"One of the fundamental problems in econometrics is the potential endogeneity in non-experimental data. This work focuses on econometric methods taking a control function approach to endogeneity. The agenda consists of two parts. In the first part, I study a general class of conditional mean regression methods with a control function, and their relative asymptotic efficiency relationship. Unlike previous results in the literature, the likelihood for the response variables can be incorrect up to the regression functions. My results provide more practical and general guidance on the choice of an estimator. In the second part, I propose a generalized Chamberlain device as a control function approach to time-invariant endogeneity in linear panel data quantile regression models with a finite time dimension. The new correlated effect (CE) estimator has substantial advantages compared to existing methods: (i) it is free of an incidental parameters problem, (ii) the correlated effect is not restricted to a linear functional form, and (iii) an arbitrary within-group dependence of regression errors is allowed. Due to the high-dimensionality of the control function, a nonconvex penalized estimator is adopted for sparse model selection. In the first chapter, I study the asymptotic relative efficiency relationship among estimators based on a quasi-limited information likelihood (QLIL). First, I show that there exists a generalized method of moments estimator (GMM-QLIML) based on all the available quasi-scores. Second, the quasi-limited information maximum likelihood estimator (QLIML) is shown to be as efficient as GMM-QLIML under a set of generalized information matrix equalities. Third, I show that in a fully robust estimation of correctly specified conditional mean functions, QLIML is efficient relative to a two-step control function approach when the generalized linear model variance assumptions hold with a scaling restriction. When a limited information structure is over-identified, the classical minimum distance (MD) estimator is often proposed as an estimation method. The purpose of the second chapter is to study its relative asymptotic efficiency relationship with respect to QLIML and two-step control function (CF) approach. First, I show that the MD estimator is asymptotically efficient relative to two other estimators. Second, I proved that the concentration of reduced form equation estimates does not affect the asymptotic efficiency of the structural parameter estimates in the MD estimation. Third, in a class of models, an if-and-only-if condition is derived for MD and other estimators to be asymptotically equivalent under the null hypothesis of exogeneity. In the third chapter, I propose a point-identifying restriction and estimation procedure for a linear panel data quantile regression model with a fixed time dimension. The proposed model restriction reasonably accounts for the -quantile-specific time-invariant heterogeneity, and allows arbitrary within-group dependence of regression errors. The generalized Chamberlain device is taken analogously as a control function to capture -quantile-specific time-invariant endogenous variations. Since the sieve-approximated control function has high-dimensionality, the estimation procedure adopts penalization techniques under the sparsity assumption. Transformation of the sieve elements into a generalized Mundlak form is considered to make the sparsity assumption more plausible in some cases. The empirical application to birth weight analysis demonstrates a convincing case where the proposed estimator works as intended in real data."--Pages ii-iii.
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