Much of the motivation for the research questions investigated by social scientists arises from the uniquely social and interactive nature of human beings. Social science phenomena are uniquely interesting, many social scientists argue, because of the consequences for both individuals and societies that are produced by the interactions between family members, co-workers, fellow citizens, tribes, and nation-states. These same interactions uniquely predispose social science data toward spatial dependence. Shared concerns between social actors combine with spatial proximity to promote familiarity. In turn, this familiarity between social actors breeds both contempt and conflict and interaction and interdependence. And by extension, it produces similar behaviors (even when these behaviors are shared conflict) between spatially proximate actors – in short, it produces spatial dependence. Alternatively, as Galton's problem recognizes, even in those instances in which actors do not interact with each other, their spatial locations can induce spatial dependence because of shared attributes that influence human behavior.

The time has never been better for social scientists to model the spatial dependence that is predicted by our theories of human behavior and that exists in the data we employ. The past two decades have witnessed a unique confluence of advances in four areas: (1) the geocoding of social science data, (2) the computational capabilities of computers, (3) the development of diagnostics and estimators for spatial autocorrelation, and (4) the inclusion of routines for spatial diagnosis and estimation in both general and more spatially specialized software packages. This book has sought to demonstrate how social scientists can diagnose and model spatial dependence. In the process, it has examined a variety of spatial diagnostics and estimators. It has also explored some of the concerns that pose challenges for spatial analysis such as the modifiable areal unit problem (MAUP) and the boundary value problem.

Going forward, three particular concerns are particularly important for future developments in spatial analysis.

Stata®: This software from Stata Corporation, widely used in the social sciences, does not include much spatial functionality in its standard release. However, there are some user-written functions that allow for the diagnosis and modeling of spatial dependence. Commands written by Maurizio Pisati allow for the importation and generation of spatial weights matrices, the diagnosis of spatial dependence in the absence of covariates, the diagnosis of spatial dependence for ordinary least squares (OLS) regressions, and maximum likelihood estimation of spatial lag and spatial error models (see Pisati 2001).Weights matrices can be imported from external objects or created in Stata using the spatwmat command. The spatgsa command provides three alternative global spatial diagnostics, Moran's I, Geary's c, and Ord and Getis's G statistic. The spatlsa command provides four local spatial diagnostics, the local Moran's I, the local Geary's c, and the Gi and statistics. Five diagnostics for spatial dependence in OLS regressions can be carried out via the spatdiag command: the Moran's I diagnostic, the Lagrange multiplier (LM) diagnostic against error dependence (LMError), the robust LM diagnostic against error dependence (Robust LMError), the LM diagnostic against spatial lag dependence (LMLag), and the robust LM diagnostic against spatial lag dependence (Robust LMLag). Finally, spatial lag and spatial error models can be estimated via maximum likelihood estimation by using the spatreg command.

Recently, David M. Drukker, Hua Peng, Ingmar Prucha, and Rafal Raciborski have created SPPACK, a Stata module for the estimation of cross-sectional spatial autoregressive models (see Drukker et al. 2013; Drukker, Prucha, and Raciborski 2013a; and Drukker, Prucha, and Raciborski 2013b). The module's spmat command provides a variety of options involving spatial weights matrices. Researchers can create contiguity matrices and inverse distance matrices or import matrices based on variables (such as distances) that are already stored in existing Stata datasets. Features of spatial weights matrices can also be summarized using the spmat summarize command.

When employing the spatial diagnostics examined in Chapter 4, social scientists will often find evidence of spatial autocorrelation (see, e.g., Eff 2004). As discussed in Chapter 1, this predisposition of social science data toward spatial autocorrelation often results from interdependence between the units studied by social scientists. In other cases, social science data exhibit spatial dependence not as a result of behavioral interdependence but as a consequence of spatial clustering in the sources of behaviors of interest to social scientists. The spatial dependence, in short, may be consistent with either a spatial lag model or a spatial error model.

