In this thesis, we provide a simple approach to identify and estimate group structure in panel models by adapting the M-estimation method. We consider both linear and nonlinear panel models where the regression coefficients are heterogeneous across groups but homogeneous within a group and the group membership is unknown to researchers. The main result of the thesis is that under certain assumptions, our approach is able to provide uniformly consistent group parameter estimator as long as the number of groups used in estimation is not smaller than the true number of groups. We also show that, with probability approaching one, our method can partition some true groups into further subgroups, but cannot mix individuals from different groups. When the true number of groups is used in estimation, all the individuals can be categorized correctly with probability approaching one, and we establish the limiting distribution for the estimates of the group parameters. In addition, we provide an information criterion to choose the number of group and established its consistency under some mild conditions. Monte Carlo simulations are conducted to examine the finite sample performance of our proposed method. Findings in the simulation confirm our theoretical results in the paper. Application to labor force participation also highlights the necessity to take into account of individual heterogeneity and group heterogeneity. iv Acknowledgement Firstly, I would like to express my special appreciation and thanks to my advisor Professor Anton Schick, he has been a tremendous mentor for me. Dr. Schick not only taught me how to conduct mathematical proof, but also helped to make mathematics fun for me. Secondly, I would like to thank my co-advisor Dr. Zuofeng Shang for the support of my research, for his patience, motivation, and immense knowledge. Next, I would like to appreciate Professor Qiqing Yu for his kindness and help. Without him, I may not be able to become a Ph.D student in Department of Mathematical Sciences. I also would like to thank other committee members, Dr Xingye Qiao and Professor Solomon Polachek for their suggestions and comments. Finally, I would like to say Thank You to my Parents. I do not think I can complete my Ph.D program without their support. v

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We propose statistical inferential procedures for panel data models with interactive fixed effects in a kernel ridge regression framework. Compared with traditional sieve methods, our method is automatic in the sense that it does not require the choice of basis functions and truncation parameters. Model complexity is controlled by a continuous regularization parameter which can be automatically selected by generalized cross validation.Based on empirical processes theory and functional analysis tools, we derive joint asymptotic distributions for the estimators in the heterogeneous setting. These joint asymptotic results are then used to construct confidence intervals for the regression means and prediction intervals for the future observations, both being the first provably valid intervals in literature.Marginal asymptotic normality of the functional estimators in homogeneous setting is also obtained. Simulation and real data analysis demonstrate the advantages of our method.