The Stability of Macroeconomic Systems with Bayesian Learners

We study abstract macroeconomic systems in which expectations play an important role. Consistent with the recent literature on recursive learning and expectations, we replace the agents in the economy with econometricians. Unlike the recursive learning literature, however, the econometricians in the analysis here are Bayesian learners. We are interested in the extent to which expectational stability remains the key concept in the Bayesian environment. We isolate conditions under which versions of expectational stability conditions govern the stability of these systems just as in the standard case of recursive learning. We conclude that Bayesian learning schemes, while they are more sophisticated, do not alter the essential expectational stability …ndings in the literature. Keywords: Expectational stability, recursive learning, learnability of rational expectations equilibrium. JEL codes: D84, E00, D83. Email: [email protected]. Any views expressed are those of the authors and do not necessarily re‡ect the views of the Federal Reserve Bank of St. Louis or the Federal Reserve System. 1 Introduction 1.1 Overview A large and expanding literature has developed over the last two decades concerning the issue of learning in macroeconomic systems. These systems have a recursive feature, whereby expectations a¤ect states, and states feed back into the expectations formation process being used by the agents. The focus of the literature has been on whether processes in this class are locally convergent to rational expectations equilibria. Evans and Honkapohja (2001), in particular, have stressed that the expectational stability condition governs the stability of real-time learning systems de…ned in this way. This line of research has so far emphasized recursive updating, including least squares learning as a special case. There has been little study of Bayesian updating in the context of expectational stability. What might one expect from an extension to Bayesian updating? There seem to be at least two lines of thought in this area. One is that Bayesian estimation is a close relative of least squares, and therefore that all expectational stability results should obtain with suitable adjustments, but without conceptual di¢ culties. A second, opposite view is that Bayesian agents are essentially endowed with rational expectations— indeed Bayesian learning is sometimes called “rational learning”in the literature— and therefore one should not expect to …nd a concept of “expectational instability”in the Bayesian case. A goal of this paper is to understand which of these views is closer to reality in abstract macroeconomic systems. It is also important to understand how Bayesian updating might repair certain apparent inconsistencies in the recursive learning literature. Cogley and Sargent (2008), for example, have noted that there are “two minds” embedded in the anticipated utility approach to learning that has become popular. According to Cogley and Sargent (2008, p. 186), “[The anticipated utility approach recommended by Kreps (1998)] is of two minds .... Parameters are treated as random