Penalized estimation principle is fundamental to high-dimensional problems. In the literature, it has been extensively and successfully applied to various models with only structural parameters. On the contrary, in this paper, we first apply this penalization principle to a linear regression model with both finite-dimensional structural parameters and high-dimensional sparse incidental parameters. For the estimated structural parameters, we derive their consistency and asymptotic distributions, which reveals an oracle property. However, the penalized estimator for the incidental parameters possesses only partial selection consistency but not consistency. This is an interesting partial consistency phenomenon: the structural parameters are consistently estimated while the incidental parameters can not. For the structural parameters, also considered is an alternative two-step penalized estimator, which improves the efficiency of the previous one-step procedure for challenging situations and is more suitable for constructing confidence regions. Further, we extend the methods and results to the case where the dimension of the structural parameters diverges with but slower than the sample size. Data-driven penalty regularization parameters are provided. The finite-sample performance of estimators for the structural parameters is evaluated by simulations and a real data set is analyzed. Supplemental materials are available online.
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