Chapter 1 Econometrics

Transcription

1 Chapter Ecoometric There are o exercie or applicatio i Chapter. 0 Pearo Educatio, Ic. Publihig a Pretice Hall

2 Chapter The Liear Regreio Model There are o exercie or applicatio i Chapter. 0 Pearo Educatio, Ic. Publihig a Pretice Hall

3 Chapter 3 Leat Square Exercie. Let x X =. x a. The ormal equatio are give by (3-), Xe = 0 (we drop the miu ig), hece for each of the colum of X,, we kow that Xe k = 0. Thi implie that Σ ei = 0 ad Σ xe i i = 0. b. Ue Σ ei to coclude from the firt ormal equatio that a = y bx. c. We kow that Σ ei = 0 ad Σ xe i i = 0. It follow the that Σ ( ) 0 xi x ei = becaue Σ xei = xσ ei = 0. Subtitute e i to obtai Σ ( xi x)( yi a bxi) = 0. or Σ ( xi x)( yi y b( xi x)) = 0. Σ ( )( ) xi x yi y The, Σ ( xi x)( yi y) = bσ ( xi x)( xi x)) o b=. Σ ( xi x) d. The firt derivative vector of e e i X e. (The ormal equatio.) The ecod derivative matrix i (e e)/ b b = X X. We eed to how that thi matrix i poitive defiite. The diagoal elemet are ad Σ xi which are clearly both poitive. The determiat i [()( Σ xi )] ( Σ xi) = 4Σ 4( ) 4 [( ) ] 4 [( xi x = Σ xi x = Σ ( xi x) ]. Note that a much impler proof appear after (3-6).. Write c a b + (c b). The, the um of quared reidual baed o c i (y Xc) (y Xc) = [y X(b + (c b))] [y X(b + (c b))] = [(y Xb) + X(c b)] [(y Xb) + X(c b)] = (y Xb) (y Xb) + (c b) X X(c b) + (c b) X (y Xb). But, the third term i zero, a (c b) X (y Xb) = (c b)x e = 0. Therefore, (y Xc) (y Xc) = e e + (c b) X X(c b) or (y Xc) (y Xc) e e = (c b) X X(c b). The right-had ide ca be writte a d d where d = X(c b), o it i ecearily poitive. Thi cofirm what we kew at the outet, leat quare i leat quare. 0 Pearo Educatio, Ic. Publihig a Pretice Hall

4 4 Greee Ecoometric Aalyi, Seveth Editio 3. I the regreio of y o i ad X, the coefficiet o X are b = (X M 0 X) X M 0 y. M 0 = I i(i i) i i the matrix which traform obervatio ito deviatio from their colum mea. Sice M 0 i idempotet ad ymmetric we may alo write the precedig a [(X M 0 )(M 0 X)] (X M 0 )(M 0 y) which implie that the regreio of M 0 y o M 0 X produce the leat quare lope. If oly X i traformed to deviatio, we would compute [(X M 0 )(M 0 X)] (X M 0 )y, but, of coure, thi i idetical. However, if oly y i traformed, the reult i (X X) X M 0 y, which i likely to be quite differet. 4. What i the reult of the matrix product M M where M i defied i (3-9) ad M i defied i (3-4)? M M = (IX (X X ) )(I X(XX) ) = M X (X X ) X M There i o eed to multiply out the ecod term. Each colum of MX i the vector of reidual i the regreio of the correpodig colum of X o all of the colum i X. Sice that x i oe of the colum i X, thi regreio provide a perfect fit, o the reidual are zero. Thu, MX i a matrix of zeroe which implie that M M = M. 5. The origial X matrix ha row. We add a additioal row, x. The ew y vector likewie ha a X y additioal elemet. Thu, X, = ad y, =. y The ew coefficiet vector i x b, = (X, X, ) (X, y, ). The matrix i X, X, = X X + x x. To ivert thi, ue (A-66); ( X X,, ) = ( XX ) ( ) ( ). + ( ) XX xx XX The vector i x X X x (X, y, ) = (X y ) + x y. Multiply out the four term to get (X, X, ) (X, y, ) = b ( ) XX xxb + ( XX ) x + x ( X y ( XX ) xx ( XX ) x X) x + ( y x XX) x x ( X X) x = b + ( XX ) x y ( XX ) ( ) xy XX xxb + x ( X X) x + x ( X X) x x ( X X) x b + ( ) ( ) y XX x XX xxb + x ( X X) x + x ( X X) x b + ( ) ( ) y + x ( X X) x XX x + x ( X X) x XX xxb b + ( XX x xb ) ( y ). + x ( X X ) x i x 0 0 yo X= =, = ad =. 0 X X X y y m (The ubcript o the part of y refer to the oberved ad miig row of X.) We will ue Frih-Waugh to obtai the firt two colum of the leat quare coefficiet vector. b = (X M X ) (X M y). Multiplyig it out, we fid that M = a idetity matrix ave for the lat diagoal elemet that i equal to X M X = XX. X X Thi jut drop the lat obervatio. X M y i computed likewie. 0 Thu, the coefficiet o the firt two colum are the ame a if y 0 had bee liearly regreed o X. 6. Defie the data matrix a follow: [ ] 0 Pearo Educatio, Ic. Publihig a Pretice Hall

