A COMPANION TO Theoretical Econometrics
Direct representation
A second commonly used approach to the formal specification of spatial autocorrelation is to express the elements of the variance-covariance matrix in a parsimonious fashion as a "direct" function of a small number of parameters and one or more exogenous variables. Typically, this involves an inverse function of some distance metric, for example,
c°v[e;, e;-] = °2f{dj (14.7)
where e; and e;- are regression disturbance terms, o2 is the error variance, d;j is the distance separating observations (locations) i and j, and f is a distance decay function such that |f < 0 and | f(d;j, ф)| < 1, with ф £ Ф as a p x 1 vector of parameters on an open subset Ф of Rp. This form is closely related to the vario - gram model used in geostatistics, although with stricter assumptions regarding stationarity and isotropy. Using (14.7) for individual elements, the full error covariance matrix follows as
E[ee'] = o2Q(d;j, ф), (14.8)
where, because of the scaling factor o2, the matrix Q.(d;j, ф) must be a positive definite spatial correlation matrix, with o;; = 1 and | a;j | < 1, V ;, j.12 Note that, in contrast to the variance for the spatial autoregressive model, the direct representation model does not induce heteroskedasticity.
In spatial econometrics, models of this type have been used primarily in the analysis of urban housing markets, e. g. in Dubin (1988, 1992), and Basu and Thibodeau (1998). While this specification has a certain intuition, in the sense that it incorporates an explicit notion of spatial clustering as a function of the distance separating two observations (i. e. positive spatial correlation), it is also fraught with a number of estimation and identification problems (Anselin, 2000a).
Nonparametric approaches
A nonparametric approach to estimating the spatial covariance matrix does not require an explicit spatial process or functional form for the distance decay. This is common in the case of panel data, when the time dimension is (considerably) greater than the cross-sectional dimension (T >> N) and the "spatial" covariance is estimated from the sample covariance for the residuals of each set of location pairs (e. g. in applications of Zellner's SUR estimator; see Chapter 5 by Fiebig in this volume).
Applications of this principle to spatial autocorrelation are variants of the well known Newey-West (1987) heteroskedasticity and autocorrelation consistent covariance matrix and have been used in the context of generalized methods of moments (GMM) estimators of spatial regression models (see Section 4.3). Conley
(1996) suggested a covariance estimator based on a sequence of weighted averages of sample autocovariances computed for subsets of observation pairs that fall within a given distance band (or spatial window). Although not presented as such, this has a striking similarity to the nonparametric estimation of a semi - variogram in geostatistics (see, e. g. Cressie, 1993, pp. 69-70), but the assumptions of stationarity and isotropy required in the GMM approach are stricter than those needed in variogram estimation. In a panel data setting, Driscoll and Kraay (1998) use a similar idea, but avoid having to estimate the spatial covariances by distance bands. This is accomplished by using only the cross-sectional averages (for each time period) of the moment conditions, and by relying on asymptotics in the time dimension to yield an estimator for the spatial covariance structure.
