Adaptive k-class estimation in high-dimensional linear models

Abstract

In this paper, we explore the k-class estimator in high-dimensional linear models with potential endogeneity issues which are very common in empirical economics studies. K-class estimator has the advantage of incorporating many popular estimators such as the OLS estimator, two stage least squares (2SLS) estimator, limited information maximum likelihood (LIML) estimator, etc., as k takes on different values. Our main innovations are: 1) In the newly proposed high-dimensional k-class estimator, we allow the value of k (hence the level of endogeneity) to be determined by the data. Therefore, our method is very useful in empirical studies where the researcher do not know the severity of endogeneity or the importance of variables in a very large pool of candidate covariates. 2) In this paper, the adaptive LASSO method is incorporated into the generalized k-class estimation for variable selection and coefficient estimation in both the structural and reduced form equations. We show that the adaptive LASSO type k-class estimator has oracle properties. In simulation studies, we show that our new estimator can choose the optimal k value as well as achieving the minimum MSE among a set of popular estimators in finite samples where the number of potential endogenous variable is large.

Keywords:

• k-class estimator
• Endogeneity
• High dimensional models
• Variable selection

Acknowledgments

The authors thank T. Cai, M. Caner, W. Lin, W. Newey, W. Zhong, L. Zhu and seminar/conference participants from Xiamen University, Renmin University etc., for their helpful comments. Thanks to Narayanaswamy Balakrishnan (Editor) and the anonymous reviewers for their insightful comments, which have helped to greatly improve the earlier version of this article.

Notes

1 In the simulations we report the variable selection correct ratio for the structural equation parameters. The variable selection performance in the reduced form equations are available upon request to the authors.

Additional information

Funding

Fan’s research, in part, was supported by the National Natural Science Foundation of China Grants 71671149 and 71631004 (Key Project) and the Fundamental Research Funds for the Central Universities (No. 20720171042).