Essays on Wavelet Estimation of the Local Average Treatment Effect
The first chapter of my dissertation makes two main contributions to the econometrics literature on program evaluation. First, we show that under very mild conditions, a policy parameter, the local average treatment effect (LATE), is identified in a class of switching regime models. The identification is achieved through either a discontinuity or a kink in the incentive assignment mechanism, depending on which the agent selects the treatment according to a threshold-crossing model. In contrast to Lee (2008) and Card, Lee, and Pei (2009), we allow for not only a (possibly) endogenous observable covariate but also a (possibly) endogenous unobservable covariate to affect program participation. Second, we introduce local constant wavelet estimators of the LATE for both discontinuous and kink incentive assignment mechanisms and establish their asymptotic properties. The finite sample performances of our local constant wavelet estimators are examined through a simulation study. In the second chapter, we introduce another new class of jump size estimators in a semiparametric mean regression model and apply it to the estimation of the LATE. We refer to members of this class as local polynomial wavelet estimators, and show that all existing jump size estimators, including estimators constructed from differencing two nonparametric estimators and partial linear estimators, belong to the class. We establish asymptotic properties of local polynomial wavelet estimators, and show that they attain the optimal convergence rate even under the presence of slope or higher-order derivative discontinuities. In addition to estimating jump sizes in level, our method automatically leads to estimators of jump sizes in both slope and higher-order derivatives. The finite sample performance of local polynomial wavelet estimators is investigated. The third chapter provides an intuitive two-step procedure for estimating sizes of the discontinuities in the nonparametric median model, instead of the ones in the semi/non-parametric mean model (Chapters 1 and 2). Our first step is to approximate the original discontinuous median model with the discontinuous mean model by local medians transformation (Zhou, 2006). The second step is to carry out local polynomial wavelets estimators (Chapter 2) for the resulting discontinuous mean model. Unlike the check function approach (Koenker, 2005), our two-step method is (1) directly coming from the least squared loss function instead of the check loss function, thus computationally efficient; (2) asymptotically normal; (3) having the optimal rate of convergence and robust to heavy tailed error distributions which may not even possess variances or means.