The Annals of Statistics

A smooth block bootstrap for quantile regression with time series

Karl B. Gregory, Soumendra N. Lahiri, and Daniel J. Nordman

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Quantile regression allows for broad (conditional) characterizations of a response distribution beyond conditional means and is of increasing interest in economic and financial applications. Because quantile regression estimators have complex limiting distributions, several bootstrap methods for the independent data setting have been proposed, many of which involve smoothing steps to improve bootstrap approximations. Currently, no similar advances in smoothed bootstraps exist for quantile regression with dependent data. To this end, we establish a smooth tapered block bootstrap procedure for approximating the distribution of quantile regression estimators for time series. This bootstrap involves two rounds of smoothing in resampling: individual observations are resampled via kernel smoothing techniques and resampled data blocks are smoothed by tapering. The smooth bootstrap results in performance improvements over previous unsmoothed versions of the block bootstrap as well as normal approximations based on Powell’s kernel variance estimator, which are common in application. Our theoretical results correct errors in proofs for earlier and simpler versions of the (unsmoothed) moving blocks bootstrap for quantile regression and broaden the validity of block bootstraps for this problem under weak conditions. We illustrate the smooth bootstrap through numerical studies and examples.

Article information

Ann. Statist., Volume 46, Number 3 (2018), 1138-1166.

Received: June 2016
Revised: February 2017
First available in Project Euclid: 3 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G09: Resampling methods
Secondary: 62G20: Asymptotic properties 62J05: Linear regression 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Kernel smoothing jackknife after bootstrap moving blocks tapering value at risk


Gregory, Karl B.; Lahiri, Soumendra N.; Nordman, Daniel J. A smooth block bootstrap for quantile regression with time series. Ann. Statist. 46 (2018), no. 3, 1138--1166. doi:10.1214/17-AOS1580.

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Supplemental materials

  • Supplement to “A smooth block bootstrap for quantile regression with time series”. Details of proofs and additional simulation results.