Abstract
Ridge Estimator (RE) has been used as an alternative estimator for Ordinary Least Squared Estimator (OLSE) to handle multicollinearity problem in the linear regression model. However, it introduces heavy bias when the number of predictors is high, and it may shrink irrelevant regression coefficients, but they are still in the model. Least Absolute Shrinkage and Selection Operator (LASSO) and Elastic net (Enet) estimator have been used to make the variable selection and shrinking the regression coefficients simultaneously. Further, the model misspecification due to excluding relevant explanatory variable in the linear regression model is considered as a severe problem in statistical research, and it will lead to bias and inconsistent parameter estimation. The performance of RE, LASSO and Enet estimators under the correctly specified regression model was well studied in the literature. This study intends to compare the performance of RE, LASSO and Enet estimators in the misspecified regression model using Root Mean Square Error (RMSE) criterion. A Monte-Carlo simulation study was used to study the performance of the estimators. In addition to that, a real-world example was employed to support the results. The analysis revealed that Enet outperformed RE and LASSO in both correctly specified model and misspecified regression model.
Highlights
Consider the linear regression model where y is the vector of observations on the predictor variable, X is the matrix of observations on non stochastic regressor variables, is a vectors of unknown parameters, ε is the normally distributed vector of disturbances, such that and Usually, by minimising the Error Sum of Squares
We examined the performance of Least Absolute Shrinkage and Selection Operator (LASSO) and Elastic net (Enet) estimator in comparison to Ridge Estimator (RE) using Root Mean Square Error (RMSE) criterion when the regression model is misspecified due to the exclusion of some important variables
Rest of the article contains the following contents: estimators under the misspecified model, a common form to represent the RE, LASSO and Enet estimators, a Monte Carlo simulation study and a numerical example to discuss the performance of the estimators, and some concluding remarks
Summary
Consider the linear regression model (1)where y is the vector of observations on the predictor variable, X is the matrix of observations on non stochastic regressor variables, is a vectors of unknown parameters, ε is the normally distributed vector of disturbances, such that and Usually, by minimising the Error Sum of Squares (2) the Ordinary Least Squared Estimator (OLSE), which is the Best Linear Unbiased Estimator (BLUE) for , is obtained as (3)It is well-known that OLSE is unstable and produces estimates having high variance when multicollinearity exists among explanatory variables, i.e., the columns ofX are highly correlated. The LASSO estimation method handles both the multicollinearity problem and best feature selection simultaneously in the high dimension linear regression model. Enet estimator in a common form in the misspecified regression model as below:
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