Method of moments estimators
Recently, a number of approaches have been outlined to estimate the coefficients in a spatial error model as an application of general principles underlying the method of moments. Kelejian and Prucha (1999a) develop a set of moment conditions that yield estimation equations for the parameter of an SAR error model. Specifically, assuming an iid error vector u, the following three conditions readily follow
E[u'u/N ] = o2
E[u' W Wu/N ] = o2(1/N)tr(W' W) (14.29)
E[u'Wu/N ] = 0
where tr is the matrix trace operator. Replacing u by e - XWe (with e as the vector of OLS residuals) in (14.29) yields a system of three equations in the parameters X, X2, and o2. Kelejian and Prucha (1999a) suggest the use of nonlinear least squares to obtain a consistent generalized moment estimator for X from this system, which can then be used to obtain consistent estimators for the в in an FGLS approach. Since the X is considered as a nuisance parameter, its significance (as a test for spatial autocorrelation) cannot be assessed, but its role is to provide a consistent estimator for the regression coefficients.26
A different approach is taken in the application of Hansen's (1982) generalized method of moments estimator (GMM) to spatial error autocorrelation in Conley
(1996) . This estimator is the standard minimizer of a quadratic form in the sample moment conditions, where the covariance matrix is obtained in nonparametric form as an application of the ideas of Newey and West (1987). Specifically, the spatial covariances are estimated from weighted averages of sample covariances for pairs of observations that are within a given distance band from each other. Note that this approach requires covariance stationarity, which is only satisfied for a restricted set of spatial processes (e. g. it does not apply to SAR error models).
Pinkse and Slade (1998) use a set of moment conditions to estimate a probit model with SAR errors. However, they focus on the induced heteroskedasticity of the process and do not explicitly deal with the spatial covariance structure.27
The relative efficiency of the new methods of moments approaches relative to the more traditional maximum likelihood techniques remains an area of active investigation.
A number of other approaches have been suggested to deal with the estimation of spatial regression models. An early technique is the so-called coding method, originally examined in Besag and Moran (1975).28 This approach consists of selecting a subsample from the data such that the relevant neighbors are removed (a non-contiguous subsample). This in effect eliminates the simultaneity bias in the spatial lag model, but at the cost of converting the model to a conditional one and with a considerable reduction of the sample size (down to 20 percent of the original sample for irregular lattice data). The advantage of this approach is that standard methods may be applied (e. g. for discrete choice models). However, it is not an efficient procedure and considerable arbitrariness is involved in the selection of the coding scheme.
Another increasingly common approach consists of the application of computational estimators to spatial models. A recent example is the recursive importance sampling (RIS) estimator (Vijverberg, 1997) applied to the spatial probit model in Beron and Vijverberg (2000).
A considerable literature also exists on Bayesian estimation of spatial models, but a detailed treatment of this is beyond the current scope.