ans_hw3_eco321_2010f

ans_hw3_eco321_2010f - Economic Statistics II Solution -...

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Unformatted text preview: Economic Statistics II Solution - Homework Assignment 3 Part-I Using the data Growth.dta, run three regressions of Growth on Regression A: T radeShare and Y earsSchool Regression B: T radeShare and ln(Y earsSchool) Regression C: T radeShare, ln(Y earsSchool), Rev Coup, Assassinations, and ln(RGDP 60) Regression D: T radeShare, T radeShare2 , T radeShare3 , ln(Y earsSchool), Rev Coup, Assassinations, and ln(RGDP 60) Assume that the error term is heteroskedastic. TABLE 1: Results Of Regressions Dependent Variable: Growth (Y ) Regressors T radeShare (X1 ) 2 T radeShare2 (X1 ) 3 T radeShare3 (X1 ) A 2.33 (.596) B 2.17 (.555) C 1.29 (.516) D -5.33 (3.23) 7.78 (4.30) -2.37 (1.43) E Y earsSchool (X2 ) ln(Y earsSchool) (ln X2 ) Rev Coup (X3 ) Assassinations (X4 ) ln(RGDP 60) (ln X5 ) T S1 (D1 ) T S1 Y earsSchool (D1 X2 ) Intercept R 2 .250 (.076) 1.03 (.201) 2.18 (.383) -2.32 (.919) .255 (.323) -1.64 (.429) 2.14 (.408) -2.03 (.950) .102 (.365) -1.59 (.453) .469 (.158) -.370 (.585) .211 -.416 (.468) .329 11.8 (3.28) .463 12.9 (3.17) .464 -.781 (.526) 3.55 (.890) -.321 (.267) 5.96 (3.65) .174 1. Construct a scatterplots of Growth(= Y ) on Y earsSchool(= X), and on ln(Y earsSchool)(= X), respectively. Add a linear regression line for each scatterplot. Using these scatterplots, explain which Regression A or B has a better fit to the sample. Is your analysis consistent with adjusted R2 ? [2 points] From the visual image of these scatterplots, the regression B looks slightly better fit to the 1 8 6 4 2 0 -2 0 2 4 yearsschool Fitted values 6 growth 8 10 -2 -2 0 2 4 6 8 -1 0 lnys Fitted values 1 growth 2 (a) vs Y earsSchool (b) vs ln(Y earsSchool) Figure 1. Scatterplots sample than the regression A. This is consistent with what an adjusted R-square tells us, as the regression B has a higher one (.329) than A (.211). 2. Find the estimated coefficient of ln(Y earsSchool) in Regression B. What is the interpretation of it? [2 points] The estimate of that coefficient is 1.03. This means that when the schooling year increases by 1%, the growth rate increases by .0103 percentage points holding other regressors constant. Note: The unit of Growth is (annual) percent, but that does NOT mean it increases by .0103 %. We must to say that it increases by .0103 percentage points. 3. In Regression B, test if ln(Y earsSchool) is statistically significant at the 1% level. [2 points] Let 2 be a coefficient of ln(Y earsSchool). In the t-test of H0 : 2 = 0 against H1 : 2 = 0, the p-value is .0000 or close to 0%, which is less than 1%. Therefore, we reject H0 , meaning that ln(Y earsSchool) is statistically significant at the 1% level. Alternatively, Since t=(1.03-0)/.201=5.1 is greater than the critical value of 2.576, we reject H0 . 4. In Regression C, find the predicted value of Growth when T radeShare is 1.3, Y earsSchool is 8, Rev Coup is .1, Assassinations is .5, RGDP 60 is 2800. [2 points] ^ From TABLE 1, we have Y = 11.8 + 1.29X1 + 2.18 ln X2 - 2.32X3 + .255X4 - 1.64 ln X5 . So, ^ Y = 11.8 + 1.29(1.3) + 2.18 ln(8) - 2.32(.1) + .255(.5) - 1.64 ln(2800) 4.86 annual percent. 