Endogenous second moments: A unified approach to fluctuations in risk, dispersion, and uncertainty

We explore a mechanism by which second moments—such as cross-sectional dispersions, risk, volatility, or uncertainty—naturally and endogenously fluctuate over time as nonlinear transformations of fundamentals. Specifically, we provide theoretical results that characterize second moments of transformed random variables when the underlying fundamentals are subject to distributional shifts that affect their means, but not their variances. We illustrate the usefulness of our results with a series of applications. Our main application concerns the cross-sectional dispersions of output, employment, and Solow residuals, which we show to become countercyclical if employment and capital are gross complements. The mechanism can account for a significant share of the empirical cyclicality patterns, without exogenous shocks to volatilities. In additional applications we use our theory to study endogenous fluctuations in the dispersion of MRPKs, in risk in security pricing, and in uncertainty in Bayesian inference problems.