On the quality of likelihood-based estimators in spatial autoregressive models when the data dependence structure is misspecified

Abstract

This paper seeks to help recast and extend existing results for the specification and the estimation of spatial statistical autoregressive models. In particular, a graph-theoretic formulation of the neighborhood structure allows an intuitive, but general, specification of the dependence structure among data. Moreover, issues of asymptotic efficiency and consistency of maximum likelihood (ML) estimators (MLE) are addressed in terms of correlational structure specification error. Parameters of interest are those for the mean response, the variance, and the spatial autocorrelation. Findings reported frequently differ from those derived for independent observations, and from those derived for time series correlation structures. Some findings are in keeping with those traditionally derived for generalized least squares (GLS) estimators. Moreover, theorems presented in this paper reveal that the quality of ML spatial autoregressive model parameter estimates for Gaussian geo-referenced data distributed across a lattice is such that, under specification errors, (1) the GLS mean response ML estimator is unbiased, consistent, and asymptotically efficient in selected situations, (2) the GLS error variance ML estimator is consistent in selected situations, but usually inefficient, and (3) spatial autocorrelation parameters MLEs fail to converge to their true values under increasing domain asymptotics. Finally, a model taking into account partial information about the data neighborhood structure is presented. © 1998 Elsevier Science B.V. All rights reserved.