  The quantile regression kink design (QRKD) is proposed by empirical
researchers as a potential method to assess heterogeneous treatment effects
under suitable research designs, but its causal interpretation remains unknown.
We propose a causal interpretation of the QRKD estimand. Under flexible
heterogeneity and endogeneity, the QRKD estimand measures a weighted average of
heterogeneous marginal effects at respective conditional quantiles of outcome
given a designed kink point. In addition, we develop weak convergence results
for the QRKD estimator as a local quantile process for the purpose of
conducting statistical inference on heterogeneous treatment effects using the
QRKD. Applying our methods to the Continuous Wage and Benefit History Project
(CWBH) data, we find significantly heterogeneous positive causal effects of
unemployment insurance benefits on unemployment durations in Louisiana between
1981 and 1983. These effects are larger for individuals with longer
unemployment durations.
