Econometric Analysis Sixth Edition

Transcription

1 Solutios ad Applicatios Maual Ecoometric Aalysis Sixth Editio William H. Greee New York Uiversity Pretice Hall, Upper Saddle River, New Jersey 07458

2 Cotets ad Notatio This book presets solutios to the ed of chapter exercises ad applicatios i Ecoometric Aalysis. There are o exercises i the text for Appedices A E. For the istructor or studet who is iterested i exercises for this material, I have icluded a umber of them, with solutios, i this book. The various computatios i the solutios ad exercises are doe with the NLOGIT Versio 4.0 computer package (Ecoometric Software, Ic., Plaiview New York, I order to cotrol the legth of this documet, oly the solutios ad ot the questios from the exercises ad applicatios are show here. I some cases, the umerical solutios for the i text examples show here differ slightly from the values give i the text. This occurs because i geeral, the derivative computatios i the text are doe usig the digits show i the text, which are rouded to a few digits, while the results show here are based o iteral computatios by the computer that use all digits. Chapter Itroductio Chapter The Classical Multiple Liear Regressio Model Chapter 3 Least Squares 3 Chapter 4 Statistical Properties of the Least Squares Estimator 0 Chapter 5 Iferece ad Predictio 9 Chapter 6 Fuctioal Form ad Structural Chage 30 Chapter 7 Specificatio Aalysis ad Model Selectio 40 Chapter 8 The Geeralized Regressio Model ad Heteroscedasticity 44 Chapter 9 Models for Pael Data 54 Chapter 0 Systems of Regressio Equatios 67 Chapter Noliear Regressios ad Noliear Least Squares 80 Chapter Istrumetal Variables Estimatio 85 Chapter 3 Simultaeous-Equatios Models 90 Chapter 4 Estimatio Frameworks i Ecoometrics 97 Chapter 5 Miimum Distace Estimatio ad The Geeralized Method of Momets 0 Chapter 6 Maximum Likelihood Estimatio 05 Chapter 7 Simulatio Based Estimatio ad Iferece 7 Chapter 8 Bayesia Estimatio ad Iferece 0 Chapter 9 Serial Correlatio Chapter 0 Models with Lagged Variables 8 Chapter Time-Series Models 3 Chapter Nostatioary Data 3 Chapter 3 Models for Discrete Choice 36 Chapter 4 Trucatio, Cesorig ad Sample Selectio 4 Chapter 5 Models for Evet Couts ad Duratio 47 Appedix A Matrix Algebra 55 Appedix B Probability ad Distributio Theory 6 Appedix C Estimatio ad Iferece 7 Appedix D Large Sample Distributio Theory 83 Appedix E Computatio ad Optimizatio 84

3 I the solutios, we deote: scalar values with italic, lower case letters, as i a, colum vectors with boldface lower case letters, as i b, row vectors as trasposed colum vectors, as i b, matrices with boldface upper case letters, as i M or Σ, sigle populatio parameters with Greek letters, as i θ, sample estimates of parameters with Roma letters, as i b as a estimate of β, sample estimates of populatio parameters with a caret, as i αˆ or β ˆ, cross sectio observatios with subscript i, as i y i, time series observatios with subscript t, as i z t ad pael data observatios with x it or x i,t- whe the comma is eeded to remove ambiguity. Observatios that are vectors are deoted likewise, for example, x it to deote a colum vector of observatios. These are cosistet with the otatio used i the text.

4 Chapter Itroductio There are o exercises or applicatios i Chapter.

5 Chapter The Classical Multiple Liear Regressio Model There are o exercises or applicatios i Chapter.

6 Chapter 3 Least Squares Exercises x. Let X = x (a) The ormal equatios are give by (3-), X'e = 0 (we drop the mius sig), hece for each of the colums of X, x k, we kow that x k e = 0. This implies that Σ e = 0 ad Σ xe = 0. (b) Use Σ e to coclude from the first ormal equatio that a = y bx. i= i (c) We kow that i= i 0 i= xei x i= ei 0 ( )( ) i= xi x yi a bxi i= i i= i i Σ e = ad Σ xe = 0. It follows the that Σ ( ) x x e = because Σ = Σ =. Substitute e i to obtai i= i i Σ =0 or Σi= ( xi x)( yi y b( xi x)) =0 Σi= ( xi x)( yi y) The, Σi= ( xi x)( yi y) = bσi= ( xi x)( xi x)) so b=. Σ ( x x) i= i i= i i 0 (d) The first derivative vector of e e is -X e. (The ormal equatios.) The secod derivative matrix is (e e)/ b b = X X. We eed to show that this matrix is positive defiite. The diagoal elemets are ad Σi= xi which are clearly both positive. The determiat is ()( Σ i= xi )-( Σ i= xi) = 4Σi= xi -4( x ) = 4[ ( Σi= xi ) x ] = 4[( Σi= ( xi x)]. Note that a much simpler proof appears after (3-6).. Write c as b + (c - b). The, the sum of squared residuals based o c is (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 is zero, as (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 side ca be writte as d d where d = X(c - b), so it is ecessarily positive. This cofirms what we kew at the outset, least squares is least squares. 3. The residual vector i the regressio of y o X is M X y = [I - X(X X) - X ]y. The residual vector i the regressio of y o Z is M Z y = [I - Z(Z Z) - Z ]y = [I - XP((XP) (XP)) - (XP) )y = [I - XPP - (X X) - (P ) - P X )y = M X y Sice the residual vectors are idetical, the fits must be as well. Chagig the uits of measuremet of the regressors is equivalet to postmultiplyig by a diagoal P matrix whose kth diagoal elemet is the scale factor to be applied to the kth variable ( if it is to be uchaged). It follows from the result above that this will ot chage the fit of the regressio. 4. I the regressio of y o i ad X, the coefficiets o X are b = (X M 0 X) - X M 0 y. M 0 = I - i(i i) - i is the matrix which trasforms observatios ito deviatios from their colum meas. Sice M 0 is idempotet ad symmetric we may also write the precedig as [(X M 0 )(M 0 X)] - (X M 0 )(M 0 y) which implies that the 3

7 regressio of M 0 y o M 0 X produces the least squares slopes. If oly X is trasformed to deviatios, we would compute [(X M 0 )(M 0 X)] - (X M 0 )y but, of course, this is idetical. However, if oly y is trasformed, the result is (X X) - X M 0 y which is likely to be quite differet. 5. What is the result of the matrix product M M where M is defied i (3-9) ad M is defied i (3-4)? M M = (I - X (X X ) - X )(I - X(X X) - X ) = M - X (X X ) - X M There is o eed to multiply out the secod term. Each colum of MX is the vector of residuals i the regressio of the correspodig colum of X o all of the colums i X. Sice that x is oe of the colums i X, this regressio provides a perfect fit, so the residuals are zero. Thus, MX is a matrix of zeroes which implies that M M = M. 6. The origial X matrix has rows. We add a additioal row, x s. The ew y vector likewise has a X y additioal elemet. Thus, Xs, = ad ys, =. s y The ew coefficiet vector is x s b,s = (X,s X,s ) - (X,s y,s ). The matrix is X,s X,s = X X + x s x s. To ivert this, use (A -66); ( X X, s, s) = ( XX ) ( ) ( ) s s + s( ) XX xx XX. The vector is x X X xs (X,s y,s ) = (X y ) + x s y s. Multiply out the four terms to get (X,s X,s ) - (X,s y,s ) = b ( XX ) xxb s s + ( XX ) x s y s ( XX ) xx s s( XX ) + x ( X X ) x + x ( X X ) x s s = x ( X X ) x b + ( XX ) x s y s + ( ) s s x s X X xs ( XX ) s s s s xy ( XX ) xxb s s + x ( X X ) x s s x s( X X) x s b + ( XX ) xsy s + x s( X ( XX ) xxb X) xs + x s( X X) xs b + ( XX ) xsy s ( XX ) xxb s s + x ( X X ) x + x ( X X ) x s s b + ( XX ) ( xs ys xb s ) + x ( X X ) x s s s s s s x s y s i x 0 0 yo 7. Defie the data matrix as follows: X=, [ ] ad. (The subscripts 0 = X = X X y = y m o the parts of y refer to the observed ad missig rows of X. We will use Frish-Waugh to obtai the first two colums of the least squares coefficiet vector. b =(X MX ) - (X My). Multiplyig it out, we fid that M = a idetity matrix save for the last diagoal elemet that is equal to X M X = XX X. This just drops the last observatio. X M y is computed likewise. Thus, X 0 the coeffiets o the first two colums are the same as if y 0 had bee liearly regressed o X. The deomoator of R is differet for the two cases (drop the observatio or keep it with zero fill ad the dummy variable). For the first strategy, the mea of the - observatios should be differet from the mea of the full uless the last observatio happes to equal the mea of the first -. For the secod strategy, replacig the missig value with the mea of the other - observatios, we ca deduce the ew slope vector logically. Usig Frisch-Waugh, we ca replace the colum of x s with deviatios from the meas, which the turs the last observatio to zero. Thus, oce agai, the coefficiet o the x equals what it is usig the earlier strategy. The costat term will be the same as well. 4

8 8. For coveiece, reorder the variables so that X = [i, P d, P, P s, Y]. The three depedet variables are E d, E, ad E s, ad Y = E d + E + E s. The coefficiet vectors are b d = (X X) - X E d, b = (X X) - X E, ad b s = (X X) - X E s. The sum of the three vectors is b = (X X) - X [E d + E + E s ] = (X X) - X Y. Now, Y is the last colum of X, so the precedig sum is the vector of least squares coefficiets i the regressio of the last colum of X o all of the colums of X, icludig the last. Of course, we get a perfect fit. I additio, X [E d + E + E s ] is the last colum of X X, so the matrix product is equal to the last colum of a idetity matrix. Thus, the sum of the coefficiets o all variables except icome is 0, while that o icome is. 9. Let RK deote the adjusted R i the full regressio o K variables icludig x k, ad let R deote the adjusted R i the short regressio o K- variables whe x k is omitted. Let R K ad R deote their uadjusted couterparts. The, R = - e e/y M 0 K y R = - e e /y M 0 y where e e is the sum of squared residuals i the full regressio, e e is the (larger) sum of squared residuals i the regressio which omits x k, ad y M 0 y = Σ i (y i - y ) The, RK = - [(-)/(-K)]( - RK ) ad R = - [(-)/(-(K-))]( - R ). The differece is the chage i the adjusted R whe x k is added to the regressio, RK - R = [(-)/(-K+)][e e /y M 0 y] - [(-)/(-K)][e e/y M 0 y]. The differece is positive if ad oly if the ratio is greater tha. After cacellig terms, we require for the adjusted R to icrease that e e /(-K+)]/[(-K)/e e] >. From the previous problem, we have that e e = e e + b K (x k M x k ), where M is defied above ad b k is the least squares coefficiet i the full regressio of y o X ad x k. Makig the substitutio, we require [(e e + b K (x k M x k ))(-K)]/[(-K)e e + e e] >. Sice e e = (-K)s, this simplifies to [e e + b K (x k M x k )]/[e e + s ] >. Sice all terms are positive, the fractio is greater tha oe if ad oly b K (x k M x k ) > s or b K (x k M x k /s ) >. The deomiator is the estimated variace of b k, so the result is proved. 0. This R must be lower. The sum of squares associated with the coefficiet vector which omits the costat term must be higher tha the oe which icludes it. We ca write the coefficiet vector i the regressio without a costat as c = (0,b * ) where b * = (W W) - W y, with W beig the other K- colums of X. The, the result of the previous exercise applies directly.. We use the otatio Var[.] ad Cov[.] to idicate the sample variaces ad covariaces. Our iformatio is Var[N] =, Var[D] =, Var[Y] =. Sice C = N + D, Var[C] = Var[N] + Var[D] + Cov[N,D] = ( + Cov[N,D]). From the regressios, we have Cov[C,Y]/Var[Y] = Cov[C,Y] =.8. But, Cov[C,Y] = Cov[N,Y] + Cov[D,Y]. Also, Cov[C,N]/Var[N] = Cov[C,N] =.5, but, Cov[C,N] = Var[N] + Cov[N,D] = + Cov[N,D], so Cov[N,D] = -.5, so that Var[C] = ( + -.5) =. Ad, Cov[D,Y]/Var[Y] = Cov[D,Y] =.4. Sice Cov[C,Y] =.8 = Cov[N,Y] + Cov[D,Y], Cov[N,Y] =.4. Fially, Cov[C,D] = Cov[N,D] + Var[D] = =.5. Now, i the regressio of C o D, the sum of squared residuals is (-){Var[C] - (Cov[C,D]/Var[D]) Var[D]} 5

