Dynamic Higher Order Expectations

In models where privately informed agents interact, agents may need to form higher order expectations, i.e. expectations of other agents' expectations. This paper explores dynamic higher order expectations in a setting where it is common knowledge that agents are rational Bayesians. This structure allows any order of expectation at any horizon to be determined recursively. The usefulness of the approach is illustrated by solving a version of Singleton's (1987) asset pricing model with disparately informed traders but without assuming that shocks can be observed perfectly with a lag. In this context, we prove that both the impact of expectations on the asset's price and the variance of expectations are decreasing as the order of expectation increases. We use these results to derive a finite dimensional state representation that can be made arbitrarily accurate. The solution method exploits the Euler-type structure of the asset pricing function and should be applicable to a variety of settings where privately informed agents optimize intertemporally.