The asymmetric Laplace likelihood is a working (misspecified) likelihood, so the naive MCMC posterior covariance of the fixed effects is not the asymptotic variance of the quantile-regression estimator. Yang, Wang and He (2016) correct this with a multiplicative sandwich that re-uses the posterior covariance as the "bread":
Usage
ywh_adjust(object, meat = c("cluster", "independence"))Value
A list with the adjusted fixed-effect covariance vcov, the naive
posterior covariance vcov_naive, the meat G, and sigma.
Details
V_adj = Sigma_post %% G %% Sigma_post,
where Sigma_post is the posterior covariance of the fixed effects and G
is the meat (variance of the ALD working-likelihood score). For the mixed
model this is the right object to correct: Sigma_post already encodes the
multilevel pooling, so the correction retains the random-effect contribution
to fixed-effect uncertainty while fixing the misspecified ALD scale. Under
correct specification G approximately equals Sigma_post^{-1} and the
correction reduces to ~Sigma_post. See compute_ywh_multiplicative().
The pure Koenker-Powell sandwich ([compute_ywh_sandwich()], valid for
fixed-effect quantile regression) was found by simulation to under-cover
the fixed effects of a mixed model, because it is computed on residuals with
the random effects removed and therefore drops the between-cluster variance.
The multiplicative form here covers at or slightly above the nominal level
across homoscedastic and heteroscedastic designs (see tools/bakeoff.R).
Validity is claimed for the fixed-effect block only; variance components keep
their model-based posterior.