Feasible pairwise pseudo-likelihood inference on spatial regressions in irregular lattice grids: the KD-T PL algorithm

Spatial regression models are central to the field of spatial statistics. Nevertheless, their estimation in case of large and irregular gridded spatial datasets presents considerable computational challenges. To tackle these computational problems, Arbia \citep{arbia_2014_pairwise} introduced a pseudo-likelihood approach (called pairwise likelihood, say PL) which required the identification of pairs of observations that are internally correlated, but mutually conditionally uncorrelated. However, while the PL estimators enjoy optimal theoretical properties, their practical implementation when dealing with data observed on irregular grids suffers from dramatic computational issues (connected with the identification of the pairs of observations) that, in most empirical cases, negatively counter-balance its advantages. In this paper we introduce an algorithm specifically designed to streamline the computation of the PL in large and irregularly gridded spatial datasets, dramatically simplifying the estimation phase. In particular, we focus on the estimation of Spatial Error models (SEM). Our proposed approach, efficiently pairs spatial couples exploiting the KD tree data structure and exploits it to derive the closed-form expressions for fast parameter approximation. To showcase the efficiency of our method, we provide an illustrative example using simulated data, demonstrating the computational advantages if compared to a full likelihood inference are not at the expenses of accuracy.