Irregular Identification, Support Conditions and Inverse Weight Estimation∗

Inverse weighted estimators are commonly used in econometrics (and statistics). Some examples include: 1) The binary response model under mean restriction introduced by Lewbel(1997) and further generalized to cover endogeneity and selection. The estimator in this class of models is weighted by the density of a special regressor. 2) The censored regression model under mean restrictions where the estimator is inversely weighted by the censoring probability (Koul, Susarla and Van Ryzin (1981)). 3) The treatment effect model under exogenous selection where the resulting estimator is one that is weighted by a variant of the propensity score. We show that point identification of the parameters of interest in these models often requires support conditions on the observed covariates that essentially guarantee “enough variation.” Under general conditions, these support restrictions, necessary for point identification, drive the weights to take arbitrary large values which we show creates difficulties for regular estimation of the parameters of interest. Generically, these models, similar to well known “identified at infinity” models, lead to estimators that converge at slower than the parametric rate, since essentially, to ensure point identification, one requires some variables to take values on sets with arbitrarily small probabilities, or thin sets. For the examples above, we derive rates of convergence under different tail conditions, analogous to Andrews and Schafgans(1998) illustrating the link between these rates and support conditions. JEL Classification: C14, C25, C13.