A simplified condition for quantile regression

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.