Inference on Higher-Order Spatial Autoregressive Models with Increasingly Many Parameters

This paper develops consistency and asymptotic normality of instrumental variables and least squares estimates for the parameters of a higher-order spatial autoregressive (SAR) model with regressors. The order of the SAR model and the number of regressors are allowed to approach infinity slowly with sample size, and the permissible rate of growth of the dimension of the parameter space relative to sample size is studied. Besides allowing the number of estimable parameters to increase with the data, this has the advantage of accommodating explicitly some asymptotic regimes that arise in practice. Illustrations are discussed, in particular settings where the need for such theory arises quite naturally. A Monte Carlo study analyses various implications of the theory in finite samples. For empirical researchers our work has implications for the choice of model. In particular if the structure of the spatial weights matrix assumes a partitioning of the data then spatial parameters should be allowed to vary over clusters.