  In Bayesian quantile regression, the most commonly used likelihood is the
asymmetric Laplace (AL) likelihood. The reason for this choice is not that it
is a plausible data-generating model but that the corresponding maximum
likelihood estimator is identical to the classical estimator by Koenker and
Bassett (1978), and in that sense, the AL likelihood can be thought of as a
working likelihood. AL-based quantile regression has been shown to produce good
finite-sample Bayesian point estimates and to be consistent. However, if the AL
distribution does not correspond to the data-generating distribution, credible
intervals based on posterior standard deviations can have poor coverage. Yang,
Wang, and He (2016) proposed an adjustment to the posterior covariance matrix
that produces asymptotically valid intervals. However, we show that this
adjustment is sensitive to the choice of scale parameter for the AL likelihood
and can lead to poor coverage when the sample size is small to moderate. We
therefore propose using Infinitesimal Jackknife (IJ) standard errors (Giordano
& Broderick, 2023). These standard errors do not require resampling but can be
obtained from a single MCMC run. We also propose a version of IJ standard
errors for clustered data. Simulations and applications to real data show that
the IJ standard errors have good frequentist properties, both for independent
and clustered data. We provide an R-package that computes IJ standard errors
for clustered or independent data after estimation with the brms wrapper in R
for Stan.
