 0.185 0.398 271.200 0.178 0.181 Age 0.013 0.014 0.014 0.029 19.950 0.014 0.014 Income 0.000 0.000 0.000 0.000 0.238 0.000 0.000 College education 0.050 0.038 0.044 0.107 71.320 0.037 0.036 Employed 0.078 0.084 0.085 0.180 121.400 0.092 0.090 Tour distance 0.013 0.013 0.013 0.027 19.100 0.013 0.012 Tour crosses highway 0.780 0.776 0.781 1.690 1174.000 0.822 0.834 Miles bike lane < 1 mile 0.009 0.013 0.010 0.022 13.590 0.015 0.063 Mile bus routes < 1/2 mile 0.014 0.015 0.015 0.031 21.950 0.011 0.003 Bus stops < 1/2 mile 0.010 0.008 0.009 0.022 14.260 0.009 0.013 Rail stops < 1 mile 0.090 0.114 0.117 0.263 177.900 0.061 0.145 Miles rail lines < 1 mile 0.040 0.051 0.053 0.117 82.400 0.035 0.039 Intersection < 1/2 mile 0.001 0.000 0.000 0.001 0.420 0.001 0.002 CBG population density 0.000 0.000 0.000 0.000 0.023 0.000 0.000 CBG employment density 0.000 0.000 0.000 0.000 0.003 0.000 0.000 Work stops 0.045 0.040 0.040 0.087 61.510 0.045 0.039 Shopping stops 0.187 0.184 0.185 0.393 271.400 0.194 0.200 Social stops 0.102 0.102 0.104 0.222 144.700 0.130 0.120 Res. low density, 0–1/4 mile 0.010 0.010 0.010 0.022 15.440 0.013 0.009 Res. med. density, 0–1/4 mile 0.009 0.010 0.009 0.020 13.920 0.012 0.011 Res. high density, 0–1/4 mile 0.010 0.011 0.011 0.024 17.230 0.011 0.009 Bus. low density, 0–1/4 mile 0.011 0.006 0.006 0.012 7.865 0.000 0.015 Bus. high density, 0–1/4 mile 0.003 0.005 0.004 0.010 7.086 0.006 0.003 Industrial, 0–1/4 mile 0.007 0.008 0.008 0.017 12.580 0.008 0.014 Res. low density, 1/4–1/2 mile 0.014 0.017 0.015 0.033 23.380 0.018 0.025 Res. med. density, 1/4–1/2 mile 0.012 0.016 0.014 0.030 21.090 0.016 0.024 Res. high density, 1/4–1/2 mile 0.015 0.019 0.018 0.040 28.340 0.013 0.027 Bus. low density, 1/4–1/2 mile 0.014 0.004 0.006 0.014 6.699 0.014 0.003 Bus. high density, 1/4–1/2 mile 0.003 0.004 0.003 0.005 3.929 0.005 0.023 Industrial, 1/4–1/2 mile 0.003 0.013 0.012 0.025 18.090 0.008 0.024 Res. low density, 1/2–3/4 mile 0.003 0.018 0.017 0.035 24.200 0.020 0.020 Res. med. density, 1/2–3/4 mile 0.002 0.023 0.021 0.049 32.560 0.033 0.024 Res. high density, 1/2–3/4 mile 0.006 0.021 0.020 0.044 31.110 0.021 0.034 Bus. low density, 1/2–3/4 mile 0.023 0.028 0.027 0.079 52.490 0.047 0.074 Bus. high density, 1/2–3/4 mile 0.001 0.016 0.015 0.030 20.700 0.018 0.049 Industrial, 1/2–3/4 mile 0.013 0.019 0.018 0.039 26.930 0.014 0.022 Res. low density, 3/4–1 mile 0.005 0.017 0.015 0.033 22.870 0.024 0.024 Res. med. density, 3/4–1mile 0.005 0.031 0.029 0.063 42.400 0.033 0.025 Res. high density, 3/4–1 mile 0.012 0.002 0.001 0.002 1.396 0.005 0.023 Bus. low density, 3/4–1 mile 0.007 0.032 0.032 0.070 50.070 0.055 0.047 Bus. high density, 3/4–1 mile 0.010 0.036 0.034 0.074 50.080 0.047 0.047 Industrial, 3/4–1 mile 0.003 0.021 0.020 0.043 30.700 0.015 0.022 q, k 0.193 0.073 0.147 0.032 0.876 0.373 Note. Significance at the 1, 5, and 10% levels shown by , , and , respectively. and BE is a matrix of built environment characteristics miles. This distance corresponds closely with the dis- including zoning variables and follows the same substi- tance bands used to calculate the zoning percentages surrounding survey respondents’ residences and there- tution for the SAR and SEM. In the SEM model, both fore is an ideal choice for D. While it is possible to q and h are set to 0 and k enters the distribution of e as described in Equation (6).In the SAR h is set to 0. estimate spatial models with some observations having no neighbors, in practice, this also causes far more problems than the benefits of having more restrictive Determination of the Spatial Weights Matrix definitions of spatial neighbors, as outlined by Bivand The spatial weight matrix, W, in the equations above is and Portnov (2004). Using this distance-based neighbor developed by a two-step process. In the first step, rule, the neighbor binary matrix is constructed, with observations are determined to be spatial neighbors if observations within D distance of each other assigned a they are within a distance D from one another. The 1, and observations further apart than D assigned a 0. bandwidth used to create the neighbor matrix was the In the second step, the neighbor matrix is trans- minimum straight line distance necessary so that each formed into a spatial weight matrix W by either row observation included at least one neighbor, D ¼ 1.076 standardizing the binary neighbor matrix so that all JOURNAL OF SUSTAINABLE REAL ESTATE 11 rows sum to 1, or applying a function based on dis- (1%) increases the probability of a non-auto trip by tance and then row standardizing the matrix. While 0.01 0.235 ¼ 0.00235 (0.235%) on average. This value there are no generally accepted procedures for deter- of q also indicates that the dissipation of the effect is mining the correct weighting structure to use for W, quite rapid, as second order neighboring observations exhibit an effect of 0.235 ¼ 0.055225 and an effect of we apply the two most commonly used weighting schemes, the binary neighbor matrix, and a weight a 1% increase is equal to 0.01 0.05225 ¼ 0.00055225 that declines with a distance where the weight of each (0.055225%). This finding is further evidence that the choice between auto and non-auto transportation is neighboring observation is set to the inverse of dis- somewhat localized to a one-mile radius surrounding tance, w ¼ 1=d , where d is the distance between ij ij observations i and j in miles. We estimate the SAR, place of residence. Using the inverse distance W SEM, and SDM models using each spatial weight which places higher weights on closer neighbors, q is 0.159. The fact that q is a lower in this weighting matrix and compare the results below. scheme gives further evidence that the effect of neigh- boring observations is weak at closer distances than Estimation 1 mile. In this model, the coefficient has a slightly dif- The SAR and SEM have been estimated in the past ferent interpretation, as the vector Wy is not a simple using maximum likelihood techniques, as well as percentage, but rather a spatially weighted percentage more recently with Bayesian techniques. The estima- of non-auto trips based on distance. tion of the model using Bayesian techniques has some The value of k in the SEM model using binary and advantages over maximum likelihood, the most inverse distance W is 0.333 and 0.069, respectively. The important being the recovery of the posterior coeffi- coefficient k is estimated using Equation (6),where u ¼ cient distributions which can be used for statistical kWu þ e, u are the errors from the normal probit equa- inference tests (LeSage, 2000). The SAR, SEM, and tion, W is the spatial weights matrix, and e are the SDM models are estimated with a Bayesian model residuals after spatial correction. The SEM model only that takes 1,000 draws with a burn-in of 100 draws. addresses the spatial correlation of errors across space, Model results are listed in Table 3. and therefore only corrects for spatial heteroskedasticity. The positive and significant values of k indicate the cor- relation between error terms that are neighbors. Spatial Dependence Parameters However, since the SAR model also demonstrates spa- The model results for both the SAR and SEM models tial autocorrelation, part of the error term spatial correl- without zoning variables included using both a binary ation may be due to missing variable bias since the neighbor row standardized and inverse distance row spatially lagged dependent variable is absent from this standardized spatial weights matrix show that there is model. The general conclusion from the significant val- spatial dependence, with the spatial parameters q and k ues of q and k indicate that a model that jointly statistically significant at the 5% level in the SAR and addresses both spatial dependence and spatial heteroske- 10% level in the SEM model using the inverse distance dasticity may be the correct model (the SDM model). weight matrix, and q and k statistically significant at The results from the SDM estimate the q parameter the 5 and 10% level in the SAR and SEM models using of 0.876 and 0.373 for the binary and inverse dis- the row standardized binary spatial weights matrix. The tance weights matrix, respectively. This value indicates value of q in the SAR using a binary W and inverse negative spatial autocorrelation when spatially distance W are 0.235 and 0.159, respectively. The lower weighted independent regressors are added to the coefficient on q in the SAR model using the inverse model. This is unexpected as positive spatial autocor- distance spatial weights matrix shows the impact of the relation was estimated in the SAR model, and the choice of W, as the inverse distance specification results from the SEM indicated that controlling for already weights closer neighbors more heavily. spatial heteroskedasticity was warranted. A negative In the model using binary W, q is the estimated spatial dependence parameter indicates that the choice parameter on the n 1 vector Wy, where y is a vector of auto by a neighbor increases the probability of of 1s and 0s indicating non-auto transportation, and non-auto use. This peculiar result seems to point to thus Wy can be understood as the percentage of non- the complex nature of transportation decisions, as auto trips of all neighboring observations. Therefore, a spatially clustered use of auto or non-auto would be value of q equal to 0.235 tells us that an increase in expected due to preferences for locating near trans- the percentage of non-auto trips of neighbors by 0.01 portation networks of choice. However, it may be the 12 A. G. MUELLER AND S. WEILER Table 4. Marginal effects: SDM model, binary W dependent case that this negative spatial autocorrelation of the variable: non-auto transportation mode ¼ 1. dependent variable is picking up a preference for non- Variable Direct Indirect Total auto travel when neighbors neighboring auto choice HH size 0.01286 0.00603 0.00683 creates congestion. HH vehicles 0.06941 0.03264 0.03677 HH bicycles 0.01032 0.00485 0.00547 Male 0.03407 0.01604 0.01802 Age 0.00267 0.00126 0.00142 Zoning Parameters Income 0.00001 0.00000 0.00000 College degree 0.00703 0.00331 0.00372 The most notable result of the SAR models is that the Employed 0.01760 0.00826 0.00934 coefficients on all three residential zoning density lev- Tour distance 0.00242 0.00114 0.00128 Tour crosses highway 0.15734 0.07399 0.08334 els are negative and statistically significant in the 0– Miles bike lanes < 1M. 0.00297 0.00140 0.00157 1/4 mile distance band. This indicates that higher lev- Miles bus routes < 0.5 M. 0.00208 0.00098 0.00110 Bus stops < 0.5 M. 0.00177 0.00083 0.00094 els of residential zoning surrounding respondents’ res- Rail stops < 0.5 M. 0.01162 0.00548 0.00614 idences are associated with a decreased likelihood of Miles rail lines < 1 M. 0.00678 0.00319 0.00359 observing non-auto transportation. The other result Intersections < 0.5 M. 0.00011 0.00005 0.00006 CBG population/sq. mile 0.00000 0.00000 0.00000 that indicates the potential association between zoning CBG jobs/sq. mile 0.00000 0.00000 0.00000 and travel behavior is the significant positive coeffi- Work stops 0.00860 0.00404 0.00456 Shopping stops 0.03715 0.01747 0.01968 cients in the 1/4–1/2 and 3/4–1 mile band for indus- Social stops 0.02484 0.01166 0.01317 trial zoning, and high density business zoning in the Res. low density, 0–1/4 mile 0.00256 0.00120 0.00136 Res. medium density, 0–1/4 mile 0.00234 0.00110 0.00124 3/4–1 mile band. This indicates that business and Res. high density, 0–1/4 mile 0.00215 0.00101 0.00114 industrial zoning moderately close to home is associ- Bus. low density, 0–1/4 mile 0.00003 0.00001 0.00002 Bus. high density, 0–1/4 mile 0.00121 0.00057 0.00064 ated with increased non-auto transportation. This Ind., 0–1/4 mile 0.00158 0.00074 0.00084 indicates that residential locations surrounded by a Res. low density, 1/4–1/2 mile 0.00339 0.00159 0.00180 Res. medium density, 1/4–1/2 mile 0.00301 0.00141 0.00159 band of residential zoning up to one half mile may Res. high density, 1/4–1/2 mile 0.00258 0.00121 0.00137 prefer to drive to shopping, employment, and recre- Bus. low density, 1/4–1/2 mile 0.00264 0.00124 0.00140 Bus. high density, 1/4–1/2 mile 0.00097 0.00045 0.00051 ation, and that zoning that precludes closer businesses Ind., 1/4–1/2 mile 0.00161 0.00075 0.00085 may be associated with more non-auto travel behav- Res. low density, 1/2–3/4 mile 0.00391 0.00184 0.00207 Res. medium density, 1/2–3/4 mile 0.00632 0.00298 0.00335 ior. The coefficients on the SEM model are all statis- Res. high density, 1/2–3/4 mile 0.00410 0.00193 0.00217 tically insignificant so no interpretation can be made Bus. low density, 1/2–3/4 mile 0.00903 0.00424 0.00479 Bus. high density, 1/2–3/4 mile 0.00337 0.00159 0.00178 for this model. Ind., 1/2–3/4 mile 0.00271 0.00127 0.00144 While the sign of the coefficients on the explana- Res. low density, 3/4–1 mile 0.00453 0.00213 0.00240 tory variables indicates the direction of association on Res. medium density, 3/4–1 mile 0.00639 0.00301 0.00339 Res. high density, 3/4–1 mile 0.00090 0.00042 0.00048 the conditional probability of non-auto transportation Bus. low density, 3/4–1 mile 0.01049 0.00493 0.00557 behavior, their magnitude cannot be interpreted in the Bus. high density, 3/4–1 mile 0.00892 0.00420 0.00473 Ind., 3/4–1 mile 0.00292 0.00138 0.00155 same way as OLS or probit models. Due to the non- (W)HH size 0.08839 0.04159 0.04681 linearity of the model, and the presence of spatial (W)HH vehicles 0.00532 0.00248 0.00284 (W)HH bicycles 0.04547 0.02137 0.02410 dependence, the impact on a change of one explana- (W)Male 0.03762 0.01767 0.01995 tory variable has a spatial feedback loop effect on the (W)Age 0.00468 0.00220 0.00248 (W)Income 0.00046 0.00021 0.00024 dependent variable due to the presence of the spatially (W)College degree 0.10217 0.04776 0.05440 lagged dependent variable in the estimated equation. (W)Employed 0.04803 0.02297 0.02507 (W)Tour distance 0.00588 0.00279 0.00309 Therefore, it is necessary to estimate the marginal (W)Tour crosses highway 0.08159 0.03795 0.04364 effects of the change in each explanatory variable in (W)Miles bike lanes < 1M. 0.01139 0.00534 0.00605 (W)Miles bus routes < 0.5 M. 0.01100 0.00517 0.00583 the model to determine the