CH6n7EPPS

CH6n7EPPS - Econ 120B Solutions to Chapters 6 and 7...

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Econ 120B Solutions to Chapters 6 and 7 Practice Problems 1. There is perfect multicollinearity present since one of the three explanatory variables can always be expressed linearly in terms of the other two ( ). Hence, there are not really three pieces of independent information contained in the three explanatory variables. Dropping one of the three will solve the problem. X 2 = X 1 X 3 2. Note: There are more than one answer to part a., and c. a. We add and subtract β 2 + 3 () X 1 i Y i = 0 + 1 X 1 i + 2 + 3 X 1 i 2 + 3 ( ) X 1 i + 2 X 2 i + 3 X 3 i + u i Y i = 0 + 1 + 2 + 3 X 1 i + 2 X 2 i X 1 i + 3 X 3 i X 1 i + u i Y i = 0 + γ 1 X 1 i + 2 W i + 3 V i + u i where , W 1 = 1 + 2 + 3 i = X 2 i X 1 i ( ) , and V i = X 3 i X 1 i ( ) We regress the last equation and obtain the OLS estimators: ) β 0 , ) γ 1 , ) 2 , ) 3 We test the following hypothesis, H 0 : 1 = 1 H 1 : 1 1 where t = ) 1 1 SE ( ) 1 ) b. This is not a linear restriction. Hence, you cannot use the t -test to test for its validity. c. We add and subtract 3 3 X 2 i Y i = 0 + 1 X 1 i + 2 X 2 i 3 3 X 2 i + 3 3 X 2 i + 3 X 3 i + u i Y i = 0 + 1 X 1 i + 2 3 3 X 2 i + 3 3 X 2 i + X 3 i + u i Y i = 0 + 1 X 1 i + 2 X 2 i + 3 W i + u i where , and 2 = 2 3 3 W i = 3 X 2 i + X 3 i ( ) We regress the last equation and obtain the OLS estimators: ) 0 , ) 1 , ) 2 , ) 3 We test the following hypothesis, H 0 : 2 = 2 H 1 : 2 2 where t = ) 2 2 SE ( ) 2 ) 3. a. For every additional inch in height, weight increases on average by 5.58 pounds, holding gender constant. Female students weigh on average 6.36 pounds less than male students, controlling for height. The regression explains 50 percent of the weight variation among students. It does not make sense to interpret the intercept, since there are no observations close to the origin, or, put differently, there are no male individuals who are zero inches tall. b. There are now observations close to the origin and you can therefore interpret the intercept. A male student who is 5ft. tall will weight roughly 105.6 pounds, on average. The two slope coefficients and their t-statistics will be unaffected, as will be the regression R 2 . Since the explanatory power of the regression is unaffected by rescaling, and the dependent variable and the total sums of squares have remained unchanged, the sums of squared residuals, and hence the SER , must also remain the same.
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c. We carry out the hypothesis test: H 0 : β 1 = 0 H 1 : 1 < 0 The t -statistic is t = ) β 1 0 SE ( ) 1 ) = 6.36 5.74 =− 1.11 . For a 10% significance level one-sided test, the critical value from the standard normal table is -1.28. Hence, you cannot reject the hypothesis that females have the same average weight as males, for a given height.
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CH6n7EPPS - Econ 120B Solutions to Chapters 6 and 7...

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