Standardized coefficients of multiple regression, also known as beta coefficients, by absolute value are usually smaller than one, but sometimes they can exceed one. This effect had been studied mostly for models with two predictors, where it was explained by the high collinearity between them. The current paper considers multiple linear regression and defines the necessary and sufficient conditions for beta coefficients to exceed one. These conditions determine a measure of each predictor's connection to the dependent variable in the model relative to the connection with other predictors. This criterion presents a new measure for diagnostics of the multicollinearity, which can be employed additionally to the commonly used variance inflation factor. A new interpretation is given to the meaning of the squared beta coefficients themselves. Numerical examples demonstrate these novel features. The obtained results are useful in applied regression analysis, helping practitioners and educators to understand and to explain the outcomes of regression modeling.

Large multiple datasets were simulated through sampling, and regression modeling results were compared with known parameters—an analysis undertaken here for the first time on such a scale. The study demonstrates that the impact of multicollinearity on the quality of parameter estimates is far stronger than commonly assumed, even at low or moderate correlations between predictors. The standard practice of assessing the significance of regression coefficients using t-statistics is compared with the actual precision of estimates relative to their true values, and the results are critically examined. It is shown that t-statistics for regression parameters can often be misleading. Two novel approaches for selecting the most effective variables are proposed: one based on the so-called reference matrix and the other on efficiency indicators. A combined use of these methods, together with the analysis of each variable’s contribution to determination, is recommended. The practical value of these approaches is confirmed through extensive testing on both simulated homogeneous and heterogeneous datasets, as well as on a real-world example. The results contribute to a more accurate understanding of regression properties, model quality characteristics, and effective strategies for identifying the most reliable predictors. They provide practitioners with better analytical tools.

This paper considers some innovative theoretical features and practical applications of the normal system of equations used for estimating parameters in multiple linear regression. The Laplace expansion of a determinant by cofactors and double Laplace expansion are employed for resolving the normal system. Additional features are described, including the ridge regularization applied directly to the normal system, geometric interpretation as a unique hyperplane through the points of special weighted means, Mahalanobis distances from observations to these means for the linear link functions, and multidimensional interpolation. The found properties are useful for a better understanding and interpretation of multiple regression, and the numerical examples demonstrate convenience and applicability of these tools in data modeling.