Asymptotic Theory For Empirical Similarity Models

We consider the stochastic process $Y_t = \sum\nolimits_{i < t} {s_w } (x_t ,x_i)Y_i /\sum\nolimits_{i < t} {s_w } (x_t ,x_i) + \varepsilon _t$, t = 2, …, n, where sw(xt, xi) is a similarity function between the tth and the ith observations and {εt} is a random disturbance term. This process was originally axiomatized by Gilboa, Lieberman, and Schmeidler (2006, Review of Economics and Statistics 88, 433–444) as a way by which agents, or even nature, reason. In the present paper, consistency and the asymptotic distribution of the quasi-maximum likelihood estimator of the parameters of the model are established. Connections to other models and techniques are drawn. In its general form, the model does not fall within any class of nonstationary econometric models for which asymptotic theory is available. For this reason, the developments in this paper are new and nonstandard.