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Perfect multicollinearity refers to a situation where the predictive variables have an exact linear relationship. When there is perfect collinearity, the design matrix X {\displaystyle X} has less than full rank , and therefore the moment matrix X T X {\displaystyle X^{\mathsf {T}}X} cannot be inverted .
Analyze the magnitude of multicollinearity by considering the size of the (^). A rule of thumb is that if (^) > then multicollinearity is high [5] (a cutoff of 5 is also commonly used [6]). However, there is no value of VIF greater than 1 in which the variance of the slopes of predictors isn't inflated.
Perfect multicollinearity refers to a situation in which k (k ≥ 2) explanatory variables in a multiple regression model are perfectly linearly related, according to = + + + + (), for all observations i. In practice, we rarely face perfect multicollinearity in a data set.
Lack of perfect multicollinearity in the predictors. For standard least squares estimation methods, the design matrix X must have full column rank p; otherwise perfect multicollinearity exists in the predictor variables, meaning a linear relationship exists between two or more predictor variables. This can be caused by accidentally duplicating ...
Multicollinearity (as long as it is not "perfect") can be present resulting in a less efficient, but still unbiased estimate. The estimates will be less precise and highly sensitive to particular sets of data. [12] Multicollinearity can be detected from condition number or the variance inflation factor, among other tests.
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This is the problem of multicollinearity in moderated regression. Multicollinearity tends to cause coefficients to be estimated with higher standard errors and hence greater uncertainty. Mean-centering (subtracting raw scores from the mean) may reduce multicollinearity, resulting in more interpretable regression coefficients.