# Valid inference under the asymmetric Laplace likelihood

## The problem

The asymmetric Laplace distribution (ALD) is a convenient working likelihood for quantile regression: its mode-as-quantile and the check-loss connection make Bayesian computation straightforward (Yu and Moyeed, 2001). But it is misspecified for almost any real data-generating process, and a misspecified likelihood produces a posterior whose spread is the wrong asymptotic variance for the quantile-regression estimator. Naive credible intervals from such a posterior do not have correct frequentist coverage.

## The correction

Yang, Wang and He (2016) restore validity with a multiplicative sandwich that re-uses the posterior covariance as the “bread”:

\[ V_\text{adj} = \Sigma_\text{post}\, G\, \Sigma_\text{post}, \]

where \(\Sigma_\text{post}\) is the posterior covariance of the fixed effects and \(G\) is the meat — the variance of the asymmetric-Laplace working-likelihood score. With score \(s_i = \sigma^{-1} x_i\,(\tau - \mathbf{1}\{r_i<0\})\) on the conditional residuals \(r_i\), the meat is \(G = \sigma^{-2}\sum_g\big(\sum_{i\in g} x_i\psi_i\big)\big(\cdot\big)'\) (cluster-robust on the grouping factor; the default), or its independence analogue.

Using \(\Sigma_\text{post}\) as the bread is what makes this correct for a mixed model: the posterior covariance already encodes the multilevel pooling, so the adjustment keeps the random-effect contribution to fixed-effect uncertainty while fixing the misspecified ALD scale. Under correct specification \(G \approx \Sigma_\text{post}^{-1}\) and the correction reduces to \(\approx \Sigma_\text{post}\).

    vcov(fit, adjusted = TRUE)    # corrected (multiplicative, cluster meat)
    vcov(fit, adjusted = FALSE)   # naive posterior covariance
    confint(fit, adjusted = TRUE)

## Why not the plain Koenker sandwich?

The textbook fixed-effects sandwich \(\tau(1-\tau)D_1^{-1}D_0D_1^{-1}/n\) (available internally as `compute_ywh_sandwich()` and validated against `quantreg`) is computed on residuals with the random effects removed, so it drops the between-cluster variance and under-covers the mixed-model fixed effects. A simulation bake-off (`tools/bakeoff.R`) confirmed this: across homoscedastic and heteroscedastic two-level designs at several quantiles, the Koenker form covered the fixed intercept at only 0.72–0.92, while the multiplicative form above covered at 0.95–1.00 — at or just above nominal everywhere.

## Scope and caveats

- Validity is claimed for the fixed-effect block. Variance components retain their model-based posterior summaries.
- The correction is mildly conservative (slightly over-nominal) under weak misspecification — the price of guaranteed validity.
- It is a large-sample / many-clusters argument; with very few clusters the cluster-robust meat is noisy.

## References

Yang, Y., Wang, H. J. and He, X. (2016). Posterior inference in Bayesian quantile regression with asymmetric Laplace likelihood. International Statistical Review, 84(3), 327-344.