A direct approach to inference in nonparametric and semiparametric quantile models

Abstract

This paper makes two main contributions to inference for conditional quantiles. First, we construct a generic con…dence interval for a conditional quantile from any given estimator of the conditional quantile via the direct approach. Our generic con…dence 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 con…dence interval, ours circumvents the need to estimate the conditional density function of the dependent variable given the covariate. We show that our new con…dence 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 con…dence band for the conditional quantile function across a range of covariate values. Second, we use a speci…c 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 speci…c model structure, one may further explore the ‡exibility and simplicity of the direct approach. For instance, by using the Yang-Stute estimator, we construct con…dence 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 compute. The advantages of our new con…dence intervals are borne out in a simulation study.

Robust uniform inference for quantile treatment effects in regression discontinuity designs

“…To cover uniform inference for local polynomial estimators over a general index set with bias correction of any arbitrary order, uniform validity of Bahadur representation over the set is essential. For classes of nonparametric kernel regressions on which our method relies, Masry (1996), Kong, Linton, and Xia (2010) and Fan and Liu (2016) develop uniform Bahadur representations over regressors. Furthermore, Guerre and Sabbah (2012), Qu and Yoon (2015a), Lee, Song, and Whang (2015), Fan and Guerre (2016) develop uniform validity over quantiles as well.…”

Fractional order statistic approximation for nonparametric conditional quantile inference

“…The supplemental appendix includes a similar figure but with a nonparametric (instead of quadratic) conditional quantile estimate along with joint CIs from Fan and Liu (2016).…”

“…We compare further in our simulations. One open question is whether using our beta reference and interpolation can improve accuracy for the general Fan and Liu (2016) method beyond the local constant estimator with a uniform kernel; our Lemma 3 shows this at least retains first-order accuracy.…”

“…Recent complementary work of Fan and Liu (2016) also concerns a "direct method" of nonparametric inference on conditional quantiles. They use a limiting Gaussian process to derive first-order accuracy in a general setting, whereas we use the finite-sample Dirichlet process to achieve high-order accuracy in an iid setting.…”

“…One limitation of our approach is that it cannot incorporate these other estimators, whereas Assumption GI(iii) in Fan and Liu (2016) includes any estimator that weakly converges (over a range of quantiles) to a Gaussian process with a particular structure. We compare further in our simulations.…”