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Estimation of Spatial Panel-Data Models Using a Minimum-Distance Estimator

Spatial autocorrelation (more generally, spatial dependence) occurs when a regression's error term at one location is correlated with that at another location. Ignoring the resulting non-diagonal disturbance covariance matrix results in misspecification and bias. While most studies focus on cross-sectional specifications, spatial panel data models have not received much attention. Here I consider modelling and estimating panel-data autoregressive spatial processes and show that minimum distance methods provide consistent estimates. I begin with a mixed regressive spatial autoregressive model and define a class of random fields with models derived from processes indexed with space, time, and cross-sectional dimensions. I use a row-standardized contiguity matrix, i.e., the spatial weight matrix is normalized so the rows sum to unity. This standardization produces a spatially dependent lagged variable that represents a vector of average values from neighbouring dependent observations. The specification is assumed to be the true data generating process relating sample data collected with reference to points in space and time. By relying on existing methods, we estimate the model in two stages. First, assuming the errors to be normally distributed, the cross-section parameters can be efficiently estimated by the concentrated maximum likelihood. Under suitable regularity conditions, this provides both the unrestricted consistent estimates (including the spatial coefficient) and some elements of scores, which are used to compute the consistent asymptotic covariance matrix for the second stage. Then, two cases are considered: the fixed parameters case and the time-varying one. The minimum distance estimator for each case is derived by stacking the estimates from the first stage into a block vector on which linear moment restrictions are imposed. Then, in the second stage, we minimize the norm of the inverse of the block asymptotic covariance matrix. The resulting minimum-distance estimator is asymptotically consistent. The restrictions imposed are to be tested. This process is used to examine the spatial aspects of the joint residential water and electricity demand in the French department of "Moselle". The data are a lattice sample of 115 neighbouring communities that represent a balanced panel of 1380 observations. Initial results indicate that this approach is feasible and, although it seems straightforward to perform, it relies on computationally demanding procedures. To the best of my knowledge, this process is novel in its application to spatial processes for panel data.