Spatial autoregressions with an extended parameter space and similarity-based weights

We provide in this paper asymptotic theory for a spatial autoregressive model (SAR, henceforth) in which the spatial coefficient is allowed to be less than or equal to unity, as well as consistent with a local to unit root (LUR) model and of the moderate integration (MI) from unity type, and the spatial weights are allowed to be similarity-based and data driven. Other special cases of our setting include the random walk, a model in which all the weights are equal, the standard SAR model and the similarity based autoregression in which data do not display a natural order. As the norming rates for the asymptotic theory are very different when the spatial parameter is strictly less than unity - compared with the unit root and LUR cases, we resort to random norming that treats all cases in a uniform manner. It turns out that standard CLT results prevail in a large class of models in which the infinity norm of the inverse of the weighting structure that characterizes the reduced-form process is of order strictly smaller than n, and is non-standard in case it is of order n. We use a shifted profile likelihood to obtain results which are valid for all cases. A small simulation experiment supports our findings and the usefulness of our model is illustrated with an empirical application of the Boston housing data set in which the estimate of the spatial parameter appeared to be very close to unity.