New Directions in Spatial Econometrics

Since Paelinck coined the term ‘spatial econometrics’ in the early 1970s to refer to a set of methods that deal with the explicit treatment of space in multiregional models, the field has come a long way. The early results established in regional economics by Blommestein, Hordijk, Klaassen, Nijkamp, Paelinck and others [e.g., Hordijk and Nijkamp (1977), Hordijk (1979), Paelinck and Klaassen (1979), Blommestein (1983)], as well as the path breaking work of Cliff and Ord in geography [Cliff and Ord (1973, 1981), Ord (1975)] have grown into a broad set of models, tests and estimation techniques that incorporate space more effectively in econometric modeling [for recent overviews, see Anselin (1988, 1992a), Haining (1990), Cressie (1991)]. In spite of these important methodological developments, it would be an overstatement to suggest that spatial econometrics has become accepted practice in current empirical research in regional science and regional economics. On the positive side, the sad State of affairs reflected in the literature surveys of Anselin and Griffith (1988) and Anselin and Hudak (1992) seems to have taken a turn for the better, since there is evidence of an increased awareness of the importance of space in recent empirical work in ‘mainstream’ economics. For example, this is indicated by the use of spatial models in the study of fiscal spill-overs in Case et al. (1993), the analysis of the productivity effects of public sector capital in Holtz-Eakin (1994), and the assessment of land price volatility in Benirschka and Binkley (1994), among others.

It has now been more than two decades since Cliff and Ord (1972) and Hordijk (1974) applied the principle of Moran’s Itest for spatial autocorrelation to the residuals of regression models for cross-sectional data. To date, Moran’sIstatistic is still the most widely applied diagnostic for spatial dependence in regression models [e.g., Johnston (1984), King (1987), Case (1991)]. However, in spite of the well known consequences of ignoring spatial dependence for inference and estimation [for a review, see Anselin (1988a)], testing for this type of misspecification remains rare in applied empirical work, as illustrated in the surveys of Anselin and Griffith (1988) and Anselin and Hudak (1992). In part, this may be due to the rather complex expressions for the moments of Moran’s I, and the difficulties encountered in implementing them in econometric Software [for detailed discussion, see Cliff and Ord (1981), Anselin (1992), Tiefelsdorf and Boots (1994)]. Recently, a number of alternatives to Moran’s Ihave been developed, such as the tests of Burridge (1980) and Anselin (1988b, 1994), which are based on the Lagrange Multiplier (LM) principle, and the robust tests of Bera and Yoon (1992) and Kelejian and Robinson (1992). These tests are all asymptotic and distributed as X ²variates. Since they do not require the computation of specific moments of the statistic, they are easy to implement and straightforward to interpret. However, they are all large sample tests and evidence on their finite sample properties is still limited.

Issues relating to spatially autocorrelated disturbance terms are often considered in regional econometric models.¹ Although various models have been suggested to describe such spatial correlation, one of the most widely used models is a spatial autoregressive (AR) model which was originally suggested by Whittle (1954) and then extensively studied by Cliff and Ord (1973).² In the model the regression disturbance vector is viewed as the sum of two parts. One of these parts involves the product of a spatial weighting matrix and a scalar parameter, say p; the other is a random vector whose elements are typically assumed to be independent and identically distributed (i.i.d.) with zero mean and finite variance. We will henceforth refer to this random vector as the innovation vector, so as to distinguish it from the disturbance vector.

In this chapter we use empirical examples to demonstrate the usefulness of the generalized error component framework suggested in Bolduc et al. (1992) for dealing with the problem of correlation among the errors of a regression based on travel flow data. This methodology augments Standard error component decompositions with first-order spatial autoregressive processes, i.e., SAR(l), with the purpose of allowing for the different sources of misspecification generally associated with this type of model. The error component approach splits the error term into a sum of one component related to the zones in origin, one component associated with the zones in destination and a remainder. The interdependencies among the errors are modeled with the help of SAR(l) processes. This decompositional approach extends the previous works by Brandsma and Ketellapper (1979) and Bolduc et al. (1989) which also relied on spatial autoregressive processes to model the error correlation.

The reason for undertaking the present investigation is that, in spatial econometrics, as was argued in Paelinck and Klaassen (1979, pp. 6–9), econometric relations in space result more often than not in highly non-linear specifications. This is very probably even more the case if services or enterprise functions in space are considered [Paelinck (1987)].

Spatial econometrics is a field of study concerned with the development of methods and techniques of data analysis that are appropriate for fitting models arising in regional science [Anselin (1988, p. 10)]. Regional science models relate to events occurring in geographic space and a particular focus of spatial econometrics is to develop methods and techniques that recognize the special problems that arise when fitting models to spatial cross sectional data.

In a recent paper [Getis (1990)], I develop a rationale for filtering spatially dependent variables into spatially independent variables and demonstrate a technique for changing one to the other. In that paper, the transformation is a multi-step procedure based on Ripley’s second order statistic (1981). In this chapter, I will briefly review the argument for the filtering procedure and propose a simplified method based on a spatial statistic developed by Getis and Ord (1992). The chapter is divided into four parts: 1) a short discussion of the rationale for filtering spatially dependent variables into spatially independent variables, 2) a review of a Getis-Ord statistic, 3) an outline of the filtering procedure, and 4) three examples taken from the literature on urban crime, regional inequality, and government expenditures.

