Rational Expectations Models with Higher Order Beliefs

This paper develops a general method of solving rational expectations models with higher order beliefs. Higher order beliefs are crucial in an environment with dispersed information and strategic complementarity, and the equilibrium policy depends on infinite higher order beliefs. It is generally believed that solving this type of equilibrium policy requires an infinite number of state variables (Townsend, 1983). This paper proves that the equilibrium policy rule can always be represented by a finite number of state variables if the signals observed by agents follow an ARMA process, in which case we obtain a general solution formula. We also prove that when the signals contain endogenous variables, a finite-state-variable representation of the equilibrium may not exist. For this case, we develop a tractable algorithm that can approximate the solution arbitrarily well. The key innovation in our method is to use the factorization identity and Wiener filter to solve signal extraction problems conditional on infinite observables. This method can be used in a wide range of applications. We demonstrate its strong practicability by solving several classical models featuring higher order beliefs, and also a full-blown business cycle model that is driven by confidence shocks.