  Estimating the causal effect of a treatment or exposure for a subpopulation
is of great interest in many biomedical and economical studies. Expected
shortfall, also referred to as the super-quantile, is an attractive effect-size
measure that can accommodate data heterogeneity and aggregate local information
of effect over a certain region of interest of the outcome distribution. In
this article, we propose the ComplieR Expected Shortfall Treatment Effect
(CRESTE) model under an instrumental variable framework to quantity the CRESTE
for a binary endogenous treatment variable. By utilizing the special
characteristics of a binary instrumental variable and a specific formulation of
Neyman-orthogonalization, we propose a two-step estimation procedure, which can
be implemented by simply solving weighted least-squares regression and weighted
quantile regression with estimated weights. We develop the asymptotic
properties for the proposed estimator and use numerical simulations to confirm
its validity and robust finite-sample performance. An illustrative analysis of
a National Job Training Partnership Act study is presented to show the
practical utility of the proposed method.

Methodology