# Quantile regression: Which standard errors?

The summary.rq function from the quantreg vignette provides a multitude of choices for standard error estimates of quantile regression coefficients. What are the special scenarios where each of these becomes optimal/desirable?

• "rank" which produces confidence intervals for the estimated parameters by inverting a rank test as described in Koenker (1994). The default option assumes that the errors are iid, while the option iid = FALSE implements the proposal of Koenker Machado (1999). See the documentation for rq.fit.br for additional arguments.

• "iid" which presumes that the errors are iid and computes an estimate of the asymptotic covariance matrix as in KB(1978).

• "nid" which presumes local (in tau) linearity (in x) of the the conditional quantile functions and computes a Huber sandwich estimate using a local estimate of the sparsity.

• "ker" which uses a kernel estimate of the sandwich as proposed by Powell(1990).

• "boot" which implements one of several possible bootstrapping alternatives for estimating standard errors.

I have read at least 20 empirical papers where this is applied either in the time-series or the cross-sectional dimension and haven't seen a mention of standard error choice.

• I hope you get many responses to this excellent question. We need some guidance in this area. Another approach, made easy by the R rms package's bootcov function is to save the bootstrap replicate regression coefficients ($\beta$s) and to use the bootstrap nonparametric percentile confidence interval approach to get confidence intervals for any contrast (combination of $\beta$s) of interest. – Frank Harrell Dec 22 '12 at 14:19
• Excellent question, I was told in class "always use bootstrapping" but I'm not sure exactly why as I'm not familiar with the theory behind the other methods. – Max Gordon Dec 22 '12 at 22:31
• Did you go through the paper Koenker and Hallock(2000): Quantile Regression: An Introduction (econ.uiuc.edu/~roger/research/intro/rq.pdf)? Bootstrap is preferable because it makes no assumption about the distribution of response (p. 47, Quantile regressions, Hao and Naiman, 2007). Also, note that the "...assumptions for the asymptotic procedure usually do not hold, and even if these assumptions are satisfied, it is complicated to solve for the standard error of the constructed scale and skewness shifts (p. 43)..." – Metrics Dec 31 '12 at 5:16
• Doesn't bootstrap resampling assume that the uniform prior is non-informative? – EngrStudent Jun 29 '13 at 14:16
• @Metrics: Maybe you should post that as an answer? – naught101 Sep 11 '13 at 5:47