9 based o the geeral regressio result Σe = Σ(y i - y ) - b Σ(x i -x ). All of the ecessary figures were obtaied above. Isertig these ad - = 0 produces a sum of squared residuals of 5.. The relevat submatrices to be used i the calculatios are Ivestmet Costat GNP Iterest Ivestmet * Costat GNP Iterest The iverse of the lower right 3 3 block is (X X) -, (X X) - = The coefficiet vector is b = (X X) - X y = ( ,.356, ). The total sum of squares is y y =.6365, so we ca obtai e e = y y - b X y. X y is give i the top row of the matrix. Makig the substitutio, we obtai e e = = To compute R, we require Σ i (x i - y ) = (3.05/5) = , so R = / = The results caot be correct. Sice log S/N = log S/Y + log Y/N by simple, exact algebra, the same result must apply to the least squares regressio results. That meas that the secod equatio estimated must equal the first oe plus log Y/N. Lookig at the equatios, that meas that all of the coefficiets would have to be idetical save for the secod, which would have to equal its couterpart i the first equatio, plus. Therefore, the results caot be correct. I a exchage betwee Leff ad Arthur Goldberger that appeared later i the same joural, Leff argued that the differece was simple roudig error. You ca see that the results i the secod equatio resemble those i the first, but ot eough so that the explaatio is credible. Further discussio about the data themselves appeared i subsequet idscussio. [See Goldberger (973) ad Leff (973).] 4. A proof of Theorem 3. provides a geeral statemet of the observatio made after (3-8). The couterpart for a multiple regressio to the ormal equatios precedig (3-7) is b + bσ ixi + b3σ ixi bkσ ixik =Σiyi bσ ixi + bσ ixi + b3σ ixixi bkσ ixixik =Σixiyi... bσ i xik + bσ i xik xi + b3σ i xik xi bk Σ i xik =Σ i xik yi. As before, divide the first equatio by, ad maipulate to obtai the solutio for the costat term, b = y bx... bk xk. Substitute this ito the equatios above, ad rearrage oce agai to obtai the equatios for the slopes, bσi( xi x) + b3σi( xi x)( xi3 x3) bkσi( xi x)( xik xk) =Σi( xi x)( yi y) bσi( xi3 x3)( xi x) + b3σi( xi3 x3) bkσi( xi3 x3)( xik xk) =Σi( xi3 x3)( yi y)... bσi ( xik xk )( xi x) + b3σi ( xik xk )( xi3 x3) bk Σi ( xik xk ) =Σi ( xik xk )( yi y). If the variables are ucorrelated, the all cross product terms of the form Σi( x ij x j)( x x ) ik k will equal zero. This leaves the solutio, bσi( xi x) =Σi( xi x)( yi y) b Σ ( x x ) =Σ ( x x )( y y) 3 i i3 3 i i3 3 i... b Σ x x =Σ x x y y), K i ( ik K ) i ( ik K )( i which ca be solved oe equatio at a time for [ ( )( )] ( ) b = Σ x x y y Σ x x k i ik k i i ik k, k =,...,K. 6

10 Each of these is the slope coefficiet i the simple of y o the respective variable. Applicatio?====================== =================================================? Chapter 3 Applicatio?======================================================================= Read $ (Data appear i the text.) Namelist ; X = oe,educ,exp,ability$ Namelist ; X = mothered,fathered,sibs$?====================================== =================================? a.?======================================================================= Regress ; Lhs = wage ; Rhs = x$ Ordiary least squares regressio LHS=WAGE Mea = Stadard deviatio = WTS=oe Number of observs. = 5 Model size Parameters = 4 Degrees of freedom = Residuals Sum of squares = Stadard error of e = Fit R-squared =.8335 Adjusted R-squared = E-0 Model test F[ 3, ] (prob) =.8 (.5080) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat EDUC EXP ABILITY ?=======================================================================? b.?======================================================================= Regress ; Lhs = wage ; Rhs = x,x$ Ordiary least squares regressio LHS=WAGE Mea = Stadard deviatio = WTS=oe Number of observs. = 5 Model size Parameters = 7 Degrees of freedom = 8 Residuals Sum of squares =.4566 Stadard error of e = Fit R-squared =.5634 Adjusted R-squared = Model test F[ 6, 8] (prob) =.4 (.340) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat EDUC EXP ABILITY MOTHERED FATHERED SIBS ?=======================================================================? c.?======================================================================= 7

11 Regress ; Lhs = mothered ; Rhs = x ; Res = meds $ Regress ; Lhs = fathered ; Rhs = x ; Res = feds $ Regress ; Lhs = sibs ; Rhs = x ; Res = sibss $ Namelist ; XS = meds,feds,sibss $ Matrix ; list ; Mea(XS) $ Matrix Result has 3 rows ad colums D D D-6 The meas are (essetially) zero. The sums must be zero, as these ew variables are orthogoal to the colums of X. The first colum i X is a colum of oes, so this meas that these residuals must sum to zero.?=======================================================================? d.?======================================================================= Namelist ; X = X,X $ Matrix ; i = iit(,,) $ Matrix ; M0 = ide() - /*i*i' $ Matrix ; b = <X'X>*X'wage$ Calc ; list ; ym0y =(N-)*var(wage) $ Matrix ; list ; cod = /ym0y * b'*x'*m0*x*b $ Matrix COD has rows ad colums Matrix ; e = wage - X*b $ Calc ; list ; cod = - / ym0y * e'e $ COD =.5634 The R squared is the same usig either method of computatio. Calc ; list ; RsqAd = - (-)/(-col(x))*(-cod)$ RSQAD =.5335? Now drop the costat Namelist ; X0 = educ,exp,ability,x $ Matrix ; i = iit(,,) $ Matrix ; M0 = ide() - /*i*i' $ Matrix ; b0 = <X0'X0>*X0'wage$ Matrix ; list ; cod = /ym0y * b0' *X0'*M0*X0*b0 $ Matrix COD has rows ad colums ; e0 = wage - X0*b0 $ ; list ; cod = - /ym0y * e0'e0 $ Matrix Calc Listed Calculator Results COD = The R squared ow chages depedig o how it is computed. It also goes up, completely artificially.?=======================================================================? e.?======================================================================= The R squared for the full regressio appears immediately below.? f. Regress ; Lhs = wage ; Rhs = X,X $ Ordiary least squares regressio WTS=oe Number of observs. = 5 Model size Parameters = 7 Degrees of freedom = 8 Fit R-squared =

12 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat EDUC EXP ABILITY MOTHERED FATHERED SIBS Regress ; Lhs = wage ; Rhs = X,XS $ Ordiary least squares regressio WTS=oe Number of observs. = 5 Model size Parameters = 7 Degrees of freedom = 8 Fit R-squared =.5634 Adjusted R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat EDUC EXP ABILITY MEDS D-4 FEDS D-4 SIBSS D-6 I the first set of results, the first coefficiet vector is - b = (X M X ) X M y ad - b = (X M X ) X M y I the secod regressio, the secod set of regressors is M X, so - b )] - = (X M X ) X M y where M = I (M X )[(M X ) (M X (M X ) Thus, because the M matrix is differet, the coefficiet vector is differe t. The secod set of coefficiets i the secod regressio is b = [(M X ) M (M X )] - (M X )M y = (X M X ) - X M y because M is idempotet. 9

13 Chapter 4 Statistical Properties of the Least Squares Estimator Exercises. Cosider the optimizatio problem of miimizig the variace of the weighted estimator. If the estimate is to be ubiased, it must be of the form c ˆθ + c ˆθ where c ad c sum to. Thus, c = - c. The fuctio to miimize is Mi c L * = c v + ( - c ) v. The ecessary coditio is L * / c = c v - ( - c )v = 0 which implies c = v / (v + v ). A more ituitively appealig form is obtaied by dividig umerator ad deomiator by v v to obtai c = (/v ) / [/v + /v ]. Thus, the weight is proportioal to the iverse of the variace. The estimator with the smaller variace gets the larger weight.. First, ˆβ = c y = c x + c ε. So E[ ˆβ ] = βc x ad Var[ ˆβ ] = σ c c. Therefore, MSE[ ˆβ ] = β [c x - ] + σ c c. To miimize this, we set MSE[ ˆβ ]/ c = β [c x - ]x + σ c = 0. Collectig terms, β (c x - )x = -σ c Premultiply by x to obtai β (c x - )x x = -σ x c or c x = β x x / (σ + β x x). The, c = [(-β /σ )(c x - )]x, so c = [/(σ /β + x x)]x. The, ˆβ = c y = x y / (σ /β + x x). The expected value of this estimator is E[ ˆβ ] = βx x / (σ /β + x x) so E[ ˆβ ] - β = β(-σ /β ) / (σ /β + x x) = -(σ /β) / (σ /β + x x) while its variace is Var[x (xβ + ε) / (σ /β + x x)] = σ x x / (σ /β + x x) The mea squared error is the variace plus the squared bias, MSE[ ˆβ ] = [σ 4 /β + σ x x]/[σ /β + x x]. The ordiary least squares estimator is, as always, ubiased, ad has variace ad mea squared error MSE(b) = σ /x x. The ratio is take by dividig each term i the umerator MSE βˆ 4 ( σ / β )/( σ / xx ' ) + σ xx/ ' ( σ / xx ' ) = ΜSΕ( b) σ / β + xx ' ( ) = [σ x x/β + (x x) ]/(σ /β + x x) = x x[σ /β + x x]/(σ /β + x x) = x x/(σ /β + x x) Now, multiply umerator ad deomiator by β /σ to obtai MSE[β ˆ ]/MSE[b] = β x x/σ /[ + β x x/σ ] = τ /[ + τ ] As τ, the ratio goes to oe. This would follow from the result that the biased estimator ad the ubiased estimator are covergig to the same thig, either as σ goes to zero, i which case the MMSE estimator is the same as OLS, or as x x grows, i which case both estimators are cosistet. 0

