Quick but not so Dirty ML Estimation of Spatial Autoregressive Models

Abstract

Positive spatial autocorrelation is a tendency for similar values of a single variable Y to be present in nearby locations on a map; it is displayed when observations contained in data sets are locationally tagged to the earth’s surface (i.e., georeferenced data sets). The prevailing nature and degree of spatial autocorrelation may be denoted by ρ, while the self-covariation of n geographically neighboring values within a variable may be represented with the n × n matrix V⁻¹ρ², which is a function of ρ. This geographic dependency feature of georeferenced data is captured by the auto-Gaussian log-likelihood function:

$${\rm constant - }(n/2)\ln (\sigma ^2 ) + \ln [\det (V)] - (Y - X\beta )^T V(Y - X\beta )/(2\sigma ^2 )$$

(9.1)

where det(V), superscript T, and ln, respectively, denote the matrix determinant and transpose operations and the natural logarithm, Y is an nx1 vector of georeferenced values, X is an n x (p+1) matrix of p corresponding predictor variables coupled with a vector of ones, and vector β?and scalar ρ, respectively, denote the standard nonconstant mean and constant variance. The parameters of (9.1) most often are estimated using maximum likelihood (ML) techniques.