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Concept
Multicollinearity
Summary
In statistics, multicollinearity (also collinearity) is a phenomenon in which one predictor variable in a multiple regression model 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 multivariable regression model with collinear 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.
Note that in statements of the assumptions underlying regression analyses such as ordinary least squares, the phrase "no multicollinearity" usually refers to the absence of multicollinearity, which is an exact (non-stochastic) linear relation among the predictors. In such a case, the design matrix has less than full rank, and therefore the moment matrix cannot be inverted. Under these circumstances, for a general linear model , the ordinary least squares estimator does not exist.
In any case, multicollinearity is a characteristic of the design matrix, not the underlying statistical model.
Multicollinearity leads to non-identifiable parameters.
Collinearity is a linear association between explanatory variables. Two variables are perfectly collinear if there is an exact linear relationship between them. For example, and are perfectly collinear if there exist parameters and such that, for all observations ,
Multicollinearity refers to a situation in which explanatory variables in a multiple regression model are highly linearly related. There is perfect multicollinearity if, for example as in the equation above, the correlation between two independent variables equals 1 or −1.
Official source
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