Substantive theory will often lead scholars to believe that a spatial lag specification or a spatial error specification is more appropriate for their particular substantive application. Scholars may, for example, expect that a spatial diffusion process is at work and thus believe that a spatial lag model is warranted. Although such a specification may seem appropriate, such a theoretical expectation should not go untested. It would be inappropriate to estimate a diffusion model with a spatially lagged dependent variable if the spatial dependence diagnosed via, for example, the univariate Moran's I, is instead produced by spatial clustering in the sources of otherwise independent behaviors. This model misspecification will lead the researcher to inappropriate substantive inferences about the nature of the spatial dependence in her data.

Inappropriate spatial model specification is all the more problematic because of the close mathematical relationship between a spatial lag model and a spatial error model with spatial autoregressive error dependence. As this chapter will discuss, a spatial autoregressive error model can be rewritten as a spatial Durbin model with both spatially lagged dependent and independent variables if a set of nonlinear common factor constraints are valid. Because of this close relationship between spatial autoregressive dependence in a spatial lag model and spatial autoregressive dependence in a spatial error model, a significant spatial parameter in a spatial lag model may reflect spatial clustering in omitted sources of the behavior of interest rather than true spatial lag dependence consistent with a diffusion process.

If the ordinary least squares (OLS) diagnostics discussed in the previous chapter indicate the existence of spatial lag or spatial error dependence, the researcher will wish to model the type of dependence indicated by these diagnostics. If the OLS diagnostics indicate the presence of a diffusion process, the researcher will wish to estimate a spatial lag model via maximum likelihood (ML) estimation or an instrumental variables specification incorporating instruments for the spatially lagged dependent variable. Alternatively, if the OLS diagnostics indicate the existence of spatial error dependence, the researcher may choose to estimate a more fully specified OLS model to model the spatial dependence or may choose to employ a ML or generalized method of moments (GMM) approach incorporating the spatial dependence in the errors.

The spatial dependence diagnosed via the diagnostics discussed in Chapter 5 may alternatively be produced by spatial heterogeneity in the effects of covariates. If this is the only source of spatial dependence, modeling this heterogeneity will be sufficient to capture the spatial dependence. As a consequence, any specification search should also consider the possibility of spatial heterogeneity, which is the focus of Chapter 7. This chapter will first, however, examine alternative approaches for modeling spatial dependence if spatial heterogeneity is not present.

This chapter begins by examining ML estimation of spatial lag models that derives from Ord (1975). Next, I explore alternative instrumental variables and GMM estimators for spatial lag dependence. Next, I turn to approaches for estimating spatial error models. I conclude by considering areas of concern in the estimation of spatial models. These include estimators for large sample sizes and diagnostics for continued spatial dependence.

MAXIMUM LIKELIHOOD SPATIAL LAG ESTIMATION

The mixed regressive, spatial autoregressive model, or spatial lag model, extends the pure spatial autoregressive model considered in Section 3.2 to include also the set of covariates and associated parameters:

y = ρWy+Xβ+ε

where X is again an N by K matrix of observations on the covariates, β is a K by 1 vector of parameters, and the remaining notation is as discussed in Section 3.2.

Thus far, this book has examined models that are appropriate when the phenomenon of interest is measured as a continuous dependent variable. Often, however, social scientists are interested in explaining phenomena that do not lend themselves to linear models for continuous dependent variables. Spatial autocorrelation also affects phenomena typically studied through these more advanced models, such as limited dependent variable models, count models, or survival models. However, although spatial dependence is likely to be present in many of these more complex phenomena, it is rarely modeled even by applied researchers employing these advanced modeling strategies. Instead, for example, event counts at one location are often assumed to be spatially independent of counts at neighboring locations. Clearly, such an assumption is untenable. This chapter examines how social scientists can incorporate spatial dependence in more advanced modeling approaches beyond the linear model. It examines several types of advanced models familiar to applied researchers in the social sciences: binary dependent variable models, multinomial models, ordinal models, count models, and survival models.

