Fast maximum likelihood estimation of very large spatial autoregressive models: A characteristic polynomial approach

The maximization of the log-likelihood function required in the estimation of spatial autoregressive linear regression models is a computationally intensive procedure that involves the manipulation of matrices of dimension equal to the size of the data set. Common computational approaches applied to this problem include the use of the eigenvalues of the spatial weights matrix (W), the application of Cholesky decomposition to compute the Jacobian term I - _pW, and various approximations. These procedures are computationally intensive and/or require significant amounts of memory for intermediate data structures, which becomes problematic in the analysis of very large spatial data sets (tens of thousands to millions of observations). In this paper, we outline a new method for evaluating the Jacobian term that is based on the characteristic polynomial of the spatial weights matrix W. In numerical experiments, this algorithm approaches linear computational complexity, which makes it the fastest direct method currently available, especially for very large data sets.