ans_hw1_eco321_2010f

ans_hw1_eco321_2010f - ECO321 by H. Morita Economic...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 n-1 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 ) n-1 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 1960-1995 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 heteroskedastic-robust 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 well-being, 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 well-being. 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 R-s q u a r e d A d j R-s 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 R-s 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 ...
View Full Document

Ask a homework question - tutors are online