Automatic Model Structure Selection
for Partially Linear Models
Partial linear models provide good
compromises between linear and nonparametric models. However, how to decide
which covariates are linear and which are nonlinear is a long-standing problem
and not completely solved yet. Two methods are commonly used in practice: the
first method is based on hypotheses testing, which is theoretically challenging
due to multiple nonparametric tests. The second method is preliminary screening
based on univariate analysis such as scatter plots to decide the proper
regression form for each term, which is kind of very ad hoc. In this paper, we
tackle this problem in the model selection perspective. A unified estimation
and selection approach is proposed, and it can automatically determine the
linearity or nonlinearity of each covariate and estimate these components
consistently at the same time. Both theoretical and numerical properties of the
proposed estimators are presented.