An approximation of the distribution of learning estimates in macroeconomic models

Adaptive learning under constant-gain allows persistent deviations of beliefs from equilib- rium 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 un- derstanding 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 dy- namic macroeconomic models under constant-gain learning. Our approximation provides renewed convergence conditions that depend on the learning gain and the model’s struc- tural parameters. We validate the accuracy of our approximation with numerical simu- lations 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 anal- ysis of learning stability and the effects of alternative specifications of interest rate policy rules on the distribution of agents’ beliefs.