5 Chapter 3 Leat Square 5 The deomiator of R i differet for the two cae (drop the obervatio or keep it with zero fill ad the dummy variable). For the firt trategy, the mea of the obervatio hould be differet from the mea of the full ule the lat obervatio happe to equal the mea of the firt. For the ecod trategy, replacig the miig value with the mea of the other obervatio, we ca deduce the ew lope vector logically. Uig Frich-Waugh, we ca replace the colum of x with deviatio from the mea, which the tur the lat obervatio to zero. Thu, oce agai, the coefficiet o the x equal what it i uig the earlier trategy. The cotat term will be the ame a well. 7. For coveiece, reorder the variable o that X = [i, P d, P, P, Y]. The three depedet variable are E d, E, ad E, ad Y = E d + E + E. The coefficiet vector are The um of the three vector i b d = (X X) X E d, b = (X X) X E, ad b = (X X) X E. b = (X X) X [E d + E + E ] = (X X) X Y. Now, Y i the lat colum of X, o the precedig um i the vector of leat quare coefficiet i the regreio of the lat colum of X o all of the colum of X, icludig the lat. Of coure, we get a perfect fit. I additio, X [E d + E + E ] i the lat colum of X X, o the matrix product i equal to the lat colum of a idetity matrix. Thu, the um of the coefficiet o all variable except icome i 0, while that o icome i. 8. Let R deote the adjuted R i the full regreio o variable icludig, ad let R deote the adjuted R i the hort regreio o - variable whe i omitted. Let R ad R deote their uadjuted couterpart. The, R = e e/y M 0 y R = e e /y M 0 y where e e i the um of quared reidual i the full regreio, e e i the (larger) um of quared reidual i the regreio which omit, ad y M 0 y = Σ i (y i y ). The, R = [( )/( )]( R ) ad R = [( )/(-( ))]( R ). The differece i the chage i the adjuted R whe i added to the regreio, R R = [( )/( + )][e e /y M 0 y] [( )/( )][e e/y M 0 y]. The differece i poitive if ad oly if the ratio i greater tha. After cacellig term, we require for the adjuted R to icreae that e e /( + )]/[( )/e e] >. From the previou problem, we have that e e = e e + b M ), where M i defied above ad b k i the leat quare coefficiet i the full regreio of y o X ad. Makig the ubtitutio, we require [(e e + b M )) ( )]/[( )e e + e e] >. Sice e e = ( ), thi implifie to [e e + b M )]/ [e e + ] >. Sice all term are poitive, the fractio i greater tha oe if ad oly b M ) > or b M / ) >. The deomiator i the etimated variace of b k, o the reult i proved. 0 Pearo Educatio, Ic. Publihig a Pretice Hall

