  We study identification in nonparametric regression models with a
misclassified and endogenous binary regressor when an instrument is correlated
with misclassification error. We show that the regression function is
nonparametrically identified if one binary instrument variable and one binary
covariate satisfy the following conditions. The instrumental variable corrects
endogeneity; the instrumental variable must be correlated with the unobserved
true underlying binary variable, must be uncorrelated with the error term in
the outcome equation, but is allowed to be correlated with the
misclassification error. The covariate corrects misclassification; this
variable can be one of the regressors in the outcome equation, must be
correlated with the unobserved true underlying binary variable, and must be
uncorrelated with the misclassification error. We also propose a mixture-based
framework for modeling unobserved heterogeneous treatment effects with a
misclassified and endogenous binary regressor and show that treatment effects
can be identified if the true treatment effect is related to an observed
regressor and another observable variable.
