Problem Set 2 Solutions

# Problem Set 2 Solutions - Problem Set 2 Answers Question 1...

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Unformatted text preview: Problem Set 2 Answers Question 1 RSS _ T55 2. F. The varimlee of Y.- is the dispersion of Y around ilH mean, while the varime in the error term is the dispersion of Y around the regmuion line. 191131 U >i’. 17 >355 ’I'SS m2 1 U, 3. F. Need lo divide by the sample variance of X. :1. F. Only lrue if our (’Sliillalﬂﬁ of are unhiawd. m: for example, under the CLRM. 5. F. We need [he awnnlptions of the CLRM. 5. F. This is not linear in the variables. It is linear in the paranmtem. The linearity misnmplion in the CLRM is ahoul Iinearll y in llu'. I)Hl'alllt’lt'rl-i: not. in [he variable-s. 7. F. At this level. we will reject the null. B. T. We do not need any awumpLions to derive the OLS estimators, only to perform inference. Question] 2 '1'0 [grove unbiacsedness, we must SlIOW E [3”] If" and E 31. I will start Willl 53.: We know _ _ r~ ZJX; ~ X10? — Y} .3. —_., (I) Zi(XF — X)" Flom our model. Y.- = :30 I .3ng I r1- I:-: [hen easily shown that _ _ Y: 53:. I SIX I F Subtracting these equations, we obtain Yi—V=,'i.(X,—FX’I(r,—? (‘2) Snlmlitul iug equation (2] inlo [1). 3] Z_(X.- — Yip-34X,- —:) + (e.- — a) 24X, —X)2 3.21%— T)? + ZIIX. — Tm, _ r} Zr£Xi —Yl2 _ 31 + Z,(X.- —Y}{_c. — E) ' SAX. —X}‘-’ } ZBIX,—Y){.—;;i{X.—7i SILK! “Xi? __ 3| } 21'in _El‘i ' EIX, —X)= -. ,3. The last equality holds since XXX. — X) 0. i Taking l'XpH’I inns: \w haw- 3“ __ Z:(Xa _ X)" EM] "£131 + —Z£(Xr _m2] 3' .. -' SAX]. _ X)2 ,3. silm- by amumptimh Elm] '- : D. So we have firm-WI unbiasvdnvss of “71‘ know go - - V — 3’17 1 AI =— K—S— X, n. 1n: r 'l‘aklng PXpec‘tnl-louﬁ and interchanging the summation and cxplfClilllDlll-i operamr, Emu} E Z EIY, — 31M 1 DH — Mi) ‘H. a 1 =30 I'J. a — 3‘. where WP han used the unbiaserlnosﬁ of Ii'l. So we have proved our result for Fig. Question 3 l. A unit- (‘hangt' in average pari-nlal heighl cansm a .73 change in student height. ll" pawnth heighl is [J (not intniliw-L student height is 19.6. 2. The variation in sLudenl heigit accounted for by variation in m-‘t-rage parent-a] height. 3. I? [9.6 + .73 )< TUJJG - - HIT-1. :1. For very tall parents, on average, the student will be shorter. For very short. parents. on average, the student will he lallur. The (:1ILOIT iH m1 avvragt‘ parental height, of 72.6. 'l‘r_\_' plugging in numbers to see the intutiun. I ﬁnd 72.6 by solving 19.6 + .73 x X, . X, for X,. Problems 4, 5, and 6 are all in the attached STATA output. log: F:\econ103\hw2.log log type: text opened on: 2 Feb 2009, 23:36:30 *Question 4 . use data2, clear {SAS creation date 27SEP00:12:40) *Part 1 sum wage if citizen== Variable | Obs Mean Std. Dev. Min Max _ _ _ _ _ _ _ _ _ _ _ __+________________________________________________________ wage I 584 14.36223 11.14626 1.25 97.758 *Part 2 sum wage if citizen== Variable | Obs Mean Std. Dev. Min Max wage | 236 11.45792 21.60437 1.785714 250.6615 *Parts 3 and 4 {uses 80f significance for part 4) ttest wage, by(citizen} unequal level(80} Two-sample t test with unequal variances Group | Obs Mean Std. Err. Std. Dev. [80% Conf. Interval] _ _ _ _ _ _ _ __+_______________________________________________________________ 0 | 236 11.45792 1.406325 21.60437 9.650564 13.26528 1 | 584 14.36223 .4612354 11.14626 13.77046 14.95399 _ _ _ _ _ _ _ __+_______________________________________________________________ combined | 320 13.52635 .52277 14.96935 12.85586 14.19685 _ _ _ _ _ _ _ __+_______________________________________________________________ diff | -2.904305 1.480029 -4.805416 - diff 1.9623 Ho: diff 286.937 mean[0} - mean(1} t 0 Satterthwaite's degrees of freedom Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr('I‘ < t.) = 0.0253 Pr{|T| > ltl) = 0.050".