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# Soln HW5_ - Answers to selected problems in 12 and 13 CH 10...

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Answers to selected problems in chapters 10, 11, 12 and 13 CH 10: MULTICOLLINEARITY: WHAT HAPPENS IF THE REGRESSORS ARE CORRELATED? 10.1 If X k is a perfect linear combination of the remaining explanatory variables, then there are ( k -1) equations with k unknowns. With more unknowns than equations, unique solutions are not possible. 10.2 ( a ) No. Variable X 3i is an exact linear combination of X 2i , because X 3 i = 2 X 2 i - 1. ( b ) Rewriting the equation yields, Therefore, we can estimate α 1 and α 2 uniquely, but not the original betas because we have two equations to solve the three unknowns. 10.3 ( a ) Although the numerical values of the intercept and the slope coefficients of PGNP and FLR have changed, their signs have not. Also, these variables are still statistically significant. These changes are due to the addition of the TFR variable, suggesting that there may be some collinearity among the regressors. ( b ) Since the t value of the TFR coefficient is very significant ( the p value is only .0032), it seems TFR belongs in the model. The positive sign of this coefficient also makes sense in that the larger the number of children born to a woman, the greater the chances of increased child mortality. ( c ) This is one of those “happy” occurrences where despite possible collinearity, the individual coefficients are still statistically significant. 10.5 ( a ) Yes. Economic time series data tend to move in the same direction. Here, the lagged variables of income will generally move in the same direction. 1

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( b ) As discussed briefly in Chapter 10 and further discussed in Chapter 17, the first difference transformation may alleviate the problem. 10.10 ( a ) No. Multicollinearity refers to linear association among variables. Here the association is nonlinear. ( b ) There is no reason to drop them. They are theoretically as well as statistically significant in the present example. ( c ) If one of the variables is dropped, there will be specification bias that will show up in the coefficient(s) of the remaining variable(s). 10.12 ( a ) False. If exact linear relationship(s) exist among variables, we cannot even estimate the coefficients or their standard errors. ( b ) False. One may be able to obtain one or more significant t values. ( c ) False. As noted in the chapter (see Eq. 7.5.6), the variance an OLS estimator is given by the following formula: As can be seen from this formula, a high can be counterbalanced by a low or high . ( d ) Uncertain . If a model has only two regressors, high pairwise correlation coefficients may suggest multicollinearity. If one or more regressors enter non-linearly, the pairwise correlations may give misleading answers. ( e
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Soln HW5_ - Answers to selected problems in 12 and 13 CH 10...

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