Abstract
We study estimation and inference in settings where the interest is in the effect of a potentially endogenous regressor on some outcome. To address the endogeneity, we exploit the presence of additional variables. Like conventional instrumental variables, these variables are correlated with the endogenous regressor. However, unlike conventional instrumental variables, they also have direct effects on the outcome, and thus are “invalid” instruments. Our novel identifying assumption is that the direct effects of these invalid instruments are uncorrelated with the effects of the instruments on the endogenous regressor. We show that in this case the limited-information-maximum-likelihood (liml) estimator is no longer consistent, but that a modification of the bias-corrected two-stage-least-square (tsls) estimator is consistent. We also show that conventional tests for over-identifying restrictions, adapted to the many instruments setting, can be used to test for the presence of these direct effects. We recommend that empirical researchers carry out such tests and compare estimates based on liml and the modified version of bias-corrected tsls. We illustrate in the context of two applications that such practice can be illuminating, and that our novel identifying assumption has substantive empirical content.
Highlights
In this paper we study estimation and inference in settings where the interest is in the effect of a potentially endogenous regressor on some outcome
To allow for the possible endogeneity, we exploit the presence of additional variables. These variables have some of the features of conventional instrumental variables, in the sense that they are correlated with the endogenous regressor
For each of the four estimators we calculate the bias as the average difference between the estimate and the true value, the median absolute deviation, and coverage rates based on confidence intervals using the four different standard errors: conventional standard errors, Bekker standard errors which are robust to the presence of many instruments, standard errors robust to the presence of many instruments and many exogenous regressors, and standard errors robust to the presence of direct effects of the instruments on the outcome
Summary
In this paper we study estimation and inference in settings where the interest is in the effect of a potentially endogenous regressor on some outcome. We contribute to the literature studying properties of instrumental variables methods allowing for direct effects of the instruments. This literature has largely focused on the case with a fixed number of instruments. The focus of this literature has been on correcting size distortions of tests, biases of estimators, sensitivity analyses, and bounds in the presence of direct effects.
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