On the Predictive Risk in Misspecified Quantile Regression

TL;DR

This paper investigates the predictive risk of misspecified quantile regression models, providing asymptotic bias characterizations, estimation methods, and empirical validation against cross-validation.

Contribution

It introduces simple asymptotic characterizations of optimism bias and proposes consistent estimators for predictive risk in misspecified quantile regression models.

Findings

Estimators for optimism bias are consistent under mild conditions.

Predictive risk estimates compare favorably with cross-validation.

Results hold for models of moderately growing size.

Abstract

In the present paper we investigate the predictive risk of possibly misspecified quantile regression functions. The in-sample risk is well-known to be an overly optimistic estimate of the predictive risk and we provide two relatively simple (asymptotic) characterizations of the associated bias, also called expected optimism. We propose estimates for the expected optimism and the predictive risk, and establish their uniform consistency under mild conditions. Our results hold for models of moderately growing size and allow the quantile function to be incorrectly specified. Empirical evidence from our estimates is encouraging as it compares favorably with cross-validation.