A Direct Approach to Inference in Nonparametric and Semiparametric Quantile Models

This paper makes two main contributions to inference for conditional quantiles. First, we construct a generic confidence interval for a conditional quantile from any given estimator of the conditional quantile via the direct approach. Our generic confidence interval makes use of two estimates of the conditional quantile function evaluated at two appropriately chosen quantile levels. In contrast to the standard Wald type confidence interval, ours circumvents the need to estimate the conditional density function of the dependent variable given the covariate. We show that our new confidence interval is asymptotically valid for any quantile function (parametric, nonparametric, or semiparametric), any conditional quantile estimator (standard kernel, local polynomial or sieve estimates), and any data structure (random samples, time series, or censored data), provided that certain weak convergence of the conditional quantile process holds for the preliminary quantile estimator. In the same spirit, we also construct a generic confidence band for the conditional quantile function across a range of covariate values. Second, we use a specific estimator, the Yang-Stute (also known as the symmetrized k-NN) estimator for a nonparametric quantile function, and two popular semiparametric quantile functions to demonstrate that oftentimes by a judicious choice of the quantile estimator combined with the specific model structure, one may further take advantage of the flexibility and simplicity of the direct approach. For instance, by using the Yang-Stute estimator, we construct confidence intervals and bands for a nonparametric and two semiparametric quantile functions that are free from additional bandwidth choices involved in estimating not only the conditional but also the marginal density functions and that are very easy to implement. The advantages of our new confidence intervals are borne out in a simulation study.