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Online Econometrics Textbook - Regression Extensions - Multicollinearity

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When two or more exogenous observation series are highly correlated or have the same predictive power with respect to the endogenous variable they are called multicollinear.

Online Econometrics Textbook - Regression Extensions - Multicollinearity


which means that both vectors are perfectly linearly dependent (see eq. (I.IV-12)).

Multicollinearity is a very serious problem, for instance if the researcher is interested in calculating elasticities. If the only aim of the researcher would be to generate forecasts, and if it would be reasonable to assume that the multicollinearity problem would not be different for the forecast period or cross-section, then multicollinearity might be considered not to be a problem at all. This is because multicollinearity will not affect the forecasts of a model but only the weights of the explanatory variables in the model.

It is easy to show this mathematically. Assume a multiple regression model with, among others, the two exogenous vectors from (III.IV-1)


(see eq. (II.II.1-1) and eq. (II.II.1-3)), then it is obvious to see that X'X must be a non singular symmetric K*K matrix (see property (I.IV-14)).

Since we assumed that


For this reason X'X is not non-singular as well, and cannot be inverted (see properties of section I.IV)! Therefore the b vector of the OLS estimation cannot be computed. It should be noted that other estimation methods like GLS, MLE, etc... will also fail.

So far we 've discussed the effects of perfect multicollinearity. But what will happen when there is near perfect multicollinearity?

(like in eq. (I.IV-51)). The columns are assumed to be the eigenvectors (corresponding to different latent roots) of the square matrix A (see eq. (I.IV-48)).

Since GG' = I (see eq. (I.IV-47)) we can write the linear model as


where the columns of Z are called the principal components, which are orthogonal with respect to each other and have a length equal to their eigenvalue.

Since Z is symmetric and pos. semi def., it follows that the eigenvalues are real (eq. (I.IV-37)) and nonnegative (see properties under section I.IV).

It can easily be verified that


which follows from (I.IV-54), and agrees with eq. (I.IV-30). Now we are able to write the formula for the parameter vector


and more importantly, the parameter covariance matrix and variance


from which it can be deduced that if the eigenvalue corresponding to a specific exogenous variable becomes small, the respective delta coefficient will have a very large variance. In the extreme case where the eigenvalue were to be zero, the variance would be infinitely large (exact multicollinearity).

From (III.IV-4) it follows


and from (III.IV-5)


Now the covariance matrix of the original b parameters can be written as


which not necessarily implies that low values for lambda yield high variances (it also depends on the other parameters). Therefore, in general, all is not lost when lambda is low (but it is possible).

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