14 3. The OLS estimator fit without a costat term is b = x y / x x. Assumig that the costat term is, i fact, zero, the variace of this estimator is Var[b] = σ /x x. If a costat term is icluded i the regressio, the, b = Σ ( x x)( y y) / Σ ( x x ) i= i i The appropriate variace is σ / Σ ( x x as always. The ratio of these two is i= i ) i= Var[b]/Var[b ] = [σ /x x] / [σ / ( ) But, Σ ( x x = x x + x i= i ) i Σ x x ] so the ratio is Var[b]/Var[b ] = [x x + x ]/x x = - x /x x = - { x /[S xx + x ]} < It follows that fittig the costat term whe it is uecessary iflates the variace of the least squares estimator if the mea of the regressor is ot zero. 4. We could write the regressio as y i = (α + λ) + βx i + (ε i - λ) = α * + βx i + ε i *. The, we kow that E[ε i * ] = 0, ad that it is idepedet of x i. Therefore, the secod form of the model satisfies all of our assumptios for the classical regressio. Ordiary least squares will give ubiased estimators of α * ad β. As log as λ is ot zero, the costat term will differ from α. 5. Let the costat term be writte as a = Σ i d i y i = Σ i d i (α + βx i + ε i ) = ασ i d i + βσ i d i x i + Σ i d i ε i. I order for a to be ubiased for all samples of x i, we must have Σ i d i = ad Σ i d i x i = 0. Cosider, the, miimizig the variace of a subject to these two costraits. The Lagragea is L * = Var[a] + λ (Σ i d i - ) + λ Σ i d i x i where Var[a] = Σ i σ d i. Now, we miimize this with respect to d i, λ, ad λ. The (+) ecessary coditios are L * / d i = σ d i + λ + λ x i, L * / λ = Σ i d i -, L * / λ = Σ i d i x i The first equatio implies that d i = [-/(σ )](λ + λ x i ). Therefore, Σ i d i = = [-/(σ )][λ + (Σ i x i )λ ] ad Σ i d i x i = 0 = [-/(σ )][(Σ i x i )λ + (Σ i x i )λ ]. We ca solve these two equatios for λ ad λ by first multiplyig both equatios by -σ the writig the Σ x i i λ resultig equatios as The solutio is Σixi Σix i = λ - σ σ λ. λ 0 = Σ ixi. Σixi Σixi 0 Note, first, that Σ i x i = x. Thus, the determiat of the matrix is Σ i x i - ( x ) = (Σ i x i - x ) = S xx λ Σixi x σ where S xx Σi= ( xi x ). The solutio is, therefore, = λ Sxx x 0 0 or λ = (-σ )(Σ i x i /)/S xx λ = (σ x )/S xx The, d i = [Σ i x i / - x x i ]/S xx This simplifies if we writeσx i = S xx + x, so Σ i x i / = S xx / + x. The, d i = / + x ( x - x i )/S xx, or, i a more familiar form, d i = / - x (x i - x )/S xx. Σ x x y /S xx = y - b x which was to be show. This makes the itercept term Σ i d i y i = (/)Σ i y i - x ( ) i= i= i i 6. Let q = E[Q]. The, q = α + βp, or P = (-α/β) + (/β)q. Usig a well kow result, for a liear demad curve, margial reveue is MR = (-α/β) + (/β)q. The profit maximizig output is that at which margial reveue equals margial cost, or 0. Equatig MR to 0 ad solvig for q produces q = α/ + 5β, so we require a cofidece iterval for this combiatio of the parameters. The least squares regressio results are ˆQ = The estimated covariace matrix of the coefficiets is. The estimate of q is The estimate of the variace of ˆq is (/4) ( ) + 5( ) or , so the estimated stadard error is i

15 The 95% cutoff value for a t distributio with 3 degrees of freedom is.6, so the cofidece iterval is (.576) to (.576) or 5.04 to a. The sample meas are (/00) times the elemets i the first colum of X'X. The sample covariace matrix for the three regressors is obtaied as (/99)[(X X) ij -00 x i x j ] Sample Var[x] = The simple correlatio matrix is b. The vector of slopes is (X X) - X y = [-.40, 6.3, 5.90, -7.55]. c. For the three short regressios, the coefficiet vectors are () oe, x, ad x : [-.3,.8,.] () oe, x, ad x 3 [-.0696,.9, 4.05] (3) oe, x, ad x 3 : [-.067, -.098, 4.358] d. The magificatio factors are for x : [(/(99(.077)) /.9] =.094 for x : [(/99(.75596)) /.] =.09 for x 3 : [(/99( ))/ 4.9] =.068. e. The problem variable appears to be x 3 sice it has the lowest magificatio factor. I fact, all three are highly itercorrelated. Although the simple correlatios are ot excessively high, the three multiple correlatios are.99 for x o x ad x 3,.988 for x o x ad x 3, ad.99 for x 3 o x ad x. 8. We cosider two regressios. I the first, y is regressed o K variables, X. The variace of the least squares estimator, b = (X X) - X y, Var[b] = σ (X X) -. I the secod, y is regressed o X ad a additioal variable, z. Usig results for the partitioed regressio, the coefficiets o X whe y is regressed o X ad z are b.z = (X M z X) - X M z y where M z = I - z(z z) - z. The true variace of b.z is the upper left K K matrix i XX ' Xz ' Var[b,c] = s zx ' zx '. But, we have already foud this above. The submatrix is Var[b.z ] = s (X M z X) -. We ca show that the secod matrix is larger tha the first by showig that its iverse is smaller. (See (A-0).) Thus, as regards the true variace matrices (Var[b]) - - (Var[b.z ]) - = (/σ )z(z z) - z which is a oegative defiite matrix. Therefore Var[b] - is larger tha Var[b.z ] -, which implies that Var[b] is smaller. Although the true variace of b is smaller tha the true variace of b.z, it does ot follow that the estimated variace will be. The estimated variaces are based o s, ot the true σ. The residual variace estimator based o the short regressio is s = e e/( - K) while that based o the regressio which icludes z is s z = e.z e.z /( - K - ). The umerator of the secod is defiitely smaller tha the umerator of the first, but so is the deomiator. It is ucertai which way the compariso will go. The result is derived i the previous problem. We ca coclude, therefore, that if t ratio o c i the regressio which icludes z is larger tha oe i absolute value, the s z will be smaller tha s. Thus, i the compariso, Est.Var[b] = s (X X) - is based o a smaller matrix, but a larger scale factor tha Est.Var[b.z ] = s z (X M z X) -. Cosequetly, it is ucertai whether the estimated stadard errors i the short regressio will be smaller tha those i the log oe. Note that it is ot sufficiet merely for the result of the previous problem to hold, sice the relative sizes of the matrices also play a role. But, to take a polar case, suppose z ad X were ucorrelated. The, XNM z X equals XNX. The, the estimated variace of b.z would be less tha that of b without z eve though the true variace is the same (assumig the premise of the previous problem holds). Now, relax this assumptio while holdig the t ratio o c costat. The matrix i Var[b.z ] is ow larger, but the leadig scalar is ow smaller. Which way the product will go is ucertai. 9. The F ratio is computed as [b X Xb/K]/[e e/( - K)]. We substitute e = Mε, ad

16 b = β + (X X) - X ε = (X X) - X ε. The, F = [ε X(X X) - X X(X X) - X ε/k]/[ε Mε/( - K)] = [ε (I - M)ε/K]/[ε Mε/( - K)]. The exact expectatio of F ca be foud as follows: F = [(-K)/K][ε (I - M)ε]/[ε Mε]. So, its exact expected value is (-K)/K times the expected value of the ratio. To fid that, we ote, first, that Mε ad (I - M)ε are idepedet because M(I - M) = 0. Thus, E{[ε (I - M)ε]/[ε Mε]} = E[ε (I- M)ε] E{/[ε Mε]}. The first of these was obtaied above, E[ε (I - M)ε] = Kσ. The secod is the expected value of the reciprocal of a chi-squared variable. The exact result for the reciprocal of a chi-squared variable is E[/χ (-K)] = /( - K - ). Combiig terms, the exact expectatio is E[F] = ( - K) / ( - K - ). Notice that the mea does ot ivolve the umerator degrees of freedom. 0. We write b = β + (X X) - X ε, so b b = β β + ε X(X X) - (X X) - X ε + β (X X) - X ε. The expected value of the last term is zero, ad the first is ostochastic. To fid the expectatio of the secod term, use the trace, ad permute ε X iside the trace operator. Thus, E[β β] = β β + E[ε X(X X) - (X X) - X ε] = β β + E[tr{ε X(X X) - (X X) - X ε}] = β β + E[tr{(X X) - X εε X(X X) - }] = β β + tr[e{(x X) - X εε X(X X) - }] = β β + tr[(x X) - X E[εε ]X(X X) - ] = β β + tr[(x X) - X (σ I)X(X X) - ] = β β + σ tr[(x X) - X X(X X) - ] = β β + σ tr[(x X) - ] = β β + σ Σ k (/λ k ) The trace of the iverse equals the sum of the characteristic roots of the iverse, which are the reciprocals of the characteristic roots of X X.. The F ratio is computed as [b X Xb/K]/[e e/( - K)]. We substitute e = M, ad b = β + (X X) - X ε = (X X) - X ε. The, F = [ε X(X X) - X X(X X) - X ε/k]/[ε Mε/( - K)] = [ε (I - M)ε/K]/[ε Mε/( - K)]. The deomiator coverges to σ as we have see before. The umerator is a idempotet quadratic form i a ormal vector. The trace of (I - M) is K regardless of the sample size, so the umerator is always distributed as σ times a chi-squared variable with K degrees of freedom. Therefore, the umerator of F does ot coverge to a costat, it coverges to σ /K times a chi-squared variable with K degrees of freedom. Sice the deomiator of F coverges to a costat, σ, the statistic coverges to a radom variable, (/K) times a chi-squared variable with K degrees of freedom.. We ca write e i as e i = y i - b x i = (β x i + ε i ) - b x i = ε i + (b - β) x i We kow that plim b = β, ad x i is uchaged as icreases, so as, e i is arbitrarily close to ε i. 3. The estimator is y = (/)Σ i y i = (/)Σ i (μ + ε i ) = μ + (/)Σ i ε i. The, E[ y ] = μ+ (/)Σ i E[ε i ] = μ ad Var[ y ]= (/ )Σ i Σ j Cov[ε i,ε j ] = σ /. Sice the mea equals μ ad the variace vaishes as, y is mea square cosistet. I additio, sice y is a liear combiatio of ormally distributed variables, y has a ormal distributio with the mea ad variace give above i every sample. Suppose that ε i were ot ormally distributed. The, ( y -μ) = (/ )(Σ i ε i ) satisfies the requiremets for the cetral limit theorem. Thus, the asymptotic ormal distributio applies whether or ot the disturbaces have a ormal distributio. For the alterative estimator, ˆμ = Σ i w i y i, so E[ ˆμ ] = Σ i w i E[y i ] = Σ i w i μ = μσ i w i = μ ad Var[ ˆμ ]= Σ i w i σ = σ Σ i w i. The sum of squares of the weights is Σ i w i = Σ i i /[Σ i i] = [(+)(+)/6]/[(+)/] = [( + 3/ + /)]/[.5( + + )]. As, the fractio will be domiated by the term (/) ad will ted to zero. This establishes the cosistecy of this estimator. The last expressio also provides the asymptotic variace. The large sample variace ca be foud as Asy.Var[ ˆμ ] = (/)lim Var[ ( ˆμ - μ)]. For the estimator above, we ca use Asy.Var[ ˆμ ] = (/)lim Var[ ˆμ - μ] = (/)lim σ [( + 3

