Academic journal article Geographical Analysis

Unconditional Maximum Likelihood Estimation of Linear and Log-Linear Dynamic Models for Spatial Panels

Academic journal article Geographical Analysis

Unconditional Maximum Likelihood Estimation of Linear and Log-Linear Dynamic Models for Spatial Panels

Article excerpt

This article hammers out the estimation of a fixed effects dynamic panel data model extended to include either spatial error autocorrelation or a spatially lagged dependent variable. To overcome the inconsistencies associated with the traditional least-squares dummy estimator, the models are first-differenced to eliminate the fixed effects and then the unconditional likelihood function is derived taking into account the density function of the first-differenced observations on each spatial unit. When exogenous variables are omitted, the exact likelihood function is found to exist. When exogenous variables are included, the pre-sample values of these variables and thus the likelihood function must be approximated. Two leading cases are considered: the Bhargava and Sargan approximation and the Nerlove and Balestra approximation. As an application, a dynamic demand model for cigarettes is estimated based on panel data from 46 U.S. states over the period from 1963 to 1992.



In recent years, there has been a growing interest in the estimation of econometric relationships based on panel data. In this article, we focus on dynamic models for spatial panels, a family of models for which, according to Elhorst (2001) and Hadinger, Muller, and Tondl (2002), no straightforward estimation procedure is yet available. This is (as will be explained later) because existing methods developed for spatial but non-dynamic and for dynamic but non-spatial panel data models produce biased estimates when these methods/models are put together.

A dynamic spatial panel data model takes the form of a linear regression equation extended with a variable intercept, a serially lagged dependent variable and either a spatially lagged dependent variable (known as spatial lag) or a spatially autoregressive process incorporated in the error term (known as spatial error). To avoid repetition, we apply to the spatial error specification in this article. The spatial lag specification is explained in a working paper (Elhorst 2003a). (1) The model is considered in vector form for a cross-section of observations at time t:

[Y.sub.t] = [tau][Y.sub.t-1] + [X.sub.t][beta] + [mu] + [[phi].sub.t], [[phi].sub.t] = [delta]W[[phi].sub.t] + [[epsilon].sub.t], E([[epsilon].sub.t]) = 0, E([[epsilon].sub.t][[epsilon]'.sub.t]) = [[sigma].sup.2][I.sub.N] (1)

where [Y.sub.t] denotes an N X 1 vector consisting of one observation for every spatial unit (i = 1,..., N) of the dependent variable in the tth time period (t = 1,..., T) and [X.sub.t] denotes an N X K matrix of exogenous explanatory variables. It is assumed that the vector [Y.sub.0] and matrix [X.sub.0] of initial observations are observable. The scalar [tau] and the K X 1 vector [beta] are the response parameters of the model. The disturbance term consists of [mu] = ([[mu].sub.1],..., [[mu].sub.N])', [[phi].sub.t] = ([[phi].sub.1t],..., [[phi].sub.Nt])', and [[epsilon].sub.t] = ([[epsilon].sub.1t],..., [[epsilon].sub.Nt])', where [[epsilon]] are independently and identically distributed error terms for all i and t with zero mean and variance [[sigma].sup.2]. [I.sub.N] is an identity matrix of size N, W represents an N X N non-negative spatial weight matrix with zeros on the diagonal, and [delta] represents the spatial autocorrelation coefficient. The properties of [mu] are explained below.

The reasons for considering serial and spatial dynamic effects, either directly as part of the specification or indirectly as part of the disturbance term, have been published earlier (Elhorst 2001, 2004). A standard space-time model, even if it is dynamic, still assumes that the spatial units are completely homogeneous, differing only in their explanatory variables. Standard space-time models include the STARMA/STARIMA (Space Time AutoRegressive [Integrated] Moving Average) model (Hepple 1978; Pfeifer and Deutsch 1980), spatial autoregression space-time forecasting model (Griffith 1996), and the serial and spatial autoregressive distributed lag model (Elhorst 2001). …

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