Essays in Semiparametric Econometrics

Abstract

This dissertation studies how the estimation of first-stage nuisance parameters associated with control and instrumental variables affects second-stage estimation of treatment-effect parameters. Chapter 1 approaches the analysis of experimental data as a mechanism-design problem that acknowledges that researchers choose between estimators according to their own preferences. Specifically, I focus on covariate adjustments, which can increase the precision of a treatment-effect estimate, but open the door to bias when researchers engage in specification searches. I establish that unbiasedness is a requirement on the estimation of the average treatment effect that aligns researchers’ preferences with the minimization of the mean-squared error relative to the truth, and that fixing the bias can yield an optimal restriction in a minimax sense. I then provide a characterization of unbiased treatment-effect estimators as sample-splitting procedures. Chapter 2 gives two examples in which we can improve estimation by shrinking in high-dimensional nuisance parameters while avoiding or even reducing the bias in a low-dimensional target parameter. I first consider shrinkage estimation of the nuisance parameters associated with control variables in a linear model, and show that for at least three control variables the standard least-squares estimator is dominated with respect to variance in the treatment effect even among unbiased estimators when treatment is exogenous. Second, I consider shrinkage in the estimation of first-stage instrumental variable coefficients in a two-stage linear regression model. For at least four instrumental variables, I establish that the standard two-stage least-squares estimator is dominated with respect to bias. Chapter 3 (with Alberto Abadie) considers regression analysis of treatment effects after nearest-neighbor matching on control variables. We show that standard errors that ignore the matching step are not generally valid if the second-step regression model is misspecified. We offer two easily implementable alternatives, (i) clustering the standard errors at the level of the matches, or (ii) a nonparametric block bootstrap procedure, that produce approximations to the distribution of the post-matching estimator that are robust to misspecification, provided that matching is done without replacement.