This dissertation consists of three chapters covering the following topics in spatial econometrics: estimation, specification and the bootstrap.
In Chapter 1, we first generalize an approximate measure of spatial dependence, the APLE statistic in Li et al (2007), to a spatial Durbin (SD) model. This generalized APLE takes into account exogenous variables directly and can be used to detect spatial dependence originating from either a spatial autoregressive (SAR), spatial error (SE) or SD process. However, that measure is not consistent. Secondly, by examining carefully the first order condition of the concentrated log likelihood of the SD (or SAR) model, whose first order approximation generates the APLE, we construct a moment equation quadratic in the autoregressive parameter that generalizes an original estimation approach in Ord (1975) and yields a closed-form consistent root estimator of the autoregressive parameter. With a specific moment equation constructed from an initial consistent estimator, the root estimator can be as efficient as the MLE under normality. Furthermore, when there is unknown heteroskedasticity in the disturbances, we derive a modified APLE and a root estimator which can be robust to unknown heteroskedasticity. The root estimators are computationally much simpler than the quasi-maximum likelihood estimators.
In Chapter 2, we consider the Cox-type tests of non-nested hypotheses for spatial autoregressive (SAR) models with SAR disturbances. We formally derive the asymptotic distributions of the test statistics. In contrast to regression models, we show that the Cox-type and J-type tests for non-nested hypotheses in the framework of SAR models are not asymptotically equivalent under the null hypothesis. The Cox test in non-spatial setting has been found often to have large size distortion, which can be removed by the bootstrap. Cox-type tests for SAR models with SAR disturbances may also have large size distortion. We show that the bootstrap is consistent for Cox-type tests in our framework. Performances of the Cox-type and J-type tests as well as their bootstrapped versions in finite samples are compared via a Monte Carlo study. These tests are of particular interest when there are competing models with different spatial weights matrices. Using bootstrapped p-values, the Cox tests have relatively high power in all experiments and can outperform J-type and several other related tests in some cases.
Chapter 3 is concerned with the use of the bootstrap for spatial econometric models. We show that the bootstrap for spatial econometric models can be studied based on linear-quadratic (LQ) forms of disturbances. By proving the uniform convergence of the cumulative distribution function for LQ forms to that of a normal distribution, we show that the bootstrap is generally consistent for test statistics that can be approximated by LQ forms, which include Moran's I, Cox-type and spatial J-type test statistics. Possible asymptotic refinements of the bootstrap for spatial econometric models may be studied based on some asymptotic expansions for LQ forms. We discuss two cases: when the disturbances are normal, we directly show the existence of Edgeworth expansions for LQ forms and apply the result to show that the bootstrap for Moran's I can provide asymptotic refinements; when the disturbances are not normal, we show the existence of a one-term asymptotic expansion of LQ forms based on martingales, which sheds light on the second-order correctness of the bootstrap for LQ forms.