Interpreting Unconditional Quantile Regression with Conditional Independence

This note provides additional interpretation for the counterfactual outcome distribution and corresponding unconditional quantile "effects" defined and estimated by Firpo, Fortin, and Lemieux (2009) and Chernozhukov, Fern\'andez-Val, and Melly (2013). With conditional independence of the policy variable of interest, these methods estimate the policy effect for certain types of policies, but not others. In particular, they estimate the effect of a policy change that itself satisfies conditional independence.
Comment: The main result is essentially the same as Proposition 1 of Rothe (2010), https://doi.org/10.1016/j.jeconom.2009.09.001