A restricted-perceptions equilibrium exists in which risk-averse agents believe stock prices follow a random walk with a conditional variance that is self-fulfilling. When agents estimate risk, bubbles and crashes arise. These effects are stronger when agents allow for ARCH in excess returns.

Analytical expectational stability results are obtained for both Euler-equation and infinite-horizon adaptive learning in a simple stochastic growth model. The rational expectations equilibrium is stable under both types of learning, though there are important differences in the learning dynamics.