Estimation for Varying Coefficient Models with Hierarchical Structure

The varying coefficient (VC) model is a generalization of ordinary linear model, which can not only retain strong interpretability but also has the flexibility of the nonparametric model. In this paper, we investigate a VC model with hierarchical structure. A unified variable selection method for VC model is proposed, which can simultaneously select the nonzero effects and estimate the unknown coefficient functions. Meanwhile, the selected model enforces the hierarchical structure, that is, interaction terms can be selected into the model only if the corresponding main effects are in the model. The kernel method is employed to estimate the varying coefficient functions, and a combined overlapped group Lasso regularization is introduced to implement variable selection to keep the hierarchical structure. It is proved that the proposed penalty estimators have oracle properties, that is, the coefficients are estimated as well as if the true model were known in advance. Simulation studies and a real data analysis are carried out to examine the performance of the proposed method in finite sample case.

Beyond Lasso: A Survey of Nonconvex Regularization in Gaussian Graphical Models

Studying complex relations in multivariate datasets is a common task in psychological science. Recently, the Gaussian graphical model has emerged as an increasingly popular model for characterizing the conditional dependence structure of random variables. Although the graphical lasso ($\ell_1$-regularization) is the most well-known estimator across the sciences, it has several drawbacks that make it less than ideal for model selection. There are now alternative forms of regularization that were developed specifically to overcome issues inherent to the $\ell_1$-penalty.To date, this information has not been synthesized. This paper provides a comprehensive survey of nonconvex regularization that spans from the smoothly clipped absolute deviation penalty to continuous approximations of the $\ell_0$-penalty (i.e., best subset) for directly estimating the inverse covariance matrix. A common thread shared by these penalties is that they all enjoy the oracle properties, that is, they perform as though the \emph{true} generating model were known in advance. To ensure their theoretical properties are general, I conducted extensive numerical experiments that indicated superior and more than competitive performance when compared to glasso and non-regularized model selection, respectively, all the while being computationally feasible for many variables. In addition, the important topics of tuning parameter selection and statistical inference in regularized models are reviewed.The penalties are employed to estimate the dependence structure of post-traumatic stress disorder symptoms. The discussion includes several ideas for future research, including a plethora of information to facilitate their study. I have implemented the methods in the

IDENTIFYING LATENT GROUPED PATTERNS IN COINTEGRATED PANELS

We consider a panel cointegration model with latent group structures that allows for heterogeneous long-run relationships across groups. We extend Su, Shi, and Phillips (2016, Econometrica 84(6), 2215–2264) classifier-Lasso (C-Lasso) method to the nonstationary panels and allow for the presence of endogeneity in both the stationary and nonstationary regressors in the model. In addition, we allow the dimension of the stationary regressors to diverge with the sample size. We show that we can identify the individuals’ group membership and estimate the group-specific long-run cointegrated relationships simultaneously. We demonstrate the desirable property of uniform classification consistency and the oracle properties of both the C-Lasso estimators and their post-Lasso versions. The special case of dynamic penalized least squares is also studied. Simulations show superb finite sample performance in both classification and estimation. In an empirical application, we study the potential heterogeneous behavior in testing the validity of long-run purchasing power parity (PPP) hypothesis in the post–Bretton Woods period from 1975–2014 covering 99 countries. We identify two groups in the period 1975–1998 and three groups in the period 1999–2014. The results confirm that at least some countries favor the long-run PPP hypothesis in the post–Bretton Woods period.

Efficient robust doubly adaptive regularized regression with applications

We consider the problem of estimation and variable selection for general linear regression models. Regularized regression procedures have been widely used for variable selection, but most existing methods perform poorly in the presence of outliers. We construct a new penalized procedure that simultaneously attains full efficiency and maximum robustness. Furthermore, the proposed procedure satisfies the oracle properties. The new procedure is designed to achieve sparse and robust solutions by imposing adaptive weights on both the decision loss and the penalty function. The proposed method of estimation and variable selection attains full efficiency when the model is correct and, at the same time, achieves maximum robustness when outliers are present. We examine the robustness properties using the finite-sample breakdown point and an influence function. We show that the proposed estimator attains the maximum breakdown point. Furthermore, there is no loss in efficiency when there are no outliers or the error distribution is normal. For practical implementation of the proposed method, we present a computational algorithm. We examine the finite-sample and robustness properties using Monte Carlo studies. Two datasets are also analyzed.

Variable selection in rank regression for analyzing longitudinal data

In this paper, we consider variable selection in rank regression models for longitudinal data. To obtain both robustness and effective selection of important covariates, we propose incorporating shrinkage by adaptive lasso or SCAD in the Wilcoxon dispersion function and establishing the oracle properties of the new method. The new method can be conveniently implemented with the statistical software R. The performance of the proposed method is demonstrated via simulation studies. Finally, two datasets are analyzed for illustration. Some interesting findings are reported and discussed.