-' Pr{'I‘ > t) = 0.9747 *Part 6 keep if female== {351 observations deleted) reg wage exp Source | SS df MS Number of obs = 469 ----------- --+-----------------------—------ F( 1, 467} = 0.02 Model | 7.39027117 1 7.39027117 Prob ) F = 0.8785 Residual | 147557.836 467 315.96967 R—squared = 0.0001 ----------- --+------------------------------ Adj R-squared = - 0 0021 Total | 147565.226 46B 315.310313 Root MSE = 17.776 wage | Coef. Std- Err. t P>|t| [95“ Conf. Interval] ___________ __+___________________________________________________________ exp | .0795038 .5198527 0.15 0.879 —.9420363 1.101044 _cons | 13.61724 6.543623 2.08 0.033 .7586509 26.47583 *Part 8 . predict wage_hat (option xb assumed; fitted values) *Part 9 . gen eps_hat=wage—wage_hat sum eps_hat Variable | Obs Mean Std. Dev. Min Max eps hat I 469 1.32e-08 17.75653 -13-45644 236.0346 *Part 11 . gen logwage=ln(wage) reg logwage educatio Source | SS df MS Number of obs = 469 ----------- --+------------------------------ F( l, 467) = 142.01 Model | 54.0923555 l 54.0923555 Prob > F = 0.0000 Residual | 17?.803046 46? .380905881 R—squared = 0.2332 ----------- --+------------------------------ Adj R-squared = 0 2315 Total | 231.9?5402 463 .495673936 Root MSE = 61713 logwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] _ _ _ _ _ _ _ _ _ _ _ __+___._________.______..________.________________.________________ educatio | .1023657 .00859 11.92 0.000 .0854858 .119245? _cons | 1.104338 .1122531 9.84 0.000 .8837541 1.324921 *Question 5 . use econ103, clear *Part 1 . count if ca== 99 sum colgpa if ca==1 Variable | Obs Mean Std. Dev. Min Max colgpa | 95 3.528547 .2569961 2.8 3.94 *Part 2 . count if ca== 2? sum colgpa if ca== Variable | Obs Mean Std. Dev. Min Max oolgpa | 2? 3.612481 .2987732 3 4 *Parts 3 and 4 (uses 90a significance for part 4) ttest colgpa,by(ca] 1evel{90) Two—sample t test with equal variances Group I Obs Mean Std. Err. Std. Dev. [903 Conf. Interval] _ _ _ _ _ _ _ __+_______________________________________________________________ 0 | 27 3.612481 .0574989 .2987732 3.51441 3.710553 1 | 95 3.52854? .0263672 .2569961 3.484745 3.572349 _______ __+_______________________________________________________________ combined | 122 3.547123 .024245 .2677953 3.506936 3.58731 _ _ _ _ _ _ _ __+_______________________________________________________________ diff | .0839341 .053143? —.0124479 .1303161 diff = mean[0} - mean(1} t = 1.4436 Ho: diff = 0 degrees of freedom = 120 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 Pr(T < t) = 0.9243 Pr{|T| > Itl) = 0.1515 Pr{T > t} = 0.0757 *Part 5 . drop if colgp ==. | momedu ==. {6 observations deleted) reg colgpa momeduc Source | 55 df MS Number of obs = 130 ----------- --+--—--—--------—--—--——-—--——-- F[ 1, 123} 1.04 Model | .074237848 1 .074237848 Prob > F = 0.3089 Residual | 9.11215261 123 .071188692 R—squared = 0.0081 ----------- --+--—--—--—--—------—-——-—--——-— Adj R—squared = 0 0003 Total | 9.18644046 129 .071212717 Root MSE = 26681 colgpa | Coef. Std. Err. t P>|t| [95% Conf. Interval] ___________ __+___________________________________________________________ momeduc | .0137684 .0134782 1.02 0.309 -.0129004 .0404373 cons | 3.485981 .0583921 59.70 0.000 3.370442 3.60151§ *Part ? predict colgpa_hat {option xb assumed; fitted values} . gen eps_hat=colgpa-colgpa_hat sum eps_hat Variable | Obs Mean Std. Dev. Min Max eps hat I 130 -6.24e-OB .2657?59 -.7685914 .4864824 *Question 6 . clear all input Y X Y X 1. 155 4 2. 185 6 3. 205 7 4. end *Part 1 reg Y X Source | 55 df MS Number of obs 3 ----------- --+------------------------------ F[ 1, 1) = 176.33 Model | 1259.52381 1 1259.52381 Prob > F = 0.0479 Residual | 7.14235714 1 7.14285714 R—squared = 0.9944 ——————————— --+--—-——--——-—------—-——-—--——-— Adj R—squared = 0 9887 Total | 1266.66667 2 633.333333 Root MSE = 2 6726 Y | Coef Std. Err. t P>|t| [95 Conf Interval] _ _ _ _ _ _ _ _ _ _ _ __+___________________________________________________________ X | 16.42857 1.237179 13.28 0.048 .7087199 32.14842 _cons | 88.57143 7.178483 12.34 0.051 -2.639841 179.7827 *Part 2 . predict Y_hat (option Kb assumed; fitted values} . gen eps_hat=Y-Y_hat *Part 3 . gen eps=Y-(80+17*X) *Part 4 sum eps_hat Variable | Obs Mean Std. Dev. Min Max eps hat | 3 0 1.889819 -2.142853 1.428574 . gen xeps_hat=x*eps_hat sum xeps_hat Variable | Obs Mean Std. Dev. Min Max xeps hat I 3 5.099-06 11.69335 ~12.S5712 10.00002 sum eps Variable | Obs Mean Std. Dev. Min Max eps | 3 5.333333 2.081666 3 7 log Close log: F:\econ103\hw2.log log type: text closed on: 2 Feb 2009.r 23:36:30 ...
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