We derive bounds on the rational variation in option‐implied beliefs about market returns. While risk preferences distort option‐implied (or risk‐neutral ) beliefs away from subjective beliefs, one can nonetheless bound risk‐neutral belief movement under a general assumption on the stochastic discount factor. The resulting test requires no knowledge of the objective distribution and allows significantly more flexibility in discount rates than standard volatility tests. Implementing our test empirically using index options, we find that there is so much movement in risk‐neutral beliefs that the bounds are routinely violated. Our results imply significant excess index‐price volatility.

When people receive new information, sometimes they revise their beliefs too much, and sometimes too little. We show that a key driver of whether people overinfer or underinfer is the strength of the information. Based on a model in which people know which direction to update in, but not exactly how much to update, we hypothesize that people will overinfer from weak signals and underinfer from strong signals. We then test this hypothesis across four different environments: abstract experiments, a naturalistic experiment, sports betting markets, and financial markets. In each environment, our consistent and robust finding is overinference from weak signals and underinference from strong signals. Our framework and findings can help harmonize apparently contradictory results from the experimental and empirical literatures.

We propose a duration‐based explanation for the premia on major equity factors, including value, profitability, investment, low‐risk, and payout factors. These factors invest in firms that earn most of their cash flows in the near future and could therefore be driven by a premium on near‐future cash flows. We test this hypothesis using a novel data set of single‐stock dividend futures, which are claims on dividends of individual firms. Consistent with our hypothesis, the expected Capital Asset Pricing Model alpha on individual cash flows decreases in maturity within a firm, and the alpha is not related to the above characteristics when controlling for maturity.

Heteroskedasticity‐ and autocorrelation‐robust (HAR) inference in time series regression typically involves kernel estimation of the long‐run variance. Conventional wisdom holds that, for a given kernel, the choice of truncation parameter trades off a test's null rejection rate and power, and that this tradeoff differs across kernels. We formalize this intuition: using higher‐order expansions, we provide a unified size‐power frontier for both kernel and weighted orthonormal series tests using nonstandard “fixed‐ b ” critical values. We also provide a frontier for the subset of these tests for which the fixed‐ b distribution is t or F . These frontiers are respectively achieved by the QS kernel and equal‐weighted periodogram. The frontiers have simple closed‐form expressions, which show that the price paid for restricting attention to tests with t and F critical values is small. The frontiers are derived for the Gaussian multivariate location model, but simulations suggest the qualitative findings extend to stochastic regressors.