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

Abstract

Many important statistics in macroeconomics and finance-such as cross-sectional dispersions, risk, volatility, or uncertainty-are second moments. In this paper, we explore a mechanism by which second moments naturally and endogenously fluctuate over time as nonlinear transformations of fundamentals. Specifically, we provide general 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 to (1) the cyclicality of the cross-sectional dispersions of macroeconomic variables, (2) the dispersion of MRPKs, (3) security pricing, and (4) endogenous uncertainty in Bayesian inference problems.

Endogenous Uncertainty and Credit Crunches

“…In a related contribution (Straub and Ulbricht, 2015), we show that with very few assumptions on the distribution of ν 1 i,t such a "concave" signal structure generally implies an investors' uncertainty that is increasing in a i,t−1 and decreasing in āi,t−1 . Here we instead use a specific linear approximation to the signal extraction from s 1 i,t that ensures that investors' posteriors are Gaussian while preserving the state-dependency in investors' ability to learn from s 1 i,t .…”

Complementarity and Macroeconomic Uncertainty

“…The paper closest to ours is the recent article by Straub and Ulbricht (2019). They show firmspecific productivity shocks create endogenous fluctuations in the cross-sectional dispersion of output when deviating from Cobb-Douglas production.…”

“…A separate segment of the literature examines the implications of incomplete information. Some of the papers feature learning with aggregate shocks (Fajgelbaum et al, 2017;Saijo, 2017;Van Nieuwerburgh and Veldkamp, 2006), while others focus on firm-specific shocks (Ilut and Saijo, 2020;Straub and Ulbricht, 2015). In these models, adverse shocks under asymmetric learning reduce economic activity and make it harder for households and firms to learn about the economy, which amplifies first moment shocks.…”

Herding Through Booms and Busts

“…Our work also contributes to a literature in which the aggregation of private information leads to nonlinear aggregate dynamics (Fajgelbaum et al, 2017). Straub and Ulbricht (2019) explore in a general setup the informativeness of nonlinear public signals and Straub and Ulbricht (2017) studies a particular application with financial constraints. None of these works consider the emergence of endogenous boom-bust cycles.…”