We provide a model-free framework for studying the dynamics of the state vector and its risk prices. Specifically, we derive a frequency domain decomposition of the unconditional asset return premium in a general setting with a log-affine stochastic discount factor (SDF). Importantly, we show that the cospectrum between returns and the SDF only displays frequency dependencies through the state vector and that its dynamics and risk prices can be inferred from covariances between asset (portfolio) returns, that is, from the cross-section. Empirically, we find low and high-frequency state vector risk to be differentially priced for US equities.

The anomaly zoo has caused many to question whether researchers are using the right tests of statistical significance. But even if researchers are using the right tests, they will still draw the wrong conclusions from their econometric analyses if they start out with the wrong priors (i.e., if they start out with incorrect beliefs about the ex ante probability of encountering a tradable anomaly, the “anomaly base rate”). We propose a way to estimate it by combining two key insights: Empirical Bayes methods capture the implicit process by which researchers form priors about the likelihood that a new variable is a tradable anomaly based on their past experience, and under certain conditions, a one-to-one mapping exists between these prior beliefs and the best-fit tuning parameter in a penalized regression. The anomaly base rate varies substantially over time, and we study trading-strategy performance to verify our estimation results.

We propose a new methodology for forming arbitrage portfolios that utilizes the information contained in firm characteristics for both abnormal returns and factor loadings. The methodology gives maximal weight to risk-based interpretations of characteristics’ predictive power before any attribution is made to abnormal returns. We apply the methodology to simulated economies and to a large panel of U.S. stock returns. The methodology works well in our simulation and when applied to stocks. Empirically, we find the arbitrage portfolio has (statistically and economically) significant alphas relative to several popular asset pricing models and annualized Sharpe ratios ranging from 1.31 to 1.66.