Quantile Regression

Abstract. Classical least squares regression may be viewed as a natural way of extending the idea of estimating an unconditional mean parameter to the problem of estimating conditional mean functions; the crucial link is the formulation of an optimization problem that encompasses both problems. Likewise, quantile regression oers an extension of univariate quantile estimation to estimation of conditional quantile functions via an optimization of a piecewise linear objective function in the residuals. Median regression minimizes the sum of absolute residuals, an idea introduced by Boscovich in the 18th century. The asymptotic theory of quantile regressionclosely parallels the theory of the univariate sample quantiles. Computation of quantile regression estimators may be formulated as a linear program-ming problem and eciently solved by simplex or barrier methods. A close link to rank based inference has been forged from the theory of the dual regression quantile process, or regression rankscore process. Quantile regression is a statistical technique intended to estimate, and conduct inference about, conditional quantile functions. Just as classical linear regression methods based on minimizing sums of squared residuals enable one to estimate models for conditional mean functions, quantile regressio