Homework 6 Solutions | Econometrics and Applications | ECON 4950, Quizzes of Introduction to Econometrics

Download Homework 6 Solutions | Econometrics and Applications | ECON 4950 and more Quizzes Introduction to Econometrics in PDF only on Docsity! Homework #6 Solutions 7.2 The following equations were estimated using the data in BWGHT.RAW: 0493.R 1,191n (.0026) (.0030) (.015) (.011) (.006) (.0085) (.0010) (.38) 0032.0030.045.034.017.)minlog(0110.0052.65.4)log( 0472.R 1,388n (.013) (.010) (.006) (.0059) (.0009) (.22) 055.027.016.)minlog(0093.0044.66.4)log( 2 ^ 2 ^ == +−++++−= == ++++−= fatheducmotheducwhitemaleparitycfaigsbwght whitemaleparitycfaigsbwght The variables are defined as in Example 4.9, but we have added a dummy variable for whether the child is male and a dummy variable indicating whether the child is classified as white. (i) In the first equation, interpret the coefficient ton the variable cigs. In particular, what is the effect on birth weight from smoking 10 more cigarettes per day. The first equation estimates suggest that an increase in cigarettes smoked by 10 units is predicted to decrease birth weight by 4.4%, holding all other factors constant. (ii) How much more is a white child predicted to weigh than a nonwhite child, holding the other factors in the first equation fixed? Is the difference statistically significant? The first equation suggests that a white child is predicted to weigh .55% more than a nonwhite child, holding all other factors constant. To determine if the difference is statistically significant, we need to compute the t-statistic on white: .055/.013=4.23. For an alpha=5%, and two-sided alternative, with n-k-1=1,388-5-1=1,382 degrees of freedom, the critical value in Table G.2 is c=1.960. Our rejection/fail to reject rule is: Fail to reject null if |t|<c and Reject null if |t|>c Since |4.23|>1.96, we reject the null of insignificance. That is, the difference between white and nonwhite is statistically different. (iii) Comment on the estimated effect and statistical significance of motheduc. The coefficient on motheduc suggests one more year of mother’s education is predicted to decrease birth weight by .30%, holding all other factors constant. (iv) From the given information, why are you unable to compute the F statistic for joint significance of motheduc and fatheduc? What would you have to do to compute the F statistic? We can not compute the F statistic because the two regressions use different sets of observations. The second regression uses fewer observations because motheduc and fatheduc are missing for some observations. We would have to re-estimate the first equation using just only the 1,191observations that are in the second regression, and get the R2. 7.3 Using the data in GPA2. RAW, the following equation was estimated: 0858.R 4,137,n (18.15) (12.71) (4.29) (.53) (3.83) (6.29) *31.6281.16909.4519.230.1910.028,1 2 2 ^ == +−−−+= blackfemaleblackfemalehsizehsizesat The variable sat is the combined SAT score, hsize is size of the student’s high school graduating class, in hundreds, female is a gender dummy variable, and black is a race dummy variable equal to one for blacks and zero otherwise. (i) Is there strong evidence that hsize2 should be included in the model? From this equation, what is the optimal high school size? The t-statistic on hsize2 is -2.19/.53=-4.13. For alpha=.05 and a two-sided test with 4137-5-1=4131 degrees of freedom, the critical value is c=1.96. Using our rule, we know that since |-4.13|>1.96, we reject the null hypothesis of insignificance. That is, hsize2 should be included in the model. The turn around point can be calculated by using the formula: 41.4|41.4||-2.19)*.19.30/(2| |)*2/(| ^^ * 2 =−=== hsizehsizehszie ββ Since size is measured in hundreds, this suggests the optimal amount of students is 441. (ii) Holding hsize fixed, what is the estimated difference in SAT score between nonblack females and nonblack males? How statistically significant is this estimated difference? The difference in SAT score between nonblack females and nonblack males is given by the coefficient on female: -45.09 (s.e.=4.29). The t-statistic is -10.51 which suggests the estimate is statistically significant from zero, suggesting there variables are relative to female non athletes. The coefficient on femaleathlete is .1751 (s.e.