Interpretation and semiparametric efficiency in quantile regression under misspecification

Allowing for misspecification in the linear conditional quantile function, this paper provides a new interpretation and the semiparametric efficiency bound for the quantile regression parameter b(t) in Koenker and Bassett (1978). The first result on interpretation shows that under a mean-squared loss function, the probability limit of the Koenker-Bassett estimator minimizes a weighted distribution approximation error, defined as FY(X'b(t)/X)-t, i.e., the deviation of the conditional distribution function, evaluated at the linear quantile approximation, from the quantile level. The second result implies that the Koenker-Bassett estimator semiparametrically efficiently estimates the quantile regression parameter that produces parsimonious descriptive statistics for the conditional distribution. Therefore, quantile regression shares the attractive features of ordinary least squares: interpretability and semiparametric efficiency under misspecification.