Exact and higher-order properties of the MLE in spatial autoregressive models, with applications to inference

The quasi-maximum likelihood estimator for the autoregressive parameter in a spatial autoregression usually cannot be written explicitly in terms of the data. A rigorous analysis of the first-order asymptotic properties of the estimator, under some assumptions on the evolution of the spatial design matrix, is available in Lee (2004), but very little is known about its exact or higher-order properties. In this paper we first show that the exact cumulative distribution function of the estimator can, under mild assumptions, be written in terms of that of a particular quadratic form. Simple examples are used to illustrate important exact properties of the estimator that follow from this representation. In general models a complete exact analysis is not possible, but a higher-order (saddlepoint) approximation is made available by the main result. We use this approximation to construct confidence intervals for the autoregressive parameter. Coverage properties of the proposed confidence intervals are studied by Monte Carlo simulation, and are found to be excellent in a variety of circumstances.

spatial autoregression, maximum likelihood estimation, group interaction, social networks, complete bipartite graph, saddlepoint

10.1016/j.jeconom.2018.01.008

0304-4076

402-422

Hillier, Grant

3423bd61-c35f-497e-87a3-6a5fca73a2a1

Martellosio, Federico

30407632-7b34-4b01-b46c-b5870c9a4dd3

August 2018

Hillier, Grant

3423bd61-c35f-497e-87a3-6a5fca73a2a1

Martellosio, Federico

30407632-7b34-4b01-b46c-b5870c9a4dd3

Hillier, Grant and Martellosio, Federico (2018) Exact and higher-order properties of the MLE in spatial autoregressive models, with applications to inference. Journal of Econometrics, 205 (2), 402-422. (doi:10.1016/j.jeconom.2018.01.008).