17 3/ + /)]/[.5( + + )] =.3333σ. Notice that this is uambiguously larger tha the variace of the sample mea, which is the ordiary least squares estimator. 4. To obtai the asymptotic distributio, write the result already i had as b = (β + Q - γ) + (X X) - X ε - Q - ε. We have established that plim b = β + Q - γ. For coveiece, let θ β deote β + Q - γ = plim b. Write the precedig i the form b - θ = (X X/) - (X ε/) - Q - γ. Sice plim(x X/) = Q, the large sample behavior of the right had side is the same as that of plim (b - θ) = Q - plim(x ε/) - Q - γ. That is, we may replace (X X/) with Q i our derivatio. The, we seek the asymptotic distributio of (b - θ) which is the same as that of [Q - plim(x ε/) - Q - γ] = Q - (/ ) Σi= ( xiεi - γ ). From this poit, the derivatio is exactly the same as that whe γ = 0, so there is o eed to redevelop the result. We may proceed directly to the same asymptotic distributio we obtaied before. The oly differece is that the least squares estimator estimates θ, ot β. 5. a. To solve this, we will use a extesio of Exercise 6 i Chapter 3 (addig oe row of data), ad the ecessary matrix result, (A-66b) i which B will be X m ad C will be I. Bypassig the matrix algebra, which will be essetially idetical to the earlier exercise, we have b c,m = b c + [I + X m (X c X c ) - X m ] - (X c X c ) - X m (y m X m b c ) But, i this case, y m is precisely X m b c, so the edig vector is zero. Thus, the coefficiet vector is the same. b. The model applies to the first c observatios, so b c is the least squares estimator for those observatios. Yes, it is ubiased. c. The residuals at the secod step are e c ad (X m b c X m b c ) = (e c, 0 ). Thus, the sum of squares is the same at both steps. d. The umerator of s is the same i both cases, however, for the secod oe, the degrees of freedom is larger. The first is ubiased, so the secod oe must be biased dowward. Applicatios?=======================================================================? Chapter 4 Applicatio?======================================================================= Read $ Year GasExp Pop Gasp Icome PNC PUC PPT PD PN PS Sample ; - 5 $ Create ; G = *gasexp/(gasp*pop)$ Create ; t = year - 95 $ Namelist ; X = oe,icome, gasp,pc,puc,ppt,pd,p,ps,t$?=======================================================================? a. Basic regressio?======================================================================= Regress ; Lhs = g ; Rhs = X $ Ordiary least squares regressio LHS=G Mea = Stadard deviatio = WTS=oe Number of observs. = 5 Model size Parameters = 0 Degrees of freedom = 4 Residuals Sum of squares = Stadard error of e = Fit R-squared =.9985 Adjusted R-squared = Model test F[ 9, 4] (prob) = (.0000) 4

18 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat INCOME D GASP PNC PUC PPT PD PN PS T ?=======================================================================? b. Hypothesis that b(nc) = b(uc) $?======================================================================= Calc ; list ; (b(4)-b(5))/sqr(varb(4,4)+varb(5,5)-*varb(4,5)) $ Listed Calculator Results Result = ?=======================================================================? c. Elasticities. I each case, elasticity = b*xbar/ybar?======================================================================= Calc ; g004 = g(5)$ Calc ; i004 = icome(5)$ Calc ; pg004 = gasp(5)$ Calc ; ppt004 = ppt(5)$ Calc ; list ; ei = b()*i004/g004 ; ep = b(3)*pg004/g004 ; eppt = b(6)*ppt004/g004$ Listed Calculator Results EI = EP = -.79 EPPT =.343?=======================================================================? d. Log regressio?======================================================================= Create ; logg = log(g) ; logpg = log(gasp) ; logi = log(icome) ; logpc=log(pc) ; logpuc = log(puc) ; logppt = log(ppt) ; logpd = log(pd) ; logp = log(p) ; logps = log(ps) $ Namelist ; LogX = oe,logi,logpg,logpc,logpuc,logppt,logpd,logp,logps,t$ Regress ; lhs = logg ; rhs = logx $ Ordiary least squares regressio LHS=LOGG Mea = Stadard deviatio =.3885 WTS=oe Number of observs. = 5 Model size Parameters = 0 Degrees of freedom = 4 Residuals Sum of squares =.3887E-0 Stadard error of e =.30994E-0 Fit R-squared = Adjusted R-squared = Model test F[ 9, 4] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGI LOGPG LOGPNC LOGPUC LOGPPT

19 LOGPD LOGPN LOGPS T ?=======================================================================? e. Correlatios of Price Variables?======================================================================= Namelist ; Prices = pc,puc,ppt,pd,p,ps$ Matrix ; list ; xcor(prices) $ Correlatio Matrix for Listed Variables PNC PUC PPT PD PN PS PNC PUC PPT PD PN PS ?=======================================================================? f. Reormalizatios of price variables?======================================================================= /* I the liear case, the coefficiets would be divided by the same scale factor, so that x*b would be uchaged, where x is a variable ad b is the coefficiet. I the logliear case, sice log(k*x)= log(k)+log(x), the reomalizatio would simply affect the costat term. The price coefficiets woulde be uchaged. */?=======================================================================? g. Oaxaca decompositio?======================================================================= Dates ; 953 $ Period ; $ Matrix ; xb0 = Mea(logx)$ Regress ; lhs = logg ; rhs = logx $ Matrix ; b0 = b ; v0 = varb $ Calc ; yb0 = ybar $ Period ; $ Matrix ; xb = mea(logx) $ Regress ; lhs = logg ; rhs = logx $ Matrix ; b = b ; v = varb $ Calc ; yb = ybar $? Now the decompositio Calc ; list ; dybar = yb - yb0 $ Total Calc ; list ; dy_dx = b'xb - b'xb0 $ Chage due to chage i x Calc ; list ; dy_db = b'xb0 - b0'xb0 $ Matrix ; vdb = v+v0 ; vdb = xb0'[vdb]xb0 $ Calc ; sdb = sqr(vdb) ; list ; lower = dy_db -.96*sqr(vdb) ; upper = dy_db +.96*sqr(vdb) $ Listed Calculator Results DYBAR = DY_DX =.745 DY_DB =.763 LOWER = UPPER =

20 ?=======================================================================? Chapter 4 Applicatio?======================================================================= Create ; lc = log(cost/pf) ; lpl=log(pl/pf) ; lpk=log(pk/pf)$ Create ; lq = log(q) ; lqq =.5*lq*lq $ Namelist ; x = oe,lq,lqq,lpk,lpl $? a. Cost fuctio Regress; lhs = lc ; rhs = x ; pritvc $ Ordiary least squares regressio LHS=LC Mea = Stadard deviatio = WTS=oe Number of observs. = 58 Model size Parameters = 5 Degrees of freedom = 53 Residuals Sum of squares = Stadard error of e = Fit R-squared =.99 Adjusted R-squared = Model test F[ 4, 53] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LQ LQQ LPK LPL D D D D D ?=======================================================================? b. capital price coefficiet?======================================================================= Wald ; f = - b_lpk - b_lpl $ WALD procedure. Estimates ad stadard errors for oliear fuctios ad joit test of oliear restrictios. Wald Statistic = Prob. from Chi-squared[ ] = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Fc() ?=======================================================================? c. efficiet scale?======================================================================= Wald ; f = exp((-b_lq)/b_lqq) $ WALD procedure. Estimates ad stadard errors for oliear fuctios ad joit test of oliear restrictios. Wald Statistic = Prob. from Chi-squared[ ] = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Fc() Calc ; qstar = waldfs() ; vqstar = varwald(,) 7

21 ; list ; lower = qstar -.96*sqr(vqstar) ; upper = qstar +.96*sqr(vqstar) $?=======================================================================? d. Raw data?======================================================================= Listed Calculator Results LOWER = UPPER = Create ; output = q $ Sort ; lhs = output $ /* The estimated efficiet scale is 877. There are 5 firms i the sample that have output larger tha this. As oted i the problem, may of the largest firms i the sample are aggregates of smaller oes, so it is difficult to draw a coclusio here. However, some of the largest firms (Souther, America Electric power) are sigly couted, ad are much larger tha this scale. The importat poit is that much of the output i the sample is produced by firms that are smaller tha this efficiet scale. There are uexploited ecoomies of scale i this idustry. */ 8

22 Chapter 5 Iferece ad Predictio Exercises. The estimated covariace matrix for the least squares estimator is 3900 / s (X X) - 0 = = where s = 50/(9-3) = 0. The, the test may be based o t = ( )/[ (.05)] / =.399. This is smaller tha the critical value of.056, so we would ot reject the hypothesis.. I order to compute the regressio, we must recover the origial sums of squares ad cross products for y. These arex y = X Xb = [6, 9, 76]. The total sum of squares is foud usig R = - e e/y M 0 y, so y M 0 y = 50 / (5/60) = 600. The meas are x = 0, x = 0, y = 4, so, y y = (4 ) = 064. The slope i the regressio of y o x aloe is b = 76/80, so the regressio sum of squares is b (80) = 7., ad the residual sum of squares is = The test based o the residual sum of squares is F = [( )/]/[50/6] =.390. I the regressio of the previous problem, the t-ratio for testig the same hypothesis would be t =.4/(.40) / =.64 which is the square root of For the curret problem, R = [0,I] where I is the last K colums. Therefore, R(X X) - RN is the lower right K K block of (X X) -. As we have see before, this is (X M X) -. Also, (X X) - R is the last K colums of (X X) -. These are (X X) - -( X' X) X' X( X' MX) R = Fially, sice q = 0, Rb - ( X' MX) q = (0b + Ib ) - 0 = b. Therefore, the costraied estimator is b b * = (X M X )b, where b ad b are the multiple regressio b - - X X X X X M X ( ' ) ' ( ' ) ( X' MX) coefficiets i the regressio of y o both X ad X. Collectig terms, this produces b * = b. But, we have from Sectio that b = (X X ) - X y - (X X ) - b - -( X' X) X' X b b ( X ' X ) X ' y X X b so the precedig reduces to b * = which was to be show. 0 If, istead, the restrictio is β = β 0 the the precedig is chaged by replacig Rβ - q = 0 with Rβ - β 0 = 0. Thus, Rb - q = b - β 0. The, the costraied estimator is b b * = (X M X )(b - β 0 ) b - - X X X X X M X ( ' ) ' ( ' ) ( X' MX) or b + 0 ( X β ' X) X' X( b ) b * = 0 b (β - b) Usig the result of the previous paragraph, we ca rewrite the first part as b * = (X X ) - X y - (X X ) - X X β 0 = (X X ) - X (y - X β 0 ) which was to be show. 9

