CHAPTER 4. REGRESSION DIAGNOSTIC I: MULTICOLLINEARITY. MULTICOLLINEARITY. One of the assumptions of the classical linear regression (CLRM) is that there is no exact linear relationship among the regressors.
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CHAPTER 4 REGRESSION DIAGNOSTIC I: MULTICOLLINEARITY Damodar Gujarati Econometrics by Example
MULTICOLLINEARITY • One of the assumptions of the classical linear regression (CLRM) is that there is no exact linear relationship among the regressors. • If there are one or more such relationships among the regressors, we call it multicollinearity, or collinearity for short. • Perfect collinearity: A perfect linear relationship between the two variables exists. • Imperfect collinearity: The regressors are highly (but not perfectly) collinear. Damodar Gujarati Econometrics by Example
CONSEQUENCES • If collinearity is not perfect, but high, several consequences ensue: • The OLS estimators are still BLUE, but one or more regression coefficients have large standard errors relative to the values of the coefficients, thereby making the t ratios small. • Even though some regression coefficients are statistically insignificant, the R2 value may be very high. • Therefore, one may conclude (misleadingly) that the true values of these coefficients are not different from zero. • Also, the regression coefficients may be very sensitive to small changes in the data, especially if the sample is relatively small. Damodar Gujarati Econometrics by Example
VARIANCE INFLATION FACTOR • For the following regression model: • It can be shown that: • and • where σ2 is the variance of the error term ui, and r23 is the coefficient of correlation between X2 and X3. Damodar Gujarati Econometrics by Example
VARIANCE INFLATION FACTOR (CONT.) • is the variance-inflating factor. • VIF is a measure of the degree to which the variance of the OLS estimator is inflated because of collinearity. Damodar Gujarati Econometrics by Example
DETECTION OF MULTICOLLINEARITY • 1. High R2 but few significant t ratios • 2. High pair-wise correlations among explanatory variables or regressors • 3. High partial correlation coefficients • 4. Significant F test for auxiliary regressions (regressions of each regressor on the remaining regressors) • 5. High Variance Inflation Factor (VIF) and low Tolerance Factor (TOL, the inverse of VIF) Damodar Gujarati Econometrics by Example
REMEDIAL MEASURES • What should we do if we detect multicollinearity? • Nothing, for we often have no control over the data. • Redefine the model by excluding variables may attenuate the problem, provided we do not omit relevant variables. • Principal components analysis: Construct artificial variables from the regressors such that they are orthogonal to one another. • These principal components become the regressors in the model. • Yet the interpretation of the coefficients on the principal components is not as straightforward. Damodar Gujarati Econometrics by Example