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Answer Set 3 - Fall 2008

# Answer Set 3 - Fall 2008 - Econ 371 Problem Set#3 Answer...

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Unformatted text preview: Econ 371 Problem Set #3 Answer Sheet 4.1 In this question, you are told that a OLS regression analysis of third grade test scores as a function of class size yields the following estimated model. d TestScore = 520 . 4- 5 . 82 × CS,R 2 = 0 . 08 ,SER = 11 . 5 (1) a. The first part of the question asks what the regression’s prediction would be for the average test score in a class of 22 students. Our model implies that d TestScore = 520 . 4- 5 . 82 × 22 = 392 . 36 (2) b. The second part of the question then asks how the test scores would change in a class that has seen an increase in class size from 19 students to 23 students. We know that, based on our model, the expected change would be given by E [ TestScore | CS = 23]- E [ TestScore | CS = 19] = (520 . 4- 5 . 82 × 23)- (520 . 4- 5 . 82 × 19) = 386 . 54- 409 . 82 =- 23 . 28 That is, test scores would be predicted to drop by 23.28 points. c. You are told that the sample average for the class size across the 100 classrooms is 21.4 and asked to compute the sample average of the test scores across the 100 classrooms. The hint suggests how to proceed. Specifically, from the formula for the OLS estimator of the intercept in equation (4.8) of the text, we know that: ˆ β = ¯ Y- ˆ β 1 ¯ X. (3) Rearranging this equation (solving for ¯ Y ) we have that ¯ Y = ˆ β + ˆ β 1 ¯ X (4) or in terms of our current set of variables TestScore = ˆ β + ˆ β 1 CS. (5) Using our parameter estimates and the information about the mean class size, we then have TestScore = 520 . 4- 5 . 82 × 21 . 4 = 395 . 85 (6) d. Finally, you are asked to compute the standard deviation of the test scores. In solving this problem, you want to think about what it is you are looking for. Specifically, we want to know: s 2 Y = 1 n- 1 n X i =1 ( Y i- ¯ Y ) 2 (7) = TSS n- 1 . (8) What we need to do is come up with a value for TSS . However, we also know the value of the R 2 and SER , and we know that R 2 is related to the TSS . Specifically, from the definition of the R 2 in equation (4.18) in the text, we have that: R 2 = 1- SSR TSS (9) We can solve for TSS , yielding: TSS = SSR 1- R 2 . (10) 1 Now, if only we knew SSR. But we do know SSR, since from equation (4.19) in the text: SER = r SSR n- 2 . (11) Rewriting the above equation, we have that SSR = SER 2 × ( n- 2) (12) Using the numbers for the problem at hand, we then have that: SSR = (11 . 5) 2 × (100- 2) = 12961 . (13) Substituting this into equation equation (10) above yields: TSS = SSR 1- R 2 = 12961 1- . 08 = 14088 . (14) Finally, from equation (8), we then have: s 2 Y = TSS n- 1 = 14088 100- 1 = 142 . 3 . (15) so that s Y = √ 142 . 3 = 11 . 9. 4.7 This question asks you to show that ˆ β is an unbiased estimator of β . It is suggested that you use the fact that ˆ β 1 is an unbiased estimator of β 1 . From the formula for ˆ β in equation (4.8) of the text, we know that: E h ˆ β i = E h ¯ Y- ˆ β 1 ¯ X i = E "ˆ β + β 1 ¯ X + 1 n n X i =1 u i !...
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Answer Set 3 - Fall 2008 - Econ 371 Problem Set#3 Answer...

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