Variable selection for partially linear models via Bayesian subset modeling with diffusing prior

Most existing methods of variable selection in partially linear models (PLM) with ultrahigh dimensional covariates are based on partial residuals, which involve a two-step estimation procedure. While the estimation error produced in the first step may have an impact on the second step, multicollinearity among predictors adds additional challenges in the model selection procedure. In this paper, we propose a new Bayesian variable selection approach for PLM. This new proposal addresses those two issues simultaneously as (1) it is a one-step method which selects variables in PLM, even when the dimension of covariates increases at an exponential rate with the sample size, and (2) the method retains model selection consistency, and outperforms existing ones in the setting of highly correlated predictors. Distinguished from existing ones, our proposed procedure employs the difference-based method to reduce the impact from the estimation of the nonparametric component, and incorporates Bayesian subset modeling with diffusing prior (BSM-DP) to shrink the corresponding estimator in the linear component. The estimation is implemented by Gibbs sampling, and we prove that the posterior probability of the true model being selected converges to one asymptotically. Simulation studies support the theory and the efficiency of our methods as compared to other existing ones, followed by an application in a study of supermarket data.