Econ 371 Problem Set #3 Answer Sheet


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1 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. T estscore = CS, R 2 = 0.08, SER = 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 T estscore = = 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 [T estscore CS = 23] E [T estscore CS = 19] = = = That is, test scores would be predicted to drop by 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: ˆβ 0 = Ȳ ˆβ 1 X. 3 Rearranging this equation solving for Ȳ we have that or in terms of our current set of variables Ȳ = ˆβ 0 + ˆβ 1 X 4 T estscore = ˆβ 0 + ˆβ 1 CS. 5 Using our parameter estimates and the information about the mean class size, we then have T estscore = = 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 Yi Ȳ 2 i=1 = T SS n 1. 8 What we need to do is come up with a value for T SS. However, we also know the value of the R 2 and SER, and we know that R 2 is related to the T SS. Specifically, from the definition of the R 2 in equation 4.18 in the text, we have that: R 2 = 1 SSR 9 T SS We can solve for T SS, yielding: T SS = 7 SSR 1 R
2 Now, if only we knew SSR. But we do know SSR, since from equation 4.19 in the text: SSR SER = n Rewriting the above equation, we have that Using the numbers for the problem at hand, we then have that: Substituting this into equation equation 10 above yields: SSR = SER 2 n 2 12 SSR = = T SS = Finally, from equation 8, we then have: so that s Y = = SSR 1 R 2 = = s 2 Y = T SS n 1 = = This question asks you to show that ˆβ 0 is an unbiased estimator of β 0. It is suggested that you use the fact that ˆβ 1 is an unbiased estimator of β 1. From the formula for ˆβ 0 in equation 4.8 of the text, we know that: [ ] ] E ˆβ0 = E [Ȳ ˆβ1 X [ ] 1 n = E β 0 + β 1 X + u i n ˆβ 1 X i=1 [ ] = E β n u i + β 1 n ˆβ 1 X = Eβ n = β 0 i=1 n Eu i + Eβ 1 ˆβ 1 X 5.2 In this question, you are told that a OLS regression analysis of wages on a gender Male dummy variable yields. i=1 W age = Male, R 2 = 0.06, SER = a. The first part of the question asks you what the estimated gender gap. This is given by: E[W age Male = 1] E[W age Male = 0] = [ ] [ ] = b. The second part of the question asks you to construct a pvalue for the twosided test of the null hypothesis H 0 : β 1 = 0. p value = 2Φ t act ˆβ 1 0 SE ˆβ < = We would clearly reject the null hypothesis in this case. 2
3 c. The third part of the question asks you to construct a 95% confidence interval for the gender gap. This is given by ˆβ 1 ± 1.96[SE ˆβ 1 ] = 2.12 ± = 1.41, d. The fourth part of the question asks you to compute the mean wage of men and the mean wage of women. However, we know from our regression model that: E[W age Men] = E[W age Male = 1] = β 0 + β 1 E[W age W omen] = E[W age Male = 0] = β 0 Using our estimated model, our estimates of these means are then: W age Men = = W age W omen = e. Finally, it is noted that another researcher uses the same data, but chooses to estimate the model where F emale i =1 for women and =0 for men. In this model W age i = γ 0 + γ 1 F emale i + v i 17 E[W age Men] = E[W age F emale = 0] = γ 0 E[W age W omen] = E[W age F emale = 1] = γ 0 + γ 1 Comparing these results with those obtained for the original specification, it is clear that so that β 0 + β 1 = γ 0 β 0 = γ 0 + γ 1 Our OLS estimates are then γ 0 = β 0 + β 1 γ 1 = β 0 γ 0 = β 1. ˆγ 0 = ˆβ 0 + ˆβ 1 = ˆγ 1 = ˆβ 1 = Due to the relationship among coefficient estimates, for each individual observation, the OLS residual is the same under the two regression equations. Thus the sum of squared residuals is the same under the two regressions. This implies that both R 2 and SER are unchanged. 5.5 This question reports on a study in Tennessee, with a regression of test scores on a dummy variable identifying small class sizes. Specifically, the study finds T estscore = SmallClass, R 2 = 0.01, SER = a. You are asked to decide whether small classes improve test scores and by how much and to discern whether this effect is large. From the regression, we know that the estimated gain from being in a small class is 13.9 points. This is less than 1 5 of the standard deviation in test scores 75, a moderate increase. b. You are then asked whether this effect is statistically significant using a 5% significance level. The null hypothesis in this case is H 0 : β 1 = 0, with a two sided alternative hypothesis. The corresponding 3