23 4. By factorig the result i (5-4), we obtai b * = [I - CR]b + w where C = (X X) - R [R(X X) - R ] - ad w = Cq. The covariace matrix of the least squares estimator is Var[b * ] = [I - CR]σ (X X) - [I - CR] = σ (X X) - + σ CR(X X) - R C - σ CR(X X) - - σ (X X) - R C. By multiplyig it out, we fid that CR(X X) - = (X X) - R (R(X X) - R ) - R(X X) - = CR(X X) - R C so Var[b * ] = σ (X X) - - σ CR(X X) - R C = σ (X X) - - σ (X X) - R [R(X X) - R ] - R(X X) - This may also be writte as Var[b * ] = σ (X X) - {I - R (R(X X) - R ) - R(X X) - } = σ (X X) - {[σ (X X) - ] - - R [Rσ (X X) - R ] - R}σ (X X) - Sice Var[Rb] = Rσ (X X) - R this is the aswer we seek. 5. The variace of the restricted least squares estimator is give i the secod equatio i the previous exercise. We kow that this matrix is positive defiite, sice it is derived i the form B σ (X X) - B, ad σ (X X) - is positive defiite. Therefore, it remais to show oly that the matrix subtracted from Var[b] to obtai Var[b * ] is positive defiite. Cosider, the, a quadratic form i Var[b * ] z Var[b * ]z = z Var[b]z - σ z (X X) - (R [R(X X) - R ] - R)(X X) - z = z Var[b]z - w [R(X X) - R ] - w where w = σr(x X) - z. It remais to show, therefore, that the iverse matrix i brackets is positive defiite. This is obvious sice its iverse is positive defiite. This shows that every quadratic form i Var[b * ] is less tha a quadratic form i Var[b] i the same vector. 6. The result follows immediately from the result which precedes (5-9). Sice the sum of squared residuals must be at least as large, the coefficiet of determiatio, COD = - sum of squares / Σ i (y i - y ), must be o larger. 7. For coveiece, let F = [R(X X) - R ] -. The, λ = F(Rb - q) ad the variace of the vector of Lagrage multipliers is Var[λ] = FRσ (X X) - R F = σ F. The estimated variace is obtaied by replacig σ with s. Therefore, the chi-squared statistic is χ = (Rb - q) F (s F) - F(Rb - q) = (Rb - q) [(/s )F](Rb - q) = (Rb - q) [R(X X) - R ] - (Rb - q)/[e e/( - K)] This is exactly J times the F statistic defied i (5-9) ad (5-0). Fially, J times the F statistic i (5-0) equals the expressio give above. 8. We use (5-9) to fid the ew sum of squares. The chage i the sum of squares is e * e * - e e = (Rb - q) [R(X X) - R ] - (Rb - q) For this problem, (Rb - q) = b + b 3 - =.3. The matrix iside the brackets is the sum of the 4 elemets i the lower right block of (X X) -. These are give i Exercise, multiplied by s = 0. Therefore, the required sum is [R(X X) - R ] = (/0)( (.05)) =.08. The, the chage i the sum of squares is.3 /.08 = 3.5. Thus, e e = 50, e * e * = 53.5, ad the chi-squared statistic is 6[53.5/50 - ] =.6. This is quite small, ad would ot lead to rejectio of the hypothesis. Note that for a sigle restrictio, the Lagrage multiplier statistic is equal to the F statistic which equals, i tur, the square of the t statistic used to test the restrictio. Thus, we could have obtaied this quatity by squarig the.399 foud i the first problem (apart from some roudig error). 9. First, use (5-9) to write e * e * = e e + (Rb - q) [R(X X) - R ] - (Rb - q). Now, the result that E[e e] = ( - K)σ obtaied i Chapter 6 must hold here, so E[e * e * ] = ( - K)σ + E[(Rb - q) [R(X X) - R ] - (Rb - q)]. Now, b = β + (X X) - X ε, so Rb - q = Rβ - q + R(X X) - X ε. But, Rβ - q = 0, so uder the hypothesis, Rb - q = R(X X) - X ε. Isert this i the result above to obtai E[e * e * ] = (-K)σ + E[ε X(X X) - R [R(X X) - R ] - R(X X) - X ε]. The quatity i square brackets is a scalar, so it is equal to its trace. Permute ε X(X X) - R i the trace to obtai E[e * e * ] = ( - K)σ + E[tr{[R(X X) - R ] - R(X X) - X εε X(X X) - R ]} We may ow carry the expectatio iside the trace ad use E[εε ] = σ I to obtai E[e * e * ] = ( - K)σ + tr{[r(x X) - R ] - R(X X) - X σ IX(X X) - R ]} 0

24 Carry the σ outside the trace operator, ad after cacellatio of the products of matrices times their iverses, we obtai E[e * e * ] = ( - K)σ + σ tr[i J ] = ( - K + J)σ. 0. Show that i the multiple regressio of y o a costat, x, ad x, while imposig the restrictio β + β = leads to the regressio of y - x o a costat ad x - x. For coveiece, we put the costat term last istead of first i the parameter vector. The costrait is Rb - q = 0 where R = [ 0] so R = [] ad R = [,0]. The, β = [] - [ - β ] = - β. Thus, y = ( - β )x + β x + αi + ε or y - x = β (x - x ) + αi + ε. Applicatios?=======================================================================? Applicatio 5. Wage Equatio?======================================================================= Read;File="F:\Text-Revisio\editio6\Solutios-ad-Applicatios\time_var.dat"; var=5;obs=799$? This creates the group cout variable. Regress ; Lhs = oe ; Rhs = oe ; Str = ID ; Pael $? This READ merges the smaller file ito the larger oe. Read;File="F:\Text-Revisio\editio6\Solutios-ad-Applicatios\time_ivar.dat"; ames=ability,med,fed,bh,sibs? ; group=_groupti ;var=5;obs=78$ Names=id,educ,lwage,pexp,t; amelist ; x=oe,educ,pexp,ability$ amelist ; x=med,fed,bh,sibs$?=======================================================================? a. Log regressio?======================================================================= regress ; lhs= lwage ; rhs = x,x $ Ordiary least squares regressio LHS=LWAGE Mea =.968 Stadard deviatio = WTS=oe Number of observs. = 799 Model size Parameters = 8 Degrees of freedom = 79 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 7, 79] (prob) = (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat EDUC PEXP ABILITY MED D FED BH SIBS ?=======================================================================? b. F test?======================================================================= Calc ; list ; fstat = Rsqrd/(kreg-)/((-rsqrd)/(-kreg)) $ FSTAT = Calc ; r = rsqrd ; df=-kreg$ Matrix ; b = b ; v = varb $ Matrix ; b =b(5:8) ; v=varb(5:8,5:8)$ Regress ; lhs = lwage ; rhs = x $

25 Ordiary least squares regressio LHS=LWAGE Mea =.968 Stadard deviatio = WTS=oe Number of observs. = 799 Model size Parameters = 4 Degrees of freedom = 795 Residuals Sum of squares = Stadard error of e = Fit R-squared =.7347 Adjusted R-squared = Model test F[ 3, 795] (prob) =5.94 (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat EDUC PEXP ABILITY ?=======================================================================? c. F test for hypothesis that coefficiets o X are zero?======================================================================= Calc ; list ; fstat = (r-rsqrd)/(col(x))/((-r)/(df)) $ FSTAT = ?=======================================================================? c. Wald test for hypothesis that coefficiets o X are zero?======================================================================= Matrix ; List ; Wald = b'<v>b $ Matrix WALD has rows ad colums Note Wald = 4*F, as expected.?=======================================================================? Applicatio 5. Traslog Cost Fuctio?=======================================================================? First prepare the data? Create ; lpk=log(pk);lpl=log(pl);lpf=log(pf)$ create ; lpk=.5*lpk^ ; lpl=.5*lpl^ ; lpf=.5*lpf^$ Create ; lpkf=lpk*lpf ; lplf=lpl*lpf ; lpkl=lpk*lpl $ Create ; lq = log(q) ; lq =.5*lq^ $ Create ; lqk=lq*lpk ; lql=lq*lpl ; lqf=lq*lpf $ Create ; lc = log(cost) $ Create ; lcpf = log(cost/pf) $ Create ; lpkpf=log(pk/pf) ; lplpf=log(pl/pf) $ Create ; lpkpf=.5*lpkpf^ ; lplpf=.5*lplpf^ ; lplfpkf=lplpf*lpkpf $ Create ; lqlpkf=lq*lpkpf ; lqlplf=lq*lplf $?=======================================================================? a. Beta is a,b,dk,dl,df,pkk,pll,pff,pkl,pkf,plf,c,tqk,tql,tqf?======================================================================= Restrictios are 0,0,,,,0,0,0,0,0,0,0,0,0,0 0,0,0,0,0,,0,0,,,0,0,0,0,0 0 R = 0,0,0,0,0,0,,0,,0,,0,0,0,0 q = 0 0,0,0,0,0,0,0,,0,,,0,0,0,0 0 0,0,0,0,0,0,0,0,0,0,0,0,,, 0?=======================================================================? b. Testig the theory?======================================================================= Namelist ; X=oe,lq,lpk,lpl,lpf,lpk,lpl,lpf,lpkl,lpkf,lplf,lq,lqk,lq... Namelist ; X0=oe,lq,lpkf,lplf,lpkpf,lplpf,lplfpkf,lq,lqlpkf,lqlplf$ Regress ; lhs = lc ; rhs=x0 $

26 Ordiary least squares regressio LHS=LC Mea = Stadard deviatio = WTS=oe Number of observs. = 58 Model size Parameters = 0 Degrees of freedom = 48 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared =.995 Model test F[ 9, 48] (prob) =36.03 (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LQ LPKF LPLF LPKPF LPLPF LPLFPKF LQ LQLPKF LQLPLF Calc ; ee0 = sumsqdev $ Regress ; lhs = lcpf ; rhs = x $ Ordiary least squares regressio LHS=LCPF Mea = Stadard deviatio = WTS=oe Number of observs. = 58 Model size Parameters = 5 Degrees of freedom = 43 Residuals Sum of squares = Stadard error of e =.3753 Fit R-squared = Adjusted R-squared = Model test F[ 4, 43] (prob) =537.8 (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LQ LPK LPL LPF LPK LPL LPF LPKL LPKF LPLF LQ LQK LQL LQF Calc ; ee = sumsqdev $ Calc ; list ; Fstat = ((ee0 - ee)/5)/(ee/(58-5))$ FSTAT = > Calc ; list ; ftb(.95,5,43)$ Result = The F statistic is small; the theory is ot rejected. 3

27 ?=======================================================================? c. Testig homotheticity?======================================================================= Ordiary least squares regressio LHS=LCPF Mea = Stadard deviatio = WTS=oe Number of observs. = 58 Model size Parameters = 0 Degrees of freedom = 48 Residuals Sum of squares =.6343 Stadard error of e =.334 Fit R-squared = Adjusted R-squared = Model test F[ 9, 48] (prob) =35.08 (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LQ LPKF LPLF LPKPF LPLPF LPLFPKF LQ LQLPKF LQLPLF Regress ; lhs = lcpf ; Rhs = x0 ; cls:b(9)=0,b(0)=0$ Liearly restricted regressio Ordiary least squares regressio LHS=LCPF Mea = Stadard deviatio = WTS=oe Number of observs. = 58 Model size Parameters = 8 Degrees of freedom = 50 Residuals Sum of squares =.8967 Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 7, 50] (prob) =74.96 (.0000) Restricts. F[, 48] (prob) = 7.36 (.0009) Not usig OLS or o costat. Rsqd & F may be < 0. Note, with restrictios imposed, Rsqd may be < 0. Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LQ LPKF LPLF LPKPF LPLPF LPLFPKF LQ LQLPKF D D LQLPLF D Calc ; list ; ftb(.95,,48)$ Result = The F statistic of 7.36 is larger tha the critical value of The hypothesis is rejected. 4

