What Matters When? Time-Varying Sparsity in Expected Returns

We provide a measure of sparsity for expected returns within the context of classical factor models. Our measure is inversely related to the percentage of active predictors. Empirically, sparsity varies over time and displays an apparent countercyclical behavior. Proxies for financial conditions and for liquidity supply are key determinants of the variability in sparsity. Deteriorating financial conditions and illiquid times are associated with an increase in the number of characteristics that are useful to predict anomaly returns (i.e., the forecasting model becomes more dense). Looking at specific categories of characteristics, we find that variables classified as value, trading frictions and, in particular, profitability are robustly present throughout the sample. A strategy that exploits the dynamics of sparsity to time factors delivers substantial economic gain out-of-sample relative to both a random walk and a simple rolling window shrinkage estimator as well as standard models based on preselected, well-know characteristics like size, momentum, book-to-market, investment and accruals.