A conditional moment based approach for partially linear models with applications in treatment effects

In the first chapter of the thesis, we propose a new estimator for the slope parameter of the endogenous variable of interest in a partially linear conditional moment model, which combines a Robinson transformation (Robinson (1988)), to partial out the non-linear part of the model with a smooth minimum distance (SMD) approach (Lavergne and Patilea (2013)), to exploit all the information of the conditional mean independence restriction. Our estimator only depends on one tuning parameter, is easy to compute, consistent and $\sqrt{n}$-asymptotically normal under standard regularity conditions. Simulations show that our estimator is competitive with GMM-type estimators, and often displays a smaller bias and variance, as well as better coverage rates for confidence intervals. We revisit and extend some of the empirical results in Dinkelman (2011) who estimates the impact of electrification on employment growth in South Africa. Overall, we obtain estimates that are smaller in magnitude, more precise, and still economically relevant. In the second chapter, we develop a new estimator for heterogeneous treatment effects in a partially linear model (PLM) with endogenous treatment. The PLM has a parametric part that includes the treatment and the interactions between the treatment and exogenous characteristics, and a nonparametric part that contains those characteristics and many other covariates. The new estimator is a combination of the estimator proposed in the first chapter and a Neyman-Orthogonalized first-order condition (NOFOC). Our estimator, using only one valid binary instrument, identifies both parameters. With the sparsity assumption, using regularised machine learning methods (i.e., the Lasso method) allows us to choose a relatively small number of polynomials of covariates. Our new estimator is less biased, consistent, and $\sqrt{n}$-asymptotically normal under standard regularity conditions. Simulations show that our estimator behaves well with different sets of instruments, but the GMM-type estimators do not. We use the Card application to show the differences between estimators using various sets of instruments. It shows that our new method generates more precise estimates in comparison to GMM. In the third chapter, we estimate the heterogeneous treatment effects of Medicaid on individual outcome variables from the Oregon Health Insurance Experiment. In this experiment, our method from the previous chapter produces more significant and more reliable results for heterogeneous effects of health coverage on economic outcomes.