Variable Selection With the Strong Heredity Constraint and Its Oracle Property

In this paper, we extend the LASSO method (Tibshirani 1996) for simultaneously fitting a regression model and identifying important interaction terms. Unlike most of the existing variable selection methods, our method automatically enforces the heredity constraint, that is, an interaction term can be included in the model only if the corresponding main terms are also included in the model. Furthermore, we extend our method to generalized linear models, and show that it performs as well as if the true model were given in advance, that is, the oracle property as in Fan and Li (2001) and Fan and Peng (2004). The proof of the oracle property is given in online supplemental materials. Numerical results on both simulation data and real data indicate that our method tends to remove irrelevant variables more effectively and provide better prediction performance than previous work (Yuan, Joseph, and Lin 2007 and Zhao, Rocha, and Yu 2009 as well as the classical LASSO method).