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PS3solutions - Econ 103 UCLA Fall 2010 Problem Set 3...

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Econ 103 UCLA, Fall 2010 Problem Set 3 Solutions by Anthony Keats and Sarolta Laczó Part 1: True or False and explain briefly why. 1. The assumption that E ( u i | X i = x i ) = 0 says that the expected value of u i changes depending on the value of X i . FALSE . This assumption says that the expected value of u i does not change depending on the value of X i , that is, X i and u i are uncorrelated. This is the first assumption of OLS. If E ( u i | X i = x i ) 6 = 0 , then X i and u i are correlated and the coefficient b β associated with this regressor will be inconsistent. 2. If Cov ( X i , u i ) > 0 , then the OLS estimator ˆ β 1 will tend to be higher than β 1 . TRUE . To see this, refer to the formula for omitted variable bias: b β 1 p -→ β 1 + ρ X,u σ u σ X and to the forumula for the correlation coefficient ρ : ρ X,u = Cov ( X i , u i ) σ X σ u Substituting into the first equation we have: b β 1 p -→ β 1 + Cov ( X i , u i ) σ 2 X Since the term in the denominator is just the variance of X i , it is always positive. Thus, if Cov ( X i , u i ) > 0 the direction of the bias will be positive as well. 3. Consider an omitted variable V i that is negatively correlated with X i . Also suppose that V i positively affects Y i . Then the OLS estimator ˆ β 1 is negatively biased. TRUE . Since V i is omitted from the regression, it enters the error term u i . The formula given above (question 2) for the omitted variable bias tells us that when ρ X,u < 0 then the OLS estimator b β 1 will be negatively biased. 4. Suppose you run a regression and obtain the estimate ˆ β 1 = 3 . 4 . STATA tells you that the t -statistic for the null hypothesis that β 1 = 0 is equal to 1.7. This implies that SE ( ˆ β 1 ) is equal to 2. TRUE . The equation for the t -statistic is: t = b β 1 - β 1 , 0 SE ( b β 1 ) 1
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Econ 103 UCLA, Fall 2010 where β 1 , 0 is the value under the null. In this case the null hypothesis is β 1 , 0 = 0 so the equation becomes: t = b β 1 SE ( b β 1 ) and rearranging this gives us: SE ( b β 1 ) = b β 1 t = 3 . 4 1 . 7 = 2 5. In the regression model Y i = β 0 + β 1 Female i + β 2 Education i + u i , β 1 represents the intercept for females. FALSE . β 0 + β 1 represents the intercept for females. The coefficient β 1 on the dummy Female i represents the difference between the intercept for males and the intercept for females. 6. In the regression model Y i = β 0 + β 1 Female i + β 2 Education i + β 3 Education i · Female i + u i , β 2 + β 3 represents the return to education for females. TRUE . β 2 + β 3 represents the change in Y i associated with a marginal change in Education for females. β 1 represents the associated change in Y i given a marginal change in Education for men. As with the intercept coefficients in the previous question, β 3 represents the difference between men and women in the return to education 7. Under perfect multicollinearity, the OLS estimator cannot be computed.
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