28 ?=======================================================================? d. Testig geeralized Cobb-Douglas agaist full traslog.?======================================================================= Regress ; lhs = lcpf ; rhs = x0 ;cls:b(5)=0,b(6)=0,b(7)=0,b(9)=0,b(0)=0$ Liearly restricted regressio Ordiary least squares regressio LHS=LCPF Mea = Stadard deviatio = WTS=oe Number of observs. = 58 Model size Parameters = 5 Degrees of freedom = 53 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared =.9930 Model test F[ 4, 53] (prob) = (.0000) Restricts. F[ 5, 48] (prob) = 6.7 (.0000) Not usig OLS or o costat. Rsqd & F may be < 0. Note, with restrictios imposed, Rsqd may be < 0. Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LQ LPKF LPLF LPKPF D D LPLPF D D LPLFPKF.46436D D LQ LQLPKF -.555D D LQLPLF D D Calc ; list ; ftb(.95,5,48)$ Listed Calculator Results Result =.7539 The F statistic of 6.7 is larger tha the critical value of.75. The hypothesis is rejected.?=======================================================================? e. Testig Cobb-Douglas agaist full traslog.?======================================================================= Matrix ; b=b(5:0) ; v=varb(5:0,5:0) $ Matrix ; list ; Fcd = /6 * b'<v>b $ Matrix FCD has rows ad colums Calc ; list ; ftb(.95,6,48)$ Listed Calculator Results Result =.6035 The F statistic of 8.87 is larger tha the critical value of.6. The hypothesis is rejected.?=======================================================================? f. Testig geeralized Cobb-Douglas agaist homothetic traslog.?======================================================================= Regress ; Lhs = lcpf ; rhs = oe,lq,lpkf,lplf,lpkpf,lplpf,lplfpkf,lq ; cls:b(5)=0,b(6)=0,b(7)=0$ Liearly restricted regressio 5

29 Ordiary least squares regressio LHS=LCPF Mea = Stadard deviatio = WTS=oe Number of observs. = 58 Model size Parameters = 5 Degrees of freedom = 53 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared =.9930 Model test F[ 4, 53] (prob) = (.0000) Restricts. F[ 3, 50] (prob) = 5. (.00) Not usig OLS or o costat. Rsqd & F may be < 0. Note, with restrictios imposed, Rsqd may be < 0. Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LQ LPKF LPLF LPKPF D D LPLPF D D LPLFPKF.4066D-4.75D LQ Calc ; list ; ftb(.95,3,50) $ Listed Calculator Results Result = ??=======================================================================? g. We have ot rejected the theory, but we have rejected all the? fuctioal forms? except the ohomothetic traslog. Just like Christese ad Greee.?=======================================================================?=======================================================================? Applicatio 5.3 Noliear restrictios?======================================================================= sample;-5$ ame;x=oe,logpg,logi,logpc,logpuc,logppt,t,logpd,logp,logps$?=======================================================================? a. Simple hypothesis test?======================================================================= Regr;lhs=logg;rhs=x$ Ordiary least squares regressio LHS=LOGG Mea = Stadard deviatio =.3885 WTS=oe Number of observs. = 5 Model size Parameters = 0 Degrees of freedom = 4 Residuals Sum of squares =.3887E-0 Stadard error of e =.30994E-0 Fit R-squared = Adjusted R-squared = Model test F[ 9, 4] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGPG

30 LOGI LOGPNC LOGPUC LOGPPT T LOGPD LOGPN LOGPS Calc;r=rsqrd$ Regr;lhs=logg;rhs=oe,logpg,logi,logpc,logpuc,logppt,t$ Ordiary least squares regressio LHS=LOGG Mea = Stadard deviatio =.3885 WTS=oe Number of observs. = 5 Model size Parameters = 7 Degrees of freedom = 45 Residuals Sum of squares = Stadard error of e = E-0 Fit R-squared = Adjusted R-squared = Model test F[ 6, 45] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGPG LOGI LOGPNC LOGPUC LOGPPT T Calc;r0=rsqrd$ Calc;list;f=((r-r0)/)/((-r)/(-0))$ Listed Calculator Results F = The critical value from the F table is.87, so we would reject the hypothesis.?=======================================================================? b. Noliear restrictio?======================================================================= Sice the restricted model is quite oliear, it would be quite cumbersome to estimate ad examie the loss i fit. We ca test the restrictio usig the urestricted model. For this problem, f = [γ c - γ uc, γ c δ s - γ pt δ d ] The matrix of derivatives, usig the order give above ad " to represet the etire parameter vector, is f / α f G = / α = δ s δd γ pt γc. The parameter estimates are Thus, f = [-.7399,.009]. The covariace matrix to use for the tests is Gs (X X) - G The statistic for the joit test is χ = f [Gs (X X) - G ] - f =.477. This is less tha the critical value for a chi-squared with two degrees of freedom, so we would ot reject the joit hypothesis. For the idividual hypotheses, we eed oly compute the equivalet of a t ratio for each elemet of f. Thus, z = ad z =.898 Neither is large, so either hypothesis would be rejected. (Give the earlier result, this was to be expected.) 7

31 ?=======================================================================? c. Computatios for oliear restrictio?======================================================================= sample;-5$ ame;x=oe,logpg,logi,logpc,logpuc,logppt,t,logpd,logp,logps$ Regr;lhs=logg;rhs=x$ Ordiary least squares regressio LHS=LOGG Mea = Stadard deviatio =.3885 WTS=oe Number of observs. = 5 Model size Parameters = 7 Degrees of freedom = 45 Residuals Sum of squares = Stadard error of e = E-0 Fit R-squared = Adjusted R-squared = Model test F[ 6, 45] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGPG LOGI LOGPNC LOGPUC LOGPPT T Calc;r=rsqrd$ Regr;lhs=logg;rhs=oe,logpg,logi,logpc,logpuc,logppt,t$ Ordiary least squares regressio LHS=LOGG Mea = Stadard deviatio =.3885 WTS=oe Number of observs. = 5 Model size Parameters = 7 Degrees of freedom = 45 Residuals Sum of squares = Stadard error of e = E-0 Fit R-squared = Adjusted R-squared = Model test F[ 6, 45] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGPG LOGI LOGPNC LOGPUC LOGPPT T Calc;r0=rsqrd$ Calc;list;fstat=((r-r0)/)/((-r)/(-0))$ FSTAT = Calc;list;ftb(.95,3,4)$ Result = REGR;Lhs=logg;rhs=x$ Calc ; ds=b(0);dd=-b(8);gpt=-b(6);gc=b(4)$ Matr;gm=[0,0,0,,-,0,0,0,0,0 / 0,0,0,ds,0,dd,0,gpt,0,gc]$ Calc;f=b(4)-b(6) ; f=b(4)*b(0)-b(6)*b(8)$ Matrix;list;f=[f/f]$ 8

32 Matrix F has rows ad colums Matrix;list;vf=gm*varb*gm'$ Matrix VF has rows ad colums Matrix;list;Wald=f'<vf>f$ Matrix WALD has rows ad colums Calc;list;z=f()/sqr(vf(,))$ Z = Calc;list;z=f()/sqr(vf(,))$ Z =

33 Chapter 6 Fuctioal Form ad Structural Chage Exercises. T he F statistic could be computed as F = {[45 - ( )] / (70-6)}/[( ) / (570-70)] =.343 The 95% critical value for the F distributio with 54 ad 500 degrees of freedom is a. Usig the hit, we seek the c * which is the slope o d i the regressio of q = y - cd - e o y ad d. The yy yd y ( y-cd-e) yy yd yy- cyd-ye regressio coefficiets are ( c ) = dy dd d y- d-e c. I the precedig, dy dd dy- dd-de ote that (y y,d y) is the first colum of the matrix beig iverted while c(y d,d d) is c times the secod. A iverse matrix times the first colum of the origial matrix is the first colum of a idetity matrix, ad likewise for the secod. Also, sice d was oe of the origial regressors i (), d e = 0, ad, of course, y e = e e. If we combie all of these, the coefficiet vector is 0 yy yd ee 0 yy yd c 0 = c dy dd 0 0 ee. We are iterested i the secod dy dd 0 (lower) of the two coefficiets. The matrix product at the ed is e e times the first colum of the iverse matrix, ad we wish to fid its secod (bottom) elemet. Therefore, collectig what we have thus far, the desired coefficiet is c * = -c - e e times the off diagoal elemet i the iverse matrix. The off diagoal elemet is -d y / [(y y)(d d) - (y d) ] = -d y / {[(y y)(d d)][ - (y d) /[(y y)(d d)]]} = -d y / [(y y)(d d)( - r yd )]. Therefore, c * = [(e e)(d y)] / [(y y)(d d)( - r yd )] - c (The two egative sigs cacel.) This ca be further reduced. Sice all variables are i deviatio form, e e/y y is ( - R ) i the full regressio. By multiplyig it out, you ca show that d = P so that d d = Σ i (d i - P) = P(-P) ad d y = Σ i (d i - P)(y i - y ) = Σ i (d i - P)y i = ( y - y ) where is the umber of observatios which have d i =. Combiig terms oce agai, we have c * = {[ ( y - y )( - R )} / {P(-P)( - r yd )} - c Fially, sice P = /, this further simplifies to the result claimed i the problem, c * = {( y - y )( - R )} / {(-P)( - r yd )} - c The problem this creates for the theory is that i the preset settig, if, ideed, c is egative, ( y - y ) will almost surely be also. Therefore, the sig of c * is ambiguous. 30

34 x * y α β 0 3. We first fid the joit distributio of the observed variables. = + ε x 0 so [y,x] have a 0 u μ* y α β 0 α+βμ* joit ormal distributio with mea vector E = + 0 = x 0 ad covariace 0 μ 0 σ* 0 0 β y β 0 * ε * matrixvar 0 ε 0 βσ +σ βσ = x σ 0 =, The probability limit of the 0 βσ* σ * +σu 0 0 σ u 0 slope i the liear regressio of y o x is, as usual, plim b = Cov[y,x]/Var[x] = β/( + σ u /σ * ) < β. The probability limit of the itercept is plim a = E[y] - (plim b)e[x] = α + βμ * - βμ * /( + σ u /σ * ) = α + β[μ * σ u / (σ * + σ u )] > α (assumig β > 0). If x is regressed o y istead, the slope will estimate plim[b ] = Cov[y,x]/Var[y] = βσ * /(β σ * + σ ε ). The,plim[/b ] = β + σ ε /β σ * > β. Therefore, b ad b will bracket the true parameter (at least i their probability limits). Ufortuately, without more iformatio about σ u, we have o idea how wide this bracket is. Of course, if the sample is large ad the estimated bracket is arrow, the results will be strogly suggestive. 4. I the regressio of y o x ad d, if d ad x are idepedet, we ca ivoke the familiar result for least squares regressio. The results are the same as those obtaied by two simple regressios. It is istructive to xx/ xd/ xy/ σ 0 * ( u / +σ β/ + σ σ u βσ ) verify this. plim = =. Therefore, although dx/ dd/ dy / 0 π γπ γ the coefficiet o x is distorted, the effect of iterest, amely, γ, is correctly measured. Now cosider what happes if x * ad d are ot idepedet. With the secod assumptio, we must replace the off diagoal zero above with plim(x d/). Sice u ad d are still ucorrelated, this equals Cov[x *,d]. This is Cov[x *,d] = E[x * d] = πe[x * d d=] + (-π)e[x * d d=0] = πμ. Also, plim[y d/] is ow βcov[x *,d] + γplim(d d/) = βπμ + γπ ad plim[y x * /] equals βplim[x * x * /] + γplim[x * d/] = βσ * + γπμ. The, the probability limits of the least squares coefficiet estimators is xx/ xd/ xy/ σ * ( u / +σ β/ + σ σ u πμ βσ +γπμ ) plim = = dx/ dd/ dy / πμ π βπμ + γπ γ π πμ βσ * + γπμ = πσ ( * +σ u ) +π( μ) πμ σ * +σu βπμ + γπ β(πσ * + π ( μ ) ) =. πσ ( * +σ u ) +π( μ) γ(π( σ * + σ u) + π ( μ ) ) + βπσu The secod expressio does reduce to plim c = γ + βπμ σ u /[π(σ * + σ u ) - π (μ ) ], but the upshot is that i the presece of measuremet error, the two estimators become a uredeemable hash of the uderlyig parameters. Note that both expressios reduce to the true parameters if σ u equals zero. Fially, the two meas are estimators of E[y d=] = βe[x * d=] + γ = βμ + γ ad E[y d=0] = βe[x * d=0] = βμ 0, so the differece is β(μ - μ 0 ) + γ, which is a mixture of two effects. Which oe will be larger is etirely idetermiate, so it is reasoable to coclude that this is ot a good way to aalyze the problem. If γ equals zero, this differece will merely reflect the differeces i the values of x *, which may be etirely urelated to the issue uder examiatio here. (This is, ufortuately, what is usually reported i the popular press.) 3

