Quantile regression (QR) methods fit a linear model for conditional quantiles, just as ordinary least squares (OLS) regression estimates a linear model for conditional means. An attractive feature of the OLS estimator is that it gives a minimum mean square error approximation to the conditional expectation function even when the linear model is mis-specified. Empirical research on quantile regression with discrete covariates suggests that QR has a similar property, but the exact nature of the linear approximation has remained elusive. In this paper, we show that QR can be interpreted as minimizing a weighted mean-squared error loss function for the specification error. We derive the weighting function and show that it is approximately equal to the conditional density of QR residuals. The paper goes on to derive the limiting distribution of QR estimators under very general conditions allowing for mis-specification of the conditional quantile function. Finally, we develop methods for the use of QR as a modelling tool for the entire conditional distribution of a random variable. Testable hypotheses include location-scale models, proportional heteroscedasticity, and stochastic dominance. These ideas are illustrated with a human capital earnings function
