ProblemSet2 Answers - Professor Mumford Econ 360 Fall 2010...

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Unformatted text preview: Professor Mumford Econ 360 - Fall 2010 [email protected] Problem Set 2 Answers True/False (20 points) Please write the entire word. No explanations are required. 1. TRUE If E( u | x ) = E( u ) then we say that u is mean independent of x . 2. FALSE The goodness-of-fit measure for OLS regressions, R-Squared, does not depend on the unit of measurement of the variables. 3. FALSE Under assumption SLR.1 - SLR.4, the OLS estimates are unbiased . This does not mean that they are equal to the true population parameters. 4. TRUE The average value of the residuals is zero. 5. FALSE The zero conditional mean assumption implies that E( u i | x i ) = 0 for all i . Another way to say this is that our best guess is that u i = 0 regardless of the value of x i . However, this does not mean that u i actually is zero for any particular i . 1 Multiple Choice Questions (20 points) 6. In the following estimated regression equation ln( wage ) = 0 . 584 + 0 . 083 educ where educ is measured in years of education, how does one more year of education affect the predicted wage? (a) wage increases by \$0.083 (b) wage increases by \$8.30 (c) wage increases by 0.083 percent (d) wage increases by 8.3 percent 7. Which is NOT a correct representation of R-squared? (a) ∑ u 2 i (b) SSE SST (c) 1- SSR SST (d) square of the sample correlation between y i and ˆ y i 8. When an explanatory variable is correlated with the error term then it is said to be (a) independent (b) endogenous (c) statistically significant (d) observational 9. Which of the following assumptions is NOT one of the four assumptions needed to show that the OLS estimates are unbiased? (a) independence (b) zero conditional mean (c) linear in parameters (d) sample variation in the explanatory variable 10. Which of the following population models violates Assumption SLR.1 (Linear in Pa- rameters)? (a) y = β + β 1 ln( x ) + u (b) y 2 = β + β 1 x 2 + u (c) y = β + x β 1 + u (d) ln( y ) = β + β 1 x + u 2 Long Answer Questions (60 points) 11. Deriving the OLS Estimators The problem is finding ˆ β 1 and ˆ β that minimize the sum of squared residuals: min ˆ β ˆ β 1 n X i =1 y i- ˆ β- ˆ β 1 x i 2 The first order conditions are:- 2 n X i =1 y i- ˆ β- ˆ β 1 x i = 0 (1)- 2 n X i =1 x i y i- ˆ β- ˆ β 1 x i = 0 (2) We can rewrite equation (1) as ¯ y- ˆ β- ˆ β 1 ¯ x = 0 and therefore ˆ β = ¯ y- ˆ β 1 ¯ x We can then plug this value for ˆ β into equation (2) and rewrite it as n X i =1 x i y i- ¯ y + ˆ β 1 ¯ x- ˆ β 1 x i = 0 Be rearranging the equation we obtain n X i =1 x i ( y i- ¯ y ) = ˆ β 1 n X i =1 x i ( x i- ¯ x ) Thus ˆ β 1 = n X i =1 x i ( y i- ¯ y ) n X i =1 x i ( x i- ¯ x ) Using the proofs from problem set 1 that ∑ n i =1 x i ( x i- ¯ x ) = ∑ n i =1 ( x i- ¯ x ) 2 and ∑ n i =1 x i ( y i- ¯ y ) = ∑ n i =1 ( x i- ¯ x...
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ProblemSet2 Answers - Professor Mumford Econ 360 Fall 2010...

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