SPATIAL BINARY DEPENDENT VARIABLE MODELS

Many phenomena of interest to social scientists such as interstate conflict, marital status, recividism, and the like take the form of binary variables. As is well known, estimators for continuous dependent variables such as ordinary least squares (OLS) are not applicable for these dichotomous dependent variables. Estimators such as OLS suffer from heteroskedasticity and non-normality, assume incorrect functional form, and can produce nonsensical predictions when the dependent variable is binary (see, e.g., Long 1997, 38–40). As a consequence, scholars utilize maximum likelihood (ML) estimators such as probit or logit. However, although these estimators are widely applied in the social sciences, rarely do social scientists model spatial dependence in the binary dependent variables of interest to them. This is problematic given the propensity of social science data toward spatial dependence. Interestingly, while spatial binary dependent variable models have been largely absent in the social science literature, these models represent a critical area of interest among current spatial analysis scholars, with many new estimators being developed in the past two decades. In this section, I examine these advances in modeling spatial dependence in dichotomous dependent variables.

Thus far this book has examined diffusion processes from a cross-sectional perspective. Diffusion processes, however, are inherently temporal: behavior diffuses across spatial locations over time. As a consequence, cross-sectional evidence can only be said to be consistent with a diffusion process; it cannot definitively demonstrate that diffusion has occurred. To gain greater leverage in the diagnosis of spatial diffusion we ideally would wish to have observations arrayed over both space and time (see also Franzese and Hays 2007).

Social scientists are increasingly employing such space–time data in which multiple observations are available for each unit over several time periods. Examples include demographic datasets where researchers have data on the population makeup of the same countries for many different time periods or political science datasets where researchers have data on voter turnout for the same counties in many different elections.

As Beck (2001) notes, there is a critical distinction between two forms of space–time data, time-series cross-sectional (TSCS) data and panel data, even though the latter term is frequently used to refer to both types of data in econometrics. In the former, the units are fixed and of interest in themselves, and appeals to asymptotics involve the time dimension (T) (since the number of units in any given time period is fixed). In contrast, in panel data, the cross-sectional observations (N) strongly outnumber the number of time periods, which often are only two or three. True panel data in this sense typically take the form of survey sample data (and thus asymptotics can be based on N). Given this distinction, most spatial models of interest involve TSCS data as the spatial locations of survey respondents are typically not publicly provided due to confidentiality concerns in survey sample panel data. The focus of this chapter therefore is largely on TSCS data rather than on panel data.

This book is designed for social scientists who seek to model the spatial dimension of behavior that is inherent in substantive social science theories. The social sciences are united by their focus on subjects that are inherently social and interdependent. Unlike more individuated concerns, the units studied by social scientists often interact with each other and affect each other's behavior as a consequence. This interaction is promoted by spatial proximity – units that are more geographically proximate are more likely to interact with each other and influence each other as a consequence. Shared concerns combine with spatial proximity to promote familiarity. Just as familiarity has long been observed to breed contempt, so also does it promote cooperation. Even when spatially proximate units do not interact with each other, they often exhibit similar behaviors as a result of shared environmental influences. In short, there is a strong spatial component to many of the behaviors studied by social scientists and this is reflected in many of our theories in the social sciences.

It might surprise social scientists who have not seriously considered the spatial dimension of behavior to learn that all social science data are, in fact, spatial data. The behaviors, processes, and events of interest to social scientists occur at specific geographic locations. The past two decades have seen an explosion in social science data that are geocoded – coded to include the geographic locations of the observations. This increase in the availability of geocoded data has been matched by advances in geographic information system (GIS) software, such as ESRI's (Environmental Systems Research Institute) popular ArcGIS package. At the same time, significant advancements have been made in the development of spatial diagnostics and estimators, many of which are now included in dedicated spatial software such as GeoDa as well as standard statistical packages such as R, Stata, and WinBUGS. The time has never been better for scholars wishing to model spatial relationships in their data.

