The Empirical Content of Bayesian Updating under Misspecification

TL;DR

This paper characterizes when belief updating under misspecification aligns with Bayesian rules, revealing conditions on posteriors and prior support, and explores implications in Gaussian environments and diagnostic expectations.

Contribution

It provides a precise characterization of misspecified Bayesian updating, clarifying when belief sequences are consistent with Bayesian rules under model misspecification.

Findings

Belief sequences are consistent with misspecified Bayesian updating if and only if certain partition and support conditions are met.

In Gaussian environments, posterior uncertainty cannot surpass prior uncertainty.

Diagnostic expectations can be consistent with misspecified Bayesianism, unlike some smooth diagnostic expectations.

Abstract

An agent is a misspecified Bayesian if she updates her belief using Bayes' rule given a subjective, possibly misspecified model of her signals. This paper shows that a belief sequence is consistent with misspecified Bayesian updating if and only if the set of posteriors admits a countable partition such that the prior contains a grain of the conditional average posterior on each cell. The condition imposes essentially no restrictions on posteriors given a full-support prior over a finite state space and reduces to a support inclusion condition on compact state spaces under mild regularity assumptions. However, it rules out posterior beliefs with heavier tails than the prior on unbounded state spaces. In Gaussian environments, it implies that posterior uncertainty cannot exceed prior uncertainty. The results delineate the boundary between updating rules that are observationally equivalent to Bayesian updating under misspecification and genuinely non-Bayesian rules. As an application, the paper shows that diagnostic expectations are consistent with misspecified Bayesianism, whereas some parameterizations of smooth diagnostic expectations are not.