In this paper, we propose a Bayesian estimation and forecasting procedure for noncausal autoregressive (AR) models. Specifically, we derive the joint posterior density of the past and future errors and the parameters, yielding predictive densities as a by-product. We show that the posterior model probabilities provide a convenient model selection criterion in discriminating between alternative causal and noncausal specifications. As an empirical application, we consider US inflation. The posterior probability of noncausality is found to be high—over 98%. Furthermore, the purely noncausal specifications yield more accurate inflation forecasts than alternative causal and noncausal AR models. Copyright © 2010 John Wiley & Sons, Ltd.