Sensitivity analysis of SAR estimators: a numerical approximation

Abstract

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.

Keywords:

• spatial autoregressive models
• least squares estimators
• sensitivity analysis
• Taylor approximations
• Kantorovich inequality

JEL Classifications:

• C11
• C15
• C52
• E17
• R12

Acknowledgements

The research was conducted when Wolfgang Polasek visited Shuangzhe Liu at University of Canberra and when Shuangzhe Liu took his OSP leave. The authors are grateful to Matthias Koch, the editor and the referees for their constructive remarks and suggestions on an early version of this paper.

Notes

Since the 2SLS estimator has the inherent danger of applying weak instruments, we follow the approach suggested by Kelley Pace et al. 14, thus fixing our signal-to-noise ratio defined by .

An Intel(R) Core(TM) i5-2410M CPU with 2.30 GHz and 4 GB of RAM was used for computation with Matlab 7.10.