Spatial data require some minimal work at the early stages of analysis to get these data ready for a spatial analysis. This work is not onerous and can typically be completed in just a few minutes. However, the steps in this work are unique to spatial data and many researchers in the social sciences are not familiar with these steps. As a consequence, it is useful to briefly discuss them. Researchers interested in exploring the details of these steps will find additional, detailed information on them in manuals for geographic information systems (GIS) and other software.

The main issue in setting up data for a spatial analysis is that datasets in the social sciences do not include geometric information on the units the researcher is seeking to examine. To be sure, these data do typically include identifiers and use of these geographic identifiers, such as Federal Information Processing Standards (FIPS) codes, is very helpful in linking social science data up to files that contain features of the polygons examined, such as their size, boundaries, and locations. But typically, social science data do not include these latter features themselves, and thus researchers must link, or join, their datasets to a file that contains this information.

The standard such geographic file in spatial analysis is the shapefile developed by Environmental Systems Research Institute (ESRI) for use with its Arcview GIS software in the early 1990s. Today, ESRI's principal GIS product is ArcGIS, which includes a suite of component applications, including ArcMap, ArcCatalog, ArcScene, and ArcGlobe. One of the more efficient ways to join one's data is to do so in ArcMap, utilizing an identifier variable that is included in both the shapefile and in the researcher's dataset. Often this variable will be the units’ FIPS codes.

Shapefiles are readily available for a variety of polygons of interest via the Internet. Many colleges and universities have GIS units on campus that can aid researchers in finding shapefiles if a search on theWeb proves unsuccessful.

Several helpful websites and listservs for spatial analysis have been developed as the use of spatial techniques has increased over the past two decades. This appendix considers some of the most helpful Web resources on spatial analysis in these two categories.

WEBSITES

AI-GEOSTATS (http://www.ai-geostats.org/): This website provides a variety of resources for geostatistics and spatial analysis. Among these are links to software, papers, books, conferences, and job openings. The site also provides links to the AI-GEOSTATS mailing list.

Center for Spatially Integrated Social Science (CSISS) (http://www.csiss.org/): Funded in 1999 by the National Science Foundation, The Center for Spatially Integrated Social Science (CSISS) (housed at the University of California, Santa Barbara) is designed to advance the dissemination of spatial techniques and perspectives in the social sciences. The CSISS website provides a variety of resources for social scientists wishing to apply spatial analysis in their research. Among these are a searchable database of more than 17,000 references featuring spatial applications in the social sciences from 1990 to 2004. The website also includes video clips of workshops, descriptions of “classic” geographically oriented studies in the social sciences, and links to software tools.

GeoDa Center (http://geodacenter.asu.edu/): Founded by Luc Anselin, the GeoDa Center for Geospatial Analysis and Computation at Arizona State University has become a leading repository for spatial studies, the dissemination of spatial software tools, and the provision of training and support. The website provides a link for downloading GeoDa and also provides tutorials for GeoDa and for spatial packages in R. The site also provides an extensive set of e-talks, lectures on spatial analysis, tools, and techniques. The website also provides links to recent working papers by scholars affiliated with the GeoDa Center as well as to spatial data that can be downloaded for analysis.

GISpopsci.org (http://gispopsci.org): This project is a collaboration between the Population Research Institute (The Pennsylvania State University) and the Center for Spatially Integrated Social Science (University of California, Santa Barbara). The site, whose development was funded in part by the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD), provides a variety of resources for research and instruction on advanced spatial analysis in the population sciences and spatial demography. The website maintains an extensive bibliography of citations applicable to research in spatial demography and the population sciences. The site also contains links to course syllabi on spatial topics.