Uniform Inference for Parameters Identified by Conditional Quantile Restrictions

Many structural and dynamic economic models imply that key parameters are identified by conditional quantile restrictions. Building on the exponential-weighting approach of Bierens (1990) and penalized maximum statistics for conditional moment restrictions developed by Chen et al. (2025), we develop a framework for uniform inference on such parameters. We transform the conditional restriction into a continuum of unconditional moment conditions and construct an adaptive $\ell_1$-penalized supremum statistic that aggregates evidence across quantile indices. The penalty regularizes maximization over the weighting direction. Under the stated conditions, adaptive penalty selection asymptotically preserves or improves worst-case local power relative to the unpenalized test over a prespecified set of alternatives. The improvement is strict whenever a positive candidate penalty raises the population maximin criterion above its value at zero. We extend the theory to settings with pre-estimated nuisance parameters, characterizing the additional terms induced by the plug-in step in the limiting process. We derive an analytically corrected variance estimator that accounts for this estimation uncertainty and establish the validity of a Gaussian multiplier bootstrap for null inference and its shifted counterpart for analysis of local power. Monte Carlo comparisons favor CvM-KS for size control under pre-estimation and show that adaptive penalization can improve empirical power relative to the unpenalized test.