High-dimensional Panel Data Model Covariates Selection And Heteroscedasticity Test

Panel data are important data types in the fields of econometrics and statistics. Apanel data set usually contains observations of individuals at several time points. Asthe observations at different time points from an individual are probably correlatedand possess some common characteristics, treating panel data as a simple randomsample from some population is generally inappropriate. To analyze panel data, mixedmodels with random effects are frequently employed, among which linear mixedmodels are mostly focused. By introducing random effects, these models can explainthe correlation between observations from a common individual and their commoncharacteristics. Recently, the modeling of panel data has received a lot of attentionand the methodologies of statistical inference for panel data have been appliedextensively. On the other hand, the modeling of panel data is also faced with somechallenges, for example, how to select covariates when a plenty of covariatecandidates exist, how to conduct diagnosis for the modeling of panel data, how toconduct model comparison and selection for panel data, and so on. Hence, it isimportant to study the modeling of panel data.As panel data are generally clustered, their response data are frequentlyheteroscedastic. Although random effects can be introduced into the model, theycannot always explain all the heteroscedasticity. That means how to detect and dealwith the heteroscedasticity in the random errors is an important problem in themodeling of panel data. When a lot of covariate candidates exist in the model of paneldata, this issue will become more complicated. In this article, covariate selectionwithout the assumption of homoscedasticity and test for heteroscedasticity of randomerrors are studied in the model of panel data with a diverging number of covariatecandidates. In the model of interest, no assumption is made about distribution type ofthe random effects and errors. Firstly, a shrinkage estimate based on modeltransformation and SCAD-type penalty is investigated to select covariates andestimate coefficients simultaneously. The consistency and oracle property of this shrinkage estimate are proved. The conditions needed for these proofs are differentfrom the assumptions generally made for other SCAD-type variable selection methods,in that the random errors may be heteroscedastic and the samples of covariates areassumed to be non-random rather than independently and identically distributed. Forthe test of heteroscedasticity, a test statistic is proposed on the basis of fixed effectselection results and its asymptotical distribution is given under null hypothesiswithout assumption on the distribution type of random effects and errors. The abovemethodologies and properties, including consistency and oracle properties ofshrinkage estimate and empirical size and power of the test, are all illustrated by asimulated example.