A new stable solution to the linear regression problem under multicollinearity

Abstract

The main shortcomings of the OLS (Ordinary Least Squares) solution to the multiple linear regression problem under multicollinearity which prevent from obtaining an adequate solution to the economic problem of evaluation of each regressor's contribution to the regressand have been considered. The main causes of the OLS incorrectness of the economic problem solution have been revealed, these causes being related to a great variability in the OLS solution under considerable data multicollinearity. The research has also shown that mathematically correct standard OLS solutions can become economically incorrect with data collinearity increasing which leads to diminishing of the OLS matrix codomain of physical correctness. The current methods for overcoming the OLS solutions' great variability have been considered in both the economic and mathematical aspects. The current methods have been proved inefficient in overcoming multicollinearity by either mathematical or economic methods such as choosing the best regressions, lasso, etc. The analysis has brought to a conclusion that the only way out is to create a new method of solving the OLS equation which would give a stable solution with small variability, as for example in the ridge method, but with a small bias. Precisely such method is the Modified OLS (MOLS) proposed in the paper. The MOLS is an approximate method which uses the known Tikhonov's regularization principle and a new solution to the regularized OLS equation, based on the modified Cramer's rule which is proposed in the paper. The MOLS method has proved to give a stable and practically unbiassed solution to the linear regression problem regardless of the near-collinearity level of the data used. Unlike the ridge method, the MOLS method gives a negligible bias and does not require optimization of the regularization constant. The proposed MOLS method has been verified for adequacy with the aid of the artificial data population (ADP), which is based on the Monte Carlo simulation method. Using the ADP, the new MOLS method has been checked for biassedness and stability for both small and large samples.