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chapter12 ANSWERS - CHAPTER 12 MULTICOLLINEARITY WHAT...

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CHAPTER 12 MULTICOLLINEARITY: WHAT HAPPENS IF EXPLANATORY VARIABLES ARE CORRELATED? QUESTIONS 12.1. An exact linear relationship between two or more (explanatory) variables; more than one exact linear relationship between two or more explanatory variables. 12.2. In perfect collinearity there is an exact linear relationship between two or more variables, whereas in imperfect collinearity this relationship is not exact but an approximate one. 12.3. Since 1 foot = 12 inches, there is an exact linear relationship between the variables “height in inches” and “height in feet”, if both variables are included in the same regression. In this case, we have only one independent explanatory variable and not two. 12.4. Disagree. The variables 2 X and 3 X are nonlinear functions of X . Hence, their inclusion in the regression model does not violate the assumption of the classical linear regression model (CLRM) of “no exact linear relationship among explanatory variables.” 12.5. Consider, for instance, Eq. (8.21). Let i i x x 2 3 2 = .Substituting this into Equation (8.21), we obtain: 0 0 ) 2 ( ) 4 )( ( ) 2 )( 2 ( ) 4 )( ( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 = - - = x x x x yx x yx b which is an indeterminate expression. The same is true of Equations (8.22), (8.25), and (8.27). 12.6. OLS estimators are still BLUE. 12.7. (1) Large variances and covariances of OLS estimators (2) Wider confidence intervals (3) Insignificant t ratios 101

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(4) A high 2 R but few significant t ratios (5) Sensitivity of OLS estimators and their standard errors to changes in the data. (6) Wrong signs for regression coefficients. (7) Difficulty in assessing the individual contributions of the explanatory variables to the ESS or 2 R . 12.8. The VIF measures the increase in the variances OLS estimators as the degree of collinearity, as measured by 2 R , increases. If the (explanatory) variables are uncorrelated, the least value of VIF is 1, but if the variables are perfectly correlated, VIF is infinite. 12.9. ( a ) large; small ( b ) undefined; undefined ( c ) variances 12.10. ( a ) False . In cases of perfect multicollinearity, OLS estimators are not even defined. ( b ) True . ( c ) Uncertain . A high 2 R can be offset by a low 2 σ or a high variance of the relevant explanatory variable included in the model, or both. ( d ) True . A simple correlation between two explanatory variables may be high, but when account is taken of other explanatory variables in the model, the partial correlation between those two variables may be low. ( e ) Uncertain . It is true only if the collinearity observed in the given sample continues to hold in the post sample period. If that is not the case, then this statement is false. 12.11. This is because business cycles and or trends (reflecting growth rates) dominate most economic time series. Therefore, in a regression of the consumer price index (CPI) on the money supply and the unemployment rate, the latter two variables are likely to exhibit collinearity.
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