Estimation and Inference in Adaptive Learning Models with Slowly Decreasing Gains

This paper develops techniques of estimation and inference in a prototypical macroeconomic adaptive learning model with slowly decreasing gains. A sequential three-step procedure based on a ‘super-consistent’ estimator of the rational expectations equilibrium parameter is proposed. It is shown that this procedure is asymptotically equivalent to first estimating the structural parameters jointly via ordinary least-squares (OLS) and then using the so-obtained estimates to form a plug-in estimator of the rational expectations equilibrium parameter. In spite of failing Grenander’s conditions for well-behaved data, a limiting normal distribution of the estimators centered at the true parameters is derived. Although this distribution is singular, it can nevertheless be used to draw inferences about joint restrictions by applying results from Andrews (1987) to show that Wald-type statistics remain valid when equipped with a pseudo-inverse. Monte-Carlo evidence confirms the accuracy of the asymptotic theory for the finite sample behaviour of estimators and test statistics discussed here.