direct, indirect, and total (W)Bus stops < 0.5 M. 0.00205 0.00097 0.00108 effects. We list the marginal effects of the SDM in (W)Rail stops < 0.5 M. 0.01590 0.00753 0.00837 (W)Miles rail lines < 1 M. 0.00506 0.00239 0.00267 Table 4. (W)Intersections < 0.5 M. 0.00045 0.00021 0.00024 While the SAR and SEM models both show signifi- (W)CBG population/sq. mile 0.00002 0.00001 0.00001 (W)CBG jobs/sq. mile 0.00000 0.00000 0.00000 cance in some of the zoning variables in determining (W)Work stops 0.06333 0.02960 0.03372 mode choice, the SDM shows significance in the low, (W)Shopping stops 0.07852 0.03718 0.04134 medium, and high residential zoning types for the (W)Social stops 0.13294 0.06236 0.07058 (W)Res. low density, 0–1/4 mile 0.00601 0.00283 0.00318 binary spatial weights matrix, significant negative (W)Res. medium density, 0–1/4 mile 0.00114 0.00054 0.00059 impacts for all spatially weighted zoning variables in (W)Res. high density, 0–1/4 mile 0.00340 0.00159 0.00181 (W)Bus. low density, 0–1/4 mile 0.00506 0.00239 0.00266 the 1/2–3/4 mile zoning band, and positive associa- (continued) tions of spatially weighted high density business and JOURNAL OF SUSTAINABLE REAL ESTATE 13 Table 4. Continued. Table 5. Log likelihood tests. Variable Direct Indirect Total Log Degrees of Model likelihood freedom AIC BIC (W)Bus. high density, 0–1/4 mile 0.00363 0.00170 0.00193 (W)Ind., 0–1/4 mile 0.01885 0.00885 0.01000 Base models: no zoning variables (W)Res. low density, 1/4–1/2 mile 0.00937 0.00440 0.00497 Probit 1840.061 22 3724.122 3868.035 (W)Res. medium density, 1/4–1/2 mile 0.00202 0.00094 0.00109 SAR 1838.980 23 3723.960 3874.415 (W)Res. high density, 1/4–1/2 mile 0.00101 0.00048 0.00053 SAR(1/d) 1839.615 23 3725.230 3875.685 (W)Bus. low density, 1/4–1/2 mile 0.01733 0.00809 0.00925 SEM 1860.135 24 3768.270 3925.266 (W)Bus. high density, 1/4–1/2 mile 0.00488 0.00228 0.00260 SEM(1/d) 1843.571 24 3735.143 3892.139 (W)Ind., 1/4–1/2 mile 0.00618 0.00291 0.00327 SDM 1821.144 44 3730.288 4018.114 (W)Res. low density, 1/2–3/4 mile 0.06475 0.03043 0.03432 SDM(1/d) 1817.655 44 3723.309 4011.135 (W)Res. medium density, 1/2–3/4 mile 0.10647 0.05002 0.05645 Zoning models (W)Res. high density, 1/2–3/4 mile 0.04537 0.02132 0.02405 Probit 1811.343 46 3714.687 4015.595 (W)Bus. low density, 1/2–3/4 mile 0.11828 0.05553 0.06275 SAR 1808.999 47 3711.999 4019.449 (W)Bus. high density, 1/2–3/4 mile 0.04730 0.02224 0.02506 SAR(1/d) 1810.157 47 3714.314 4021.764 (W)Ind., 1/2–3/4 mile 0.08333 0.03914 0.04420 SEM 1865.249 48 3826.498 4140.490 (W)Res. low density, 3/4–1 mile 0.06179 0.02904 0.03275 SEM(1/d) 1861.738 48 3819.475 4133.467 (W)Res. medium density, 3/4–1 mile 0.12995 0.06106 0.06890 SDM 1772.528 92 3729.057 4330.874 (W)Res. high density, 3/4–1 mile 0.03219 0.01514 0.01704 SDM(1/d) 1778.245 92 3740.491 4342.308 (W)Bus. low density, 3/4–1 mile 0.09916 0.04657 0.05259 (W)Bus. high density, 3/4–1 mile 0.05024 0.02356 0.02667 of the model parameters. These sample distributions Notes. Significance at the 1, 5, and 10% levels shown by , , and , respectively. of coefficients can be used to compute average mar- (W) preceding variable names indicates variables pre-multiplied by the ginal effects across observations of a change in an spatial weights matrix in SDM models. independent variable of the model on the probability of the independent variable, non-auto travel mode industrial zoning in the 3/4–1 mile zoning band pro- choice (LeSage & Pace, 2009). While the SEM model viding some evidence of association of association coefficients can be interpreted as marginal effects as between zoning and mode choice. in ordinary least squares because the spatial variation The SDM with binary spatial weights matrix was is only present in the error term, for the SAR and the best model using the log likelihood test. The SAR SDM models which include spatially lagged dependent with binary spatial weights matrix is indicated as the or independent variables, the impacts of a change in best model using Akaike Information Criteria (AIC) an explanatory variable can have an impact on all (Akaike, 1974) and the standard probit model is indi- other neighboring dependent variables, creating a cated as the best model using the Bayesian feedback loop with several orders of magnitude. Thus, Information Criteria (BIC) (Schwarz, 1978). A sum- these spatial models exhibit direct, indirect, and total mary of the log likelihood, AIC, and BIC of each of impacts. LeSage and Pace (2009) propose summary the models tested is shown in Table 5. All models measures of the marginal effects of a change in an show relatively similar results from the log likelihood, explanatory variable x by using the average change in AIC, and BIC criteria. However, the aggregate results r the expected value of the dependent variable y and of the Moran’s I, Geary’s C, and the significance of changing the multiplier matrix S (W) based on the both SAR and SEM models indicate that there may be spatial model. The expected value of a change is listed both spatial autocorrelation and spatial heterogeneity in Equation (10), where X is an n p matrix of n in the data generating process of the underlying data- observations and p explanatory variables. set. Therefore, the only model that produces unbiased results is the SDM (LeSage & Pace, 2009), Comparing EðyÞ¼ S ðWÞx þ aI (10) r r the use of the two spatial weights matrices in each i1 model, the binary row standardized spatial weights matrix leads to a better posterior distribution fit to S ðWÞ for the SAR and SDM model are given in the data, indicating that the binary matrix is preferred Equations (11) and (12). to the inverse distance matrix. This indicates that the The diagonal elements of the trace of the S ðWÞ matrix multiplied by the change in independent vari- spatial effects may be strong within the distance used able x give the direct impacts (Equation 13), while to specify spatial neighbors, just over one mile. ir the trace of the entire S (W) matrix multiplied by the change in independent variable x gives the total ir Marginal Effects and Elasticities impacts (Equation 14). Indirect impacts are the differ- The Bayesian Markov Chain Monte Carlo (MCMC) ence between total and direct impacts (Equation 15). estimation technique used to estimate the models Marginal direct effects for individual observations above produces samples of the posterior distribution are contained in the diagonal elements of S (W) r 14 A. G. MUELLER AND S. WEILER Table 6. Elasticities: SDM model, binary W dependent vari- (Equation 16) and indirect marginal effects are con- able: non-auto transportation mode ¼ 1. tained in the off diagonal elements of S (W)(Equation Variable Direct Indirect Total 17) (LeSage & Pace, 2009). HH size 0.077 0.036 0.041 HH vehicles 0.417 0.196 0.221 S ðWÞ¼ ðI qWÞ b (11) r n HH bicycles 0.062 0.029 0.033 S ðÞ W ¼ I qW I b þ Wh (12) ðÞðÞ Age 0.016 0.008 0.009 r n n r Income 0.000 0.000 0.000 MðrÞdirect ¼ n trðS ðWÞÞ (13) Tour distance 0.015 0.007 0.008 1 1 Miles bike lanes < 1M. 0.018 0.008 0.009 Mr total ¼ n I SðÞ W I (14) ðÞ r n Miles bus routes < 0.5 M. 0.012 0.006 0.007 ** Bus stops < 0.5 M. 0.011 0.005 0.006 MrðÞindirect ¼ MrðÞtotal MrðÞdirect (15) Rail stops < 0.5 M. 0.070 0.033 0.037 @y i Miles rail lines < 1 M. 0.041 0.019 0.022 ¼ S W (16) r ii Intersections < 0.5 M. 0.001 0.000 0.000 @x ir CBG population/sq. mile 0.000 0.000 0.000 @y i CBG jobs/sq. mile 0.000 0.000 0.000 ¼ S W (17) r ij Work stops 0.052 0.024 0.027 @x jr Shopping stops 0.223 0.105 0.118 Social stops 0.149 0.070 0.079 To calculate the elasticities, the change of each vari- Res. low density, 0–1/4 mile 0.015 0.007 0.008 Res. medium density, 0–1/4 mile 0.014 0.007 0.007 able is taken at the mean of the posterior distribution Res. high density, 0–1/4 mile 0.013 0.006 0.007 and the mean of the expected probability of the binary Bus. low density, 0–1/4 mile 0.000 0.000 0.000 Bus. high density, 0–1/4 mile 0.007 0.003 0.004 dependent variable, which is 16.63%. Marginal effects Ind., 0–1/4 mile 0.010 0.004 0.005 are reported for the direct, indirect, and total marginal Res. low density, 1/4–1/2 mile 0.020 0.010 0.011 Res. medium density, 1/4–1/2 mile 0.018 0.008 0.010 effects of a change in each independent variable. Res. high density, 1/4–1/2 mile 0.016 0.007 0.008 Direct effects are the change in the probability of Bus. low density, 1/4–1/2 mile 0.016 0.007 0.008 observing non-auto mode choice attributed to the Bus. high density, 1/4–1/2 mile 0.006 0.003 0.003 Ind., 1/4–1/2 mile 0.010 0.005 0.005 change in the independent variable. Indirect effects Res. low density, 1/2–3/4 mile 0.024 0.011 0.012 represent the spatially lagged effect on the autocorre- Res. medium density, 1/2–3/4 mile 0.038 0.018 0.020 Res. high density, 1/2–3/4 mile 0.025 0.012 0.013 lated dependent variable of a change in one of the Bus. low density, 1/2–3/4 mile 0.054 0.025 0.029 independent variables after the feedback loop from a Bus. high density, 1/2–3/4 mile 0.020 0.010 0.011 Ind., 1/2–3/4 mile 0.016 0.008 0.009 change in an independent variable has affected the Res. low density, 3/4–1 mile 0.027 0.013 0.014 spatially lagged dependent variable of spatial neighbor Res. medium density, 3/4–1 mile 0.038 0.018 0.020 Res. high density, 3/4–1 mile 0.005 0.003 0.003 observations. The sum of direct and indirect effects Bus. low density, 3/4–1 mile 0.063 0.030 0.033 equals the total effect of a change in the independent Bus. high density, 3/4–1 mile 0.054 0.025 0.028 Ind., 3/4–1 mile 0.018 0.008 0.009 variables after the feedback loop of the change has (W)HH size 0.531 0.250 0.281 run its course. Dummy variable elasticities are not (W)HH vehicles 0.032 0.015 0.017 (W)HH bicycles 0.273 0.128 0.145 reported. Results for the SDM model are reported for (W)Age 0.028 0.013 0.015 the binary spatial weights matrix in Table 6. (W)Income 0.003 0.001 0.001 Several of the statistically significant variables in the (W)Tour distance 0.035 0.017 0.019 (W)Miles bike lanes < 1M. 0.068 0.032 0.036 SAR model have small marginal effects on the expected (W)Miles bus routes < 0.5 M. 0.066 0.031 0.035 sign. The largest of these is the number of household (W)Bus stops < 0.5 M. 0.012 0.006 0.007 (W)Rail stops < 0.5 M. 0.096 0.045 0.050 vehicles, with a marginal effect of 9.123%. This is not (W)Miles rail lines < 1 M. 0.030 0.014 0.016 surprising considering this variable indicates preference (W)Intersections < 0.5 M. 0.003 0.001 0.001 (W)CBG population/sq. mile 0.000 0.000 0.000 for owning an asset that encourages automobile trans- (W)CBG jobs/sq. mile 0.000 0.000 0.000 portation. Miles of bike lanes have an unexpected nega- (W)Work stops 0.381 0.178 0.203 (W)Shopping stops 0.472 0.224 0.249 tive marginal effect but is very small. One possible (W)Social stops 0.799 0.375 0.424 reason for the unexpected sign on this variable may be (W)Res. low density, 0–1/4 mile 0.036 0.017 0.019 (W)Res. medium density, 0–1/4 mile 0.007 0.003 0.004 that areas that are more dense, such as the CBD, may (W)Res. high density, 0–1/4 mile 0.020 0.010 0.011 have an overall lower mileage of bike lanes, while areas (W)Bus. low density, 0–1/4 mile 0.030 0.014 0.016 (W)Bus. high density, 0–1/4 mile 0.022 0.010 0.012 that lack access to goods and services within a non- (W)Ind., 0–1/4 mile 0.113 0.053 0.060 auto distance have a high mileage of bike lanes that are (W)Res. low density, 1/4–1/2 mile 0.056 0.026 0.030 (W)Res. medium density, 1/4–1/2 mile 0.012 0.006 0.007 intended for recreational use. (W)Res. high density, 1/4–1/2 mile 0.006 0.003 0.003 Mileage of bus routes and number of bus stops (W)Bus. low density, 1/4–1/2 mile 0.104 0.049 0.056 both have the expected sign but the effects are also (W)Bus. high density, 1/4–1/2 mile 0.029 0.014 0.016 (W)Ind., 1/4–1/2 mile 0.037 0.017 0.020 small. The estimated coefficient for number of rail (continued) stops is unexpectedly negative, while the coefficient JOURNAL OF SUSTAINABLE REAL ESTATE 15 Table 6. Continued. though they are slightly farther than other statistically Variable Direct Indirect Total significant variables would suggest for encouraging (W)Res. low density, 1/2–3/4 mile 0.389 0.183 0.206 non-auto transportation. (W)Res. medium density, 1/2–3/4 mile 0.640 0.301 0.339 Elasticities calculated from the marginal effects at (W)Res. high density, 1/2–3/4 mile 0.273 0.128 0.145 (W)Bus. low density, 1/2–3/4 mile 0.711 0.334 0.377 the means of the coefficient distributions indicate that (W)Bus. high density, 1/2–3/4 mile 0.284 0.134 0.151 non-auto transportation mode preferences are highly (W)Ind., 1/2–3/4 mile 0.501 0.235 0.266 (W)Res. low density, 3/4–1 mile 0.372 0.175 0.197 inelastic. This result in part captures the sample distri- (W)Res. medium density, 3/4–1 mile 0.781 0.367 0.414 bution which indicates that people use auto for their (W)Res. high density, 3/4–1 mile 0.194 0.091 0.102 (W)Bus. low density, 3/4–1 mile 0.596 0.280 0.316 mode of transportation at a much higher frequency (W)Bus. high density, 3/4–1 mile 0.302 0.142 0.160 than all other modes combined. Elasticities for num- (W)Ind., 3/4–1 mile 0.508 0.238 0.269 ber of household vehicles and age are above one, sig- Note. Significance at the 1, 5, and 10% levels shown by , , and , respectively. naling that these variables are a good indicator of (W) preceding variable names indicates variables pre-multiplied by the transportation mode preference. The most interesting spatial weights matrix in SDM models. result is that low density residential zoning is more elastic than all other zoning types. This is the expected for miles of rail lines has the expected sign but a small result for the zoning band within a quarter mile of positive coefficient. The study area has a more mature place of residence, indicating that altering this zoning bus system than rail system, and the rail stops are type may have the most potential of all possible zon- spread more evenly between dense urban locations ing changes in promoting non-auto transportation. near downtown and suburban locations. Perhaps the Marginal effects in the SDM model with binary spa- reason for the unexpected sign on the rail stops coeffi- tial weights are similar to those in the SAR model, with cient is capturing the propensity of most suburban many of the same variables being statistically significant residents to use auto