This dissertation is divided in three self-contained chapters. The first extends the GMM redundancy results of Prokhorov and Schmidt (2009) for nonsmooth objective functions, giving sharp guidelines about how to obtain efficient estimates of parameters of interest in the presence of nuisance parameters. The use of one-step GMM estimators for both sets of parameters is asymptotically more efficient than two-step procedures. These results are applied to Wooldridge's (2007) inverse probability weighted estimator (IPW), generalizing the framework to deal with missing-data in this context. Even though two-step estimation of the parameters of interest is more efficient than using known probabilities of selection, this is dominated by a one-step joint estimation procedure. Examples for quantile regression with missing data and instrumental variable quantile regression are provided.The second chapter analyzes the asymptotic distribution of local polynomial estimators in the context of regression discontinuity designs. The standard "small-h" approach in the literature (Hahn et al., 2001; Porter, 2003; Imbens and Lemieux, 2008; Lee and Lemieux, 2009) is to assume the bandwidth, h, around the discontinuity shrinks towards zero as the sample size increases. However, in practice, the researcher has to choose an h>0 to implement the estimator. This chapter derives the fixed-h asymptotic distribution that allows for the bandwidth to be positive, providing refined approximations for the estimator's behavior. When h>0, the small-h asymptotic variance is equivalent to assuming that the density of the running variable and the conditional variance of the dependent variable are constant around the cutoff. Simulations provide evidence that fixed-h asymptotic distributions better describe the behavior of both bias and variance of the estimator, leading to improved inference. Estimators for fixed-h standard errors are proposed and incorporate the theoretical gains of the improved approximations. The fixed-h variance estimators improve markedly over small-h estimators in the presence of some forms of heteroskedasticity. Interestingly, in the special case of homoskedastic errors using a local linear estimator, the variance estimators based on small-h asymptotics produce tests with similar size to the fixed-h variance estimators proposed in this chapter.Chapter 3 develops the asymptotic properties of quantile regression estimators under standard stratification sampling, following Wooldridge (2001). Formulas for the asymptotic variance and feasible estimators are provided. Under exogenous stratification the usual quantile regression estimators and standard errors are still valid.
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