A sufficient condition for the MSE dominance of the positive-part shrinkage estimator when each individual regression coefficient is estimated in a misspecified linear regression model

ABSTRACT

In this paper, assuming that there exist omitted explanatory variables in the specified model, we derive the exact formula for the mean squared error (MSE) of a general family of shrinkage estimators for each individual regression coefficient. It is shown analytically that when our concern is to estimate each individual regression coefficient, the positive-part shrinkage estimators have smaller MSE than the original shrinkage estimators under some conditions even when the relevant regressors are omitted. Also, by numerical evaluations, we showed the effects of our theorem for several specific cases. It is shown that the positive-part shrinkage estimators have smaller MSE than the original shrinkage estimators for wide region of parameter space even when there exist omitted variables in the specified model.

KEYWORDS:

• Preliminary test estimator
• shrinkage estimator
• mean squared error
• positive-part estimator

Acknowledgements

The authors thank the editor and two anonymous referees for their extensively valuable comments. We are also grateful to Professors Aman Ullah, Tae-Hwy Lee and other participants of the seminar held at University of California-Riverside.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Japan Society for the Promotion of Science [Grant No. 23243038, 26780136], the Fundamental Research Funds for the Central Universities [Grant No. 20720171022] and the Key Projects of Philosophy and Social Sciences Research of Ministry of Education of China [Grant No. 16JZD016].