Two-step series estimation and specification testing of (partially) linear models with generated regressors

Abstract

This paper studies three semiparametric models that are useful and frequently encountered in applied econometric work—a linear and two partially linear specifications with generated regressors, i.e., the regressors that are unobserved, but can be nonparametrically estimated from the data. Our framework allows for generated regressors to appear in linear or nonlinear components of partially linear models. We propose two-step series estimators for the finite-dimensional parameters, establish their $\sqrt{n}$-consistency (with sample size n) and asymptotic normality, and provide the asymptotic variance formulae that take into account the estimation error of generated regressors. Moreover, we develop a nonparametric specification test for the models considered. Numerical performances of the proposed estimators and test via simulation experiments and an empirical application illustrate the utility of our approach.

Keywords:

• Linear and nonlinear generated regressors
• partially linear model
• semiparametric linear model
• series estimation
• specification tests

JEL CLASSIFICATION:

• C01
• C13
• C14

Acknowledgments

We thank the Editor Esfandiar Maasoumi and two anonymous referees for their constructive comments on earlier versions of this article. Yu-Chin Hsu gratefully acknowledges the research support from Ministry of Science and Technology of Taiwan (MOST 110-2628-H-001-007), Academia Sinica Investigator Award (AS-IA-110-H01) and Center for Research in Econometric Theory and Applications (110L9002) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education of Taiwan. Jen-Che Liao gratefully acknowledges the research support from Ministry of Science and Technology of Taiwan (MOST 102-2410-H-001-100).

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 The generated regressor can also be viewed as a technique to reduce dimensionality in the setting of nonparametric regression models with many regressors. See the discussion in Rilstone (1996).

2 We note that there is an enormous literature on semiparametric estimation problems (e.g., Andrews (1994), Newey (1994), Ai and Chen (2003, 2007), and Ichimura and Lee (2010), among others) that involve both finite- and infinite-dimensional parameters, without explicitly considering the nonparametric component in a form of generated regressors.

3 For example, Pendakur and Sperlich (2010) consider a semiparametric partially linear model of consumer demand over prices and expenditure, where the uncompensated expenditure-share system is linear in prices and is an unspecified function of log real expenditure. In their context, the problem of a nonparametrically generated regressor arises since real expenditure is not observed, but can be estimated under the model.

4 We note that this statement holds for local-linear kernel estimators. The rate of convergence of kernel estimators can be improved by using the local-polynomial kernel estimation, provided the regression function is sufficiently continuously differentiable. We thank an anonymous referee for pointing this out to us.

5 Although in our design, x_i, $z_{1i},$ and $z_{2i}$ do not satisfy the common support that is required by our assumption, the simulation result will indicate the robustness of the proposed methods to noncompact support.

6 For the selection of the number of series terms, the BIC has been used by Hoshino (2014) in a partially linear additive quantile regression model. Another alternative is the generalized cross validation suggested by Ozabaci et al. (2014) for additive nonparametric regression with generated regressors.

Additional information

Funding

We thank the Editor Esfandiar Maasoumi and two anonymous referees for their constructive comments on earlier versions of this article. Yu-Chin Hsu gratefully acknowledges the research support from Ministry of Science and Technology of Taiwan (MOST 110-2628-H-001-007), Academia Sinica Investigator Award (AS-IA-110-H01) and Center for Research in Econometric Theory and Applications (110L9002) from the Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education of Taiwan. Jen-Che Liao gratefully acknowledges the research support from Ministry of Science and Technology of Taiwan (MOST 102-2410-H-001-100).