Estimating nonseparable models with mismeasured endogenous variables

We study the identification and estimation of covariate-conditioned average marginal effects of endogenous regressors in nonseparable structural systems when the regressors are mismeasured. We control for the endogeneity by making use of covariates as control variables; this ensures conditional independence between the endogenous causes of interest and other unobservable drivers of the dependent variable. Moreover, we recover distributions of the underlying true causes from their error-laden measurements to deliver consistent estimators. We obtain uniform convergence rates and asymptotic normality for estimators of covariate-conditioned average marginal effects, faster convergence rates for estimators of their weighted averages over instruments, and root-n consistency and asymptotic normality for estimators of their weighted averages over control variables and regressors. We investigate their finite-sample behavior using Monte Carlo simulation and apply new methods to study the impact of family income on child achievement measured by math and reading scores, using a matched mother–child subsample of the National Longitudinal Survey of Youth. Our findings suggest that these effects are considerably larger than previously recognized, and depend on parental abilities and family income. This underscores the importance of measurement errors, endogeneity of family income, nonlinearity of income effects, and interactions between causes of child achievement.