Efficient estimation of generalized nonparametric model under additive structure - XIA, Ying (Institutional Knowledge at Singapore Management University, 2023)
In this thesis, we develop novel nonparametric estimation techniques for two distinct classes of models: (1) Generalized Additive Models with Unknown Link Functions (GAMULF) and (2) Generalized Panel Data Transformation Models with Fixed Effects. Both models avoid parametric assumptions on their respective link or transformation functions, as well as the distribution of the idiosyncratic error terms. The first chapter aims to provide an in-depth and systematic introduction to cross- sectional and panel-data nonparametric transformation models, encompassing practical applications, a diverse range of estimation techniques, and the study of asymptotic properties. We discuss the advantages and limitations of these models and estimation methods, delving into the latest advancements and innovations in the field. Furthermore, we propose a potential approach to mitigate the curse of dimensionality in the context of fully nonparametric transformation models with fixed effects in panel-data settings. The second chapter proposes a three-stage nonparametric least squares (NPLS) estimation procedure for the additive functions in the GAMULF. In the first stage, we estimate conditional expectation by the local-linear kernel regression and then apply matching method to the splines series to obtain initial estimators. In the second stage, we use the local-polynomial kernel regression to estimate the link function. In the third stage, given the estimators in Stages 1 and 2, we apply the local-linear kernel regression to refine the initial estimator. The great advantage of such a procedure is that the estimators obtained at all stages have closed-form expressions, which overcomes the computational hurdle for existing estimators of the GAMULF model. The third chapter proposes a multiple-stage Local Maximum Likelihood Estimator (LMLE) for the structural functions in the generalized panel data transformation model with fixed effects. In the first stage, we apply the regularized logistic sieve method to estimate the sieve coefficients associated with the approximation of a composite function and then apply a matching method to obtain initial consistent estimators of the additive structural functions. In the second stage, we apply the local polynomial method to estimate certain composite function and its derivatives to be used later on. In the third stage, we apply the local linear method to obtain the refined estimator of the additive structural functions based on the estimators obtained in Steps 1 and 2. The greatest advantage is that all minimization problems are convex and thus overcome the computational hurdle for existing approaches to the generalized panel data transformation model. The final estimates of the additive terms in two models achieve the optimal one-dimensional convergence rate, asymptotic normality and oracle efficiency. The Monte Carlo simulations demonstrate that our new estimator performs well in finite samples. The thesis demonstrates the effectiveness of the proposed nonparametric estimation techniques in addressing the complexities of generalized additive models with unknown link functions and panel data transformation models with fixed effects.