Abstract
The problem of multicollinearity in multiple linear regression models arises when the predictor variables are correlated among each other. The variance of the ordinary least squared estimator become unstable in such situation. In order to mitigate the problem of multicollinearity, Liu regression is widely used as a biased method of estimation with shrinkage parameter 'd'. The optimal value of shrinkage parameter plays a vital role in bias-variance trade-off.
Several estimators are available in literature for the estimation of shrinkage parameter. But the existing estimators do not perform well in terms of smaller mean squared error when the problem of multicollinearity is high or severe.
In this paper, some new estimators for the shrinkage parameter are proposed. The proposed estimators are the class of estimators that are based on quantile of the regression coefficients. The performance of the new estimators is compared with the existing estimators through Monte Carlo simulation. Mean squared error and mean absolute error is considered as evaluation criteria of the estimators. Tobacco dataset is used as an application to illustrate the benefits of the new estimators and support the simulation results.
The new estimators outperform the existing estimators in most of the considered scenarios including high and severe cases of multicollinearity. 95% mean prediction interval of all the estimators is also computed for the Tobacco data. The new estimators give the best mean prediction interval among all other estimators.
We recommend the use of new estimators to practitioners when the problem of high to severe multicollinearity exists among the predictor variables.