An O(N) parallel method of computing the Log-Jacobian of the variable transformation for models with spatial interaction on a lattice

A parallel method for computing the log of the Jacobian of variable transformations in models of spatial interactions on a lattice is developed. The method is shown to be easy to implement in parallel and distributed computing environments. The advantages of parallel computations are significant even in computer systems with low numbers of processing units, making it computationally efficient in a variety of settings. The non-iterative method is feasible for any sparse spatial weights matrix since the computations involved impose modest memory requirements for storing intermediate results. The method has a linear computational complexity for datasets with a finite Hausdorff dimension. It is shown that most geo-spatial data satisfy this requirement. Asymptotic properties of the method are illustrated using simulated data, and the method is deployed for obtaining maximum likelihood estimates for the spatial autoregressive model using data for the US economy.