6 6 Greee Ecoometric Aalyi, Seveth Editio 9. Thi R mut be lower. The um of quare aociated with the coefficiet vector which omit the cotat term mut be higher tha the oe which iclude it. We ca write the coefficiet vector i the regreio without a cotat a c = (0,b * ) where b * = (W W) W y, with W beig the other colum of X. The, the reult of the previou exercie applie directly. 0. We ue the otatio Var[.] ad Cov[.] to idicate the ample variace ad covariace. Our iformatio i Var[N] =, Var[D] =, Var[Y] =. Sice C = N + D, Var[C] = Var[N] + Var[D] + Cov[N, D] = ( + Cov[N, D]). From the regreio, we have Cov[C, Y]/Var[Y] = Cov[C, Y] = 0.8. But, Cov[C, Y] = Cov[N, Y] + Cov[D, Y]. Alo, Cov[C, N]/Var[N] = Cov[C, N] = 0.5, but, Cov[C, N] = Var[N] + Cov[N, D] = + Cov[N, D], o Cov[N, D] = 0.5, o that Var[C] = ( + 0.5) =. Ad, Cov[D, Y]/Var[Y] = Cov[D, Y] = 0.4. Sice Cov[C, Y] = 0.8 = Cov[N, Y] + Cov[D, Y], Cov[N, Y] = 0.4. Fially, Cov[C, D] = Cov[N, D] + Var[D] = = 0.5. Now, i the regreio of C o D, the um of quared reidual i ( ){Var[C] (Cov[C,D]/ Var[D]) Var[D]} baed o the geeral regreio reult Σe = Σ(y i y) b Σ (x i x ). All of the eceary figure were obtaied above. Iertig thee ad = 0 produce a um of quared reidual of 5.. The relevat ubmatrice to be ued i the calculatio are Ivetmet Cotat GNP Iteret Ivetmet * Cotat GNP Iteret The ivere of the lower right 3 3 block i (X X), (X X) = The coefficiet vector i b = (X X) X y = ( ,.356, ). The total um of quare i y y =.6365, o we ca obtai e e = y y b X y. X y i give i the top row of the matrix. Makig the ubtitutio, we obtai e e = = To compute R, we require Σ i (y i y ) = (3.05/5) = , o R =.0036/ = The reult caot be correct. Sice log S/N = log S/Y + log Y/N by imple, exact algebra, the ame reult mut apply to the leat quare regreio reult. That mea that the ecod equatio etimated mut equal the firt oe plu log Y/N. Lookig at the equatio, that mea that all of the coefficiet would have to be idetical ave for the ecod, which would have to equal it couterpart i the firt equatio, plu. Therefore, the reult caot be correct. I a exchage betwee Leff ad Arthur Goldberger that appeared later i the ame joural, Leff argued that the differece wa a imple roudig error. You ca ee that the reult i the ecod equatio reemble thoe i the firt, but ot eough o that the explaatio i credible. Further dicuio about the data themelve appeared i a ubequet dicuio. [See Goldberger (973) ad Leff (973).] 0 Pearo Educatio, Ic. Publihig a Pretice Hall