transportation even when they and thus leading to many of the same conclusions. The live in close proximity to rail stops. This phenomenon inclusion of the spatially weighted variables in the may be due to the rail lines not going to locations regression make some of the variables that were statis- that meet suburban household needs since many of tically significant in the SAR regression insignificant. the rail lines were built to service commuting to Elasticities in the SDM model are also similar to the downtown from the suburbs, but not to perform SAR model. One interesting observation is that house- everyday shopping or recreational tasks close to home. hold vehicles have a negative direct marginal effect, but Shopping and social stops along a tour are both a positive indirect impact, indicating that having many negative and relatively large compared with many of cars may encourage auto usage, but discourage neigh- the other variables in the regression. These are the boring respondents to use auto transportation. The expected sign and indicate a propensity to drive when effects are still small, however, with the indirect elasti- shopping for goods that may need to be carried home city being less than half of direct elasticity. or to social gatherings that are located in recreational Elasticities of residential zoning variables are nega- or residential areas. Residential zoning within a quar- tive but very small, indicating that the response to ter mile from place of residence has the expected residential zoning is highly inelastic. Although the negative sign, although the effects are small. Given the marginal effects are non-linear, in general most of the value people place on their own time, it is not surpris- estimates follow a normal distribution. Thus, when ing that higher residential density, and therefore lower considering that the marginal effects are capturing a business density, may encourage people to drive to locations that are not in the immediate vicinity of one-percent increase in a specific zoning type, it may be more appropriate to consider that, for example, a their residence. Residential zoning in the quarter to 10% increase in a zoning type would have roughly ten half mile range has the opposite sign with similarly times the impact on the probability of non-auto trans- small marginal effects. It is uncertain what explains the positive marginal effect on non-auto transporta- portation modes being chosen. For example, if resi- tion for higher residential zoning levels within this dential low-density zoning was to increase by 10% band. Finally, high density business within the three within a quarter mile of a survey respondents’ resi- quarter to one-mile band has a positive marginal dence, using this rough measure we would expect to see a 0.136% decrease in non-auto transportation effect on non-auto transportation. This may indicate that if a high level of businesses are located within mode choices. While still a small impact, this change this band, survey respondents are willing to travel by is not insignificant when considering the magnitude non-auto modes to reach these destinations even of trips away from home taken by city residents across 16 A. G. MUELLER AND S. WEILER the United States each year. Even small percentage of long-term path dependence that followed from an reductions in auto trips could add up to large overall early preference for auto transportation in the devel- reductions in vehicle miles driven. opment of transportation infrastructure. The wide- spread building of roads may have led to a long chain of city planning decisions that have shaped the built Conclusion environment to accommodate automobile transporta- In comparing the current results to those of the naïve tion to the detriment of alternative modes of transpor- non-spatial empirical approach by Mueller and Trujillo tation that use less energy and decrease congestion. (2019), this paper has shown that explicit spatial model- Further research will be needed to determine if drastic ing is clearly an important factor in a fuller understand- changes in built environment design that focus on ing of travel choice behavior. This work has shown that alternatives to automobile transportation can change both spatial autocorrelation and heterogeneity are pre- society’s preference for the automobile towards trans- sent in the data and formal modeling techniques give portation usage that is more environmentally and cul- insight into the spatial aspects of associations between turally sustainable for the long-term future. In addition to urban form, there are many other factors built environment variables and mode choice. The spa- that determine travel behavior, such as weather, phys- tial results here reconfirm the overall inelasticity of travel mode choice, yet also offer explicit clues as to ical health, and availability of transportation options. zoning’s subtle influences on mode choice. The current dataset precludes the testing of such rela- The evidence provided by the log likelihood test tionships, but in cities where sweeping changes to more flexible zoning policies have been implemented, indicate that the Bayesian models of the association event studies could shed light on the causal relation- between zoning on travel mode preference favor the ships between zoning mix and transportation behav- Spatial Durbin Model with a binary spatial weights ior. This study has provided some direction on the matrix indicated in Table 5. Residential low, medium, utility of using spatial models in transportation stud- and high density within one quarter mile of residences ies, but more research is needed to incorporate holis- are all statistically significant and associated with a tic solutions to urban transportation problems that lower propensity for non-auto travel. Using AIC to include vehicle electrification, driverless cars, bike, assess the models considered in this chapter, the SAR and scooter sharing programs, and alternative trans- model with a binary spatial weights matrix was the portation infrastructure. best model overall. In this model, residential zoning variables of low, medium, and high density in the 0– 1/4 mile zoning band all have statistically significant Note and negative impacts on survey respondents’ probabil- 1. All estimations are implemented in the software system R ity of choosing non-auto transportation. Using BIC, the (R Core Team, 2022). The spatial weights matrix was standard probit model had the best fit to the data. One constructed and standardized using the R add-on package potential conclusion from these results is that zoning spdep (Bivand et al., 2013;Bivand & Piras, 2015). 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Publisher: Taylor & Francis
Copyright: © 2023 The Author(s). Published with license by Taylor & Francis Group, LLC
ISSN: 1949-8284
DOI: 10.1080/19498276.2023.2174661
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Abstract

JOURNAL OF SUSTAINABLE REAL ESTATE 2023, VOL. 15, NO. 1, 2174661 ARES https://doi.org/10.1080/19498276.2023.2174661 American Real Estate Society a b Andrew G. Mueller and Stephan Weiler a b Burns School of Real Estate and Construction Management, University of Denver, Denver, CO, USA; Department of Economics, Colorado State University, Fort Collins, CO, USA KEYWORDS ABSTRACT Land use restrictions; spatial This paper develops a spatial econometric model of transportation mode choice and tests econometrics; transportation the association between zoning and other built environment variables and the choice of mode choice; travel auto and non-auto transportation. We provide an extensive review of spatial econometrics behavior; walkability; and demonstrate the importance of using models that treat space formally when investigat- zoning ing urban transportation behavior. Using a unique combination of travel, employment, and built environment datasets from Denver, Colorado, we confirm previous results that built environment variables have a small association with choice of transportation mode and show the benefits of formal spatial modeling to the traditional probit model. Introduction logit, or multinomial logit models to estimate impacts on mode choice (Ewing & Cervero, 2010). To investi- Location of economic activity has always been an gate the importance of formal spatial modeling, we use important aspect of regional economics and real estate. the dataset from a previous study on the impacts of Spatial proximity plays a key role in many decisions zoning on mode choice (Mueller & Trujillo, 2019)and made by individuals when it comes to weighing the formally model spatial dependence and spatial hetero- benefits and costs of purchase decisions, allocation of geneity to test whether the results are significantly dif- resources, and other economic behavior. Transportation ferent from the standard econometric approach where choices are particularly affected by location, and distan- space is dealt with informally. It is particularly import- ces between origins and destinations undoubtedly influ- ant to investigate the presence of spatial dependence ence travel decisions among individuals. and heterogeneity in regard to land use restrictions The development of formal econometric techniques because each individual faces a unique set of transpor- to address location is a more recent development in tation choices based on their residential location and the field. Of particular importance to the field of spa- the proximity of this residential location to available tial econometrics is the treatment of spatial depend- goods, services, recreation, transportation, and employ- ence (spatial autocorrelation) and spatial heterogeneity ment opportunities. While built environment variables (spatial structure). Spatial dependence and spatial het- have been studied for their impact on driving behavior erogeneity are important in applied economic models (Vance & Hedel, 2008), their role in land use segmen- because the presence of these phenomena may invali- tation (Levkovich et al., 2018), development and dens- date or bias results from mainstream approaches. In ity (Butsic et al., 2011;Newburn & Ferris, 2016), and addition, these issues have been largely ignored in the sprawl (Vyn, 2012), to the author’s knowledge this is mainstream literature (Anselin, 1988). This paper focuses on consumer transportation the first study that employs formal spatial modeling to investigate the association of zoning restrictions with mode choice in a spatial context to demonstrate the travel mode choice. From a policy perspective, models importance of formal spatial modeling techniques in transportation behavior research. Previous studies of that inform the impact that a change in zoning laws transportation choice largely implement OLS, probit, might have on transportation choice can further inform CONTACT Andrew G. Mueller andrew.mueller@du.edu Burns School of Real Estate and Construction Management, University of Denver, Denver, CO 80208, USA. 2023 The Author(s). Published with license by Taylor & Francis Group, LLC This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 A. G. MUELLER AND S. WEILER alternative transportation agendas. Non-auto forms of construction, see Mueller and Trujillo (2019)and the transportation and built environments that support descriptive statistics in Table 1. their use can provide traffic congestion relief, and so In this work, we formally test for spatial autocorrel- are the primary focus of this research, although other ation and spatial heterogeneity in the data and apply agendas, such as the transition to electric vehicles may spatial econometric methods to correct for spatial com- have other benefits. ponents found in the data generating process. The Previous work has used a choice-specific multi- paper reviews the relevant spatial econometric theory, nomial logit model to estimate the effect of built and applies the Moran’s I and Geary’s C tests for spatial environment, socioeconomic, and land restriction processes at work in the data generating process. We measures on the propensity of survey respondents to then explore several models that address spatial auto- use four mode choice alternatives; auto, transit, bike, correlation, spatial heterogeneity, and both spatial auto- and walk (Ewing & Cervero, 2010; Mueller & Trujillo, correlation and heterogeneity jointly and compare them 2019). Due to the lack of observations in the data for with the results of the canonical probit approach. walk (7.42%), bike (1.5%), and transit (7.71%) which Estimates of each model are performed with the two individually represent a small subset of the dataset, we most commonly used spatial weights matrices in the lit- condense the dependent variable choice set into a bin- erature, the row standardized binary weights matrix and ary choice between auto and non-auto transportation the row standardized inverse distance matrix. The (16.6%). This paper adds to the previous work by for- results from each model show that there is a high likeli- mally implementing econometric techniques that expli- hood of the presence of spatial processes in the data citly deal with bias and inefficiency in the estimation of generating process and that these models are preferable effects that are introduced if spatial autocorrelation to canonical approaches to estimating travel mode and/or spatial heterogeneity are present in the underly- choice behavior in this travel survey sample. We pro- ing data generating process. The dataset does not allow vide evidence that the spatial Durbin model (SDM) for testing of causality in the relationship between the with a binary spatial weights matrix is the superior built environment and transportation choice, and we model that accounts for both spatial dependence and acknowledge that there may be simultaneity and self- heterogeneity, and accounts for spatial processes that selection present in the dynamic process that deter- are not modeled in the traditional probit approach. mines the relationship between mode preference and Particularly, if spatially lagged forms of the dependent zoning variables by respondents choosing to locate in and independent variables show signs of spatial depend- areas that align with their transportation preferences, ence or heterogeneity, only the