Two types of hypothesis testing problems in regression analysis

In this paper, we provide a brief overview of two types of hypothesis testing problems in regression analysis, i.e., model specification testing and variable significance testing. We introduce classic and recently developed testing methods for these two types of problems. We present the construction principles of relevant statistics and evaluate the possible problems which may exist in different methods. Classic methods mainly focus on low-dimensional data and are often greatly influenced by the dimensionality of predictors. The recently developed methods are mainly applicable to high-dimensional data, aiming to effectively overcome the curse of dimensionality in these two types of testing problems. The use of machine learning algorithms is an important feature of recently developed testing methods. We conclude with a brief discussion of some potential issues.