Specification Errors in Spatial Models: Impacts on Modeling and Estimation.

A principal motivation for the research reported in this paper is to seek to understand what consequences can result from an erroneous specification of the lattice topology when an autoregressive spatial model is employed for inferential purposes. The first part of the discussion concerns impacts of such a specification error on the resulting dependence structure among data in SAR and CAR models. Although some ideas on this matter can be found in popular monographs about spatial statistics (Cliff and Ord 1981; Griffith 1988; Cressie 1993; Guyon 1995), the problem does not seem to have been attacked directly in the literature, two exceptions being Griffith (1995) and Griffith and Lagona (1997). In this paper an additional step is made in this direction, extensively using the theory of power series for matrices (see, for instance, Cooke 1950) and basic graph theory (see, for instance, Ore 1960). The second part of the discussion concerns impacts of a misspecified lattice topology on the quality of popular estimation procedures for SAR and CAR models. Because estimator quality can be measured by sampling distribution properties, unbiasedness and efficiency are considered here as basic measures of it. For this problem, a useful starting point is furnished by Oksanen (1991), and Cordey and Griffith's (1993) work is extended here. Much of the finite sample mathematical statistics summarized in this paper is numerically demonstrated in Griffith (1995). And, the Jacobian work outlined in Griffith and Sone (1995) is exploited to attain selected analytical results.