1、財(cái)經(jīng)大試題命題紙課程名稱：計(jì)量經(jīng)濟(jì)學(xué)（I（ 經(jīng)濟(jì)學(xué)院各專業(yè)）（A命題教師Givenarandomvariablesy,andrandomvectorsx,z.Please tvar財(cái)經(jīng)大試題命題紙課程名稱：計(jì)量經(jīng)濟(jì)學(xué)（I（ 經(jīng)濟(jì)學(xué)院各專業(yè)）（A命題教師Givenarandomvariablesy,andrandomvectorsx,z.Please tvary Evary| xvar(E(y| vary|xEvary| x,z| xvar(E(y| x,z)| x) Evary| x Evary| x,z(d)Considerthetwoy1(x,z)u1y2(x)u2 E(u1|x,z)E(

2、u2 | x)t Var(y|x,z) and Var(y|x) are both constant, what can you Aabout n Var(u1)andVar(u2) relationshipt the R2 2. Prove the ted restricted least squares estimator is never nt ted with the unrestricted least squares estimator. t ing restrictions improvesthefitofthe.3Forthelineary X (a) Definethelea

3、stsquareestimator bm0columnX is i, (b) Ifttheleastsquaremto(c) Let (1 ,2 )RK1 2 ,b (b ,b )and X (X1,X2), derive the s b1 and b24. In the least squares of Y on constant and X, to compute the coefficients on X, we transform Y to deviations from the mean Y and, transform each column of X to deviations

4、form the respective column; second, regress the transformed Y on the transformed X without a constant. Do we get the same result if we only transform Y? what ifweonlytransform5. ConsidertheleastsquaresofYon K constant)X.consideralternative set of regressors Z XP ,wher a mixture of some of the column

5、s of X. Proves a nonsingularweonlytransform5. ConsidertheleastsquaresofYon K constant)X.consideralternative set of regressors Z XP ,wher a mixture of some of the columns of X. Proves a nonsingular matrix. Thus,eachcolumn of Z ttheresidualhesof YonXYonZareidentical.Whatrelevancedoesthehavetothequesti

6、onofchangingthefitofachangingtheunitsofmeasurementoftheindependent6.UsingthedatainGPA2.RAWon4,137collegestudents,thefollowingequationwasestimatedby colgpa1.3920.0135hspercn4137, R2 wherecolgpaismeasuredonafour-scale,hspercisthehehighschoolclass(definedcombinedt,forle, hsperc=5meansthe topfivepercent

7、oftheclass),andsatisndverbalscoresonthestudentachievementWhydoesitmakesenseforthecoefficientonhsperctobeWhatisthepredictedcollegeGPAwhenhsperc=20andsat=(c)ettwohighschoolgraduates,AandB,hesamepercentilefromschool,butStudentAsSATscorewas140shigher(aboutonestandardheWhatisthepredicteddifferenceincolle

8、geGPAforthesetwostudents?Isthedifference(d) Holdinghspercfixed,whatdifferenceinSATscoresleadstoapredictedcolgpadifferenceof.50, or one-half of a grade po? Comment on your answer.Solution1Evary| xvar(E(y| x) EE(| x)E(y| x)2 EE(y| x)2E(Solution1Evary| xvar(E(y| x) EE(| x)E(y| x)2 EE(y| x)2E( E(y2)EE(y

9、|x)2EE(y| x)2E(y)2 var(Evary| x,z| x EE(y2 | x,z)E(y| x,z)2 | E(y2 |x)EE(y| x,z)2 |var(E(y|x,z)| x) EE(y|x,z)2 | xEE(y| x,z)| Evary| x,z| xvar(E(y| x,z)| E(y2 |x)EE(y|x,z)2 | xEE(y|x,z)2 |xE(y|x)2 var(y|vary|xEvary|x,z| x.Bypart(b)Evary| x EEvary|x,z|x Evary|x,zVar(u1 |x,z)Var(y|x,Var(u2 |x)Var(y|By

10、 . 22Var(u1) Var(u2) amption, there are constant and equal 2 respectively. But then part (c) tThis t, 21errorvarianceareconstant,theerrorvariancefallsasmoreexplanatoryvariablesareconditionedSolutionFor the linear regres the residual is e . Also sup e* , then there is:y X , e the unrestricted estimat

11、ion of is b e the restricted estimation of on R q is b* and the residual b*bXTX1 RT RXTX1 RT Rbe*e X(b*eTe eTe(b b)T XT X(b * *R2 R2 (b* b)T XTX(b* b) *y y2ias XT X itiveSolution(a) bargmin (Y X)(YR2 R2 (b* b)T XTX(b* b) *y y2ias XT X itiveSolution(a) bargmin (Y X)(Y X(b) order condition of the leas

12、t square is: X (Y Xb)0X e0. Since 0columnX is i,sotheleastsquaremtoorderconditionoftheleastsquareis: X (Y Xb)0,(c)X Xb X X1(1 1 X1 1bXX1XX,X Y X 2 b b Y X 2 2 2 2 2 X1X1b1X1X2b2 X1X b X X b X 1 2 Thesolutionofthissystemofequationsb (X M )1 M Y , b (X M )1 M 12 1 Where M2 I X2(X2X2) X2, M1 I X1(X1X1)

13、 X1Solution heof Yon i and X, the coefficientson X are b(X M0X)1X M0Y M0 I i(ii)1i is the matrix which transforms o deviations from their I i(ii)1iis idempotent and symmetric we may also write the preceding Mmeans. (XM0)(M0X)1(XM0M0Y)which of M0Y on M0Xtthe the least squares slopes. If only X is tra

14、nsformed to deviations, we would (XM0)(M0X)1(X M0)Y . but , course, this is identical.(XM0)(M0X)1(X M0)Y . but , course, this is identical. However, if only y transformed,theresultis (X X)1X M0Y whichislikelybequiteSolution The residual he of Y on X is MXY I X(X X) X Y . residualheofYonZMZY I Z(ZZ)

15、ZI XP(XP)(XP)1(XP)MXYSince the residual vectors are identical, the fits must be as well. Changing the units of measurementtheregressorsisequivalenttmultiplyingbyadiagonalPmatrixwhosekthdiagonalelementisscale factor to be d to the kth variable(1 ifit is to be unchanged). It followsfrom theresulttthis

16、willnotchangethefitofthe .Solution (a) hsperc isdefined t the smaller it is, the lower the students standing in high Everything else equal, the worse the students standing in high school, the lower is his/her college(b)Justplugtheseothecolgpa =1.392.0135(20)+.00148(1050)=(c)The differencenAandBissimply140timesthe coefficientonsat,because hsperc thesameforbothSoAispredictedtohaveascore.00148(140)