Nonparametric identification of a binary random factor in cross section data

Suppose V and U are two independent mean zero random variables, where V has an asymmetric distribution with two mass points and U has some zero odd moments (having a symmetric distribution suffices). We show that the distributions of V and U are nonparametrically identified just from observing the sum V + U , and provide a pointwise rate root n estimator. This can permit point identification of average treatment effects when the econometrician does not observe who was treated. We extend our results to include covariates X , showing that we can nonparametrically identify and estimate cross section regression models of the form Y = g ( X , D ∗ ) + U , where D ∗ is an unobserved binary regressor.