Seminario de Econometría – Departamento de Economía Universidad Carlos III

A Non-Crossing Quantile and Expected Shortfall Regression

Timo Dimitriadis

(University Frankfurt)

«A Non-Crossing Quantile and Expected Shortfall Regression»

Abstract:

Recently proposed joint and two-step estimators for Expected Shortfall (ES) and Value-at-Risk (VaR) regressions often exhibit crossings of the estimated regression functions in finite samples, analogous to the classical quantile crossing problem. We propose an M-estimator for joint VaR and ES regression at multiple probability levels that prevents such crossings by incorporating non-crossing constraints into the optimization problem. We establish that, when crossings arise from finite-sample variability or mild model misspecification, the constrained estimator is asymptotically equivalent to its unconstrained counterpart, a result corroborated by simulation studies. The constraints can be imposed over either a subset or a superset of the covariate space, with the chosen domain acting as a regularization parameter that shrinks the regression functions toward common slopes. We illustrate the practical utility of the proposed method in two applications. First, in recently proposed macroeconomic regressions for Inflation-at-Risk, the non-crossing estimator not only eliminates crossings but also stabilizes parameter estimates and produces smoother predictions. Second, for forecasting VaR and ES of global financial indices using CAViaR-type models, imposing non-crossing constraints improves out-of-sample forecast performance compared to unconstrained CAViaR as well as classical benchmark models.