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Unformatted text preview: ECO321 by H. Morita Economic Statistics II Solution  HW1
Part I [9 points] Suppose that you collected the following sample.
X Y 3 4 8 7 5 6 4 3 (1) Construct a scatterplot. [1 point] Answer: (2) In the regression Yi = 0 + 1 Xi + ui , find the estimate of intercept (0 ) and the estimate of slope (1 ). [2 points] Answer: Since n = 4, X = (3 + 4 + 5 + 6)/4 = 4.5, and Y = (8 + 7 + 4 + 3)/4 = 5.5, V ar(X) = 1 n1
n (Xi  X)2
i=1 = (3  4.5)2 + (4  4.5)2 + (5  4.5)2 + (6  4.5)2 /3 = 1.667 n 1 Cov(X, Y ) = (Xi  X)(Yi  Y ) n1
i=1 = [(3  4.5)(8  5.5) + (4  4.5)(7  5.5) + (5  4.5)(4  5.5) + (6  4.5)(3  5.5)] /3 = 3 Thus, ^ 1 = ^ 0 Cov(X, Y ) 3 = = 1.8 V ar(X) 1.667 ^ = Y  1 X = 5.5  (1.8)(4.5) = 13.6 (3) Find SSR, ESS, and T SS. [3 points] ^ ^ ^ ^ Answer: We found Yi = 0 + 1 Xi = 13.6  1.8Xi and ui = Yi  Yi , so, ^
X Y ^ Y u ^ 3 4 5 8 7 4 8.2 6.4 4.6 .2 .6 .6
1 6 3 2.8 .2 n T SS =
i=1 (Yi  Y )2 = (8  5.5)2 + (7  5.5)2 + (4  5.5)2 + (3  5.5)2 = 17
n SSR =
i=1 (ui )2 = (.2)2 + (.6)2 + (.6)2 + (.2)2 = .8
n ESS =
i=1 (Yi  Y )2 = (8.2  5.5)2 + (6.4  5.5)2 + (4.6  5.5)2 + (2.8  5.5)2 = 16.2 Note that T SS = ESS + SSR. (4) Find R2 and SER. Interpret the number in R2 . [3 points] Answer: SER = R
2 SSR/(n  2) = .8/2 = .632 = ESS/T SS = 16.2/17 = .953 This means that 95.3% of the variation in Y can be explained by the regression, indicating that the regression fits to the sample very well. Part II [7 points] Growth.dta contains data on average growth rates over 19601995 for 65 countries, along with variables that are potentially related to growth. A detailed description is given in Growth Description.pdf. Note: Both files Growth.dta and Growth Description.pdf are available on the course web site.) 1. Construct a table that shows sample means, standard deviations, and minimum/maximum values for average annual growth rates (growth) and schooling years (yearsschool). [1 point] Answer: The sample means, the standard deviations, and the minimum/maximum values for average annual growth rates (growth) and schooling years (yearsschool) are reported in Table . Table 1. Data Summary Variables growth [annual %] yearsschool [years] Mean 1.94 3.99
2 Std.Dev. 1.90 2.54 Max 7.16 .20 Min 2.81 10.07 2. Construct a scatter plot of growth (in the vertical axis) on yearsschool (in the horizontal axis). [1 point] Answer: FIGURE 1. Scatterplot of growth on yearsschool
8 2 0 0 2 4 6 2 4 yearsschool Fitted values 6 growth 8 10 3. Using all observations, run a regression of growth on yearsschool ( i.e., growthi = 0 + 1 yearsschooli + ui ). Find the estimate of intercept (0 ) and the estimate of slope (1 ). [1 point] Answer: 1 = .247 and 0 = .958. 4. Using the regression in 3., find the heteroskedasticrobust standard errors of 0 and 1 . [1 point] Answer: SE(1 ) = .081 and SE(0 ) = .443. 5. Using the regression in 3., find the predicted values of the growth rates for a country with years in school of 6. (Hint: To answer this question, compute it manually using the results of the regression above) [1 point] Answer: Growth = 0 + 1 Y earsSchool = .958 + (.247)(6) = 2.44 [annual %] 6. In 1960, some countries contemplated that an education policy that will increase average years of schooling from 4 to 6 years would raise a country's economic wellbeing, that is a real GDP. Based on the results of the regression above, what do you think of this idea. [2 points] Answer: When country's average years of schooling is 4 in 1960, the predicted value of the growth rate of real GDP is Growth = .958 + (.247)(4) = 1.95 [annual %]. We found that it is 2.44 annual % for 6 years in 5). So, the difference is .45 % each year. From these evidence, we conclude that the increase in average years of schooling would raise a country's economic wellbeing. Note: .46% difference looks too small, but it's not so small for the growth rate of real GDP. In fact, this .46% difference each year becomes 4.7%[= 1  (1 + .0046)10 ] difference in 10 years, and 25.8%[= 1  (1 + .0046)50 ] difference in 50 years (from 1960 to 2010). 3 Appendix: Stata Outputs . sum growth yearsschool Variable  Obs Mean Std . Dev . Min Max       +                                                               growth  65 1.942715 1.89712 2.811944 7.156855 yearsschool  65 3.985077 2.542 .2 10.07 . reg growth yearsschool Source  SS df MS       +                                     Model  25.236075 1 25.236075 Residual  205.103982 63 3.25561876       +                                     Total  230.340057 64 3.59906339 N u m b e r of obs F( 1, 63) Prob > F Rs q u a r e d A d j Rs q u a r e d Root MSE = = = = = = 65 7.75 0.0071 0.1096 0.0954 1.8043                                                                               growth  Coef . Std . Err . t P >t  [95% C o n f . I n t e r v a l ]       +                                                                       yearsschool  .2470275 .0887261 2.78 0.007 .0697226 .4243324 _ cons  .9582918 .4184559 2.29 0.025 .1220743 1.794509                                                                               . reg growth yearsschool , r Linear regression N u m b e r of obs F( 1, 63) Prob > F Rs q u a r e d Root MSE = = = = = 65 9.28 0.0034 0.1096 1.8043                                                                                Robust growth  Coef . Std . Err . t P >t  [95% C o n f . I n t e r v a l ]       +                                                                       yearsschool  .2470275 .0810945 3.05 0.003 .084973 .409082 _ cons  .9582918 .4431176 2.16 0.034 .072792 1.843792                                                                               4 ...
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 Spring '11
 HiroshiMorita

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