Spatial regressive and autoregressive models with SARMA error terms

Abstract

The aim of this paper is to define and to analyse classes of models satisfactory for the consideration of the spatial effects of dependence and heterogeneity. In particular, we introduce the ADL-N (Autoregressive Distributed Lags - Noise) class in order to model sets of spatial data. This class consists of a generalization of the well known class of the ADL (Hendry et al., 1984) models, introduced in the analysis of time series. This generalization is obtained by means of the hypothesis that the error component is generated by a spatial ARMA process (Cressie, 1993). It is well known that the multilateral dependence, which characterizes spatial processes, results in correlation between the endogenous variable and the disturbance term in the model, causing the inconsistency of ordinary least squares estimators (Anselin, 1988). The model we propose is based upon the use of the conditional likelihood function to obtain consistent and asymptotically efficient estimators for the model's parameters. In the paper we report the gradient and the Hessian matrix of the log-likelihood function for the model proposed. To perform hypothesis testing, we have coded two procedures to test ML (Maximum Likelihood) estimation results, termed respectively OLT (Omitted Lags Test) and DLT (Deleted Lags Test). The latter tests the hypothesis that a component introduced in the model is inconsistent with the data; the former tests that a component not still introduced is - on the contrary - consistent with the data. All these procedures need the use of the Fisher Information matrix, which is reported for the ADL-N class of models.