7 Chapter 3 Leat Square 7 Applicatio? Chapter 3 Applicatio Read $ (Data appear i the text.) Namelit ; X = oe,educ,exp,ability$ Namelit ; X = mothered,fathered,ib$? a. Regre ; Lh = wage ; Rh = x$ Ordiary leat quare regreio LHS=WAGE Mea = Stadard deviatio = WTS=oe Number of oberv. = 5 Model ize Parameter = 4 Degree of freedom = Reidual Sum of quare = Stadard error of e = Fit R-quared =.8335 Adjuted R-quared = E-0 Model tet F[ 3, ] (prob) =.8 (.5080) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Cotat EDUC EXP ABILITY ? b. Regre ; Lh = wage ; Rh = x,x$ Ordiary leat quare regreio LHS=WAGE Mea = Stadard deviatio = WTS=oe Number of oberv. = 5 Model ize Parameter = 7 Degree of freedom = 8 Reidual Sum of quare =.4566 Stadard error of e = Fit R-quared =.5634 Adjuted R-quared = Model tet F[ 6, 8] (prob) =.4 (.340) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Cotat EDUC EXP ABILITY MOTHERED FATHERED SIBS Pearo Educatio, Ic. Publihig a Pretice Hall

8 8 Greee Ecoometric Aalyi, Seveth Editio? c. Regre ; Lh = mothered ; Rh = x ; Re = med $ Regre ; Lh = fathered ; Rh = x ; Re = fed $ Regre ; Lh = ib ; Rh = x ; Re = ib $ Namelit ; XS = med,fed,ib $ Matrix ; lit ; Mea(XS) $ Matrix Reult ha 3 row ad colum D D D-6 The mea are (eetially) zero. The um mut be zero, a thee ew variable are orthogoal to the colum of X. The firt colum i X i a colum of oe, o thi mea that thee reidual mut um to zero.? d. Namelit ; X = X,X $ Matrix ; i = iit(,,) $ Matrix ; M0 = ide() - /*i*i' $ Matrix ; b = <X'X>*X'wage$ Calc ; lit ; ym0y =(N-)*var(wage) $ Matrix ; lit ; cod = /ym0y * b'*x'*m0*x*b $ Matrix COD ha row ad colum Matrix ; e = wage - X*b $ Calc ; lit ; cod = - /ym0y * e'e $ COD =.5634 The R quared i the ame uig either method of computatio. Calc RSQAD =.5335? Now drop the cotat Namelit ; X0 = educ,exp,ability,x $ Matrix ; i = iit(,,) $ Matrix ; M0 = ide() - /*i*i' $ ; lit ; RqAd = - (-)/(-col(x))*(-cod)$ Matrix ; b0 = <X0'X0>*X0'wage$ Matrix ; lit ; cod = /ym0y * b0'*x0'*m0*x0*b0 $ Matrix COD ha row ad colum Matrix ; e0 = wage - X0*b0 $ Calc ; lit ; cod = - /ym0y * e0'e0 $ Lited Calculator Reult COD = The R quared ow chage depedig o how it i computed. It alo goe up, completely artificially. 0 Pearo Educatio, Ic. Publihig a Pretice Hall

9 Chapter 3 Leat Square 9? e. The R quared for the full regreio appear immediately below.? f. Regre ; Lh = wage ; Rh = X,X $ Ordiary leat quare regreio WTS=oe Number of oberv. = 5 Model ize Parameter = 7 Degree of freedom = 8 Fit R-quared =.5634 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Cotat EDUC EXP ABILITY MOTHERED FATHERED SIBS Regre ; Lh = wage ; Rh = X,XS $ Ordiary leat quare regreio WTS=oe Number of oberv. = 5 Model ize Parameter = 7 Degree of freedom = 8 Fit R-quared =.5634 Adjuted R-quared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Cotat EDUC EXP ABILITY MEDS D-4 FEDS D-4 SIBSS D-6 I the firt et of reult, the firt coefficiet vector i b = (X M X ) X M y ad b = (X M X ) X M y. I the ecod regreio, the ecod et of regreor i M X, o b = (X M X ) X M y where M = I (M X )[(M X ) (M X )] (M X ). Thu, becaue the M matrix i differet, the coefficiet vector i differet. The ecod et of coefficiet i the ecod regreio i b = [(M X ) M (M X )] (M X )M y = (X M X ) X M y becaue M i idempotet. 0 Pearo Educatio, Ic. Publihig a Pretice Hall