Dynamic Higher Order Expectations

In models where privately informed agents interact, they may need to form higher-order expectations, i.e. expectations about other agents' expectations. In this paper we prove that there exists a unique equilibrium in a class of linear dynamic rational expectations models in which privately informed agents form higher order expectations. We propose an iterative procedure that recursively computes increasing orders of expectations. The algorithm is a contraction mapping, and the implied dynamics of the endogenous variables converge to the unique equilibrium of the model. The contractive property of the algorithm implies that, in spite of the fact that the model features an infinite regress of expectations, the equilibrium dynamics of the model can be approximated to an arbitrary accuracy with a finite-dimensional state. We provide explicit bounds on the approximation errors. These results hold under quite general conditions: It is sufficient that agents discount the future and that the exogenous processes follow stationary (but otherwise unrestricted) VARMA processes.