Forecasts and Conditionally I.I.D. Models

Résumé

We characterize when one-step-ahead forecasts are consistent with a conditionally i.i.d. (CIID) model, i.e., Bayesian learning about a stable but unknown i.i.d. data-generating process. For two periods and binary outcomes, symmetry (pairwise exchangeability) and reinforcement (realized outcomes become more likely) are necessary and sufficient. For two periods and arbitrary finite outcomes, forecasts admit a CIID representation if and only if a forecast-derived matrix of joint probabilities is completely positive; with at most four outcomes, complete positivity reduces to positive semidefiniteness. Two-period forecasts cannot detect beliefs in positively autocorrelated outcomes, but some negatively autocorrelated beliefs can be identified. For multi-period forecasts with binary outcomes, we derive an easily checked characterization of CIID representations by linking to the truncated moment problem, and show how the minimal-support rationalizations depend on the number of periods. With multiple periods and outcomes, CIID holds exactly when forecasts satisfy pairwise exchangeability and the associated hierarchy of moment tensors is completely positive.