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Semiparametric Maximum Likelihood Estimates of Spatial Dependence

We semiparametrically model spatial dependence via a combination of simpler weight matrices (termed spatial basis matrices) and fit this model via maximum likelihood. Estimation of the model relies on the intuition that bounds to the log-determinant term in the log-likelihood can provide penalties to overfitting both the level and pattern of spatial dependence. By relying on symmetric and doubly stochastic spatial basis matrices that reflect different weight specifications assigned to neighboring observations, we are able to derive a mathematical expression for bounds on the log-determinant term that appears in the likelihood function. These bounds can be conveniently calculated allowing us to solve for maximum likelihood estimates at the bounds using a simple optimization over two quadratic forms that involve small matrices. An intuitively pleasing aspect of our approach is that the objective function for the bounded log-likelihoods contains one quadratic form equal to the sum-of-squared errors measuring the quality of fit, and another quadratic form reflecting a penalty to overfitting spatial dependence. We apply our semiparametric estimation method to a housing model using 57,647 U.S. census tracts.