5. In Regression D, find the partial effect of T radeShare on Growth when T radeShare increases from 0.5 to 0.6. [2 points] 2 3 ^ From TABLE 1, we have Y = 12.9- 5.33X1 + 7.78X1 - 2.37X1 . Let us ignore the other variables because they are eliminated when we hold them constant to find a partial effect. So, when the trade share (X1 ) increases from .5 to .6, the change in Growth is (-5.33(.6) + 7.78(.6)2 - 2 2.37(.6)3 ) - (-5.33(.5) + 7.78(.5)2 - 2.37(.5)3 ) = -.909 - (-1.016) .107 Growth. Thus, the partial effect is Growth/T radeShare = .107/(.6 - .5) = 1.07 2 ^ Alternatively, Y /X1 = -5.33 + 2(7.78)X1 - 3(2.37)X1 . Using the midpoint .55 for trade ^ /X1 = -5.33 + 2(7.78)(.55) - 3(2.37)(.55)2 1.08. share, we get Y 6. In Regression D, is there any evidence of linear relationship between T radeShare and Growth (holding other regressors constant) at the 5% level of significance? To answer this question, test if T radeShare2 and T radeShare3 are jointly statistically significant? [2 points] Let 2 and 3 be a coefficient of T radeShare2 and T radeShare3 , respectively. In the F -test of H0 : 2 = 0 and 3 = 0 against H1 : 2 = 0 or 2 = 0, the p-value is .1198 or close to 11.98%, which is greater than 5%. So, we fail to reject H0 , meaning that T radeShare2 and T radeShare3 are not jointly statistically significant at the 5% level. Therefore, there is no any evidence of nonlinear relationship between T radeShare and Growth. Alternatively, Since F =2.20 is less the critical value of 3.00 (d.f.=2 at 5% level), we fail to reject H0 . NOTE: The formula of F using SSR or R2 is only for homoskedastic errors. That doesn't apply to heteroskedastic errors. 7. Which Regression C or D is a better model to predict Growth? Explain. [2 points] Since T radeShare2 and T radeShare3 are not jointly statistically significant, Regression C is better model to predict Growth than D. Part II UUsing the same data above, create a new binary variable T S1 which takes 1 if a country's trade share is more than 1, and 0 otherwise. NOTE: Stata command: generate ts1 = (tradeshare>1) Then, run the regression of Growth on Regression E: T S1, Y earsSchool, (T S1 Y earsSchool), and ln(RGDP 60) Assume that the error term is heteroskedastic. 8. Test if (T S1 Y earsSchool) is statistically significant at the 5% level. [2 points] In the t-test of H0 : 3 = 0 against H1 : 3 = 0, the p-value is .238 or 23.5%, which is greater than 5%. Therefore, we fail to reject H0 , meaning that (T S1 Y earsSchool) is not statistically significant at the 5% level. Alternatively, Since |t| = |(-.3205 - 0)/.2669| = 1.20 is less than the critical value of 1.96, we fail to reject H0 9. Interpret the result in Q-8. [2 points] The estimate is -.321. This is a multiplied effect on Growth having a high trade shares and high education at the same time. Although such multiplied effect is negative, it's not statistically 3 significant. Thus, there is no multiplied effect. Appendix: Stata Outputs . * Part I . g e n t s 2 = t r a d e s h a r e ^2 . g e n t s 3 = t r a d e s h a r e ^3 . gen lnys = log ( yearsschool ) . gen lnrgdp60 = log ( rgdp60 ) . . * Regression A . reg growth tradeshare yearsschool , r Linear regression N u m b e r of obs F( 2, 62) Prob > F R-s q u a r e d Root MSE = = = = = 65 11.60 0.0001 0.2359 1.6849 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | Robust growth | Coef . Std . Err . t P >|t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - tradeshare | 2.331286 .5960601 3.91 0.000 1.139779 3.522793 yearsschool | .2500282 .0755781 3.31 0.002 .0989497 .4011067 _ cons | -.3701506 .5850767 -0.63 0.529 -1.539702 .799401 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . display " Adjusted Rsquared = " _ result (8) Adjusted Rsquared = .21125556 . . * Regression B . reg g r o w t h t r a d e s h a r e lnys , r Linear regression N u m b e r of obs F( 2, 62) Prob > F R-s q u a r e d Root MSE = = = = = 65 19.39 0.0000 0.3503 1.5537 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | Robust growth | Coef . Std . Err . t P >|t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - tradeshare | 2.172501 .5551062 3.91 0.000 1.062859 3.282142 lnys | 1.030777 .2016121 5.11 0.000 .6277601 1.433794 _ cons | -.416268 .4680183 -0.89 0.377 -1.351823 .5192873 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . display " Adjusted Rsquared = " _ result (8) Adjusted Rsquared = .3293109 . 4 . * Regression C . reg growth tradeshare lnys rev_ coups assasinations lnrgdp60 , r Linear regression N u m b e r of obs F( 5, 59) Prob > F R-s q u a r e d Root MSE = = = = = 65 15.28 0.0000 0.5055 1.3894 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | Robust growth | Coef . Std . Err . t P >|t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - tradeshare | 1.287937 .5164692 2.49 0.015 .2544844 2.321389 lnys | 2.18318 .3833806 5.69 0.000 1.416037 2.950323 rev_ coups | -2.317988 .9186524 -2.52 0.014 -4.156208 -.4797693 assasinati ~s | .2545822 .3229126 0.79 0.434 -.3915644 .9007288 lnrgdp60 | -1.642083 .428589 -3.83 0.000 -2.499688 -.7844787 _ cons | 11.78525 3.278647 3.59 0.001 5.224691 18.3458 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . display " Adjusted Rsquared = " _ result (8) Adjusted Rsquared = .46360336 . . * Regression D . reg growth tradeshare ts2 ts3 lnys Linear regression rev_ coups assasinations lnrgdp60 , r N u m b e r of obs F( 7, 57) Prob > F R-s q u a r e d Root MSE = = = = = 65 223.08 0.0000 0.5228 1.3886 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | Robust growth | Coef . Std . Err . t P >|t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - tradeshare | -5.334176 3.230639 -1.65 0.104 -11.80342 1.135067 ts2 | 7.776234 4.298654 1.81 0.076 -.8316717 16.38414 ts3 | -2.366314 1.433134 -1.65 0.104 -5.236114 .5034869 lnys | 2.13558 .4078304 5.24 0.000 1.318914 2.952247 rev_ coups | -2.038818 .9499453 -2.15 0.036 -3.94105 -.1365851 assasinati ~s | .1024156 .3651838 0.28 0.780 -.6288524 .8336836 lnrgdp60 | -1.588102 .4525192 -3.51 0.001 -2.494256 -.6819479 _ cons | 12.90439 3.167876 4.07 0.000 6.560831 19.24796 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . display " Adjusted Rsquared = " _ result (8) Adjusted Rsquared = .46421867 . . * E x a m p l e o f F-t e s t . test ts2 ts3 ( 1) ( 2) ts2 = 0 ts3 = 0 F( 2, 57) = Prob > F = 2.20 0.1198 . . * Part II . g e n t s 1 = ( t r a d e s h a r e >1) . gen tsys = ts1 * yearsschool . . * Regression E . reg growth ts1 yearsschool tsys lnrgdp60 , r Linear regression N u m b e r of obs = F( 4, 60) = Prob > F = 65 8.53 0.0000 5 R-s q u a r e d Root MSE = = 0.2264 1.7234 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - | Robust growth | Coef . Std . Err . t P >|t | [95% C o n f . I n t e r v a l ] - - - - - - -+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ts1 | 3.549992 .8904249 3.99 0.000 1.768877 5.331107 yearsschool | .4695754 .1577728 2.98 0.004 .1539828 .785168 tsys | -.3204906 .2669116 -1.20 0.235 -.8543932 .213412 lnrgdp60 | -.7812846 .5255453 -1.49 0.142 -1.832532 .2699624 _ cons | 5.959975 3.652163 1.63 0.108 -1.34544 13.26539 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . display " Adjusted Rsquared = " _ result (8) Adjusted Rsquared = .17478097 6 ...
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