Efficient Combined Estimator for Parameter Estimation of Linear Regression Model with Multicollinearity

The classical linear regression model relies on several key assumptions, including homoscedasticity, normality of errors, independence of observations and the absence of multicollinearity among explanatory variables (Gujarati, 2021), These assumptions are rarely fulfilled in real life situations. Multicollinearity occurs when the assumption of independent explanatory variables is violated (Alreshidi et al., 2025), There are many sources of multicollinearity, some of which are the data collection methods, the constraints placed on the model or having an overdetermined model (Paul, 2006), When multicollinearity exists in a model using conventional parameter estimation models like the Ordinary Least Squares (OLS) often leads to unstable and unreliable parameter estimates. (Alharthi & Akhtar, 2025), To address these challenges several biased estimators have been developed. Some of these are the ridge (Hoerl and Kennard, 1970), liu (Liu, 1993), and principal components (pc) (Hotelling, 1933), estimators. Each of these existing estimators have their strengths and limitations. However, no single estimator consistently outperforms the others under all conditions. Over the years other researchers have developed combined estimators with the expectation that the combination of different estimators might inherit the individual advantages of the estimators. Following their work, in this paper effort is made to provide a combined estimator based on ridge (Hoerl and Kennard, 1970), liu (Liu, 1993), and principal components (pc) (Hotelling, 1933), estimators. This estimator- the principal component ridge liu, leverages on the strengths of these three existing estimators. The performance of the developed estimator is compared with the existing individual estimators using Mean Square Error (MSE). Results show that the developed estimator performs better than the existing ones providing more stable and accurate parameter estimates in the presence of multicollinearity. Monte Carlo experiments a robust method for assessing statistical properties under controlled conditions with varying degrees of multicollinearity, error variance, and sample size (Oduntan, 2024), were performed one thousand (1000) times on two (2) linear regression models with four (4) and seven (7) explanatory variables exhibiting five (5) degrees of multicollinearity (0.75, 0.85, 0.95, 0.99, 0.999), three(3) levels of error variance (1, 25, 100), at eight (8) sample sizes (n=10, 15, 20,30,40,50,100 and 250). The MSE criterion was used to examine the estimators and the number of times each estimator had the minimum MSE was counted at each combination of classifications. The ranking of the estimators was also done based on their MSE. Tables and figures were used to present the results of the findings. The results of the investigation revealed that when multicollinearity problems exist in linear regression models, the proposed RHKMALUMIWPC estimator is best.