spatial Durban model and possibly also influencing the political process that addresses both spatial phenomena simultaneously determines zoning in their neighborhoods. Testing of (LeSage & Pace, 2009). such relationships would require repeat observations of households in a longitudinal survey, and such a survey Reviewing the Spatial Econometric Approach is not available currently. Using transportation survey data from the City and Spatial econometrics differentiates itself from main- County of Denver in 2010 and corresponding built stream econometric approaches by applying formal environment measures during that time, we test the spatial modeling best summarized in Luc Anselin’spio- association of zoning and other built environmental neering work on the topic: “I will consider the field of variables on survey respondents’ choice of auto and spatial econometrics to consist of those methods and non-auto transportation while controlling for demo- techniques that, based on formal representation of the graphic and preference variables collected in the survey. structure of spatial dependence and spatial heterogen- The unit of observation is a “tour” taken by individuals eity, provide the means to carry out the proper specifi- where a respondent leaves the home, makes stops for cation, estimation, hypothesis testing, and prediction various activities, and returns home. The key independ- for models in regional science” (Anselin, 1988,p.10). ent variables of interest in this study are the land use restriction variables encompassed in percentages of dif- Spatial Effects ferent zoning types (low, medium, and high density residential, low and high density business, and indus- Real estate economics and regional science inherently trial) within a 1/4, 1/4–1/2, 1/2–3/4, and 3/4–1-mile deal with issues related to human behavior across radius surrounding each survey respondent’s home. For space, cities, and regions. The term spatial economet- a more thorough explanation of the dataset and its rics and its designation as a distinct branch of JOURNAL OF SUSTAINABLE REAL ESTATE 3 Table 1. Descriptive statistics. Variable N Mean St. Dev. Min Max Household vehicles 5,123 1.895 0.901 0.000 6.000 Household bikes 5,123 1.627 1.794 0.000 30.000 Household size 5,123 2.419 1.193 1.000 7.000 Male 5,123 0.468 0.499 0.000 1.000 Age 5,123 52.314 15.689 16.000 93.000 Income 5,123 79.369 50.587 0.000 160.000 College education 5,123 0.374 0.484 0.000 1.000 Employed 5,123 0.658 0.474 0.000 1.000 Driver’s license 5,123 0.945 0.228 0.000 1.000 Transit pass 5,123 0.136 0.343 0.000 1.000 Miles bike land < 1 mile 5,123 11.667 5.534 0.000 33.048 Mile bus routes < 1/2 mile 5,123 8.596 12.314 0.000 90.313 Bus stops < 1/2 mile 5,123 29.342 17.621 0.000 138.000 Rail stops < 1 mile 5,123 0.262 0.907 0.000 7.000 Miles rail lines < 1 mile 5,123 2.184 3.180 0.000 14.346 Intersection < 1/2 mile 5,123 125.574 28.072 9.000 255.000 CBG population density 5,123 7,174.166 4,497.568 27.723 33,612.000 CBG employment density 5,123 4,862.703 20,847.640 0.000 278,470.500 Home < 2 miles of downtown 5,123 0.173 0.378 0.000 1.000 Number of tours 5,123 1.486 0.773 1.000 7.000 Travel mode ¼ non-auto 5,123 0.166 0.372 0.000 1.000 Tour distance 5,123 10.816 12.269 0.012 135.415 Trip duration 5,123 49.936 37.055 2.000 452.000 Tour crosses highway 5,123 0.746 0.435 0.000 1.000 Arrival hour 5,123 11.206 4.260 0.000 23.000 Departure hour 5,123 6.706 6.565 0.000 23.000 Work stops 5,123 0.570 1.098 0.000 30.000 Shopping stops 5,123 0.710 0.984 0.000 10.000 Health stops 5,123 0.068 0.264 0.000 3.000 Social stops 5,123 0.252 0.504 0.000 4.000 Res. low density, 0–1/4 mile 5,123 63.435 36.588 0.000 100.000 Res. med. density, 0–1/4 mile 5,123 9.200 18.181 0.000 100.000 Res. high density, 0–1/4 mile 5,123 12.527 22.923 0.000 100.000 Industrial, 0–1/4 mile 5,123 1.793 8.243 0.000 73.680 Bus. low density, 0–1/4 mile 5,123 2.044 3.909 0.000 29.601 Bus. High Density, 0–1/4 mile 5,123 8.047 17.695 0.000 100.000 Res. low density, 1/4–1/2 mile 5,123 59.288 30.483 0.000 100.000 Res. med. density, 1/4–1/2 mile 5,123 10.837 15.047 0.000 98.131 Res. high density, 1/4–1/2 mile 5,123 11.964 15.999 0.000 76.586 Industrial, 1/4–1/2 mile 5,123 3.297 9.920 0.000 76.188 Bus. low density, 1/4–1/2 mile 5,123 1.968 2.294 0.000 14.610 Bus. high density, 1/4–1/2 mile 5,123 9.622 13.830 0.000 99.991 Res. low density, 1/2–3/4 mile 5,123 56.181 26.581 0.000 100.000 Res. med. density, 1/2–3/4 mile 5,123 11.910 14.836 0.000 98.207 Res. high density, 1/2–3/4 mile 5,123 11.836 13.089 0.000 83.916 Industrial, 1/2–3/4 mile 5,123 4.814 11.310 0.000 94.524 Bus. low density, 1/2–3/4 mile 5,123 2.032 2.107 0.000 12.704 Bus. high density, 1/2–3/4 mile 5,123 9.978 11.800 0.000 88.327 Res. low density, 3/4–1 mile 5,123 54.202 23.843 0.000 99.894 Res. med. density, 3/4–1mile 5,123 12.842 14.669 0.000 99.635 Res. high density, 3/4–1 mile 5,123 11.314 10.409 0.000 88.762 Industrial, 3/4–1 mile 5,123 6.535 11.893 0.000 97.455 Bus. low density, 3/4–1 mile 5,123 1.881 1.737 0.000 10.890 Bus. high density, 3/4–1 mile 5,123 9.825 10.309 0.000 68.460 econometrics dates back to the seminal work of spaces, (4) differentiation between ex post and ex ante Paelinck and Klaassen (1979) that collected a growing interaction, and (5) explicit modeling of space. While body of literature in the regional sciences that it is possible to measure and model spatial data using attempted to formally deal with the problems inherent standard econometric techniques by including variables in modeling spatial data in the context of regional in the model that have a spatial nature to their meas- econometric models. The primary characteristics that urement, the distinction to be made here is that spatial delineate the field according to Paelinck and Klaassen econometrics formally deals with specific spatial (1979) and summarized by Anselin (1988,p. 7) are: aspects of the data that preclude the use of traditional (1) the role of spatial interdependence in spatial mod- econometric techniques, and more particularly, address els, (2) the asymmetry in spatial relations, (3) the spatial dependence and spatial heterogeneity formally importance of explanatory factors located in other (Anselin, 1988; LeSage & Pace, 2009). 4 A. G. MUELLER AND S. WEILER Spatial dependence addresses the lack of mutual Formally Modeling Spatial Interaction independence across observations in cross-sectional At the center of spatial econometrics is defining spatial data sets and is often referred to as spatial autocorrel- association amongst observations (Anselin, 1988, 2010; ation following the path-breaking work of Cliff and Arbia, 2006). To formally address the spatial connect- Ord (1969, 1973). In essence, addressing spatial edness of observations across space, an approach has dependence is the development of formal statistical been developed which uses a decision rule that deter- specifications of economic models that address mines whether two observations are spatial neighbors Tobler’s first law of geography, that “everything is and thus close enough to exert influence on each other. related to everything else, but near things are more The typical convention is to formally define spatial related than distant things” (Tobler, 1970, p. 236). connectedness through the use of a symmetric matrix Spatial dependence is estimated by the relative loca- W of dimensions equal to the number of observations tion of one observation in the dataset to another, with n, whose strictly non-negative elements w indicate the ij an emphasis on the effect of distance between obser- spatial connectedness between units i and j 6¼ i: With vations. Spatial dependence is caused by a variety of the spatial neighbor matrix constructed, spatial model- measurement errors, by spatial spill-over effects, or ing proceeds by re-weighting each row to develop a spatial externalities (Anselin, 1988), by spatially auto- spatial weights matrix, then pre-multiplying either the correlated variables (Fingleton & Lopez-Bazo, 2006), dependent or independent variables by the spatial or any situation in which the covariance of observa- weights matrix and estimating a vector of coefficients tions across geographical space is not equal to zero that includes a spatial dependence parameter. This (Anselin, 2001). For example, spatial autocorrelation modeling approach formally connects variables of is often found in hedonic pricing models of residential neighboring observations through the spatial weights real estate, where the sale price of one residential matrix W and produces an estimate of spatial associ- property is influenced by housing prices in surround- ation in the data generating process through the spatial ing neighborhoods (von Graevenitz & Panduro, 2015). dependence parameter(s). To demonstrate the use of Spatial heterogeneity is the “lack of stability over the spatial weights matrix, the spatial autoregressive space of the behavioral or other relationship under model equation is illuminating. In its simplest form study. More precisely, this implies that functional with no independent regressors, the spatial autoregres- forms and parameters vary with location and are not sive model equation is: homogeneous throughout the data set” (Anselin, 1988, X y ¼ q w y þ e (1) i ij j i p. 9). This type of econometric model addresses these issues by formally modeling the variation in parame- ters across space to address the heterogeneous effect The term w y þ e gives a weighted sum of ij j i an independent variable may have in different loca- each neighboring observation j’s dependent variable tions. More importantly, when spatial dependence and y , j 6¼ i: The estimated spatial dependence parameter spatial heterogeneity are present in the data generating q gives a measure of the influence those neighboring process and not explicitly modeled, the results of observations have on each y observation. High values mainstream econometric techniques may be biased, of q indicate strong spatial autocorrelation between inefficient, or both (Anselin, 1988; LeSage & Pace, observations, while a value of 0 indicates no spatial 2009; Schnier & Felthoven, 2011). Spatial econometric autocorrelation. In addition to measuring the direct techniques address spatial processes within the data influence of neighbors j on observation i, the param- generating process and are generally preferred when eter q is sometimes referred to as the spatial decay differences due to spatial location are present in data. parameter, because it also indicates how fast the effect An example of spatial heterogeneity is the variation in of neighboring observations declines with higher order the effect income may have on travel mode preferen- neighbors, i.e. neighbors of neighbors (Anselin, 1988; ces across the urban landscape. Income may have the LeSage & Pace, 2009). For example, second order opposite effect on preferences to drive in suburban neighbors of y are first order neighbors of y ’s first i i locations than it does in central business districts order neighbors and have an influence on y equal to because higher incomes allow suburban dwellers q , their influence on y being exerted indirectly greater access to automobiles, while in urban locations through y ’s direct neighbors. Influence dissipates as higher income may allow individuals to live in areas observations become further removed from y , and k with better access to goods and services, thus increas- order neighbors have and influence equal to q . Thus, ing reliance on alternative forms of transportation. values of q closer to 1 indicate a slowly dissipating JOURNAL OF SUSTAINABLE REAL ESTATE 5 influence, while values close to 0 indicate an effect models to the choice of spatial weights matrix by esti- that quickly dissipates with higher order neighbors. mating models with both a binary row standardized The literature has yet to determine a formal spatial weights matrix and an inverse distance row approach to developing the spatial weights matrix, standardized weights matrix. although several approaches have widespread adop- The decision to standardize the spatial weights matrix is not at all clear from the literature, and deci- tion. The pioneering work of Moran (1950) and sions on how to form the spatial weights matrix are Geary (1954) developed the notion of a binary weights generally determined by a priori assumptions made by matrix W, where each element w was assigned a ij the researcher in the context of each study. Anselin value of 1 if two observational units were neighbors (1988) argues that in certain cases, such as inverse dis- and assumed to exhibit influence on each other, and 0 tance, the standardization of the spatial weights matrix otherwise. The spatial weights matrix was originally may eliminate the economic interpretation of the developed in the context of areal units and neighbors results. However, the consensus is that the standard- were defined as two observational units that shared a ization of the spatial weights matrix is the preferable common border (Cliff & Ord, 1973). When observa- approach to avoid magnitude complications amongst tional units are points in space rather than areal units variables and avoid certain spatially weighted variables (as the data in this study is), neighbors are identified dominating the results of spatial models (LeSage & based on distance. Two spatial point observations i Pace, 2009). and j are considered neighbors if 0 d D, where ij Formally, each element of a binary spatial weights d is the distance between points i and j and D is the ij matrix (spatial neighbor matrix) is calculated based on bandwidth after which interaction between observa- a decision rule. For contiguity neighbors, each element tions is considered non-existent and w is assigned a ij w ¼ 1 if the two areal units represented as polygons ij 0 weight (Anselin, 1988). Assignment of a zero weight share a common boundary, and 0 otherwise. For dis- does not preclude spatial effects occurring between tance based neighbors, w ¼ 1 if d D and 0 other- ij more distant neighbors, however. Instead, influence is wise, where d is the distance between observation i modeled as a higher order recursive effect through the and j, and D is a pre-determined distance threshold estimated spatial dependence parameter as discussed