ABSTRACT
Regression quantile estimators solve a linear program and can be computed efficiently. The finite-sample distributions of regression quantiles can be characterized (Koenker, 2005), but they are difficult to use for statistical inference. Suppose that we have data { ( X i , Y i ) $ \{(X_i, Y_i ) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315120256/9526b76e-b656-493f-8862-138dc7c55298/content/inline-math2_1.tif"/> , i = 1 , … , n } $ i=1, \ldots , n\} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315120256/9526b76e-b656-493f-8862-138dc7c55298/content/inline-math2_2.tif"/> , where the conditional quantile of Y given X is of interest and assumed to be linear. The asymptotic distributions of regression quantiles are normal under mild conditions, but the asymptotic variance depends on the conditional densities of Y given X = X i $ X=X_i $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315120256/9526b76e-b656-493f-8862-138dc7c55298/content/inline-math2_3.tif"/> , which are generally unknown. Statistical inference based on the asymptotic variance is arguably difficult for a simple reason. That is, one needs to use a nonparametric estimate of the asymptotic variance that requires the choice of a smoothing parameter, and such estimates can be quite unstable. Even if the asymptotic variance is well estimated, the accuracy of its approximation to the finite-sample variance depends on the design matrix as well as the quantile level. Resampling methods provide a reliable approach to inference for quantile regression analysis under a wide variety of settings.