4 pvalue is given by p value = 2Φ t act ˆβ 1 0 SE ˆβ < = c. Finally, you are asked to construct a 99% confidence interval for the effect of SmallClass on test score. This is given by ˆβ 1 ± 2.58[SE ˆβ 1 ] = ± = 6.74, The two empirical exercises in this homework use the same dataset: CPS04. The data can be downloaded from the Web site listed in the assignment which you can also reach from the class website. A program that carries all of the tasks for problems E4.1 and E5.1 is appended to this answer sheet. E4.1 a. The first task you are asked to do is to regress average hourly earnings AHE on age Age. The results are as follows: AHE = Age, R 2 = The specific questions you are asked to respond to are: What is the estimated intercept? 3.32 What is the estimated slope? 0.45 How much do earnings increase as workers age by one year? Earnings increase, on average, by 0.45 dollars per hour when workers age by 1 year. b. Next, you are asked to predict the earnings of Bob Age26 and Alex Age = 30. Using our regression results we have: Bob s predicted earnings is = $ Alex s predicted earnings is = $ c. Finally, you are asked whether or not age accounts for a large fraction of the variance of earnings across individuals. The R 2 is This means that age explains a small fraction of the variability in earnings across individuals. E5.1 This question uses the results from E4.1, reported above, and then expands on these results by conducting the regression for subsamples of the available data. a. Your first task is to determine whether or not the regression slope coefficient is statistically significant. Using the regression results, the corresponding pvalue for the twosided test of the null hypothesis H 0 : β 1 = 0 is given by: p value = 2Φ t act ˆβ 1 0 SE ˆβ < You could also read this information directly off of the output from Stata, as it provides pvalues for each coefficient individually being tested. The results will differ slightly here because the above calculation uses rounded values for the various estimates, whereas Stata will keep all of the available digits. 4
5 b. The second question asks that you construct a 95% confidence interval for the slope coefficient. Again, this can be read directly from the output, or you can do the calculation manually as: ˆβ 1 ± 1.96[SE ˆβ 1 ] = 0.45 ± = , c. You are then asked to estimate the model using only high school graduates. This involves using the regression command with an if option. The attached program provides the specifics of the command. The resulting parameter estimates are AHE = Age The corresponding tstatistic read directly from the Stata output is 7.43, with a corresponding p value of less than leading us to reject the null hypothesis that the age slope coefficient is zero. d. The next step is to estimate the model using only college graduates, yielding: AHE = Age The corresponding tstatistic read directly from the Stata output is 13.06, with a corresponding p value of less than leading us to, again, reject the null hypothesis that the age slope coefficient is zero. e. The final question asks that you examine the difference in the age effect on earnings for the two subpopulations. There hint i.e., to see exercise 5.15 provides the way to proceed. Instead of considering men versus women as in exercise 5.15, we are now considering differences between high school and college graduates. If we write the two models as: AHE h,i = β h,0 + β h,1 Age h,i + u h,i AHE c,i = β c,0 + β c,1 Age c,i + u c,i our null hypothesis becomes H 0 : β h,1 β c,1 = 0. Based on our regression results, our estimate of this difference is ˆβ h,1 ˆβ c,1 = = The formula for the standard error of this difference is SE ˆβh,1 ˆβ 2 2 c,1 = SE ˆβh,1 + SE ˆβc,1 = = The pvalue for our twosided hypothesis test becomes: p value = 2Φ t act = 2Φ ˆβ h,1 ˆβ c,1 0 SE ˆβh,1 ˆβ c, < Clearly, we reject the null hypothesis that the two subsamples have the same marginal effect of Age. 5
6 ; Problem Set #3 ; # delimit ; clear; cap log close; ; Specify the output file ; log using Problemset3.log,replace; set more 1; ; Read in and summarize the data ; use CPS04.dta; describe; summarize ahe age; ; Estimate the model for question E4.1 ; reg ahe age,r; ; Estimate the model for question E5.1c ; reg ahe age if bachelor==0,r; ; Estimate the model for question E5.1c ; reg ahe age if bachelor==1,r; log close; clear; exit;
7 Problemset3.log log: C:\Documents and Settings\jaherrig\My Documents\Classes\Economics 371\Stata\Problemset3.log log type: text opened on: 9 Oct 2008, 10:14:48. set more 1;. ;. > Read in and summarize the data > > ;. use CPS04.dta;. describe; Contains data from CPS04.dta obs: 7,986 vars: 4 15 Jan :16 size: 159, % of memory free storage display value variable name type format label variable label ahe float %9.0g bachelor float %9.0g female float %9.0g age float %9.0g Sorted by:. summarize ahe age; Variable Obs Mean Std. Dev. Min Max ahe age ;. > Estimate the model for question E4.1 > > ;. reg ahe age,r; Linear regression Number of obs = 7986 F 1, 7984 = Prob > F = Rsquared = Root MSE = Robust ahe Coef. Std. Err. t P> t [95% Conf. Interval] age _cons Page 1
8 Problemset3.log. ;. > Estimate the model for question E5.1c > > ;. reg ahe age if bachelor==0,r; Linear regression Number of obs = 4346 F 1, 4344 = Prob > F = Rsquared = Root MSE = Robust ahe Coef. Std. Err. t P> t [95% Conf. Interval] age _cons ;. > Estimate the model for question E5.1c > > ;. reg ahe age if bachelor==1,r; Linear regression Number of obs = 3640 F 1, 3638 = Prob > F = Rsquared = Root MSE = Robust ahe Coef. Std. Err. t P> t [95% Conf. Interval] age _cons log close; log: C:\Documents and Settings\jaherrig\My Documents\Classes\Economics 371\Stata\Problemset3.log log type: text closed on: 9 Oct 2008, 10:14: Page 2
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