Luc Anselin and Nancy Lozano-Gracia 1223with the off-diagonal elements of!being!ij=e−φdij, andγas a non-negative
scaling parameter. In order to facilitate interpretation and specification testing, the
diagonal elements of!are often set to zero (the variance is captured by the term
σ^2 I). The distance metric and parameter space must be such that the elements
ofe−φdijyield a valid spatial correlation matrix (see Anselin, 2001a, for technical
details).
Estimation in parametric spatial error models is most commonly based on the
ML principle (see Anselin, 1988; Dubin, 1988). Due to computational limitations
in very large datasets, recent attention has shifted to alternatives, such as the
generalized moments (GM) and generalized method of moments (GMM) estima-
tors suggested by Kelejian and Prucha (1998, 1999). An early application of this
approach to a hedonic specification can be found in Bell and Bockstael (2000) (see
also section 26.5 for further examples). Generalization of this approach to an error
structure that contains both spatial autocorrelation and heteroskedasticity can be
found in recent papers by Lin and Lee (2005) and Kelejian and Prucha (2006).
A different approach is to avoid the parametric specification of spatial covariance
as a function of a distance metric and to use a nonparametric perspective. This is
an extension to the spatial domain of the principle behind the heteroskedasticity
and autocorrelation consistent covariance matrix estimation of Newey and West
(1987) and Andrews (1991), among others.
As in the direct representation approach, the spatial covariance is a func-
tion of the distance separating two observations, but the functional form is left
unspecified. For example, for the regression error terms:
E[εiεj]=f(dij), (26.16)wheredij is a “proper” positive and symmetric distance metric (for regularity
conditions on the distance metric, see Conley, 1999; Kelejian and Prucha, 2007).
This estimator follows essentially the same principle as in the time series domain
by adding up sample spatial autocovariances. In order to ensure positive definite-
ness of the estimator, a kernel is applied to the cross-products. For example, in the
recent paper by Kelejian and Prucha (2007), a general covariance matrix estimator
takes the form:
Vˆ=n−^1∑i∑jxix′jεˆiεˆjK(dij/d), (26.17)whereK()is a kernel function andda suitable cutoff distance. This yields a so-
called heteroskedastic and spatial autocorrelation consistent, or HAC, estimator.
A recent application of this approach to spatial hedonic models can be found in
Anselin and Lozano-Gracia (2008).
Arguably, the treatment of spatially structured omitted variables may be
addressed without resorting to a spatial error. The most commonly used technique
in the empirical literature is to address this by means of spatial fixed-effects, e.g.,
by including a dummy variable for a larger spatial area that individual housing