Sensitivity analysis of SAR estimators: a numerical approximation

Abstract

Provided in Cooperation with:Institute for Advanced Studies, Vienna Suggested Citation: Liu, Shuangzhe; Polasek, Wolfgang; Sellner, Richard (2011) AbstractEstimators of spatial autoregressive (SAR) models depend in a highly non-linear way on the spatial correlation parameter and least squares (LS) estimators cannot be computed in closed form. We first compare two simple LS estimators by distance and covariance properties and then we study the local sensitivity behavior of these estimators using matrix derivatives. These results allow us to calculate the Taylor approximation of the least squares estimator in the spatial autoregression (SAR) model up to the second order. UsingKantorovich inequalities, we compare the covariance structure of the two estimators and we derive efficiency comparisons by upper bounds. Finally, we demonstrate our approach by an example for GDP and employment in 239 European NUTS2 regions. We find a good approximation behavior of the SAR estimator, evaluated around the non-spatial LS estimators. These results can be used as a basis for diagnostic tools to explore the sensitivity of spatial estimators.

Sensitivity analysis in linear models

“…The sensitivities in (8) or (13) can be used to nd rst-order Tylor approximations for certain estimators; see e.g. Liu et al [12,14] for relevant ideas and uses.…”

“…For local sensitivities and diagnostic tests with applications to linear and random effects models, see Magnus and Vasnev [17]. For the sensitivity matrices of least squares estimators and their relevant uses in spatial and panel-spatial autoregressive models, see Liu et al [12,14]. The local sensitivities of the posterior mean and precision matrix in the Bayesian context are established as well; see Polasek [19,20].…”

Spatial system estimators for panel models: A sensitivity and simulation study

“…Theorem 3 In the SAR-SUR model in (14), the sensitivity of the reduced form GLS estimator (16) is the kT × T matrix…”

“…Paelinck and Klaassen (1979), Anselin (1988Anselin ( , 2010, Florax and Van Der Vlist (2003), Haining (2003), LeSage and Polasek (2008), LeSage and Pace (2009), and Liu et al (2012). On the other hand, panel models have become increasingly important and different estimators in such models with spatial components have also been studied; see e.g.…”

“…A previous study of the sensitivity of spatial estimators, like the cross-sectional SAR and the SEM model, has been made in Liu et al (2012). It was shown that good approximations exists for small values of the spatial correlation parameter.…”