Exact Properties of the Maximum Likelihood Estimator in Spatial Autoregressive Models

Economics

Further details

The maximum likelihood estimator (MLE) for the autoregressive parameter in a spatial autoregressive (SAR) model is a zero of a polynomial of degree n + 1 (n = sample size), and so cannot be written explicitly in terms of the data. Thus, the only known properties of the estimator have hitherto been its standard asymptotic properties (L-F Lee, Econometrica, (2004)). In this paper we show that, notwithstanding its unavailability in closed form, the cumulative distribution function (cdf) of the estimator can be written down explicitly. The key to this result is the observation that, on the relevant parameter space, the pro…le likelihood is single-peaked, and that this implies that the cdf is defined by the event that a certain quadratic form is non-positive. Expressions for the density and moments of the MLE follow quite directly, at least in principle. A number of examples of theoretical or practical interest are analyzed in detail. Results on the existence and the support of the MLE are also obtained.