Efficiency Bounds for Semiparametric Estimation of Quantile Regression under Misspecification

Allowing for misspecification in the linear conditional quantile function (CQF), we calculate the semiparametric efficiency bound for the solution to the population quantile regression (QR) problem or the best linear predictor for a response variable under the asymmetric check loss function. The pseudo-true QR parameter that solves a non-differentiable unconditional moment restriction is interpreted explicitly by Angrist, Chernozhukov, & Fernández-Val (2006) as the best weighted meansquared linear approximation to the true CQF. Alternatively, the QR parameter for the misspecified linear projection model can be understood by the orthogonality condition of the covaritates and the distribution error, i.e., the deviation of the true conditional distribution function, evaluated at the approximated quantile, from the true probability. This suggests that even though the CQF is nonlinear in the covariates, the sample analog estimator by Koenker & Bassett (1978) semiparametrically efficiently estimates a pseudo-true parameter that offers meaningful descriptive statistics for the conditional distribution of the dependent variable. These properties, analogous with ordinary least squares, reinforce our understanding of QR.