Identification and estimation in panel models with overspecified number of groups

Abstract

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

Blocked Clusterwise Regression

“…Outside of the linear case, Zhang et al (2019) and Chen et al (2019) study clustered linear conditional quantile regression, and Bonhomme and Manresa (2019) considers discrete latent types as an approximation to continuous unobserved heterogeneity. Liu et al (2019) studies clustering in M-estimation with an over-specified number of groups. We build on their techniques and significantly sharpen their rate results for the linear case.…”

“…Thus, the behavior of estimators with a misspecified number of clusters is important both for model selection theory as well as for our understanding the finite-sample properties of clustering estimators. Here, we make some contributions to the theory of models with an over-specified number of clusters, improving the convergence rates given in Liu et al (2019) for the linear regression setting. In contrast to the well-specified case, difficulty obtaining the "fast rate" O p ( 1 N T ) when we over-specify the number of clusters suggests that over-fitting may be severe when the number of clusters is over-specified.…”

To pool or not to pool: What is a good strategy for parameter estimation and forecasting in panel regressions?

“…Second, it produces estimates with well‐understood and desired statistical properties. In particular, when there exists a group pattern of heterogeneity and the number of groups is correctly specified or over‐specified, the estimated slope coefficients are consistent, while underspecification of the number of groups leads to inconsistent estimates but gains more efficiency (Bonhomme & Manresa, ; Liu, Schick, Shang, Zhang, & Zhou, ). Therefore, the consistency–efficiency trade‐off remains valid for the post‐screening model space.…”

“…The bias–variance properties of these IC‐based screening approaches are, however, less explicit compared to the proposed screening approach. To avoid the danger of making arguments sensitive to our choice of screening procedures, we will also consider in our Monte Carlo studies alternative methods—for example, the mixture‐like iterative (M‐estimation) method proposed by Liu et al () and agglomerative hierarchical clustering. Unreported results show that our main results are not affected.…”

“…The tuning parameter for C-Lasso is chosen by trying different values and selecting the best one in terms of risk. Comparison with more alternative methods can be found in the Supporting Information Appendix.10 We also compare with an M-estimation method for the latent group structure proposed byLiu et al (2018). This method is used to directly provide a forecast and to shrink the model space as an alternative to C-Lasso.…”