A Study of Biased Estimator in Regression model in Presence of Multicollinearity

Abstract

The biased estimation is one of the most commonly used method to reduce the effect of the multicollinearity problem. Many studies proposed several estimators that were successfully applied to reduce this problem on estimation. In this thesis, the concept of biased estimation has been proposed to reduce the effect of multicollinearity on estimation of the parameter. We have studied three main methods of biased estimation which are: family of ridge regression estimator, family of principle component estimator and family of hybrids estimators. Specifically, out of them, we have studied 16 estimators which were given in the literature and compared the performance of each one with others using the simulation method based on estimated mean square error criteria. Also, we have studied 39 estimators of ridge parameter (k) which were also given in the literature and proposed other 5 estimators and compare all of them with the 16 estimators to find the best estimator that can be used to reduce the effect of multicollinearity under different cases. The results of this comparison have found that a modified unbiased ridge regression, almost unbiased generalized ridge estimator and modified (r-k) class ridge regression estimators are the best estimators out of the 16 estimators. Regrading to the ridge parameter (k), the results of the simulation study have shown that KM8 is the best estimator out of all ridge parameter. When two different estimators are available for a parameter, it is hoped that a combination of these two would inherit the advantages of both. With this unbiased two-parameter estimator and ordinary ridge regression, we have proposed a new biased estimator named as modified unbiased two-parameter estimator. The performance of the proposed estimator is compared with other existing estimators has been studied. Also, a simulation study is conducted to assess the performance of the proposed estimators. In the simulation study, we compare the modified unbiased two-parameter estimator with other three estimators; ordinary least squares estimator, ordinary ridge regression and the unbiased two-parameter estimator that were given in the literature. Clearly, the new proposed estimator has shown a better performance as compared with other estimators.