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In this article, we develop asymptotic theory for a spatial autoregressive (SAR) model where the network structure is defined according to a similarity-based weight matrix, in line with the similarity theory, which in turn has an axiomatic justification. We prove consistency of the quasi-maximum-likelihood estimator and derive its limit distribution. The contribution of this article is two-fold: on one hand, we incorporate a regression component in the data generating process while 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. On the other hand, this work complements the literature on SAR models by adopting a data-driven weight matrix which depends on a finite set of parameters that have to be estimated. The spatial parameter, which corresponds to the weight of the similarity structure, is in turn allowed to take values at the boundary of the standard SAR parameter space. In addition, our setup accommodates strong forms of cross-sectional correlation that are normally ruled out in the standard SAR literature. Our framework is general enough to include as special cases also 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.

Inference in a similarity-based spatial autoregressive model