Heteroscedasticity and autocorrelation typically are assumed to be absent in econometric models. Linear regression models are forgiving if these assumptions fail: ordinary least squares (OLS) estimates remain consistent if errors are not homoscedastic or are autocorrelated. Estimators for models with discrete data are not always as forgiving as OLS. For example, the Standard probit estimator continues to provide consistent estimates when error terms are autocorrelated, but the estimates are inconsistent as well as inefficient when errors have non-constant variances. Failure of the homoscedasticity assumption also leads to inconsistent estimates in such common models as tobit and logit. Thus, heteroscedasticity is a serious problem in models with discrete data.

Much research has been done on estimating models of spatial dependence with continuous random variables [for example, see Ripley (1981), Anselin (1988), Cressie (1991)]. However, relatively little work has been done on incorporating spatial dependencies into models with qualitative dependent variables. Boots and Kanaroglou (1988) have incorporated spatial considerations into a migration model and Anne Case (1992) has done likewise for a model of technology adoption.

In this chapter, we examine a logit model designed specifically for spatial choice among aggregate destinations. Typically, the logit and gravity models are applied to problems of spatial interaction without due consideration for the aggregation scale of the data. There is of course a general awareness that models estimated at different levels of aggregation will yield different parameter estimates but most researchers proceed directly with their analysis at the aggregate level and make no adjustments. Research on aggregation issues has been intertwined with work on the ‘modifiable areal unit problem’. This research examines how spatial Statistical models and diagnostics are affected by the aggregation scale and spatial configuration of spatial units. Choice processes per se, as are present in all problems of spatial interaction, are not specifically addressed in this body of research. Rather the focus is on the univariate or bivariate spatial processes associated with one or two spatially-referenced attributes [Openshaw (1984) and Arbia (1989)]. Aggregation issues are relevant in both scenarios but as will be illustrated in this chapter, the theoretical framework to accommodate aggregation in a choice context is quite different from that considered in analysis of the modifiable areal unit problem.

Specific instances of the general linear model (GLM) have already been implemented within spatial statistics. Griffith (1978) summarizes how to write the simple one- way ANOVA model in the presence of spatial autocorrelation, more recently extending this to N-way ANOVA and unbalanced designs [Griffith (1992b)]. This recent article highlights the need for more work on mixed and random effects unbalanced design models, which is the case in traditional statistics as well. Griffith (1979) describes how to write the one-way MANOVA model for geo-referenced data, more recently extending this to N-way MANOVA and unbalanced designs [Griffith (1992b)], too. His initial findings in this case are consistent with the perspective promoted by Haining (1991). Griffith (1989) has also outlined spatial Statistical two-groups discriminant function analysis and ANCO VA models [see also Anselin (1988), for a spatial econometric implementation of this latter model]. Other attempts along these lines are found in Switzer (1985), who has devised a spatial principal components analysis, Mardia (1988) and Griffith (1988), who have constructed multivariate geo-referenced data models, and Wartenberg (1985), who has formulated a cross-Moran Coefficient (cross-MC). And, Cressie and Hilterbrand (1993) have studied the problem of multivariate geo-statistics. What remains explicitly unaddressed is treatment of canonical correlation, and a spatial Statistical fc-groups discriminant function analysis model (which should relate to the eigenvectors of the aforementioned spatially adjusted MANOVA model).

This chapter is concerned with the measurement of parametric instability in general, and spatial parametric instability in particular. Let us begin by placing into focus some pertinent concepts and issues.

Time series forecasting for relatedunits is common practice. Examples include sales of a chain of fast-food restaurants in a metropolitan area, precipitation in neighboring sections of farm land, and, as explored in this chapter, tax revenues for the school districts of a county. Our premise is that there is information in the cross- sectional data to be exploited for forecasting. The geographic scale of our research is much smaller and therefore our data are more dynamic than those of related studies. Garcia-Ferrer et al. (1987) and Zellner and Hong (1989) used Bayesian shrinkage methods for pooling time series forecasts — similar to the one we develop in this chapter — to forecast growth in the economies of several countries. Lesage and Magura (1990) applied the same and additional methods to multi-regional data, seven metropolitan areas in Ohio. Our interest is in small local government economies at the subregional level. We forecast (and backcast) income tax revenues for forty school districts in Allegheny County, Pennsylvania.

The spatial dimension of banking markets is often discussed in the banking and finance literature, but neglected in empirical studies of behavior by depository institutions. As Anselin and Griffith (1988) point out, neglect of spatial aspects of economic behavior is not unusual, even by those working in regional science. Traditional depository institution regulatory policy, based on a structural analysis of local markets, generally operates under the assumption of the U.S. banking system as a collection of segmented markets versus an integrated national banking system. From this perspective, banks and thrifts in local regions have little or no effect on the pricing decisions of firms operating in other localities. In the deregulated banking environment of the 1980s, some bank retail deposit markets may be better characterized as operating in such a way as to generate spatial effects. As of October 1983, all depository institutions were permitted to offer competitive market rates on interest-sensitive deposits, including retail certificates of deposit (CDs). In turn, many institutions, particularly large banks, have used national advertising and brokers to attract large retail (insured) deposits from other regions. With greater deregulation and competition for deposit funds and increased reliance of banks on interest-sensitive deposits, significant spatial effects in the deposit-rate setting decisions of banks across regions might be expected. The presence or lack of spatial effects in regional bank markets is important to bank managers, analysts, and regulators in terms of defining relevant markets and measuring the competitive effects of greater deregulation.