35 Applicatios?=======================================================================? Applicatio 6.?======================================================================= a. Wage equatio Namelist ; X = oe,educ,ability,pexp,med,fed,bh,sibs$ Regress ; Lhs = lwage ; Rhs = x $ Calc ; xb = b()+b()*+b(3)*xbr(ability)+b(4)*xbr(med) +b(5)*xbr(fed)+b(6)*0+b(7)*xbr(sibs) $ Calc ; list ; mv = exp(xb) * b() $ Ordiary least squares regressio LHS=LWAGE Mea =.968 Stadard deviatio = WTS=oe Number of observs. = 799 Model size Parameters = 7 Degrees of freedom = 79 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 6, 79] (prob) = 63.0 (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat EDUC ABILITY PEXP MED FED SIBS Listed Calculator Results MV =.7576b. Step fuctio?=======================================================================? b.?======================================================================= Histogram ; Rhs = Educ $ 3

36 Create ; HS = Educ <= $ Create ; Col = (Educ>) * (educ <=6) $ Create ; Grad = Educ > 6 $ Regress ; Lhs=lwage ; Rhs = oe,col,grad,ability,pexp,med,fed,bh,sibs $ Ordiary least squares regressio LHS=LWAGE Mea =.968 Stadard deviatio = WTS=oe Number of observs. = 799 Model size Parameters = 9 Degrees of freedom = 790 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 8, 790] (prob) = (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat COL GRAD ABILITY PEXP MED FED BH SIBS c. Educatio squared Create ; educsq = educ*educ $ Regress ; Lhs = lwage;rhs=oe,educ,educsq,ability,pexp,med,fed,bh,sibs$ Ordiary least squares regressio LHS=LWAGE Mea =.968 Stadard deviatio = WTS=oe Number of observs. = 799 Model size Parameters = 9 Degrees of freedom = 790 Residuals Sum of squares = Stadard error of e = Fit R-squared =.7700 Adjusted R-squared = Model test F[ 8, 790] (prob) = 48.8 (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat EDUC EDUCSQ ABILITY PEXP MED FED BH SIBS Namelist ; x = oe,educ,educsq,ability,pexp,med,fed,bh,sibs $ Matrix ; meas = mea(x)$ Matrix ; meas()=0 $ Matrix ; meas(3)=0$ Calc ; a=meas'b $ Calc ; b=b() ; b3=b(3) $ Sample ; $ 33

37 Fplot ; fc = a + b*schoolg + b3*schoolg^ ; pts=00 ; start = ; limits =,0 ; labels=schoolg ; plot(schoolg) $ d. Iteractio. Sample ; All $ Create ; EA = Educ*ability $ Regress ; Lhs = lwage;rhs=oe,educ,ability,ea,pexp,med,fed,bh,sibs$ Calc ; abar =xbr(ability) $ Calc ; list ; me = b()+b(4)*abar $ Calc ; sdme = sqr(varb(,)+abar^*varb(4,4) + *abar*varb(,4))$ Calc ; list ; lower = me -.96*sdme ; upper = me +.96*sdme $ Ordiary least squares regressio LHS=LWAGE Mea =.968 Stadard deviatio = WTS=oe Number of observs. = 799 Model size Parameters = 9 Degrees of freedom = 790 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared =.7573 Model test F[ 8, 790] (prob) = (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat EDUC ABILITY EA PEXP MED.5477D FED BH SIBS Listed Calculator Results ME = LOWER = UPPER =

38 e. Regress ; Lhs = lwage;rhs=oe,educ,educsq,ability,ea,pexp,med,fed,bh,sibs$ Ordiary least squares regressio LHS=LWAGE Mea =.968 Stadard deviatio = WTS=oe Number of observs. = 799 Model size Parameters = 0 Degrees of freedom = 7909 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 9, 7909] (prob) = 433. (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat EDUC EDUCSQ ABILITY EA PEXP MED FED BH SIBS Listed Calculator Results AVGLOW = AVGHIGH =.7789 Create ; lowa = ability < xbr(ability) ; higha = - lowa $ Calc ; list ; avglow= lowa'ability / lowa'lowa ; avghigh=higha'ability/higha'higha $ Calc ; a = b() + b(6)*xbr(pexp)+b(7)*xbr(med)+ b(8)*xbr(fed)+b(9)*xbr(bh)+b(0)*xbr(sibs)$ Calc ; al=a+b(4)*avglow ; ah = a+b(4)*avghigh$ Samp;-0$ Create ; school = tr(9,.)$ Create ; lwlow = al + b()*school+b(3)*school^ + b(5)*avglow*school $ Create ; lwhigh = ah + b()*school+b(3)*school^ + b(5)*avghigh*school $ Plot ; lhs = school ; rhs =lwhigh,lwlow ;fill ;grid ;Title=Compariso of logwage Profiles for Low ad High Ability$ 35

39 ?=======================================================================? Applicatio 6.?======================================================================= Sample ; All $ Namelist ; X = oe,educ,ability,pexp,med,fed,sibs$ Regress ; For [bh=0] ; Lhs = lwage ; Rhs = x $ Calc ; ee0=sumsqdev $ Matrix ; b0=b ; v0=varb $ Regress ; For [bh=] ; Lhs = lwage ; Rhs = x $ Calc ; ee=sumsqdev $ Matrix ; b=b ; v=varb $ Regress ; Lhs = lwage ; Rhs = x $ Calc ; ee=sumsqdev $ Calc ; list ; chow = ((ee-ee0-ee)/col(x))/ ((ee0+ee)/(-*col(x))) $ Listed Calculator Results CHOW = Matrix ; db=b0-b ; vdb=v0+v $ Matrix ; list ; Wald = db'<vdb>db $ Matrix WALD has rows ad colums

40 ?=======================================================================? Applicatio 6.3?======================================================================= a. The least squares estimates of the four models are q/a = lk q/a = /k l(q/a) = lk l(q/a) = /k At these parameter values, the four fuctios are early idetical. A plot of the four sets of predictios from the regressios ad the actual values appears below. b. The scatter diagram is show below. The last seve years of the data set show clearly the effect observed by Solow.. 37

41 c. The regressio results for the various models are listed below. (d is the dummy variable equal to for the last seve years of the data set. Stadard errors for parameter estimates are give i paretheses.) α β γ δ R e e Model :q/a = α + βlk + γd + δ(dlk) + ε (.00903) (.0093) (.003) (.007) ( ) (.005) (.008) (.09) (.06) Model : q/a = α - β(/k) + γd + δ(d/k) + ε (.0089) (.09) (.0033) (.00849) (.0008) (.00336) (.00863) (.0354) (.097) Model 3: l(q/a) = α + βlk + γd + δ(dlk) + ε (.037) (.04) (.0064) (.0069) ( ) (.0064) (.0048) (.07) (.079) Model 4: l(q/a) = α - β(/k) + γd + δ(d/k) + ε (.03) (.0337) (.0036) (.0098) (.008) (.00366) (.0094) (.0386) (.0999) d. For the four models, the F test of the third specificatio agaist the first is equivalet to the Chow-test. The statistics are: Model : F = ( )/ / (.00003/37) = 0.6 Model : F = = 0.43 Model 3: F = = 37.0 Model 4: F = = The critical value from the F table for ad 37 degrees of freedom is 3.6, so all of these are statistically sigificat. The hypothesis that the same model applies i both subperiods must be rejected. 38

42 ?=======================================================================? Applicatio 6.4?======================================================================= Accordig to the full model, the expected umber of icidets for a ship of the base type A built i the base period 960 to 964, is 3.4. The other 9 predicted values follow from the previous results ad are left as a exercise. The relevat test statistics for differeces across ship type ad year are as follows: ( )/4 type : F[4,] = =4.8, 660.9/ ( )/3 year : F[3,] = = / The 5 percet critical values from the F table with these degrees of freedom are 3.6 ad 3.49, respectively, so we would coclude that the average umber of icidets varies sigificatly across ship types but ot across years. Regressio Coefficiets Full Model Time Effects Type Effects No Effects Costat B C D E R e e

43 Chapter 7 Specificatio Aalysis ad Model Selectio Exercises. The result cited is E[b ] = β + P. β where P. = (X X ) - X X, so the coefficiet estimator is biased. If the coditioal mea fuctio E[X X ] is a liear fuctio of X, the the sample estimator P. actually is a ubiased estimator of the slopes of that fuctio. (That result is Theorem B.3, equatio (B- 68), i aother form). Now, write the model i the form y = X β + E[X X ]β + ε + (X - E[X X ])β So, whe we regress y o X aloe ad compute the predictios, we are computig a estimator of X (β + P. β ) = X β + E[X X ]β. Both parts of the compoud disturbace i this regressio ε ad (X - E[X X ])β have mea zero ad are ucorrelated with X ad E[X X ], so the predictio error has mea zero. The implicatio is that the forecast is ubiased. Note that this is ot true if E[X X ] is oliear, sice P. does ot estimate the slopes of the coditioal mea i that istace. The geerality is that leavig out variables wil bias the coefficiets, but eed ot bias the forecasts. It depeds o the relatioship betwee the coditioal mea fuctio E[X X ] ad X P... The log estimator, b. is ubiased, so its mea squared error equals its variace, σ (X M X ) - The short estimator, b is biased; E[b ] = β + P. β. It s variace is σ (X X ) -. It s easy to show that this latter variace is smaller. You ca do that by comparig the iverses of the two matrices. The iverse of the first matrix equals the iverse of the secod oe mius a positive defiite matrix, which makes the iverse smaller hece the origial matrix is larger - Var[b. ] > Var[b ]. But, sice b is biased, the variace is ot its mea squared error. The mea squared error of b is Var[b ] + bias bias. The secod term is P. β β P.. Whe this is added to the variace, the sum may be larger or smaller tha Var[b. ]; it depeds o the data ad o the parameters, β. The importat poit is that the mea squared error of the biased estimator may be smaller tha that of the ubiased estimator. 3. The log likelihood fuctio at the maximum is ll = -/[ + lπ + l(e e/)] = -/{ + lπ + l[s yy ( R )]} = -/{ + lπ + l(s yy ) + l(-r )} where S yy = Σ ( ) i= yi y sice R = - e e/s yy. The derivative of this expressio is ll/ R = (-/){/(-R )}(-) which is always positive. Therefore, the log likelihood icreases whe R icreases. 4. A icoveiet way to obtai the result is by repeated substitutio of C t-, the C t- ad so o. It is much easier ad faster to itroduce the lag operator used i Chapter 0. Thus, the alterative model is C t = γ + γ Y t + γ 3 LC t + ε t where LC t = C t-. The, ( γ 3 L)C t = γ + γ Y t + ε t. Now, multiply both sides of the equatio by /(-γ 3 L) = + γ 3 L + γ 3 L + to obtai C t = γ /( - γ 3 ) + γ Y t + γ γ 3 Y t- + Σ s= γ γ s 3 Y t-s + Σ s= 0 γ s 3 ε t-s. 40