above which observations are said to exhibit no direct above. Thus, observations that are not direct neigh- influence on each other. The choice of the distance bors can influence each other indirectly through inter- threshold D is not well developed in the literature and mediary neighbors that connect them. Once a binary is typically based on domain knowledge or corres- spatial weights matrix is constructed which determines pondence to other distance measurements in the data- which observations are neighbors of each other, the set. Each element of a row standardized spatial spatial weight matrix is often row standardized so that weights matrix W is calculated as: each row sums to 1. Row standardization normalizes ij spatial effects across a dataset, preventing observations w ¼ P (2) ij ij that have many spatial neighbors from dominating coefficient estimates (Anselin, 1988). s with each element of W equal to 0 or 1 in a binary Additional weighting schemes have also been specification, and 1=d in an inverse distance specifica- applied to the binary weights matrix, and it is cur- s tion if the two observations are neighbors. Matrix W rently convention to row standardize the spatial is used to link neighboring observations in spatial weight matrices after applying alternative weighting regression models, which produces estimates of coeffi- schemes. If an alternative weighting scheme is applied, cients on the resulting spatially weighted variables. the construction of the spatial weights matrix becomes a two-step process, first constructing a binary spatial Measuring Spatial Dependence neighbor matrix as above, then multiplying this matrix by another measure of spatial association. Cliff Constructing a spatial weights matrix allows for for- and Ord (1973, 1981) pioneered this concept by mul- mal testing of spatial dependence in the data generat- tiplying the binary spatial neighbor matrix by the ing process. The canonical measure of spatial inverse of the distance between observations. This dependence was developed by Moran (1950) and is approach places higher weights on neighboring obser- widely used across many fields of study. Moran’s I is vations that are closer, while still placing zero weights a global test of spatial dependence. Shortly after on neighbors that are further apart than distance D. Moran, Geary (1954) developed a formal test of local- In this study, we test the sensitivity of results of all ized autocorrelation, known as Geary’s C. Moran’s I 6 A. G. MUELLER AND S. WEILER indicates the level of global spatial autocorrelation, association also being stronger across most variables while Geary’s C indicates localized spatial autocorrel- using the inverse distance spatial weights matrix. This ation and therefore the possibility that spatial hetero- result is an indication that spatial dependence may be geneity is also present in the data generating process. present both globally (spatial autocorrelation) and Moran’s I ranges between 1 and 1, with values locally (spatial heterogeneity) in the underlying data near 1 (1) indicating the positive (negative) spatial generating processes. The dataset may include cluster- autocorrelation, and values near 0 indicating weak ing that results from socioeconomic traits, political spatial dependence of the observed variables (Moran, zoning boundary determination, and transit network 1950). Moran’s I can be interpreted as a spatial ver- design among other spatial phenomena, which is not sion of a typical correlation calculation. Geary’s C surprising, considering that spatial segregation of land ranges between 0 and 2, with values <1 demonstrat- use is one of the objectives of zoning laws (Levkovich ing increasing positive spatial autocorrelation and val- et al., 2018), and socioeconomic segregation is a widely ues >1 indicating increasing negative spatial accepted phenomenon. autocorrelation (Geary, 1954). Formally, Moran’s I is The test results from Table 2 justify using spatial calculated as: econometric modeling techniques to address the spa- P P n n tial dependence and heterogeneity that is present in w ðx xÞðx xÞ n ij i j i¼1 j¼1 P P I ¼ (3) n n the data. We develop three specifications of spatial n 2 ij ðx xÞ i¼1 j¼1 i¼1 models to correct for these spatial processes; the spa- and Geary’s C is calculated as: tial autoregressive model (SAR) which addresses spa- P P n n 2 tial dependence, the spatial error model (SEM) which w ðx x Þ ðn 1Þ ij i j i¼1 j¼1 addresses spatial heterogeneity, and the spatial Durbin C ¼ P P (4) n n n 2 2 w ij ðx xÞ i¼1 j¼1 i¼1 model (SDM) which simultaneously addresses spatial dependence and spatial heterogeneity. As suggested by Using the equations above, it is possible to test LeSage and Pace (2009), we estimate each of the mod- Moran’s I and Geary’s C statistics against their theor- els using the two most common row standardized etical values under different distributional assump- spatial weights matrices and use a Lagrange multiplier tions. We test these two statistics against their test to determine which spatial weight matrix best theoretical values under a normal Gaussian distribu- fits the data. The two spatial weighting schemes tion and the results are shown as significance stars in employed in the spatial weights matrix before row Table 2. As can be seen from the equations above, standardization are the two most common in the lit- both statistics I and C are measurements of the erature, binary and inverse distance. Both spatial covariance of deviation from the mean of observations weight matrices are then row standardized before esti- of a single variable x across a dataset, linked through mating each model. the spatial weights matrix W. Thus, one can think of the measures as clustering of deviations from the mean. If neighboring observations defined through W The Spatial Autoregressive Model deviate from the mean in the same direction, high The spatial autoregressive model (SAR) formally esti- spatial clustering (autocorrelation) is present. mates the presence of spatial dependence by incorpo- Table 2 shows that Moran’s I and Geary’s C for rating a spatially lagged dependent variable on the most variables in the current dataset are statistically right-hand side of the regression equation (Cliff & significant at the 1% level, providing confidence in Ord, 1973). Thus, observations of the dependent vari- their estimated values. Comparing Moran’s I using the able are influenced by other observations of the binary and inverse distance weight matrices, one can dependent variable nearby. In the context of the pre- see that the inverse distance matrix picks up higher sent study, the SAR model is a way of controlling for values of positive spatial correlation for demographic the influence of neighboring survey respondents’ variables (household size, vehicles, bikes, age, income, transportation mode choices on the observational unit and college education). This is an indication that the under study which represents a spatial clustering closest neighbors are more spatially correlated. Urban effect. In the binomial context, the choice variable form variables have Moran’s I values close to 1 using observed (transportation mode ¼ auto or non-auto) either spatial weights matrix, indicative of neighboring observations sharing a common urban form. Geary’s C depends on the underlying utility of the choice indica- statistics also show local positive auto-correlation tor observed. The underlying latent variable y ¼ under both spatial weights matrices, with the U U is assumed to follow a normal distribution 1i 0i JOURNAL OF SUSTAINABLE REAL ESTATE 7 Table 2. Moran’s I and Geary’s C statistics. Variable Moran’s I Moran’s I (1/d) Geary’s C Geary’s C (1/d) HH size 0.084 0.449 0.915 0.555 HH vehicles 0.065 0.427 0.934 0.569 HH bikes 0.072 0.477 0.935 0.527 Male 0.000 0.228 1.000 1.229 Age 0.035 0.277 0.962 0.734 Income (000s) 0.083 0.420 0.918 0.581 College education 0.070 0.175 0.931 0.829 Employed 0.011 0.088 0.988 0.902 Tour distance 0.024 0.081 0.979 0.935 Tour crosses highway 0.039 0.117 0.963 0.876 Miles bike lanes < 1 M. 0.838 0.953 0.156 0.048 Miles of bus routes < 0.5 M. 0.639 0.903 0.354 0.111 Bus stops < 0.5 M. 0.654 0.905 0.344 0.111 Rail stops < 0.5 M. 0.473 0.846 0.530 0.173 Miles of rail lines < 0.5 M. 0.687 0.899 0.301 0.101 Intersections with 0.5 M 0.475 0.836 0.496 0.150 CBG population/sq. mile 0.423 0.758 0.590 0.244 CBG jobs/sq. mile 0.179 0.629 0.854 0.361 Work stops 0.000 0.029 1.004 0.964 Shopping stops 0.006 0.083 0.996 0.907 Social stops 0.001 0.084 1.006 0.912 Res. low density, 0–1/4 mile 0.548 0.847 0.449 0.154 Res. medium density, 0–1/4 mile 0.392 0.808 0.580 0.190 Res. high density, 0–1/4 mile 0.520 0.820 0.486 0.186 Bus. low density, 0–1/4 mile 0.175 0.697 0.819 0.288 Bus. high density, 0–1/4 mile 0.397 0.815 0.579 0.187 Ind., 0–1/4 mile 0.269 0.698 0.667 0.253 Res. low density, 1/4–1/2 mile 0.681 0.899 0.315 0.103 Res. medium density, 1/4–1/2 mile 0.649 0.895 0.326 0.104 Res. high density, 1/4–1/2 mile 0.658 0.880 0.360 0.125 Bus. low density, 1/4–1/2 mile 0.355 0.778 0.667 0.224 Bus. high density, 1/4–1/2 mile 0.527 0.848 0.454 0.148 Ind., 1/4–1/2 mile 0.469 0.797 0.449 0.167 Res. low density, 1/2–3/4 mile 0.476 0.566 0.539 0.444 Res. medium density, 1/2–3/4 mile 0.356 0.477 0.611 0.503 Res. high density, 1/2–3/4 mile 0.362 0.479 0.654 0.522 Bus. low density, 1/2–3/4 mile 0.440 0.529 0.586 0.489 Bus. high density, 1/2–3/4 mile 0.282 0.388 0.737 0.591 Ind., 1/2–3/4 mile 0.363 0.475 0.599 0.476 Res. low density, 3/4–1 mile 0.515 0.599 0.500 0.406 Res. medium density, 3/4–1 mile 0.402 0.525 0.570 0.470 Res. high density, 3/4–1 mile 0.333 0.446 0.681 0.551 Bus. low density, 3/4–1 mile 0.420 0.517 0.589 0.504 Bus. high density, 3/4–1 mile 0.303 0.430 0.708 0.557 Ind., 3/4–1 mile 0.402 0.518 0.569 0.448 Note. Significance at the 1, 5, and 10% levels shown by , , and , respectively. in the probit model estimation. The general spatial Typically, the SAR model is used to adjust for autoregressive model in a binomial context can be for- dependent variables that have a direct effect on the mally stated in the system of equations as: realization of the dependent variable in close proxim- ity. The classic example is SAR hedonic pricing mod- y ¼ qWy þ Xb þ e (5) els of residential home values (e.g., Pace & Barry, e Nð0, r I Þ 2004), where the value of a house sold has a direct y ¼ 1, if y 0 impact on other residential home prices in the area and has been shown to be a valuable addition to trad- y ¼ 0, if y < 0 itional home price models (Anselin & Lozano-Gracia, Where y is the unobserved latent utility of mode 2007). Conceptually, the SAR would be the correct choice, y ¼ 1 if the binomial choice is observed, and 0 model for the underlying data generating process if a otherwise, W is the spatial weights matrix, q is an esti- survey respondent’s choice to use auto or non-auto mated spatial dependence parameter of spatial autocorrel- transportation depended upon neighboring survey ation between observations, X is a matrix of independent respondents’ transportation mode choices, i.e. a clus- variables, and b is a vector of estimated coefficients. The tering effect of mode choice. latent utility construct implies that Prðy ¼ 1Þ¼ While theoretically, the model has justifiable merit PrðU U Þ¼ Prðy 0Þ (LeSage & Pace, 2009). in controlling for spatial dependence, it is important to 1i 0i i 8 A. G. MUELLER AND S. WEILER note that this model does not distinguish the direction and the estimation of its spatial lag parameter k. of causality, only the association of built environment Unlike the SAR model, indirect and direct effects can- characteristics with transportation mode choice. It is not be estimated because there is no feedback loop of quite possible that people who enjoy non-auto forms of changes in the dependent regressors of neighboring transportation tend to live in the same locations observations on the dependent variable since there is because these locations provide employment, leisure, no autocorrelation parameter present. The parameter and shopping in close enough proximity to make non- k represents the extent to which heterogeneous inde- auto trips more convenient. However, this model does pendent coefficient estimates vary across space. This is identify if there is spatially clustered transportation the correct model to use if neighboring respondents’ behavior, and how fast this clustering effect deteriorates transportation mode choices do not affect an individ- with distance. If the spatial dependence parameter q is ual’s mode choice, but the effect of independent varia- significant, explicitly modeling spatial dependence is bles have varying effects across space, such as the justified and therefore relevant to the study of spatial variation in the effect of income described earlier. associations between zoning laws and transportation choices. While this model cannot determine the under- Spatial Durbin Model lying cause of clustering in mode choice, it does con- The spatial Durbin model (SDM) allows for the esti- trol for the spatial phenomenon, leading to unbiased mation of both spatial autocorrelation and spatial het- estimates of the association between urban form and erogeneity simultaneously by including a spatially transportation behavior. It is also important to note that in the SAR model, lagged dependent variable as well as spatially lagged the spatial dependence parameter q incorporates a independent variables in a single model. The advan- feedback loop in the effect of neighboring observa- tages of this model are the simultaneous control of tions on the dependent variable. There is a direct both spatial dependence and spatial heterogeneity, but effect of independent variables on transportation in practice can suffer from the curse of dimensional- choice, and this transportation choice then indirectly ity. One advantage of the Bayesian approach to model effects transportation mode choices of neighboring estimation employed in this study and described observations, which in turn affect the observation below is the ability to estimate such models without under study, creating a spatial feedback loop effect. running into non-convergence problems. These prob- Thus, direct, indirect, and total effects of independent lems can be a