We propose a new modeling approach for the cross section of returns. Our method, Instrumented Principal Component Analysis (IPCA), allows for latent factors and time-varying loadings by introducing observable characteristics that instrument for the unobservable dynamic loadings. If the characteristics/expected return relationship is driven by compensation for exposure to latent risk factors, IPCA will identify the corresponding latent factors. If no such factors exist, IPCA infers that the characteristic effect is compensation without risk and allocates it to an “anomaly” intercept. Studying returns and characteristics at the stock-level, we find that five IPCA factors explain the cross section of average returns significantly more accurately than existing factor models and produce characteristic-associated anomaly intercepts that are small and statistically insignificant. Furthermore, among a large collection of characteristics explored in the literature, only ten are statistically significant at the 1% level in the IPCA specification and are responsible for nearly 100% of the model’s accuracy.

Stock momentum, long-term reversal, and other past return characteristics that predict future returns also predict future realized betas, suggesting these characteristics capture time-varying risk compensation. We formalize this argument with a conditional factor pricing model. Using instrumented principal components analysis, we estimate latent factors with time-varying factor loadings that depend on observable firm characteristics. We show that factor loadings vary significantly over time, even at short horizons over which the momentum phenomenon operates (one year), and this variation captures reliable conditional risk premia missed by other factor models commonly used in the literature. Our estimates of conditional risk exposure can explain a sizable fraction of momentum and long-term reversal returns and can be used to generate even stronger return predictions.

We propose a statistical model of heterogeneous beliefs wherein investors are represented as different machine learning model specifications. Investors form return forecasts from their individual models using common data inputs. We measure disagreement as forecast dispersion across investor-models (MFD). Our measure aligns with analyst forecast disagreement but more powerfully predicts returns. We document a large and robust association between belief disagreement and future returns. A decile spread portfolio that sells stocks with high disagreement and buys stocks with low disagreement earns a value-weighted return of 13% per year. Further analyses suggest MFD-alpha is mispricing induced by short-sale costs and limits-to-arbitrage.