i 2 σ ) i = 1,2,...,n , and = 3.01 = 4.01

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1 ECO 745, Homework 6 Le Cabrera. Assume that the followg data come from the lear model: ε ε ~ N, σ,,..., Fd the mamum lkelhood estmates of,, ad σ ε s ε k k σ. he model: α α α E E u s estmated b OLS, where E ad E are dumm varables dcatg membershp of the secod ad thrd educatoal classes, respectvel. Show that the OLS estmates are: a a a where deotes the mea value of the th educatoal class. of

2 of Cost E E Fd : Subtract rows ad from row ; realze - - Dvde row b ; row b ; row b / / / / / Subtract row from row ad row / / / / / / / / / / / / / / / / / / / / / / umber each educatoal class;

3 of / / / / / / / / / / / whch s what we wated to show. Suppose that C ad represet per capta cosumpto ad per capta dsposable come respectvel, ad that J.M. Kees thks that the are related b the equato of the form: C u b, ~ σ N u Kees wats to estmate the parameter b whch measures the margal propest to cosume. he radom varable u s uobservable, but Kees ca obta pars of observatos C,, C,,..., C,, whch relate observatos of C, over dfferet ears.

4 Fd a epresso for the ordar least squares estmator, b OLS. Use ordar least squares ad the followg data to estmate the margal propest to cosume. ear C C ad C [ ] C [ ] C C b OLS C b OLS of

5 4. he hardess,, of the shells of eggs lad b a certa breed of chckes was assumed to be roughl learl related to the amout,, of a certa food supplemet put to the det of chckes. he model assumed s the classcal lear regresso model. Data were collected ad are gve below: est the hpothess that. versus the hpothess that.. Use a pe I error probablt of 5 percet. est the hpothess that > versus the hpothess that. H :. H a :. R [ ] q [.] R q [ R R ] R q F / k { } / m ~ F m, k Reecto rego: F < FINV.975,,. or F > FINV.5,, F 45.7 > reect H ad coclude. H : >. H a :. Same set up, but ow reecto rego for.: F > FINV.5,, 5. F 45.7 > 5. do't reect H ot suffcet evdece to sa. 5 of

6 6 of 5. A producto fucto model s specfed as: u X X where log output, X log labor put, ad X log captal put. he data refer to a sample of frms, ad the observatos are measured as from the sample meas: Estmate,, ther stadard errors, ad R. est the hpothess that. Suppose ow that ou wat to mpose the restrcto that. What s the least squares estmate of ad ts stadard error? What s the value of R ths case? Compare the results wth those obtaed part ad commet. "as from the sample meas" OLS..7

7 ~, OLS N σ.4/.67 Note : e RSS see below s Note : hs s based o MLE for, but could use for ubased k estmator whch case s.4/ σ Usg s, σ σ sqrt s s sqrt.5.5 e RSS SS.4 RSS R - -.4/ R.6 SS H : H a : R [ ] q [ ] { R q } R q [ R R ] F / k / m ~ F m, k Reecto rego: F < FINV.975,,. or F > FINV.5,, of

8 of F.7.../.49 do ot reect H ~, ~ ~ ~ ~ ~.5.45 ~ ~ 6 ~ ~ ~ ~ OLS.75, ~ OLS N σ.5/ Note : e RSS see below

9 s Note : hs s based o MLE for, but could use k estmator whch case s.5/ ~ ~ σ.5.5 Usg ~ ~ s,.5.99 σ sqrt.5.9 s sqrt for ubased e RSS [ ] [ ] SS RSS R - -.5/ R.5 SS I order to eforce, we had to make both parameters bgger. As the umbers tur out, the dfferece s shared equall betwee them.5 larger tha part. Wh that's the case, I have ow dea. It does makes sese that forcg ths codto results a lower R value.6 vs..5 because we are o loger usg all the data to mmze the sum of the squared resduals we're stll mmzg them, but the altered resduals are't the same. Oe good thg about the secod verso s that the parameters are ot correlated as the were part. I kow the depedet varables ad are ot supposed to be correlated, but we dd't talk about the parameters. he strage thg s that the stadard errors for s actuall smaller the secod case although for t's hgher. 6. ou wll be e-maled a dataset o heghts, sec, mother's heght, ad father's heght. Usg ths dataset, estmate the ucostraed regresso of heght o se, mother's heght, ad father's heght. Iterpret our results. ou wsh to test the hpothess that the coeffcets o mother's heght ad father's heght sum to oe, ad that the coeffcet o se s equal to -. What s the R matr ad the r vector that correspod to these restrctos? Coduct a test of ths hpothess that has a F-statstc. What are our coclusos? 9 of

10 Heght Geder.7 Mom.46 Dad ε O tal specto t would appear geder s the bggest determat of heght, but t's ot. What geder does mpl s that wome geder are almost cm shorter tha me assumg ther paret's are the same heght. hs cm s't much whe we factor the average heghts of moms ad dads whch have a greater mpact o heght tha geder does absolute terms: 5. for average mom ad 7.4 for average dad. Father's heght s more sgfcat because father's are taller geeral, but also more sgfcat o the marg for each cm of heght a father adds.46 cm to hs offsprg versus ol.7 cm a mom passes o. H : Mom Dad ad Geder - H a : oe or both of these do t hold R q Usg Stata. test mom dad mom dad F, 9.95 Prob > F.965. test geder -, accumulate mom dad geder - F, Prob > F.5 Up to a 99.75% cofdece level, we ca reect that both Mom Dad ad Geder -. Based o the results of the frst test, t would appear Geder - s the part that causes the ot test to fal. Documetato Prof Werer showed how to fd for problem class. Prof Werer told me "as from the sample meas" problem 5. She also told me how to set t up to do part. Scott showed us how to use the sheet, regress ad test commads Stata to do problem 6. of

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