ˆ SSE SSE q SST R SST R q R R q R R q

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Derrick Daniel
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1 Bll Evas Spg 06 Sggested Aswes, Poblem Set 5 ECON a) The R meases the facto of the vaato Y eplaed by the model. I ths case, R =SSM/SST. Yo ae gve that SSM = bt ot SST. Howeve, ote that SST=SSM+SSE so SST= = Theefoe, R = 3.059/6.709 = b) =SSE/(-k-). I ths case, SSE = 3.650, =30, k=5, so -k-=4, so SSE / ( k ) / c) 99% cofdece teval s t /( k )[ se( )]. β =0.098, se( ) ad wth 4 degees of feedom ad α=0.0, the appopate ctcal vale of the t-dstbto s.797. So t ( k )[ se( )] =0.098 ±.797(0.0336) = (-0.00, 0.87). Sce the 99% cofdece / teval cotas 0, we CANNOT REJECT the ll hypothess. d) No calclato s ecessay. Sce a p-vale of s gve, p-vale < 0.0 ad we ca eject the ll. t / se( ) 0.363/ Wth 4 degees of feedom ad α=0.05, the appopate ctcal vale of the t-dstbto s.064 so, sce t t /( k ) at the 95% cofdece level, oe CAN REJECT the ll that β =0.e) ( SSE SSE) / q F. SSE =4.850, SSE =3.650, q=3, -k-=4, so SSE / ( k ) e) 5 5 f) ( SSE SSE) / q ( ) / 3 F.63. If the ll s coect, the F-test SSE / ( k ) / 4 statstc s dstbted as a F dstbto wth 3 ad 4 degees of feedom. The 95% ctcal vale wold the be 30 ad sce F F, we CANNOT REJECT the ll. g) No calclatos ae ecessay. STATA epots the f-test o ths ll hypothess as.008 so oe ca easly eject the ll that all coeffcets ae zeo.. The F-test s defed as ( SSE SSE) / q F. The R fo the estcted model s by defto SSE / ( k ) R ( SSE / SST ) so theefoe, SSE SST R ad lkewse SSE SST R. Note that ( ) ( ) SST s the same both the estcted ad estcted models. Sbstttg these vales to the defto of the F-test SSE SSE q SST R SST R q R R q R R q F SSE / ( k ) ( SST ( R )) / ( k ) ( R ) / ( k ) ( R ) / ( k ) ( ) / [ ( ) ( )]/ [( ) ( )]/ ( ) / 3. a. The cofdece teval s by defto t /( t k ) se( ). Lookg at the ptot, ad se( ) The egessos has =4 k=3 ad -k-=0. The appopate ctcal

2 vale of the t-dstbto s theefoe.086. Theefoe, the 95% cofdece teval s (3.44) (7.5,6.4). Sce the teval does ot cota zeo, we ca eject the ll. b. Gve a ll hypothess that H o:β =a, the t-statstc s defed as a t. I the poblem, we ae se( ) gve that a=0, ad se( ) 3.44 so a t.66. Sce se( 3.44 ) t t ( k ) we ca eject the ll that β =0. / c. Wth a 99% cofdece level, the ctcal vale of the t-dstbto wth 0 degees of feedom s.845. I ths case, t t /( k ) so we caot eject the ll. d. Pael A cotas the estcted model ad Pael B s the estcted model. The F-test s by ( SSE SSE) / q F SSE / ( k ) ad ote that the deomato the f-test s smple the estcted model, whch s label as the MSE o mea sqaed esdal o the ptot ( ). I ths case, SSE = , SSE = , q=, -k-=0. ( SSE SSE) / q ( ) / F 0.95 SSE / ( k ) The 95% ctcal vale of the F-dstbto wth ad 0 degees of feedom s 3.49, so sce F, we caot eject the ll hypothess. F 4. a) We ae gve the model y ad the ll H o: β =(/)β =3β 3. Note that β =β ad (/3)β =β 3 so sbsttte these vales above ad collect lke tems. y (/ 3) y ( (/ 3) ) y ( ) whee (/ 3) 5 3 b) The ll ths case s H o: β 4=- 4β - β -β 3 so sbsttte - 4β - β -β 3 fo β 4 ad collect lke tem y y ( 4 ) y ( 4 ) ( ) ( ) y ( 4 ) ( ) ( ) y * * * * whee y y, ( 4 ), ( ), ( ) * * * *

3 5. A sample pogam amed meps_005.do that geeates eslts ad the log fom ths pogam s clded o the web page. a. SSE=0,978.99, R =0.93 b. Males have 7.7 pecet lowe spedg tha female a oe t cease the BMI wll cease spedg by.6% a 0% cease come wll edce spedg by (0.)(-0.68)=-0.07 o by.7 pecet c. t o come s -.57 ad the 95% ctcal vale of the t-dstbto wth ove 3000 degees of feedom s.96 so sce t t /( k ) we caot eject the ll the te paamete s zeo. d. Afte g the estcted model, add the followg le to pefom the f-test. test mdwest soth west Yo wll see the F-statstc s 3.4. If the ll s coect, the test statstc s dstbted as a F-dstbto wth 3 ad fte degees of feedom ad the 95% ctcal vale s.60 so we ca eject the ll. e. I mst admt ths s a stpd qesto o my pat. Sce yo caot ejected the ll at the 95% level, yo ca also ot eject the ll at the 99% level. 6. a. A t cease hosepowe ceases pces by $6 b. A 00% cease MPG (MPG dobles) wll cease pce by $6,364 c. All wheel dve vehcles cost $469 moe tha o-awd vehcles d. Sedas cost $,054 less tha tcks e. SUVs cost $674 moe tha tcks 7. a. The sample pogam lottey_eample.do geeates the eslts fo ths poblem. The eslts fom the estcted model ae epoted below. Note that the coeffcet o c_ppl, K_eamak_ppl, ad ot_eamak_ppl ae 0.03, 0.78 ad 0.39, espectvely. Ths meas that f comes cease by $ the state, 3 cets eds p school spedg. I cotast, fo each addtoal dolla (pe ppl) lottey pofts hat ae geeated, 78 cets eds p school spedg. Fally, each addtoal dolla geeal lottey pofts that ae ot eamaked fo schools, 39 cets ed p edcato.. eg ep_ppl c_ppl k_eamak_ppl ot_eamak_ppl tme Soce SS df MS Nmbe of obs = F( 4, 677) = 7.5 Model Pob > F = Resdal R-sqaed = Adj R-sqaed = Total.97e Root MSE = ep_ppl Coef. Std. E. t P> t [95% Cof. Iteval] c_ppl k_eama~l ot_eama~l tme

