We investigate persistence in CRSP monthly excess stock returns, using a state space model with stable disturbances. The non-Gaussian state space model with volatility persistence is estimated by maximum likelihood, using the optimal filtering algorithm given by Sorenson and Alspach (1971 Automatica 7 465-79). The conditional distribution has a stable of 1.89, and normality is strongly rejected even after accounting for GARCH. However, stock returns do not contain a significant mean-reverting component. The optimal predictor is the unconditional expectation of the series, which we estimate to be 9.8% per annum.