Academic

In contemporary research, there exists a wide range of theories and techniques that can be utilized to estimate causal effects in observational studies when unmeasured confounding variables are present. One popular approach to address this issue of endogeneity is the implementation of instrumental variable (IV) methods, with the two-stage least squares (2SLS) procedure being the most commonly employed estimation technique. Previous studies have made significant contributions to the theory and techniques of instrumental variable estimation. However, these earlier works have primarily focused on examining the linear relationship between the instrumental variable and the endogenous variable. Therefore, the objective of this study is to relax the assumption of linearity and explore an unknown functional dependency between the endogenous variable and the instrumental variable. In this thesis, we make two contributions to the development of nonparametric instrumental two-stage model.

The first contribution is a new nonparametric model averaging approach to the instrumental variable (IV) regression where the effects of multiple instruments on the endogenous variable are modeled as nonparametric functions in the reduced form equations. Even if individual IVs may have weak and nonlinear relevance to the exposure, our proposed model averaging is able to ensemble their effects with optimal weights to produce valid inference. Our analysis covers both the case in which the number of IV is fixed and the case in which the dimension of IV is diverging with sample size. This novel framework can be especially beneficial to the practical situations involving weak IVs since in many recent observational studies we may encounter a large number of instruments and their quality could range from poor to strong. Numerical studies are carried out and comparisons are made between our proposed method and a wide range of existing alternative methods.

The second contribution also focuses on the nonparametric instrumental two-stage model, but it differs significantly from the first one. This approach employs the ratio method instead of two-stage regression and incorporates kernel regression to generate a nonparametric estimate, whereas the first method utilizes smoothing spline. The objective is to examine the variations in the performance of the two nonparametric estimates under different circumstances and to investigate whether the combination of different methods can yield notable outcomes. We analyze the theoretical properties of the estimator and conduct numerical studies to compare our proposed method with existing approaches. The results demonstrate the adaptability and reliability of our approach in various scenarios.