significant challenge with the maximiza- variables on the dependent variable are estimated. tion procedures employed in maximum likelihood and generalized method of moments estimation, and often lead to severe computational challenges. The binomial The Spatial Error Model probit SDM model can be formally stated as: In contrast to the spatial autoregressive model, the spa- y ¼ qWy þ Xb þ WXh þ e (7) tial error model (SEM) allows for heterogeneous effects e Nð0, r I Þ of independent regressors across space. This adaptation y ¼ 1, if y 0 of the traditional OLS or probit model allows for both i y ¼ 0, if y < 0 global coefficients (b) and local variation across the space of coefficients to be modeled in the error struc- where q is the estimated parameter of spatial autocor- ture. In the binomial choice context, the latent variable relation of the dependent variable, b is the estimated approach of unobserved utility of the resulting choice vector of parameters on the independent variables, indicator is used for the probit estimator similar to the and h is the vector of estimated parameters on the process described for the SAR model. The SEM bino- spatially lagged independent variables. The estimation mial choice model can be formally stated as: of the SDM is similar to that of the SAR model with y ¼ Xb þ u (6) the independent variables multiplied by the spatial weights matrix added as additional independent varia- u ¼ kWu þ e 2 bles, WX. The resulting model then produces a vector ðÞ e N 0, r In of global effects of the independent variables b and a y ¼ 1, if y 0 vector of local effects of the independent variables h. y ¼ 0, if y < 0 LeSage and Pace (2009) detail the advantages of where W is the spatial weights matrix. The SEM each spatial modeling approach, and determine that model allows for spatial variance of the error term when the correct model is unknown and not dictated JOURNAL OF SUSTAINABLE REAL ESTATE 9 by theory, only the SDM gives unbiased results even if (1992) because the posterior distributions are available the true model is SAR or SEM. More particularly, to calculate valid inference measures of the parameter when the true data generating process is the SEM estimates, thus escaping the bias inherent in model, SAR and SDM will produce unbiased but inef- McMillen’s algorithm and the necessity to specify the ficient estimates. When the true data generating pro- functional form of model variance over space a priori. cess is the SAR model, the SEM model produces The likelihood function for the SAR, SDM, and SEM biased estimates, while the SDM does not. If the true models is: data generating process is the SDM model, the other 1 1 ðÞ Ly, Wq, b, r ¼ jI qWjexp e e ðÞ models will have omitted variable bias. The SAR, n ðÞ 2 2r 2pr SEM, and SDM versions of the travel behavior—built (8) environment models are estimated in Table 3 using both a binary and inverse distance weighted row e ¼ðI qWÞ y Xb for the SAR model, standardized spatial weights matrix. e ¼ðI qWÞðy XhÞ Xb for the SDM model, e ¼ðI kWÞðy XbÞ for the SEM model (LeSage, 2000,p. 23). Estimation Techniques It is important to note that the Bayesian approach McMillen (1992) was the first to propose techniques for to modeling is fundamentally different from that of the estimating the SAR and SEM probit models. Due to the frequentist approach employed in OLS, probit, and complicated error structure of the SAR and SEM probit other canonical statistical models. The results of models, direct maximum-likelihood estimation is not Bayesian estimation produce full distributions of par- possible; however, in McMillen’s procedure, the discrete ameter estimates, and convention is to report the mean variable is replaced by the expected value of the under- of each parameter distribution. Significance tests are lying latent variable, and the expectation is calculated then the probability of the parameter estimate contain- iteratively until convergence. McMillen (1992), among ing zero calculated directly from the parameter distri- others, deems this procedure impractical for large data- bution. This approach is fundamentally different from sets. LeSage (2000) outlines several other drawbacks to the frequentist approach, which calculates the probabil- the procedure. First, the estimation procedure requires ity of the parameter estimate being zero from the the estimation of the likelihood function, which prohib- standard errors of each estimate and the underlying its the use of the information matrix for calculating the distributional assumption (often Gaussian) of the errors precision of the parameter estimates. Attempts to cir- (Albert, 2007; Albert & Chib, 1993; LeSage, 2000). cumvent this problem produces biased estimates of the covariance matrix. Second, McMillen’s approach Econometric Models and Results requires the researcher to specify a functional form of the heteroskedastic spatial variance and leads to varying Econometric Model inferences across alternative specifications. Alternatively, Three econometric models are specified following the Bayesian estimation techniques do not require these theoretical specifications for the SAR, SEM, and SDM assumptions about the functional form of the error pro- above. The binary choice indicator variable y is set to cess. We, therefore, implement a spatial Bayesian tech- 1 if the survey respondent used non-auto transporta- nique to estimate the spatial probit models in this study. tion for an observed tour, and 0 otherwise. The spatial Following the work of Albert and Chib (1993) and probit model for the SAR, SEM, and SDM is com- Chib (1992), which detail the estimation of probit and prised of the travel choice indicator variable and the logit models for discrete choices using Markov Chain independent regressors which are the same as in Monte Carlo estimation in a Bayesian context, LeSage Mueller and Trujillo (2019). The formal equation to (2000) proposes a Bayesian estimation technique be estimated for the SDM is then: based on the Gibbs sampling approach (Albert & y ¼ qWy þ Xb þ WXh þ e (9) Chib, 1993). The estimation technique specifies a i complete set of prior distributions for all parameters e Nð0, r InÞ in the model and then samples from these distribu- y ¼ 1, if y 0 tions until a large number of parameter draws are y ¼ 0, if y < 0 obtained that converge to the true joint posterior dis- tribution of the parameters. This approach overcomes where X ¼½IS BE where I is an n 1vector of the drawbacks of the approach proposed by McMillen ones, S is a matrix of sociodemographic characteristics, 10 A. G. MUELLER AND S. WEILER Table 3. Model coefficient comparison. Dependent variable: non-auto transportation mode ¼ 1 Variable Probit SAR SAR (1/d) SEM SEM (1/d) SDM SDM (1/d) Intercept 0.029 0.419 0.279 0.433 246.400 0.329 1.346 Household size 0.050 0.052 0.052 0.108 75.260 0.067 0.061 Household vehicles 0.363 0.368 0.368 0.777 531.200 0.362 0.377 Household bikes 0.057 0.052 0.053 0.116 79.000 0.054 0.060 Male 0.190 0.186 0.185 0.398 271.200 0.178 0.181 Age 0.013 0.014 0.014 0.029 19.950 0.014 0.014 Income 0.000 0.000 0.000 0.000 0.238 0.000 0.000 College education 0.050 0.038 0.044 0.107 71.320 0.037 0.036 Employed 0.078 0.084 0.085 0.180 121.400 0.092 0.090 Tour distance 0.013 0.013 0.013 0.027 19.100 0.013 0.012 Tour crosses highway 0.780 0.776 0.781 1.690 1174.000 0.822 0.834 Miles bike lane < 1 mile 0.009 0.013 0.010 0.022 13.590 0.015 0.063 Mile bus routes < 1/2 mile 0.014 0.015 0.015 0.031 21.950 0.011 0.003 Bus stops < 1/2 mile 0.010 0.008 0.009 0.022 14.260 0.009 0.013 Rail stops < 1 mile 0.090 0.114 0.117 0.263 177.900 0.061 0.145 Miles rail lines < 1 mile 0.040 0.051 0.053 0.117 82.400 0.035 0.039 Intersection < 1/2 mile 0.001 0.000 0.000 0.001 0.420 0.001 0.002 CBG population density 0.000 0.000 0.000 0.000 0.023 0.000 0.000 CBG employment density 0.000 0.000 0.000 0.000 0.003 0.000 0.000 Work stops 0.045 0.040 0.040 0.087 61.510 0.045 0.039 Shopping stops 0.187 0.184 0.185 0.393 271.400 0.194 0.200 Social stops 0.102 0.102 0.104 0.222 144.700 0.130 0.120 Res. low density, 0–1/4 mile 0.010 0.010 0.010 0.022 15.440 0.013 0.009 Res. med. density, 0–1/4 mile 0.009 0.010 0.009 0.020 13.920 0.012 0.011 Res. high density, 0–1/4 mile 0.010 0.011 0.011 0.024 17.230 0.011 0.009 Bus. low density, 0–1/4 mile 0.011 0.006 0.006 0.012 7.865 0.000 0.015 Bus. high density, 0–1/4 mile 0.003 0.005 0.004 0.010 7.086 0.006 0.003 Industrial, 0–1/4 mile 0.007 0.008 0.008 0.017 12.580 0.008 0.014 Res. low density, 1/4–1/2 mile 0.014 0.017 0.015 0.033 23.380 0.018 0.025 Res. med. density, 1/4–1/2 mile 0.012 0.016 0.014 0.030 21.090 0.016 0.024 Res. high density, 1/4–1/2 mile 0.015 0.019 0.018 0.040 28.340 0.013 0.027 Bus. low density, 1/4–1/2 mile 0.014 0.004 0.006 0.014 6.699 0.014 0.003 Bus. high density, 1/4–1/2 mile 0.003 0.004 0.003 0.005 3.929 0.005 0.023 Industrial, 1/4–1/2 mile 0.003 0.013 0.012 0.025 18.090 0.008 0.024 Res. low density, 1/2–3/4 mile 0.003 0.018 0.017 0.035 24.200 0.020 0.020 Res. med. density, 1/2–3/4 mile 0.002 0.023 0.021 0.049 32.560 0.033 0.024 Res. high density, 1/2–3/4 mile 0.006 0.021 0.020 0.044 31.110 0.021 0.034 Bus. low density, 1/2–3/4 mile 0.023 0.028 0.027 0.079 52.490 0.047 0.074 Bus. high density, 1/2–3/4 mile 0.001 0.016 0.015 0.030 20.700 0.018 0.049 Industrial, 1/2–3/4 mile 0.013 0.019 0.018 0.039 26.930 0.014 0.022 Res. low density, 3/4–1 mile 0.005 0.017 0.015 0.033 22.870 0.024 0.024 Res. med. density, 3/4–1mile 0.005 0.031 0.029 0.063 42.400 0.033 0.025 Res. high density, 3/4–1 mile 0.012 0.002 0.001 0.002 1.396 0.005 0.023 Bus. low density, 3/4–1 mile 0.007 0.032 0.032 0.070 50.070 0.055 0.047 Bus. high density, 3/4–1 mile 0.010 0.036 0.034 0.074 50.080 0.047 0.047 Industrial, 3/4–1 mile 0.003 0.021 0.020 0.043 30.700 0.015 0.022 q, k 0.193 0.073 0.147 0.032 0.876 0.373 Note. Significance at the 1, 5, and 10% levels shown by , , and , respectively. and BE is a matrix of built environment characteristics miles. This distance corresponds closely with the dis- including zoning variables and follows the same substi- tance bands used to calculate the zoning percentages surrounding survey respondents’ residences and there- tution for the SAR and SEM. In the SEM model, both fore is an ideal choice for D. While it is possible to q and h are set to 0 and k enters the distribution of e as described in Equation (6).In the SAR h is set to 0. estimate spatial models with some observations having no neighbors, in practice, this also causes far more problems than the benefits of having more restrictive Determination of the Spatial Weights Matrix definitions of spatial neighbors, as outlined by Bivand The spatial weight matrix, W, in the equations above is and Portnov (2004). Using this distance-based neighbor developed by a two-step process. In the first step, rule, the neighbor binary matrix is constructed, with observations are determined to be spatial neighbors if observations within D distance of each other assigned a they are within a distance D from one another. The 1, and observations further apart than D assigned a 0. bandwidth used to create the neighbor matrix was the In the second step, the neighbor matrix is trans- minimum straight line distance necessary so that each formed into a spatial weight matrix W by either row observation included at least one neighbor, D ¼ 1.076 standardizing the binary neighbor matrix so that all JOURNAL OF SUSTAINABLE REAL ESTATE 11 rows sum to 1, or applying a function based on dis- (1%) increases the probability of a non-auto trip by tance and then row standardizing the matrix. While 0.01 0.235 ¼ 0.00235 (0.235%) on average. This value there are no generally accepted procedures for deter- of q also indicates that the dissipation of the effect is mining the correct weighting structure to use for W, quite rapid, as second order neighboring observations exhibit an effect of 0.235 ¼ 0.055225 and an effect of we apply the two most commonly used weighting schemes, the binary neighbor matrix, and a weight a 1% increase is equal to 0.01 0.05225 ¼ 0.00055225 that declines with a distance where the weight of each (0.055225%). This finding is further evidence that the choice between auto and non-auto transportation is neighboring observation is set to the inverse of dis- somewhat localized to a one-mile radius surrounding tance, w ¼ 1=d , where d is the distance between ij ij observations i and j in miles. We estimate the SAR, place of residence. Using the inverse distance W SEM, and SDM models using each spatial weight which places higher weights on closer neighbors, q is 0.159. The fact that q is a lower in this weighting matrix and compare the results below. scheme gives further evidence that the effect of neigh- boring observations is weak at closer distances than Estimation 1 mile. In this model, the coefficient has a slightly dif- The SAR and SEM have been estimated in the past ferent interpretation, as the vector Wy is not a simple using maximum likelihood techniques, as well as percentage, but rather a spatially weighted percentage more recently with Bayesian techniques. The estima- of non-auto trips based on distance. tion of the model using Bayesian techniques has some The value of k in the SEM model using binary and advantages over maximum likelihood, the most inverse distance W is 0.333 and 0.069, respectively. The important being the recovery of the posterior coeffi- coefficient k is estimated using Equation (6),where u ¼ cient distributions which can be used for statistical kWu þ e, u are the errors from the normal probit equa- inference tests (LeSage, 2000). The SAR, SEM, and tion, W is the spatial weights matrix, and e are the SDM models are estimated with a Bayesian model residuals after spatial correction. The SEM model only that takes 1,000 draws with a burn-in of 100 draws. addresses the spatial correlation of errors across space, Model results are listed in Table 3. and therefore only corrects for spatial heteroskedasticity. The positive and significant values of k indicate the cor- relation between error terms that are neighbors. Spatial Dependence Parameters However, since the SAR model also demonstrates spa- The model results for both the SAR and SEM models tial autocorrelation, part of the error term spatial correl- without zoning variables included using both a binary ation may be due to missing variable bias since the neighbor row standardized and inverse distance row spatially lagged dependent variable is absent from this standardized spatial weights matrix show that there is model. The general conclusion from the significant val- spatial dependence, with the spatial parameters q and k ues of q and k indicate that a model that jointly statistically significant at the 5% level in the SAR and addresses both spatial dependence and spatial heteroske- 10% level in the SEM model using the inverse distance dasticity may be the correct model (the SDM model). weight matrix, and q and k statistically significant at The results from the SDM estimate the q parameter the 5 and 10% level in the SAR and SEM models using of 0.876 and 0.373 for the binary and inverse dis- the row standardized binary spatial weights matrix. The tance weights matrix, respectively. This value indicates value of q in the SAR using a binary W and inverse negative spatial autocorrelation when spatially distance W are 0.235 and 0.159, respectively. The lower weighted independent regressors are added to the coefficient on q in the SAR model using the inverse model. This is unexpected as positive spatial autocor- distance spatial weights matrix shows the impact of the relation was estimated in the SAR model, and the choice of W, as the inverse distance specification results from the SEM indicated that controlling for already weights closer neighbors more heavily. spatial heteroskedasticity was warranted. A negative In the model using binary W, q is the estimated spatial dependence parameter indicates that the choice parameter on the n 1 vector Wy, where y is a vector of auto by a neighbor increases the probability of of 1s and 0s indicating non-auto transportation, and non-auto use. This peculiar result seems to point to thus Wy can be understood as the percentage of non- the complex nature of transportation decisions, as auto trips of all neighboring observations. Therefore, a spatially clustered use of auto or non-auto would be value of q equal to 0.235 tells us that an increase in expected due to preferences for locating near trans- the percentage of non-auto trips of neighbors by 0.01 portation networks of choice. However, it may be the 12 A. G. MUELLER AND S. WEILER Table 4. Marginal effects: SDM model, binary W dependent case that this negative spatial autocorrelation of the variable: non-auto transportation mode ¼ 1. dependent variable is picking up a preference for non- Variable Direct Indirect Total auto travel when neighbors neighboring auto choice HH size 0.01286 0.00603 0.00683 creates congestion. HH vehicles 0.06941 0.03264 0.03677 HH bicycles 0.01032 0.00485 0.00547 Male 0.03407 0.01604 0.01802 Age 0.00267 0.00126 0.00142 Zoning Parameters Income 0.00001 0.00000 0.00000 College degree 0.00703 0.00331 0.00372 The most notable result of the SAR models is that the Employed 0.01760 0.00826 0.00934 coefficients on all three residential zoning density lev- Tour distance 0.00242 0.00114 0.00128 Tour crosses highway 0.15734 0.07399 0.08334 els are negative and statistically significant in the 0– Miles bike lanes < 1M. 0.00297 0.00140 0.00157 1/4 mile distance band. This indicates that higher lev- Miles bus routes < 0.5 M. 0.00208 0.00098 0.00110 Bus stops < 0.5 M. 0.00177 0.00083 0.00094 els of residential zoning surrounding respondents’ res- Rail stops < 0.5 M. 0.01162 0.00548 0.00614 idences are associated with a decreased likelihood of Miles rail lines < 1 M. 0.00678 0.00319 0.00359 observing non-auto transportation. The other result Intersections < 0.5 M. 0.00011 0.00005 0.00006 CBG population/sq. mile 0.00000 0.00000 0.00000 that indicates the potential association between zoning CBG jobs/sq. mile 0.00000 0.00000 0.00000 and travel behavior is the significant positive coeffi- Work stops 0.00860 0.00404 0.00456 Shopping stops 0.03715 0.01747 0.01968 cients in the 1/4–1/2 and 3/4–1 mile band for indus- Social stops 0.02484 0.01166 0.01317 trial zoning, and high density business zoning in the Res. low density, 0–1/4 mile 0.00256 0.00120 0.00136 Res. medium density, 0–1/4 mile 0.00234 0.00110 0.00124 3/4–1 mile band. This indicates that business and Res. high density, 0–1/4 mile 0.00215 0.00101 0.00114 industrial zoning moderately close to home is associ- Bus. low density, 0–1/4 mile 0.00003 0.00001 0.00002 Bus. high density, 0–1/4 mile 0.00121 0.00057 0.00064 ated with increased non-auto transportation. This Ind., 0–1/4 mile 0.00158 0.00074 0.00084 indicates that residential locations surrounded by a Res. low density, 1/4–1/2 mile 0.00339 0.00159 0.00180 Res. medium density, 1/4–1/2 mile 0.00301 0.00141 0.00159 band of residential zoning up to one half mile may Res. high density, 1/4–1/2 mile 0.00258 0.00121 0.00137 prefer to drive to shopping, employment, and recre- Bus. low density, 1/4–1/2 mile 0.00264 0.00124 0.00140 Bus. high density, 1/4–1/2 mile 0.00097 0.00045 0.00051 ation, and that zoning that precludes closer businesses Ind., 1/4–1/2 mile 0.00161 0.00075 0.00085 may be associated with more non-auto travel behav- Res. low density, 1/2–3/4 mile 0.00391 0.00184 0.00207 Res. medium density, 1/2–3/4 mile 0.00632 0.00298 0.00335 ior. The coefficients on the SEM model are all statis- Res. high density, 1/2–3/4 mile 0.00410 0.00193 0.00217 tically insignificant so no interpretation can be made Bus. low density, 1/2–3/4 mile 0.00903 0.00424 0.00479 Bus. high density, 1/2–3/4 mile 0.00337 0.00159 0.00178 for this model. Ind., 1/2–3/4 mile 0.00271 0.00127 0.00144 While the sign of the coefficients on the explana- Res. low density, 3/4–1 mile 0.00453 0.00213 0.00240 tory variables indicates the direction of association on Res. medium density, 3/4–1 mile 0.00639 0.00301 0.00339 Res. high density, 3/4–1 mile 0.00090 0.00042 0.00048 the conditional probability of non-auto transportation Bus. low density, 3/4–1 mile 0.01049 0.00493 0.00557 behavior, their magnitude cannot be interpreted in the Bus. high density, 3/4–1 mile 0.00892 0.00420 0.00473 Ind., 3/4–1 mile 0.00292 0.00138 0.00155 same way as OLS or probit models. Due to the non- (W)HH size 0.08839 0.04159 0.04681 linearity of the model, and the presence of spatial (W)HH vehicles 0.00532 0.00248 0.00284 (W)HH bicycles 0.04547 0.02137 0.02410 dependence, the impact on a change of one explana- (W)Male 0.03762 0.01767 0.01995 tory variable has a spatial feedback loop effect on the (W)Age 0.00468 0.00220 0.00248 (W)Income 0.00046 0.00021 0.00024 dependent variable due to the presence of the spatially (W)College degree 0.10217 0.04776 0.05440 lagged dependent variable in the estimated equation. (W)Employed 0.04803 0.02297 0.02507 (W)Tour distance 0.00588 0.00279 0.00309 Therefore, it is necessary to estimate the marginal (W)Tour crosses highway 0.08159 0.03795 0.04364 effects of the change in each explanatory variable in (W)Miles bike lanes < 1M. 0.01139 0.00534 0.00605 (W)Miles bus routes < 0.5 M. 0.01100 0.00517 0.00583 the model to determine the direct, indirect, and total (W)Bus stops < 0.5 M. 0.00205 0.00097 0.00108 effects. We list the marginal effects of the SDM in (W)Rail stops < 0.5 M. 0.01590 0.00753 0.00837 (W)Miles rail lines < 1 M. 0.00506 0.00239 0.00267 Table 4. (W)Intersections < 0.5 M. 0.00045 0.00021 0.00024 While the SAR and SEM models both show signifi- (W)CBG population/sq. mile 0.00002 0.00001 0.00001 (W)CBG jobs/sq. mile 0.00000 0.00000 0.00000 cance in some of the zoning variables in determining (W)Work stops 0.06333 0.02960 0.03372 mode choice, the SDM shows significance in the low, (W)Shopping stops 0.07852 0.03718 0.04134 medium, and high residential zoning types for the (W)Social stops 0.13294 0.06236 0.07058 (W)Res. low density, 0–1/4 mile 0.00601 0.00283 0.00318 binary spatial weights matrix, significant negative (W)Res. medium density, 0–1/4 mile 0.00114 0.00054 0.00059 impacts for all spatially weighted zoning variables in (W)Res. high density, 0–1/4 mile 0.00340 0.00159 0.00181 (W)Bus. low density, 0–1/4 mile 0.00506 0.00239 0.00266 the 1/2–3/4 mile zoning band, and positive associa- (continued) tions of spatially weighted high density business and JOURNAL OF SUSTAINABLE REAL ESTATE 13 Table 4. Continued. Table 5. Log likelihood tests. Variable Direct Indirect Total Log Degrees of Model likelihood freedom AIC BIC (W)Bus. high density, 0–1/4 mile 0.00363 0.00170 0.00193 (W)Ind., 0–1/4 mile 0.01885 0.00885 0.01000 Base models: no zoning variables (W)Res. low density, 1/4–1/2 mile 0.00937 0.00440 0.00497 Probit 1840.061 22 3724.122 3868.035 (W)Res. medium density, 1/4–1/2 mile 0.00202 0.00094 0.00109 SAR 1838.980 23 3723.960 3874.415 (W)Res. high density, 1/4–1/2 mile 0.00101 0.00048 0.00053 SAR(1/d) 1839.615 23 3725.230 3875.685 (W)Bus. low density, 1/4–1/2 mile 0.01733 0.00809 0.00925 SEM 1860.135 24 3768.270 3925.266 (W)Bus. high density, 1/4–1/2 mile 0.00488 0.00228 0.00260 SEM(1/d) 1843.571 24 3735.143 3892.139 (W)Ind., 1/4–1/2 mile 0.00618 0.00291 0.00327 SDM 1821.144 44 3730.288 4018.114 (W)Res. low density, 1/2–3/4 mile 0.06475 0.03043 0.03432 SDM(1/d) 1817.655 44 3723.309 4011.135 (W)Res. medium density, 1/2–3/4 mile 0.10647 0.05002 0.05645 Zoning models (W)Res. high density, 1/2–3/4 mile 0.04537 0.02132 0.02405 Probit 1811.343 46 3714.687 4015.595 (W)Bus. low density, 1/2–3/4 mile 0.11828 0.05553 0.06275 SAR 1808.999 47 3711.999 4019.449 (W)Bus. high density, 1/2–3/4 mile 0.04730 0.02224 0.02506 SAR(1/d) 1810.157 47 3714.314 4021.764 (W)Ind., 1/2–3/4 mile 0.08333 0.03914 0.04420 SEM 1865.249 48 3826.498 4140.490 (W)Res. low density, 3/4–1 mile 0.06179 0.02904 0.03275 SEM(1/d) 1861.738 48 3819.475 4133.467 (W)Res. medium density, 3/4–1 mile 0.12995 0.06106 0.06890 SDM 1772.528 92 3729.057 4330.874 (W)Res. high density, 3/4–1 mile 0.03219 0.01514 0.01704 SDM(1/d) 1778.245 92 3740.491 4342.308 (W)Bus. low density, 3/4–1 mile 0.09916 0.04657 0.05259 (W)Bus. high density, 3/4–1 mile 0.05024 0.02356 0.02667 of the model parameters. These sample distributions Notes. Significance at the 1, 5, and 10% levels shown by , , and , respectively. of coefficients can be used to compute average mar- (W) preceding variable names indicates variables pre-multiplied by the ginal effects across observations of a change in an spatial weights matrix in SDM models. independent variable of the model on the probability of the independent variable, non-auto travel mode industrial zoning in the 3/4–1 mile zoning band pro- choice (LeSage & Pace, 2009). While the SEM model viding some evidence of association of association coefficients can be interpreted as marginal effects as between zoning and mode choice. in ordinary least squares because the spatial variation The SDM with binary spatial weights matrix was is only present in the error term, for the SAR and the best model using the log likelihood test. The SAR SDM models which include spatially lagged dependent with binary spatial weights matrix is indicated as the or independent variables, the impacts of a change in best model using Akaike Information Criteria (AIC) an explanatory variable can have an impact on all (Akaike, 1974) and the standard probit model is indi- other neighboring dependent variables, creating a cated as the best model using the Bayesian feedback loop with several orders of magnitude. Thus, Information Criteria (BIC) (Schwarz, 1978). A sum- these spatial models exhibit direct, indirect, and total mary of the log likelihood, AIC, and BIC of each of impacts. LeSage and Pace (2009) propose summary the models tested is shown in Table 5. All models measures of the marginal effects of a change in an show relatively similar results from the log likelihood, explanatory variable x by using the average change in AIC, and BIC criteria. However, the aggregate results r the expected value of the dependent variable y and of the Moran’s I, Geary’s C, and the significance of changing the multiplier matrix S (W) based on the both SAR and SEM models indicate that there may be spatial model. The expected value of a change is listed both spatial autocorrelation and spatial heterogeneity in Equation (10), where X is an n p matrix of n in the data generating process of the underlying data- observations and p explanatory variables. set. Therefore, the only model that produces unbiased results is the SDM (LeSage & Pace, 2009), Comparing EðyÞ¼ S ðWÞx þ aI (10) r r the use of the two spatial weights matrices in each i1 model, the binary row standardized spatial weights matrix leads to a better posterior distribution fit to S ðWÞ for the SAR and SDM model are given in the data, indicating that the binary matrix is preferred Equations (11) and (12). to the inverse distance matrix. This indicates that the The diagonal elements of the trace of the S ðWÞ matrix multiplied by the change in independent vari- spatial effects may be strong within the distance used able x give the direct impacts (Equation 13), while to specify spatial neighbors, just over one mile. ir the trace of the entire S (W) matrix multiplied by the change in independent variable x gives the total ir Marginal Effects and Elasticities impacts (Equation 14). Indirect impacts are the differ- The Bayesian Markov Chain Monte Carlo (MCMC) ence between total and direct impacts (Equation 15). estimation technique used to estimate the models Marginal direct effects for individual observations above produces samples of the posterior distribution are contained in the diagonal elements of S (W) r 14 A. G. MUELLER AND S. WEILER Table 6. Elasticities: SDM model, binary W dependent vari- (Equation 16) and indirect marginal effects are con- able: non-auto transportation mode ¼ 1. tained in the off diagonal elements of S (W)(Equation Variable Direct Indirect Total 17) (LeSage & Pace, 2009). HH size 0.077 0.036 0.041 HH vehicles 0.417 0.196 0.221 S ðWÞ¼ ðI qWÞ b (11) r n HH bicycles 0.062 0.029 0.033 S ðÞ W ¼ I qW I b þ Wh (12) ðÞðÞ Age 0.016 0.008 0.009 r n n r Income 0.000 0.000 0.000 MðrÞdirect ¼ n trðS ðWÞÞ (13) Tour distance 0.015 0.007 0.008 1 1 Miles bike lanes < 1M. 0.018 0.008 0.009 Mr total ¼ n I SðÞ W I (14) ðÞ r n Miles bus routes < 0.5 M. 0.012 0.006 0.007 ** Bus stops < 0.5 M. 0.011 0.005 0.006 MrðÞindirect ¼ MrðÞtotal MrðÞdirect (15) Rail stops < 0.5 M. 0.070 0.033 0.037 @y i Miles rail lines < 1 M. 0.041 0.019 0.022 ¼ S W (16) r ii Intersections < 0.5 M. 