Missing data are a common challenge facing empirical researchers. This paper presents a general GMM framework and estimator for dealing with missing values of an explanatory variable in linear regression analysis. The GMM estimator is efficient under assumptions needed for consistency of linear-imputation methods. The estimator, which also allows for a specification test of the missingness assumptions, is compared to existing linear imputation, complete data, and dummy variable methods commonly used in empirical research. The dummy variable method is generally inconsistent even when data are missing completely at random, and the dummy variable method, when consistent, can be less efficient than the complete data method.

A unifying framework to test for causal effects in nonlinear models is proposed. We consider a generalized linear-index regression model with endogenous regressors and no parametric assumptions on the error disturbances. To test the significance of the effect of an endogenous regressor, we propose a statistic that is a kernel-weighted version of the rank correlation statistic (tau) of Kendall (1938). The semiparametric model encompasses previous cases considered in the literature (continuous endogenous regressors (Blundell and Powell (2003)) and a single binary endogenous regressor (Vytlacil and Yildiz (2007))), but the testing approach is the first to allow for (i) multiple discrete endogenous regressors, (ii) endogenous regressors that are neither discrete nor continuous (e.g., a censored variable), and (iii) an arbitrary “mix” of endogenous regressors (e.g., one binary regressor and one continuous regressor).

This paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator. The theory developed also applies to other estimators within a cube-root convergence class. For some single-parameter estimators in this class, the results suggest a simple method for inference based upon the bootstrap.