  In a multiple linear regression model, the algebraic formula of the
decomposition theorem explains the relationship between the univariate
regression coefficient and partial regression coefficient using geometry. It
was found that univariate regression coefficients are decomposed into their
respective partial regression coefficients according to the parallelogram rule.
Multicollinearity is analyzed with the help of the decomposition theorem. It
was also shown that it is a sample phenomenon that the partial regression
coefficients of important explanatory variables are not significant, but the
sign expectation deviation cause may be the population structure between the
explained variables and explanatory variables or may be the result of sample
selection. At present, some methods of diagnostic multicollinearity only
consider the correlation of explanatory variables, so these methods are
basically unreliable, and handling multicollinearity is blind before the causes
are not distinguished. The increase in the sample size can help identify the
causes of multicollinearity, and the difference method can play an auxiliary
role.
