A Closed‐Form Consistent Estimator for Linear Models with Spatial Dependence

Common techniques for approximating the maximum likelihood estimator often lead to biased and inconsistent estimates when a systematic bias is introduced into the log‐likelihood function. This article proposes to replace the true scores in the likelihood equations with quasi‐scores that are easy‐to‐compute yet have an expected value of zero. This innovation would dramatically simplify the likelihood‐based estimation while preserving a majority of desirable statistical properties of maximum likelihood. Consequently, this article introduces an easy‐to‐compute consistent estimator for models with spatial dependence and documents its computational and statistical properties. To illustrate the quality and predictability of the closed form estimator in a practical setting we use Monte Carlo simulations that address both the spatial lag and spatial error settings. The estimated variance of the estimator is elevated relative to that of maximum likelihood’s asymptotic variance and actual method’s performance. However, it is shown that inflated confidence intervals are largely inconsequential to the analysis of spatial dependence in practice. The estimator is shown to be computationally superior to the true maximum likelihood for any tested matrix sizes and levels of spatial dependence.