0.001 0.000 0.000 @x ir CBG population/sq. mile 0.000 0.000 0.000 @y i CBG jobs/sq. mile 0.000 0.000 0.000 ¼ S W (17) r ij Work stops 0.052 0.024 0.027 @x jr Shopping stops 0.223 0.105 0.118 Social stops 0.149 0.070 0.079 To calculate the elasticities, the change of each vari- Res. low density, 0–1/4 mile 0.015 0.007 0.008 Res. medium density, 0–1/4 mile 0.014 0.007 0.007 able is taken at the mean of the posterior distribution Res. high density, 0–1/4 mile 0.013 0.006 0.007 and the mean of the expected probability of the binary Bus. low density, 0–1/4 mile 0.000 0.000 0.000 Bus. high density, 0–1/4 mile 0.007 0.003 0.004 dependent variable, which is 16.63%. Marginal effects Ind., 0–1/4 mile 0.010 0.004 0.005 are reported for the direct, indirect, and total marginal Res. low density, 1/4–1/2 mile 0.020 0.010 0.011 Res. medium density, 1/4–1/2 mile 0.018 0.008 0.010 effects of a change in each independent variable. Res. high density, 1/4–1/2 mile 0.016 0.007 0.008 Direct effects are the change in the probability of Bus. low density, 1/4–1/2 mile 0.016 0.007 0.008 observing non-auto mode choice attributed to the Bus. high density, 1/4–1/2 mile 0.006 0.003 0.003 Ind., 1/4–1/2 mile 0.010 0.005 0.005 change in the independent variable. Indirect effects Res. low density, 1/2–3/4 mile 0.024 0.011 0.012 represent the spatially lagged effect on the autocorre- Res. medium density, 1/2–3/4 mile 0.038 0.018 0.020 Res. high density, 1/2–3/4 mile 0.025 0.012 0.013 lated dependent variable of a change in one of the Bus. low density, 1/2–3/4 mile 0.054 0.025 0.029 independent variables after the feedback loop from a Bus. high density, 1/2–3/4 mile 0.020 0.010 0.011 Ind., 1/2–3/4 mile 0.016 0.008 0.009 change in an independent variable has affected the Res. low density, 3/4–1 mile 0.027 0.013 0.014 spatially lagged dependent variable of spatial neighbor Res. medium density, 3/4–1 mile 0.038 0.018 0.020 Res. high density, 3/4–1 mile 0.005 0.003 0.003 observations. The sum of direct and indirect effects Bus. low density, 3/4–1 mile 0.063 0.030 0.033 equals the total effect of a change in the independent Bus. high density, 3/4–1 mile 0.054 0.025 0.028 Ind., 3/4–1 mile 0.018 0.008 0.009 variables after the feedback loop of the change has (W)HH size 0.531 0.250 0.281 run its course. Dummy variable elasticities are not (W)HH vehicles 0.032 0.015 0.017 (W)HH bicycles 0.273 0.128 0.145 reported. Results for the SDM model are reported for (W)Age 0.028 0.013 0.015 the binary spatial weights matrix in Table 6. (W)Income 0.003 0.001 0.001 Several of the statistically significant variables in the (W)Tour distance 0.035 0.017 0.019 (W)Miles bike lanes < 1M. 0.068 0.032 0.036 SAR model have small marginal effects on the expected (W)Miles bus routes < 0.5 M. 0.066 0.031 0.035 sign. The largest of these is the number of household (W)Bus stops < 0.5 M. 0.012 0.006 0.007 (W)Rail stops < 0.5 M. 0.096 0.045 0.050 vehicles, with a marginal effect of 9.123%. This is not (W)Miles rail lines < 1 M. 0.030 0.014 0.016 surprising considering this variable indicates preference (W)Intersections < 0.5 M. 0.003 0.001 0.001 (W)CBG population/sq. mile 0.000 0.000 0.000 for owning an asset that encourages automobile trans- (W)CBG jobs/sq. mile 0.000 0.000 0.000 portation. Miles of bike lanes have an unexpected nega- (W)Work stops 0.381 0.178 0.203 (W)Shopping stops 0.472 0.224 0.249 tive marginal effect but is very small. One possible (W)Social stops 0.799 0.375 0.424 reason for the unexpected sign on this variable may be (W)Res. low density, 0–1/4 mile 0.036 0.017 0.019 (W)Res. medium density, 0–1/4 mile 0.007 0.003 0.004 that areas that are more dense, such as the CBD, may (W)Res. high density, 0–1/4 mile 0.020 0.010 0.011 have an overall lower mileage of bike lanes, while areas (W)Bus. low density, 0–1/4 mile 0.030 0.014 0.016 (W)Bus. high density, 0–1/4 mile 0.022 0.010 0.012 that lack access to goods and services within a non- (W)Ind., 0–1/4 mile 0.113 0.053 0.060 auto distance have a high mileage of bike lanes that are (W)Res. low density, 1/4–1/2 mile 0.056 0.026 0.030 (W)Res. medium density, 1/4–1/2 mile 0.012 0.006 0.007 intended for recreational use. (W)Res. high density, 1/4–1/2 mile 0.006 0.003 0.003 Mileage of bus routes and number of bus stops (W)Bus. low density, 1/4–1/2 mile 0.104 0.049 0.056 both have the expected sign but the effects are also (W)Bus. high density, 1/4–1/2 mile 0.029 0.014 0.016 (W)Ind., 1/4–1/2 mile 0.037 0.017 0.020 small. The estimated coefficient for number of rail (continued) stops is unexpectedly negative, while the coefficient JOURNAL OF SUSTAINABLE REAL ESTATE 15 Table 6. Continued. though they are slightly farther than other statistically Variable Direct Indirect Total significant variables would suggest for encouraging (W)Res. low density, 1/2–3/4 mile 0.389 0.183 0.206 non-auto transportation. (W)Res. medium density, 1/2–3/4 mile 0.640 0.301 0.339 Elasticities calculated from the marginal effects at (W)Res. high density, 1/2–3/4 mile 0.273 0.128 0.145 (W)Bus. low density, 1/2–3/4 mile 0.711 0.334 0.377 the means of the coefficient distributions indicate that (W)Bus. high density, 1/2–3/4 mile 0.284 0.134 0.151 non-auto transportation mode preferences are highly (W)Ind., 1/2–3/4 mile 0.501 0.235 0.266 (W)Res. low density, 3/4–1 mile 0.372 0.175 0.197 inelastic. This result in part captures the sample distri- (W)Res. medium density, 3/4–1 mile 0.781 0.367 0.414 bution which indicates that people use auto for their (W)Res. high density, 3/4–1 mile 0.194 0.091 0.102 (W)Bus. low density, 3/4–1 mile 0.596 0.280 0.316 mode of transportation at a much higher frequency (W)Bus. high density, 3/4–1 mile 0.302 0.142 0.160 than all other modes combined. Elasticities for num- (W)Ind., 3/4–1 mile 0.508 0.238 0.269 ber of household vehicles and age are above one, sig- Note. Significance at the 1, 5, and 10% levels shown by , , and , respectively. naling that these variables are a good indicator of (W) preceding variable names indicates variables pre-multiplied by the transportation mode preference. The most interesting spatial weights matrix in SDM models. result is that low density residential zoning is more elastic than all other zoning types. This is the expected for miles of rail lines has the expected sign but a small result for the zoning band within a quarter mile of positive coefficient. The study area has a more mature place of residence, indicating that altering this zoning bus system than rail system, and the rail stops are type may have the most potential of all possible zon- spread more evenly between dense urban locations ing changes in promoting non-auto transportation. near downtown and suburban locations. Perhaps the Marginal effects in the SDM model with binary spa- reason for the unexpected sign on the rail stops coeffi- tial weights are similar to those in the SAR model, with cient is capturing the propensity of most suburban many of the same variables being statistically significant residents to use auto transportation even when they and thus leading to many of the same conclusions. The live in close proximity to rail stops. This phenomenon inclusion of the spatially weighted variables in the may be due to the rail lines not going to locations regression make some of the variables that were statis- that meet suburban household needs since many of tically significant in the SAR regression insignificant. the rail lines were built to service commuting to Elasticities in the SDM model are also similar to the downtown from the suburbs, but not to perform SAR model. One interesting observation is that house- everyday shopping or recreational tasks close to home. hold vehicles have a negative direct marginal effect, but Shopping and social stops along a tour are both a positive indirect impact, indicating that having many negative and relatively large compared with many of cars may encourage auto usage, but discourage neigh- the other variables in the regression. These are the boring respondents to use auto transportation. The expected sign and indicate a propensity to drive when effects are still small, however, with the indirect elasti- shopping for goods that may need to be carried home city being less than half of direct elasticity. or to social gatherings that are located in recreational Elasticities of residential zoning variables are nega- or residential areas. Residential zoning within a quar- tive but very small, indicating that the response to ter mile from place of residence has the expected residential zoning is highly inelastic. Although the negative sign, although the effects are small. Given the marginal effects are non-linear, in general most of the value people place on their own time, it is not surpris- estimates follow a normal distribution. Thus, when ing that higher residential density, and therefore lower considering that the marginal effects are capturing a business density, may encourage people to drive to locations that are not in the immediate vicinity of one-percent increase in a specific zoning type, it may be more appropriate to consider that, for example, a their residence. Residential zoning in the quarter to 10% increase in a zoning type would have roughly ten half mile range has the opposite sign with similarly times the impact on the probability of non-auto trans- small marginal effects. It is uncertain what explains the positive marginal effect on non-auto transporta- portation modes being chosen. For example, if resi- tion for higher residential zoning levels within this dential low-density zoning was to increase by 10% band. Finally, high density business within the three within a quarter mile of a survey respondents’ resi- quarter to one-mile band has a positive marginal dence, using this rough measure we would expect to see a 0.136% decrease in non-auto transportation effect on non-auto transportation. This may indicate that if a high level of businesses are located within mode choices. While still a small impact, this change this band, survey respondents are willing to travel by is not insignificant when considering the magnitude non-auto modes to reach these destinations even of trips away from home taken by city residents across 16 A. G. MUELLER AND S. WEILER the United States each year. Even small percentage of long-term path dependence that followed from an reductions in auto trips could add up to large overall early preference for auto transportation in the devel- reductions in vehicle miles driven. opment of transportation infrastructure. The wide- spread building of roads may have led to a long chain of city planning decisions that have shaped the built Conclusion environment to accommodate automobile transporta- In comparing the current results to those of the naïve tion to the detriment of alternative modes of transpor- non-spatial empirical approach by Mueller and Trujillo tation that use less energy and decrease congestion. (2019), this paper has shown that explicit spatial model- Further research will be needed to determine if drastic ing is clearly an important factor in a fuller understand- changes in built environment design that focus on ing of travel choice behavior. This work has shown that alternatives to automobile transportation can change both spatial autocorrelation and heterogeneity are pre- society’s preference for the automobile towards trans- sent in the data and formal modeling techniques give portation usage that is more environmentally and cul- insight into the spatial aspects of associations between turally sustainable for the long-term future. In addition to urban form, there are many other factors built environment variables and mode choice. The spa- that determine travel behavior, such as weather, phys- tial results here reconfirm the overall inelasticity of travel mode choice, yet also offer explicit clues as to ical health, and availability of transportation options. zoning’s subtle influences on mode choice. The current dataset precludes the testing of such rela- The evidence provided by the log likelihood test tionships, but in cities where sweeping changes to more flexible zoning policies have been implemented, indicate that the Bayesian models of the association event studies could shed light on the causal relation- between zoning on travel mode preference favor the ships between zoning mix and transportation behav- Spatial Durbin Model with a binary spatial weights ior. This study has provided some direction on the matrix indicated in Table 5. Residential low, medium, utility of using spatial models in transportation stud- and high density within one quarter mile of residences ies, but more research is needed to incorporate holis- are all statistically significant and associated with a tic solutions to urban transportation problems that lower propensity for non-auto travel. Using AIC to include vehicle electrification, driverless cars, bike, assess the models considered in this chapter, the SAR and scooter sharing programs, and alternative trans- model with a binary spatial weights matrix was the portation infrastructure. best model overall. In this model, residential zoning variables of low, medium, and high density in the 0– 1/4 mile zoning band all have statistically significant Note and negative impacts on survey respondents’ probabil- 1. All estimations are implemented in the software system R ity of choosing non-auto transportation. Using BIC, the (R Core Team, 2022). The spatial weights matrix was standard probit model had the best fit to the data. One constructed and standardized using the R add-on package potential conclusion from these results is that zoning spdep (Bivand et al., 2013;Bivand & Piras, 2015). 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Journal of Sustainable Real Estate – Taylor & Francis

Published: Dec 31, 2023

Keywords: Land use restrictions; spatial econometrics; transportation mode choice; travel behavior; walkability; zoning

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Spatial Models of Travel Behavior and Land Use Restriction

Spatial Models of Travel Behavior and Land Use Restriction

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Spatial Models of Travel Behavior and Land Use Restriction

Spatial Models of Travel Behavior and Land Use Restriction

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