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In a multiple linear regression model, the algebraic formula of thedecomposition theorem explains the relationship between the univariateregression coefficient and partial regression coefficient using geometry. Itwas found that univariate regression coefficients are decomposed into theirrespective partial regression coefficients according to the parallelogram rule.Multicollinearity is analyzed with the help of the decomposition theorem. Itwas also shown that it is a sample phenomenon that the partial regressioncoefficients of important explanatory variables are not significant, but thesign expectation deviation cause may be the population structure between theexplained variables and explanatory variables or may be the result of sampleselection. At present, some methods of diagnostic multicollinearity onlyconsider the correlation of explanatory variables, so these methods arebasically unreliable, and handling multicollinearity is blind before the causesare not distinguished. The increase in the sample size can help identify thecauses of multicollinearity, and the difference method can play an auxiliaryrole.

Explanation of multicollinearity using the decomposition theorem of ordinary linear regression models