In situations like these one may need to examine the

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In situations like these, one may need to examine the partial correlation coefficients. 3. Examination of Partial Correlations This is the correlation coefficient between any two regressors holding the remaining regressors constant. This can be calculated by regressing two of the X variables, say X 2 and X 3 , on the remaining X variables, say X 4 and X 5 , and then computing the correlation coefficient between the two sets of residuals. Recall that this is the partial correlation coefficient and multicollinearity may be present if it is high. If R 2 is high but the partial correlations are low, multicollinearity is a possibility. Here one or more variables may be superfluous. But if R 2 is high and the partial correlations are also high, multicollinearity may not be readily detectable. Although a study of the partial correlations may be useful, there is no guarantee that they will provide an infallible guide to multicollinearity.
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 10 4. Variance-Inflating Factor (VIF) Recall that for the case of two explanatory va riables (X’s): 2 2 2 1 2 2 2 2 2 2 2 2 12 12 1 ˆ var( ) . . 1 1 t t t t t t VIF x x x r r (7) 2 2 2 2 2 2 2 2 2 2 2 2 12 12 1 ˆ var( ) . . 1 1 t t t VIF x x x r r (8) When there are more than two explanatory variables, multicollinearity produces similar effects. The VIF takes the form: 2 1 1 i i VIF P (12) where 2 i P is the coefficient of multiple determinations from the regression of i th independent variable on the remaining independent variables. VIF tells us how much the variance of ˆ j is being inflated by its correlation with the other variables, that is why, it is called variance- inflating factor.
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ECON 301 - Introduction to Econometrics I May 2013 METU - Department of Economics Instructor: Dr. Ozan ERUYGUR e-mail: [email protected] Lecture Notes 11 Note that 2 1 i i P VIF    (VIF does not have an upper-bound) 2 0 1 i i P VIF (VIF does have a lower-bound) 2 0 i P means that, there is no linear relationship between its associated explanatory variable and the remaining explanatory variables, or when a vector of the X matrix is orthogonal to the remaining vectors. The VIF for each term in the model measures the combined effect of the dependences among the regressors (explanatory variables) on the variance of that term. Practical experience indicates that if any of the VIFs exceeds 10, it is an indication that the associated regression coefficients are poorly estimated because of serious multicollinearity.
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