In partially linear single-index models, Professor Chih-Ling Tsai and co-authors Hua Liang and Xiang Liu from the University of Rochester and Runze Li from Pennsylvania State University obtain the semiparametrically efficient profile least-squares estimators of regression coefficients. The authors also employ the smoothly clipped absolute deviation penalty (SCAD) approach to simultaneously select variables and estimate regression coefficients. The study shows that the resulting SCAD estimators are consistent and possess the oracle property.

Subsequently, the authors demonstrate that a proposed tuning parameter selector, BIC, identifies the true model consistently. Finally, they develop a linear hypothesis test for the parametric coefficients and a goodness-of-fit test for the nonparametric component, respectively. Monte Carlo studies are also presented.