  When analyzing large datasets, it is common to select a model prior to making
inferences. For reliable inferences, it is important to make adjustments that
account for the model selection process, resulting in selective inferences. Our
paper introduces an asymptotic pivot to infer about the effects of selected
variables on conditional quantile functions. Utilizing estimators from smoothed
quantile regression, our proposed pivot is easy to compute and ensures
asymptotically-exact selective inferences without making strict distributional
assumptions about the response variable. At the core of the pivot is the use of
external randomization, which enables us to utilize the full sample for both
selection and inference without the need to partition the data into independent
data subsets or discard data at either step. On simulated data, we find that:
(i) the asymptotic confidence intervals based on our pivot achieve the desired
coverage rates, even in cases where sample splitting fails due to insufficient
sample size for inference; (ii) our intervals are consistently shorter than
those produced by sample splitting across various models and signal settings.
We report similar findings when we apply our approach to study risk factors for
low birth weights in a publicly accessible dataset of US birth records from
2022.
