A Simplified Condition For Quantile Regression
Liang Peng, Yongcheng Qi
TL;DR
Addresses identification of conditional quantiles in quantile regression with heteroscedastic residuals. Develops a probability-theoretic framework based on characteristic functions and moment-generating functions to derive when conditional probability statements remain equivalent under alternative representations of the model (QR1) and (QR2). Presents three technical lemmas and three theorems showing that under conditions such as a nonzero characteristic function on a dense set or finite MGFs in a neighborhood, residuals are identically distributed, yielding equivalences like $P(U\le0|Y)=q$, $P(U\le0|Y+W)=q$, and $P(U\le0|YZ)=q$. The findings provide verifiable, simplified identification criteria for quantile regression with heteroscedasticity, with implications for risk forecasting in insurance, econometrics, and systemic risk.
Abstract
Quantile regression is effective in modeling and inferring the conditional quantile given some predictors and has become popular in risk management due to wide applications of quantile-based risk measures. When forecasting risk for economic and financial variables, quantile regression has to account for heteroscedasticity, which raises the question of whether the identification condition on residuals in quantile regression is equivalent to one independent of heteroscedasticity. In this paper, we present some identification conditions under three probability models and use them to establish simplified conditions in quantile regression.