Sensitivity analysis of SAR estimators: a numerical approximation

The estimation of a spatial autoregressive (SAR) model depends on the spatial correlation parameter ρ in a highly nonlinear way, and the least squares (LS) estimators for ρ cannot be computed in a closed form. In this paper, we propose two simple LS estimators and compare them by distance and covariance properties in order to study the local sensitivity behaviour of these estimators. Using matrix derivatives we calculate the Taylor approximation of the LS estimator in the SAR model up to the second order. In a next step, we compare the covariance structure of the two estimators by Kantorovich inequalities and derive efficiency comparisons by upper bounds. Finally, we explore the quality of our new approximations by a Monte Carlo simulation study. The simulation results show significant computation time reductions and a good approximation behaviour of the SAR LS estimator in the neighborhood of ρ=0, when using a non-spatial LS estimator. The results are encouraging and can be used for further developments like quick diagnostic tools to explore the sensitivity of spatial estimators with respect to the size of the spatial correlation.