Suggested Answers, Problem Set 4 ECON The R 2 for the unrestricted model is by definition u u u u

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1 Da Hgerma Fall 9 Sggested Aswers, Problem Set 4 ECON 333 The F-test s defed as ( SSEr The R for the restrcted model s by defto SSE / ( k ) R ( SSE / SST ) so therefore, SSE SST ( R ) ad lkewse SSEr SST ( Rr ) Note that SST s the same both the restrcted ad restrcted models Sbstttg these vales to the defto of the F-test ( SSEr [ SST ( Rr ) SST ( R )] / q [( Rr ) ( R )] / q ( R Rr SSE / ( k ) ( SST ( R )) / ( k ) ( R ) / ( k ) ( R ) / ( k ) The reslts from STATA are below Gve a ll hypothess that H o :β a, the t-statstc s defed as β a t se( β) β a 685 t 67 se( β ) 8846 β 685 ad se( β ) 8846 so We are gve a ad the prtot below,, There are 49 degrees of freedom ths model ad the 95% crtcal vale of a t wth 49 degrees of freedom s I ths case, we caot reject the ll the coeffcet s se state_cg_data keep f year988 (969 observatos deleted) reg retal_prce state_tax Sorce SS df MS Nmber of obs F(, 49) 597 Model Prob > Resdal R-sqared Adj R-sqared 7598 Total Root MSE retal_prce Coef Std Err t P> t [95% Cof Iterval] state_tax _cos test state_tax ( ) state_tax F(, 49) 6

2 Prob > 3 a) We are gve the model y β + x β + xβ + x3 β3 + x4β4 + ε 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 terms y β + x β + x β + x (/ 3) β + x β + ε y β + ( x + x + (/ 3) x ) β + x β + ε y β + ( x ) β + x β + ε where x x + x + (/ 3) x 5 3 b) The ll ths case s H o : β 4-4β - β -β 3 so sbsttte - 4β - β -β 3 for β 4 ad collect lke term y β + x β + x β + x β + x β + ε y β + x β + x β + x β + x ( 4β β β ) + ε y β + ( x 4 x ) β + ( x x ) β + ( x x ) β + x + ε y x β + ( x 4 x ) β + ( x x ) β + ( x x ) β + ε y β + x β + x β + x * * * * 3 β + ε 3 where y y x, x ( x 4 x ), x ( x x ), x ( x x ) * * * * I the smple bvarate regresso y β + xβ + ε we kow the estmate for β ca be wrtte as β ( y y)( x x) bt ths case x or There are observatos the sample ad ( x x) x observatos for whch x ad y x y ( x ) ad y y x ( x ) Work wth the merator for β frst ( x ) for whch x ad + Recall also that ( y y)( x x) ( y y) x y x y x y x y Note that y x y ad y, the sample mea of y, s smply a weghted average of y ad y where + Therefore, the merator ca be wrtte as y y y

3 y y y ( ) y y y y y y y y ad becase + the ad the merator eqals + ( y y ) Now work wth the deomator Note that Remember that x ad sce x or zero the ( x x) ( x x) x x x x x x so ( ) x x ( ) ad therefore ( y y ) β ( y y ) 5 a The cofdece terval s by defto β ± t α /( t k ) se( β) Lookg at the prtot, β 3478 ad se( β) 344 The regressos has 4 k3 ad -k- The approprate crtcal vale of the t-dstrbto s therefore 86 Therefore, the 95% cofdece terval s 3478± 86(344) (75, 64) Sce the terval does ot cota zero, we ca reject the ll b Gve a ll hypothess that H o :β a, the t-statstc s defed as β a t I the problem, we are se( β) gve that a, β 3478 ad se( β) 344 so β a 3478 t 66 Sce se( β 344 ) t > t ( k ) we ca reject the ll that β α / c Wth a 99% cofdece level, the crtcal vale of the t-dstrbto wth degrees of freedom s 845 I ths case, t /( k ) so we caot reject the ll d Pael A cotas the restrcted model ad Pael B s the restrcted model The F-test s by ( SSEr ad ote that the deomator the f-test s smple σε the restrcted model, SSE / ( k ) whch s label as the MSE or mea sqared resdal o the prtot ( ) I ths case, SSE , SSE r 77399, q, -k- ( SSEr ( ) / 95 SSE / ( k ) The 95% crtcal vale of the F-dstrbto wth ad degrees of freedom s 349, so sce 3

4 F < F α, we caot reject the ll hypothess 6 A sample program that geerates reslts ad the log from ths program are clded o the web page a SSE,97899, R 93 b Males have 77 percet lower spedg tha female a oe t crease the BMI wll crease spedg by 6% a % crease come wll redce spedg by ()(-68)-7 or by 7 percet c t o come s -57 ad the 95% crtcal vale of the t-dstrbto wth over 3 degrees of freedom s 96 so sce t /( k ) we caot reject the ll the tre parameter s zero d After rg the restrcted model, add the le test mdwest soth west To perform the f-test Yo wll see the F-statstc s 34 If the ll s correct, the test statstc s dstrbted as a F-dstrbto wth 3 ad fte degrees of freedom ad the 95% crtcal vale s 6 so we ca reject the ll e I mst admt ths s a stpd qesto o my part Sce yo rejected the ll at the 95% level, yo ca also reject the ll at the 8% level 7 a) The R measres the fracto of the varato Y explaed by the model I ths case, R -(SSE/SST)-(3344/43798) 78 b) σ ε SSE/(-k-) I ths case, SSE 3344, 9, k3, so -k-5, so ε SSE / ( k ) 3344 / σ c) 95% cofdece terval s β ± t α /( k )[ se( β)] β -449, se( β ) 64 ad wth 5 degrees of freedom ad α5, the approprate crtcal vale of the t-dstrbto s 6 So β ± t α /( k )[ se( β)] 449 ± 6(64) [ 787, ] Sce the 95% cofdece does ot cota, we CAN REJECT the ll hypothess t β / se( β ) 668 / Wth 5 degrees of freedom ad α, the approprate crtcal vale of the t-dstrbto s 787 so, sce t /( k ) at the 99% cofdece level, oe CANNOT REJECT the ll that β d) e) ( SSEr SSE / ( k ) SSE r 36736, SSE 3344, q, -k-5, so 4

5 ( SSEr ( ) / 5 If the ll s correct, the F-test SSE / ( k ) 3343 / 5 statstc s dstrbted as a F dstrbto wth ad 5 degrees of freedom The 95% crtcal vale wold the be 339 ad sce F < F α, we CANNOR REJECT the ll 5