An approximation of the distribution of learning estimates in macroeconomic models

Adaptive learning under constant-gain allows persistent deviations of beliefs from equilibrium so as to more realistically reflect agents' attempt of tracking the continuous evolution of the economy. A characterization of these beliefs is therefore paramount to a proper understanding of the role of expectations in the determination of macroeconomic outcomes. In this paper we propose a simple approximation of the first two moments (mean and variance) of the asymptotic distribution of learning estimates for a general class of dynamic macroeconomic models under constant-gain learning. Our approximation provides renewed convergence conditions that depend on the learning gain and the model's structural parameters. We validate the accuracy of our approximation with numerical simulations of a Cobweb model, a standard New-Keynesian model, and a model including a lagged endogenous variable. The relevance of our results is further evidenced by an analysis of learning stability and the effects of alternative specifications of interest rate policy rules on the distribution of agents' beliefs.