Spatial stochastic process models
The most often used approach to formally express spatial autocorrelation is through the specification of a functional form for the spatial stochastic process
(14.1) that relates the value of a random variable at a given location to its value at other locations. The covariance structure then follows from the nature of the process. In parallel to time series analysis, spatial stochastic processes are categorized as spatial autoregressive (SAR) and spatial moving average (SMA) processes, although there are several important differences between the crosssectional and time series contexts.6
For example, for an N x 1 vector of random variables, y, observed across space, and an N x 1 vector of iid random errors e, a simultaneous spatial autoregressive (SAR) process is defined as
(y – pi) = pW(y – pi) + e, or (y – pi) = (I – pW)-1e, (14.3)
where p is the (constant) mean of yi, i is an N x 1 vector of ones, and p is the spatial autoregressive parameter.
Before considering the structure of this process more closely, note the presence of the N x N matrix W, which is referred to as a spatial weights matrix. For each location in the system, it specifies which of the other locations in the system affect the value at that location. This is necessary, since in contrast to the unambiguous notion of a "shift" along the time axis (such as yt-1 in an autoregressive model), there is no corresponding concept in the spatial domain, especially when observations are located irregularly in space.7 Instead of the notion of shift, a spatial lag operator is used, which is a weighted average of random variables at "neighboring" locations.8
The spatial weights crucially depend on the definition of a neighborhood set for each observation. This is obtained by selecting for each location i (as the row) the neighbors as the columns corresponding to nonzero elements wij in a fixed (nonstochastic) and positive N x N spatial weights matrix W.9 A spatial lag for y at i then follows as
[Wy]i = I Wij • yj, (14.4)
j=1, . . . , N
or, in matrix form, as
Since for each i the matrix elements wij are only nonzero for those j £ Si (where Si is the neighborhood set), only the matching yj are included in the lag. For ease of interpretation, the elements of the spatial weights matrix are typically row – standardized, such that for each i, YjWij = 1. Consequently, the spatial lag may be interpreted as a weighted average (with the wij being the weights) of the neighbors, or as a spatial smoother.
It is important to note that the elements of the weights matrix are nonstochastic and exogenous to the model. Typically, they are based on the geographic arrangement of the observations, or contiguity. Weights are nonzero when two locations share a common boundary, or are within a given distance of each other. However, this notion is perfectly general and alternative specifications of the spatial weights (such as economic distance) can be considered as well (Anselin, 1980, ch. 8; Case, Rosen, and Hines, 1993; Pinkse and Slade, 1998).
The constraints imposed by the weights structure (the zeros in each row), together with the specific form of the spatial process (autoregressive or moving average) determine the variance-covariance matrix for y as a function of two parameters, the variance o2 and the spatial coefficient, p. For the SAR structure in (14.3), this yields (since E[y – pi] = 0)
cov[(y – pi), (y – pi)] = E[(y – pi)(y – pi)’] = o2[(I – pW)'(I – pW)]-1.
This is a full matrix, which implies that shocks at any location affect all other locations, through a so-called spatial multiplier effect (or, global interaction).10
A major distinction between processes in space compared to the time domain is that even with iid error terms e„ the diagonal elements in (14.6) are not con – stant.11 Furthermore, the heteroskedasticity depends on the neighborhood structure embedded in the spatial weights matrix W. Consequently, the process in y is not covariance-stationary. Stationarity is only obtained in very rare cases, for example on regular lattice structures when each observation has an identical weights structure, but this is of limited practical use. This lack of stationarity has important implications for the types of central limit theorems (CLTs) and laws of large numbers (LLNs) that need to be invoked to obtain asymptotic properties for estimators and specification test, a point that has not always been recognized in the literature.