4 _cos b. To test the ll that H o: β K_eamak_ppl=, we ca do ths thee ways. Fst, we ca se a t-statstc. Gve a mll hypothess that H : 0 j a, whch ca costct the t-test as t ( ) / ( j a se j) whch ths case eqals t (0.777 ) / The ctcal vale fo a t wth 677 degees f feedom at the 95% cofdece level s oghly.96 ad sce t.96 we caot eject the ll that β K_eamak_ppl=. Note as well that the 95% cofdece teval fo ths paamete cldes so sg the cofdece teval, we caot eject the ll. Fally, we ca do a f-test afte we estmate the estcted model. * test fo qesto b sg f-test. test k_eamak_ppl= ( ) k_eamak_ppl = F(, 677) = 0.83 Pob > F = c. To aswe ths qesto, we smply costct a post-estmato test STATA, whch s llstated below. Note that the p-vale o the f-test s vey low meag that we ca easly eject the ll at the 95% cofdece level. As a qck check o yo wok that yo ae makg the coect decso, ote that the 95% cofdece tevals fo c_ppl ad K_eamak_ppl dod ot ovelap so we wold epect that we wold eject the ll that they ae the same. Lookg at yo f-test table, the ctcal vale fo a f wth ad fte degees of feedom ad a alpha of 0.05 s * test fo qesto c. test k_eamak_ppl=c_ppl ( ) - c_ppl + k_eamak_ppl = 0 F(, 677) = 9.6 Pob > F = d. To aswe ths qesto, we aga costct a post-estmato test STATA, whch s llstated below. I ths case, the p-vale s whch meas that we CANNOT eject the ll that these two coeffcets ae the same. Lookg at yo f-test table, the ctcal vale fo a f wth ad fte degees of feedom ad a alpha of 0.05 s test k_eamak_ppl=ot_eamak_ppl ( ) k_eamak_ppl - ot_eamak_ppl = 0 F(, 677) = 3.4 Pob > F = e. Retg to qesto d), wth a 90% cofdece level (alpha=0.), we wold be able to eject the ll sce the p-vale (0.065) s less tha 0.0. Lookg the back of yo book, the ctcal vale fo a f wth ad fte degees of feedom ad a alpha of 0.0 s.7 so sce f.7we ca ow eject the ll. 8. Let case be the stato whee we have =00 obsevatos. I ths case, we get the followg eslt 4

5 () () t.33 () () ( ) () ( ) Whee () () ad () ae the paametes fo case. Now, we wat to cease the sample sze the hopes of cease the t-statstc absolte vale to. () () t () () ( ) () ( ) As the sample sze gows fom to, we epect that wth a fte sample we wll get dffeet estmates fo () () ad (). Howeve, we kow that () () ad () ae based estmates of the te delyg poplato vales, jst lke () () ad () ae as well. Theefoe, set () () () () ad () (). Theefoe t ( ) ( ) /.33 t ( ) ( ) Notg that =00 ad solvg fo, we get =5. Note stadad eos ae oghly popotoal to the sqae oot of sample sze. If we wat the t-statstc to cease by a facto of.5, we eed the sample sze to cease by a facto of.5 =.5. Sce =00, =5. 9. To aswe ths qesto, yo mst fst kow what the ll hypothess s. Yo boght the sadwch so the ll hypothess mst be that the sadwch s fe f yo thoght the sadwch was bad fom the stat yo wold ot have pchased t. I ths case, a Type I eo (false postve) s that yo thow away a good sadwch. A Type II eo (false egatve) s that yo eat a bad sadwch. 0. I the smple bvaate egesso y 0 we kow the estmate fo β ca be wtte as ( y y)( ) bt ths case = o 0. Thee ae obsevatos the sample ad ( ) 5

6 obsevatos fo whch = ad 0 y y ( ) ad y 0 y ( ) Wok wth the meato fo fst. ( ) fo whch =0 ad + 0=. Recall also that ( y y)( ) ( y y) y y y y Note that y y ad y, the sample mea of y, s smply a weghted aveage of y ad y 0 whee 0 y y y0. Theefoe, the meato ca be wtte as y y y ( ) y y y y y0 y y y0 ad becase the ad the meato eqals 0 y y Now wok wth the deomato. Note that ( ) ( ) Remembe that ad sce = o zeo the so ( ) 0 ( ) ad theefoe 0 y y0 y y0 0 6

Professor Wei Zhu. 1. Sampling from the Normal Population

Professor Wei Zhu. 1. Sampling from the Normal Population AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple