<Dd> λ 0 + λ 1 X 1 i + λ 2 X 2 i + ⋯ + λ k X k i + v i = 0 . (\ displaystyle \ lambda _ (0) + \ lambda _ (1) X_ (1i) + \ lambda _ (2) X_ (2i) + \ cdots + \ lambda _ (k) X_ (ki) + v_ (i) = 0 .) </Dd> <P> In this case, there is no exact linear relationship among the variables, but the X j (\ displaystyle X_ (j)) variables are nearly perfectly multicollinear if the variance of v i (\ displaystyle v_ (i)) is small for some set of values for the λ (\ displaystyle \ lambda)' s . In this case, the matrix X X has an inverse, but is ill - conditioned so that a given computer algorithm may or may not be able to compute an approximate inverse, and if it does so the resulting computed inverse may be highly sensitive to slight variations in the data (due to magnified effects of either rounding error or slight variations in the sampled data points) and so may be very inaccurate or very sample - dependent . </P> <P> Indicators that multicollinearity may be present in a model include the following: </P> <Ol> <Li> Large changes in the estimated regression coefficients when a predictor variable is added or deleted </Li> <Li> Insignificant regression coefficients for the affected variables in the multiple regression, but a rejection of the joint hypothesis that those coefficients are all zero (using an F - test) </Li> <Li> If a multivariable regression finds an insignificant coefficient of a particular explanator, yet a simple linear regression of the explained variable on this explanatory variable shows its coefficient to be significantly different from zero, this situation indicates multicollinearity in the multivariable regression . </Li> <Li> Some authors have suggested a formal detection - tolerance or the variance inflation factor (VIF) for multicollinearity: t o l e r a n c e = 1 − R j 2, V I F = 1 t o l e r a n c e, (\ displaystyle \ mathrm (tolerance) = 1 - R_ (j) ^ (2), \ quad \ mathrm (VIF) = (\ frac (1) (\ mathrm (tolerance))),) where R j 2 (\ displaystyle R_ (j) ^ (2)) is the coefficient of determination of a regression of explanator j on all the other explanators . A tolerance of less than 0.20 or 0.10 and / or a VIF of 5 or 10 and above indicates a multicollinearity problem . </Li> <Li> Condition number test: The standard measure of ill - conditioning in a matrix is the condition index . It will indicate that the inversion of the matrix is numerically unstable with finite - precision numbers (standard computer floats and doubles). This indicates the potential sensitivity of the computed inverse to small changes in the original matrix . The condition number is computed by finding the square root of the maximum eigenvalue divided by the minimum eigenvalue of the design matrix . If the condition number is above 30, the regression may have significant multicollinearity; multicollinearity exists if, in addition, two or more of the variables related to the high condition number have high proportions of variance explained . One advantage of this method is that it also shows which variables are causing the problem . </Li> <Li> Farrar--Glauber test: If the variables are found to be orthogonal, there is no multicollinearity; if the variables are not orthogonal, then at least some degree of multicollinearity is present . C. Robert Wichers has argued that Farrar--Glauber partial correlation test is ineffective in that a given partial correlation may be compatible with different multicollinearity patterns . The Farrar--Glauber test has also been criticized by other researchers . </Li> <Li> Perturbing the data . Multicollinearity can be detected by adding random noise to the data and re-running the regression many times and seeing how much the coefficients change . </Li> <Li> Construction of a correlation matrix among the explanatory variables will yield indications as to the likelihood that any given couplet of right - hand - side variables are creating multicollinearity problems . Correlation values (off - diagonal elements) of at least 0.4 are sometimes interpreted as indicating a multicollinearity problem . This procedure is, however, highly problematic and cannot be recommended . Intuitively, correlation describes a bivariate relationship, whereas collinearity is a multivariate phenomenon . </Li> </Ol>

Multicollinearity affects the interpretation of the regression coefficients