  This paper proposes a new class of M-estimators that double weight for the
twin problems of nonrandom treatment assignment and missing outcomes, both of
which are common issues in the treatment effects literature. The proposed class
is characterized by a `robustness' property, which makes it resilient to
parametric misspecification in either a conditional model of interest (for
example, mean or quantile function) or the two weighting functions. As leading
applications, the paper discusses estimation of two specific causal parameters;
average and quantile treatment effects (ATE, QTEs), which can be expressed as
functions of the doubly weighted estimator, under misspecification of the
framework's parametric components. With respect to the ATE, this paper shows
that the proposed estimator is doubly robust even in the presence of missing
outcomes. Finally, to demonstrate the estimator's viability in empirical
settings, it is applied to Calonico and Smith (2017)'s reconstructed sample
from the National Supported Work training program.
