HW9_sol

HW9_sol - ECON 414 SOLUTION TO HOMEWORK 9 Spring 2011 8.2...

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ECON 414 SOLUTION TO HOMEWORK 9 Spring 2011 8.2 With Var( u | inc,price,educ,female ) = σ 2 inc 2 , h ( x ) = inc 2 , where h ( x ) is the heteroskedas- ticity function defined in equation (8.21). Therefore, ( ) h x = inc , and so the transformed equation is obtained by dividing the original equation by inc : 0 1 2 3 4 (1/ ) ( / ) ( / ) ( / ) ( / ). beer inc price inc educ inc female inc u inc inc β = + + + + + Notice that 1 , which is the slope on inc in the original model, is now a constant in the transformed equation. This is simply a consequence of the form of the heteroskedasticity and the functional forms of the explanatory variables in the original equation. 8.6 (i) The proposed test is a hybrid of the BP and White tests. There are k + 1 regressors, each original explanatory variable and the squared fitted values. So, the number of restrictions tested is k + 1, and this is the numerator df . The denominator df is n - ( k + 2) = n k 2. (ii) For the BP test, this is easy: the hybrid test has an extra regressor, 2 ˆ y , and so the R -squared will be no less for the hybrid test than for the BP test. For the special case of the White test, the argument is a bit more subtle. In regression (8.20), the fitted values are a linear function of the regressors (where, of course, the coefficients in the linear function are the OLS estimates). So, we are putting a restriction on how the original explanatory variables appear in the regression. This means that the R -squared from (8.20) will be no greater than the R -squared from the hybrid regression. (iii) No. The F statistic for joint significance of the regressors depends on 2 2 2 2 ˆ ˆ /(1 ) u u R R - , and it is true that this ratio increases as 2 2 ˆ u R increases. But, the F statistic also depends on the
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HW9_sol - ECON 414 SOLUTION TO HOMEWORK 9 Spring 2011 8.2...

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