# Spatial autocorrelation

In a regression context, spatial effects pertain to two categories of specifications. One deals with spatial dependence, or its weaker expression, spatial autocorrelation, and the other with spatial heterogeneity.3 The latter is simply structural instabil­ity, either in the form of non-constant error variances in a regression model (heteroskedasticity) or in the form of variable regression coeffcients. Most of the methodological issues related to spatial heterogeneity can be tackled by means of the standard econometric toolbox.4 Therefore, given the space constraints for this chapter, the main focus of attention in the remainder will be on spatial dependence.

The formal framework used for the statistical analysis of spatial autocorrelation is a so-called spatial stochastic process (also often referred to as a spatial random field), or a collection of random variables y, indexed by location i,

{y, i О D}, (14.1)

where the index set D is either a continuous surface or a finite set of discrete locations. (See Cressie (1993), for technical details.) Since each random variable is "tagged" by a location, spatial autocorrelation can be formally expressed by the moment condition,

c°v[Уи y] = E[yj – E[y] ■ E[y] * 0, for і Ф j (14.2)

where i, j refer to individual observations (locations) and yi (yj) is the value of a random variable of interest at that location. This covariance becomes meaning­ful from a spatial perspective when the particular configuration of nonzero i, j pairs has an interpretation in terms of spatial structure, spatial interaction or the spatial arrangement of the observations. For example, this would be the case when one is interested in modeling the extent to which technological innovations in a county spill over into neighboring counties.

The spatial covariance can be modeled in three basic ways. First, one can specify a particular functional form for a spatial stochastic process generating the random variable in (14.1), from which the covariance structure would follow. Second, one can model the covariance structure directly, typically as a func­tion of a small number of parameters (with any given covariance structure cor­responding to a class of spatial stochastic processes). Third, one can leave the covariance unspecified and estimate it nonparametrically.5 I will review each of these approaches in turn.
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