Abstract
This paper revisits the theory of asymptotic behaviour of the well-known Gaussian Quasi-Maximum Likelihood estimator of parameters in mixed regressive, high-order autoregressive spatial models. We generalise the approach previously published in the econometric literature by weakening the assumptions imposed on the spatial weight matrix. This allows consideration of interaction patterns with a potentially larger degree of spatial dependence. Moreover, we broaden the class of admissible distributions of model residuals. As an example application of our new asymptotic analysis we also consider the large sample behaviour of a general group effects design.
Highlights
It is a broadly employed assumption in a wide range of theoretical studies on spatial econometrics that the spatial weight matrix is absolutely row and column summable
Olejnik first to formulate their assumptions as explicit requirements regarding the spatial weight matrix. Their Central Limit Theorem (CLT), which turned out to be a milestone in the development of asymptotic theories for spatial econometric models, relies on the absolute summability of the weight matrix involved
By revisiting the classical argument of Lee (2004) and, importantly, introducing a generalised CLT for linear-quadratic forms, we are able to provide a theory for consistency and asymptotic normality of Quasi Maximum Likelihood (QML) estimates for high-order spatial autoregressive models under relaxed conditions
Summary
It is a broadly employed assumption in a wide range of theoretical studies on spatial econometrics that the spatial weight matrix is absolutely row and column summable This restriction is mostly a result of the Central Limit Theorem (CLT) used in the derivation of the result on asymptotic behaviour. In that earlier work, extending the scope of spatial weight matrices beyond the standard asymptotic analysis of Lee (2004) was found to be useful, and results analogous to our Theorem 1 on consistency were independently obtained Their derivation of the asymptotic distribution of the estimates still relies on the assumption of absolute summability. The aim of this paper is to present a refinement to the asymptotic analysis of the Gaussian Quasi-Maximum Likelihood (QML) estimator for high-order, spatial autoregressive models, considering the assumptions imposed on the spatial weight matrix. Appendices contain some details of the proofs, as well as a set of Monte Carlo simulations that empirically demonstrate the validity of the theory under the relaxed conditions
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