Multicollinearity

Abstract Predictor variables that are highly correlated provide little independent explanatory ability. This pattern is known as multicollinearity or simply collinearity . Collinearity increases the variances of the regression coefficient, so that they may (i) have theoretically implausible magnitudes or signs; (ii) vary substantially with small changes in the sample of observations or the set of predictors; and (iii) be individually nonsignificant even though they explain significant amounts of variance overall. One indicator of collinearity is correlations of above 0.8 or 0.9 between predictor variables. Another is high R 2 s from regressing each predictor x i on all the other predictors The inverse of the unexplained variance, , is known as the variance inflation factor (VIF) . A VIF ≥10 indicates potentially harmful collinearity. Condition indices (CIs) – functions of eigenvalues of the scaled predictor variables – in combination with the variance proportions of the standard errors of the regression coefficients associated with each CI are especially useful indicators. Treatments include combining or omitting predictors, or constraining coefficients in theoretically justifiable ways. Ridge regression and structural equation modeling may reduce problems that occur with ordinary multiple regression. Model development and data collection procedures are often found to be the best remedy for collinearity problems.