Multicollinearity

In statistics, multicollinearity (also collinearity) is a phenomenon in which two or more predictor variables in a multiple regression model are highly correlated, meaning that one can be linearly predicted from the others with a substantial degree of accuracy. In this situation the coefficient estimates of the multiple regression may change erratically in response to small changes in the model or the data. Multicollinearity does not reduce the predictive power or reliability of the model as a whole, at least within the sample data set; it only affects calculations regarding individual predictors. That is, a multiple regression model with correlated predictors can indicate how well the entire bundle of predictors predicts the outcome variable, but it may not give valid results about any individual predictor, or about which predictors are redundant with respect to others.

In case of perfect multicollinearity the design matrix $X$ $X$ has less than full rank, and therefore the moment matrix $X^{\mathsf {T}}X$ $X^{\mathsf {T}}X$ cannot be inverted. Under these circumstances, for a general linear model $y=X\beta +\epsilon$ $y=X\beta +\epsilon$ , the ordinary least-squares estimator ${\hat {\beta }}_{OLS}=(X^{\mathsf {T}}X)^{-1}X^{\mathsf {T}}y$ ${\hat {\beta }}_{OLS}=(X^{\mathsf {T}}X)^{-1}X^{\mathsf {T}}y$ does not exist.

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