Variable selection is fundamental in high-dimensional statistical modeling, including non- and semiparametric regression. However little work has been done for variable
selection in a partially linear model. We propose and study a unified approach via double penalized least squares, retaining good features of both variable selection and
model estimation in the framework of partially linear models. The proposed method is distinguished from others in that the penalty functions combine the l_1 penalty coming
from wavelet thresholding in the nonparametric component with the l_1 penalty from the lasso in the parametric component. Simulations are used to investigate the performance of the proposed estimator in various settings, illustrating its effectiveness for simultaneous variable selection as well as estimation.