  This paper considers the instrumental variable quantile regression model
(Chernozhukov and Hansen, 2005, 2013) with a binary endogenous treatment. It
offers two identification results when the treatment status is not directly
observed. The first result is that, remarkably, the reduced-form quantile
regression of the outcome variable on the instrumental variable provides a
lower bound on the structural quantile treatment effect under the stochastic
monotonicity condition (Small and Tan, 2007; DiNardo and Lee, 2011). This
result is relevant, not only when the treatment variable is subject to
misclassification, but also when any measurement of the treatment variable is
not available. The second result is for the structural quantile function when
the treatment status is measured with error; I obtain the sharp identified set
by deriving moment conditions under widely-used assumptions on the measurement
error. Furthermore, I propose an inference method in the presence of other
covariates.
