Convergence in models of misspecified learning

We establish convergence of beliefs and actions in a class of one-dimensional learning settings in which the agent's model is misspecified, she chooses actions endogenously, and the actions affect how she misinterprets information. Our stochastic-approximation-based methods rely on two crucial features: that the state and action spaces are continuous, and that the agent's posterior admits a one-dimensional summary statistic. Through a basic model with a normal-normal updating structure and a generalization in which the agent's misinterpretation of information can depend on her current beliefs in a flexible way, we show that these features are compatible with a number of specifications of how exactly the agent updates. Applications of our framework include learning by a person who has an incorrect model of a technology she uses or is overconfident about herself, learning by a representative agent who may misunderstand macroeconomic outcomes, as well as learning by a firm that has an incorrect parametric model of demand.