=.0840), and has a t-stat (2.08); p-value ( 0.037) and 95% CI ( .0103, .3398). At alpha=.05 and n-k-1=4137-7-1=4129 degrees of freedom, the critical value is 1.96. Using either the t-stat, p-value, or CI, we can come to the conclusion that the coefficient on femaleathlete is statistically different from zero at the 5% level, suggesting there is a difference between female athletes and female nonathletes. (v) Does the effect of sat on colgpa differ by gender? Justify your answer. In order to determine whether there is a differential effect of sat on college GPA by gender, we could include the interaction of sat and female into the model in (i) or even in (iv). For simplicity, I add it to (i). The interaction is created in STATA, and denoted femalesat. See STATA output. The coefficient on the interaction term femalesat is .00005 (s.e.=.0001). It has a t- statistic (0.40), p-value (0.692) and 95% CI( -.000202, .0003044). The coefficient on the interaction is interpreted as the return to a higher SAT score for a female versus a male. Using alpha=.05, and n-k-1=4137-7-1=4129 degrees of freedom, we fail to reject the null hypothesis of insignificance. That is, there is no statistical evidence that the effect of sat on GPA differs by gender. Wednesday October 21 17:21:17 2009 Page 1 ___ ____ ____ ____ ____tm /__ / ____/ / ____/ ___/ / /___/ / /___/ Statistics/Data Analysis 1 . reg colgpa hsize hsizesq hsperc sat female athlete satfemale log: C:\Documents and Settings\user\My Documents\Teaching\Fall 2009\Econ 8740\Homework Solut log type: smcl opened on: 21 Oct 2009, 16:56:31 2 . do "C:\DOCUME~1\user\LOCALS~1\Temp\STD0d000000.tmp" 3 . /*C 6.2*/ 4 . 5 . use "C:\Documents and Settings\user\My Documents\Teaching\Fall 2009\Econ 8740\Homework Assignments\Data\ 6 . 7 . /*(i)*/ 8 . 9 . reg lwage educ exper expersq Source SS df MS Number of obs = 526 F( 3, 522) = 74.67 Model 44.5393713 3 14.8464571 Prob > F = 0.0000 Residual 103.79038 522 .198832146 R-squared = 0.3003 Adj R-squared = 0.2963 Total 148.329751 525 .28253286 Root MSE = .44591 lwage Coef. Std. Err. t P>|t| [95% Conf. Interval] educ .0903658 .007468 12.10 0.000 .0756948 .1050368 exper .0410089 .0051965 7.89 0.000 .0308002 .0512175 expersq -.0007136 .0001158 -6.16 0.000 -.000941 -.0004861 _cons .1279975 .1059323 1.21 0.227 -.0801085 .3361035 10 . 11 . 12 . /*(iv)*/ 13 . sum exper if exper>=29 Variable Obs Mean Std. Dev. Min Max exper 121 37.8595 5.997646 29 51 14 . end of do-file 15 . clear 16 . do "C:\DOCUME~1\user\LOCALS~1\Temp\STD0d000000.tmp" 17 . /*C 7.4 */ 18 . 19 . use "C:\Documents and Settings\user\My Documents\Teaching\Fall 2009\Econ 8740\Homework Assignments\Data\ 20 . 21 . /*(ii)*/ 22 . reg colgpa hsize hsizesq hsperc sat female athlete Source SS df MS Number of obs = 4137 F( 6, 4130) = 284.59 Model 524.819305 6 87.4698842 Prob > F = 0.0000 Residual 1269.37637 4130 .307355053 R-squared = 0.2925 Adj R-squared = 0.2915 Total 1794.19567 4136 .433799728 Root MSE = .5544 Wednesday October 21 17:21:18 2009 Page 2 colgpa Coef. Std. Err. t P>|t| [95% Conf. Interval] hsize -.0568543 .0163513 -3.48 0.001 -.0889117 -.0247968 hsizesq .0046754 .0022494 2.08 0.038 .0002654 .0090854 hsperc -.0132126 .0005728 -23.07 0.000 -.0143355 -.0120896 sat .0016464 .0000668 24.64 0.000 .0015154 .0017774 female .1548814 .0180047 8.60 0.000 .1195826 .1901802 athlete .1693064 .0423492 4.00 0.000 .0862791 .2523336 _cons 1.241365 .0794923 15.62 0.000 1.085517 1.397212 23 . 24 . /*(iii)*/ 25 . reg colgpa hsize hsizesq hsperc female athlete Source SS df MS Number of obs = 4137 F( 5, 4131) = 191.92 Model 338.217123 5 67.6434247 Prob > F = 0.0000 Residual 1455.97855 4131 .35245184 R-squared = 0.1885 Adj R-squared = 0.1875 Total 1794.19567 4136 .433799728 Root MSE = .59368 colgpa Coef. Std. Err. t P>|t| [95% Conf. Interval] hsize -.0534038 .0175092 -3.05 0.002 -.0877313 -.0190763 hsizesq .0053228 .0024086 2.21 0.027 .0006007 .010045 hsperc -.0171365 .0005892 -29.09 0.000 -.0182916 -.0159814 female .0581231 .0188162 3.09 0.002 .0212333 .095013 athlete .0054487 .0447871 0.12 0.903 -.0823582 .0932556 _cons 3.047698 .0329148 92.59 0.000 2.983167 3.112229 26 . 27 . /*(iv)*/ 28 . 29 . gen femaleathlete=0 30 . replace femaleathlete=1 if female==1 & athlete==1 (45 real changes made) 31 . 32 . 33 . gen maleathlete=0 34 . replace maleathlete=1 if female==0 & athlete==1 (149 real changes made) 35 . 36 . 37 . gen malenonathlete=0 38 . replace malenonathlete=1 if female==0 & athlete==0 (2128 real changes made)