A Direct Approach to Inference in Nonparametric and Semiparametric Quantile Regression Models

This paper makes two main contributions. First, we construct “density-free” confidence intervals and confidence bands for conditional quantiles in nonparametric and semiparametric quantile regression models. They are based on pairs of symmetrized k-NN quantile estimators at two appropriately chosen quantile levels. In contrast to Wald-type confidence intervals or bands based on the asymptotic distributions of estimators of the conditional quantiles, our confidence intervals and bands circumvent the need to estimate the conditional quantile density function, do not require the covariate to have a density function, and are very easy to compute. The advantages of our new confidence intervals are borne out in a simulation study. Second, we present a generic confidence interval for conditional quantiles using the rearranged quantile curves that is asymptotically valid for any quantile regression (parametric, nonparametric, or semiparametric), any method of estimation, and any data structure, provided that the conditional quantile function satisfies some mild smoothness assumptions and the original quantile estimator is such that its associated quantile process converges weakly to a Gaussian process with a covariance kernel proportional to the conditional quantile density function.