Subjective beliefs estimators and their properties

Information-theoretic methods recover investors’ subjective beliefs by minimizing the statistical discrepancy between beliefs and the data-generating process, subject to assets’ Euler constraints. We show that the estimated beliefs converge in probability to their pseudo-true value. Comparing estimators in the Cressie–Read family, we show that the exponential tilting and empirical likelihood estimators produce qualitatively similar estimates of the risk aversion levels and beliefs. The quadratic divergence estimator leads to negative subjective probabilities, implausibly large risk aversion levels, and underestimation of left tail risk. Our results suggest that large institutional investors, like pension funds, have countercyclical beliefs about the market return, while extrapolative investors have procyclical beliefs. Our results offer an alternative explanation of the momentum effect in stock returns, help reconcile procyclical beliefs reported in individual investor surveys versus countercyclical beliefs implied by rational expectations representative agent models, and establish the information-theoretic approach as a powerful methodology for the recovery of beliefs.