This paper studies the following problem. An agent takes actions based on a possibly misspecified model. The agent is large, in the sense that his actions influence the model he is trying to learn about. The agent is aware of potential model misspecification and tries to detect it, in real-time, using an econometric specification test. If his model fails the test, he formulates a new better-fitting model. If his model passes the test, he uses it to formulate and implement a policy based on the provisional assumption that the current model is correctly specified, and will not change in the future. We claim that this testing and model validation process is an accurate description of most macroeconomic policy problems. Unfortunately, the dynamics produced by this process are not at all well understood. We make progress on this problem by relating it to a problem that is well understood. In particular, we relate it to the dynamics of constant-gain stochastic approximation algorithms. Doing this enables us to appeal to well known results from the large deviations literature to help us understand the dynamics of testing and model revision. We show that as the agent applies an increasingly stringent specification test, the large deviation properties of the discrete model validation dynamics converge to those of the continuous learning dynamics. This sheds new light on the recent constant-gain learning literature.

This article develops a dynamic asset pricing model with persistent heterogeneous beliefs. The model features competitive traders who receive idiosyncratic signals about an underlying fundamentals process. We adapt Futia's (1981) frequency domain methods to derive conditions on the fundamentals that guarantee non-invertibility of the mapping between observed market data and the underlying shocks to agents' information sets. When these conditions are satisfied, agents remain asymmetrically informed in equilibrium and must ‘forecast the forecasts of others’. An econometrician, who incorrectly imposes a homogeneous beliefs equilibrium, will find that the asset price displays violations of variance bounds, predictability of excess returns, and rejections of cross-equation restrictions.