Inference in a similarity-based spatial autoregressive model

In this paper we develop asymptotic theory for a similarity-based spatial autoregressive (SAR) model. The model is hybrid in the terminology of Gilboa et al. (2006), with the data generating process for a dependent variable containing a rule-based linear component and a case-based term with a similarity structure. The weight of the similarity structure is allowed to vary in the unit interval and to be estimated explicitly. We prove consistency of the quasi-maximum-likelihood estimator and derive its limit distribution. This paper contributes to the literature on SAR and empirical similarity by incorporating a regression-type component in the data generating process, by allowing the similarity structure to accommodate non-ordered data and by estimating explicitly the weight of the similarity, allowing it to be equal to unity. The model we consider is formally similar to a standard SAR model with exogenous regressors and a data-driven weight matrix which depends on a finite set of parameters that have to be estimated. Our setup accommodates strong forms of cross-sectional correlation that are normally ruled out in the standard literature on spatial autoregressions, and also includes as special cases the random walk with a drift model, the local to unit root model (LUR) with a drift and the model for moderate integration with a drift.