44 Applicatio The J test i Example is carried out usig over 50 years of data. It is optimistic to hope that the uderlyig structure of the ecoomy did ot chage i 50 years. Does the result of the test carried out i Example 8. persist if it is based o data oly from 980 to 000? Repeat the computatio with this subset of the data.?====================================? Example 7. ad Applicatio 7.?==================================== Dates ; 950. $ Period ; $ Create ; Ct = Realcos ; Yt = RealDPI $ Create ; Ct = Ct[-] ; Yt = Yt[-] $? Example 7. Period ; $ Regress; Lhs = Ct ; Rhs = oe,yt,yt ; Keep = CY $ Regress; Lhs = Ct ; Rhs = oe,yt,ct ; Keep = CC $ Regress; Lhs = Ct ; Rhs = oe,yt,yt,cc $ Ordiary least squares regressio Model was estimated May, 007 at 08:56:9AM LHS=CT Mea = Stadard deviatio = WTS=oe Number of observs. = 03 Model size Parameters = 4 Degrees of freedom = 99 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 3, 99] (prob) =******* (.0000) Diagostic Log likelihood = Restricted(b=0) = Chi-sq [ 3] (prob) =760.5 (.0000) Ifo criter. LogAmemiya Prd. Crt. = Akaike Ifo. Criter. = Autocorrel Durbi-Watso Stat. =.0560 Rho = cor[e,e(-)] = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat YT YT CC Regress; Lhs = Ct ; Rhs = oe,yt,ct,cy $ Ordiary least squares regressio Model was estimated May, 007 at 08:56:9AM LHS=CT Mea = Stadard deviatio = WTS=oe Number of observs. = 03 Model size Parameters = 4 Degrees of freedom = 99 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 3, 99] (prob) =******* (.0000) Diagostic Log likelihood = Restricted(b=0) = Chi-sq [ 3] (prob) =760.5 (.0000) Ifo criter. LogAmemiya Prd. Crt. = Akaike Ifo. Criter. =

45 Autocorrel Durbi-Watso Stat. =.0560 Rho = cor[e,e(-)] = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat YT CT CY ? Applicatio 7.. We use oly the 980 data, so we? start i quarter of 980 eve though data are? available for the last quarter of 979. Period ; $ Regress; Lhs = Ct ; Rhs = oe,yt,yt ; Keep = CY $ Regress; Lhs = Ct ; Rhs = oe,yt,ct ; Keep = CC $ Regress; Lhs = Ct ; Rhs = oe,yt,yt,cc $ Ordiary least squares regressio Model was estimated May, 007 at 08:58:9AM LHS=CT Mea = Stadard deviatio = WTS=oe Number of observs. = 83 Model size Parameters = 4 Degrees of freedom = 79 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 3, 79] (prob) =******* (.0000) Diagostic Log likelihood = Restricted(b=0) = Chi-sq [ 3] (prob) = (.0000) Ifo criter. LogAmemiya Prd. Crt. = Akaike Ifo. Criter. = Autocorrel Durbi-Watso Stat. =.8534 Rho = cor[e,e(-)] = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat YT YT CC Regress; Lhs = Ct ; Rhs = oe,yt,ct,cy $ Ordiary least squares regressio Model was estimated May, 007 at 08:58:9AM LHS=CT Mea = Stadard deviatio = WTS=oe Number of observs. = 83 Model size Parameters = 4 Degrees of freedom = 79 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 3, 79] (prob) =******* (.0000) Diagostic Log likelihood = Restricted(b=0) = Chi-sq [ 3] (prob) = (.0000) Ifo criter. LogAmemiya Prd. Crt. = Akaike Ifo. Criter. = Autocorrel Durbi-Watso Stat. =.8534 Rho = cor[e,e(-)] =

46 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat YT CT CY ?? The results are essetially the same. This suggests? that either model is right. The regressios are based o real cosumptio ad real disposable icome. Results for 950 to 000 are give i the text. Repeatig the exercise for 980 to 000 produces: for the first regressio, the estimate of α is.03 with a t ratio of 3.7 ad for the secod, the estimate is -.4 with a t ratio of Thus, as before, both models are rejected. This is qualitatively the same results obtaied with the full 5 year data set. 43

47 Chapter 8 The Geeralized Regressio Model ad Heteroscedasticity Exercises. Write the two estimators as ˆβ = β + (X Ω - X) - X Ω - ε ad b = β + (X X) - X ε. The, ( ˆβ - b) = [(X Ω - X) - X Ω - - (X X) - X ]ε has E[ ˆβ - b] = 0 sice both estimators are ubiased. Therefore, Cov[ ˆβ, ˆβ - b] = E[( ˆβ - β)( ˆβ - b) ]. The, E{(X Ω - X) - X Ω - εε [(X Ω - X) - X Ω - - (X X) - X ] } = (X Ω - X) - X Ω - (σ Ω)[Ω - X(X Ω - X) - - X(X X) - ] = σ (X Ω - X) - X Ω - ΩΩ - X(X Ω - X) - - (X Ω - X) - X Ω - ΩX(X X) - = (X Ω - X) - (X Ω - X)(X Ω - X) - - (X Ω - X) - (X X)(X X) - = 0 oce the iverse matrices are multiplied. First, (R ˆβ - q) = R[β + (X Ω - X) - X Ω - ε)] - q = R(X Ω - X) - X Ω - ε if Rβ - q = 0. Now, use the iverse square root matrix of Ω, P = Ω -/ to obtai the trasformed data, X * = PX = Ω -/ X, y * = Py = Ω -/ y, ad ε * = Pε = Ω -/ ε. The, E[ε * ε * ] = E[Ω -/ εε Ω - ] = Ω -/ (σ Ω)Ω -/ = σ I, ad, ˆβ = (X Ω - X) - X Ω - y = (X * X * ) - X * y * = the OLS estimator i the regressio of y * o X *. The, R ˆβ - q = R(X * X * ) - X * ε * ad the umerator is ε * X * (X * X * ) - R [R(X * X * ) - R ] - R(X * X * ) - X * ε * / J. By multiplyig it out, we fid that the matrix of the quadratic form above is idempotet. Therefore, this is a idempotet quadratic form i a ormally distributed radom vector. Thus, its distributio is that of σ times a chi-squared variable with degrees of freedom equal to the rak of the matrix. To fid the rak of the matrix of the quadratic form, we ca fid its trace. That is tr{x * (X * X * ) - R [R(X * X * ) - R ] - R(X * X * ) - X * } = tr{(x * X * ) - R [R(X * X * ) - R ] - R(X * X * ) - X * X * } = tr{(x * X * ) - R [R(X * X * ) - R ] - R} = tr{[r(x * X * ) - R ][R(X * X * ) - R ] - } = tr{i J } = J, which might have bee expected. Before proceedig, we should ote, we could have deduced this outcome from the form of the matrix. The matrix of the quadratic form is of the form Q = X * ABA X * where B is the osigular matrix i the square brackets ad A = (X * X * ) - R, which is a K J matrix which caot have rak higher tha J. Therefore, the etire product caot have rak higher tha J. Cotiuig, we ow fid that the umerator (apart from the scale factor, σ ) is the ratio of a chi-squared[j] variable to its degrees of freedom. We ow tur to the deomiator. By multiplyig it out, we fid that the deomiator is (y * - X * ˆβ ) (y * - X * ˆβ )/( - K). This is exactly the sum of squared residuals i the least squares regressio of y * o X *. Sice y * = X * β + ε * ad ˆβ = (X * X * ) - X * y * the deomiator is ε * M * ε * /( - K), the familiar form of the sum of squares. Oce agai, this is a idempotet quadratic form i a ormal vector (ad, agai, apart 44

48 from the scale factor, σ, which ow cacels). The rak of the M matrix is - K, as always, so the deomiator is also a chi-squared variable divided by its degrees of freedom. It remais oly to show that the two chi-squared variables are idepedet. We kow they are if the two matrices are orthogoal. They are sice M * X * = 0. This completes the proof, sice all of the requiremets for the F distributio have bee show. 3. First, we kow that the deomiator of the F statistic coverges to σ. Therefore, the limitig distributio of the F statistic is the same as the limitig distributio of the statistic which results whe the deomiator is replaced by σ. It is useful to write this modified statistic as W * = (/σ )(R ˆβ - q) [R(X * X * ) - R ] - (R ˆβ - q)/j. Now, icorporate the results from the previous problem to write this as W * = ε * X * (X * X * ) - R [Rσ (X * X * ) - R ] - R(X * X * ) - X * ε/j Let ε 0 = R(X * X * ) - X * ε *. Note that this is a J vector. By multiplyig it out, we fid that E[ε 0 ε 0 ] = Var[ε 0 ] = R{σ (X * X * ) - }R. Therefore, the modified statistic ca be writte as W * = ε 0 Var[ε 0 ] - ε 0 /J. This is the full rak quadratic form discussed i Appedix B. For coveiece, let C = Var[ε 0 ], T = C -/, ad v = Tε 0. The, W * = v v. By costructio, v = Var[ε 0 ] -/ ε 0, so E[v] = 0 ad Var[v] = I. The limitig distributio of v v is chi-squared J if the limitig distributio of v is stadard ormal. All of the coditios for the cetral limit theorem apply to v, so we do have the result we eed. This implies that as log as the data are well behaved, the umerator of the F statistic will coverge to the ratio of a chi-squared variable to its degrees of freedom. 4. The developmet is uchaged. As log as the limitig behavior of (/) ˆX ˆX = (/)X ˆΩ - X is the same as that of (/)X * X *, the limitig distributio of the test statistic will be the same as if the true Ω were used istead of the estimate ˆΩ. 5. First, i order to simplify the algebra somewhat without losig ay geerality, we will scale the colums of X so that for each x k, x k x k =. We do this by begiig with our origial data matrix, say, X 0 ad obtaiig X as X = X 0 D -/, where D is a diagoal matrix with diagoal elemets D kk = x k 0 x k 0. By multiplyig it out, we fid that the GLS slopes based o X istead of X 0 are ˆβ = [(X 0 D -/ ) Ω - (X 0 D -/ )] - [(X 0 D -/ ) Ω - y] = D / [X Ω - X](D ) / (D ) -/ X Ω - y = D / ˆβ 0 with variace Var[ ˆβ ] = D / σ [X Ω - X] - (D ) / = D / Var[ ˆβ 0 ](D ) /. Likewise, the OLS estimator based o X istead of X 0 is b = D / b 0 ad has variace Var[b] = D / Var[b 0 ](D ) /. Sice the scalig affects both estimators idetically, we may igore it ad simply assume that X X = I. If each colum of X is a characteristic vector of Ω, the, for the kth colum, x k, Ωx k = λ k x k. Further, x k Ωx k = λ k ad x k Ωx j = 0 for ay two differet colums of X. (We eglect the scalig of X, so that X X = I, which we would usually assume for a set of characteristic vectors. The implicit scalig of X is absorbed i the characteristic roots.) Recall that the characteristic vectors of Ω - are the same as those of Ω while the characteristic roots are the reciprocals. Therefore, X ΩX = Λ K, the diagoal matrix of the K characteristic roots which correspod to the colums of X. I additio, X Ω - X = Λ - K, so (X Ω - X) - = Λ K, adx Ω - y = Λ - K X y. Therefore, the GLS estimator is simply ˆβ = X y with variace Var[ ˆβ ] = σ Λ K. The OLS estimator is b = (X X) - X y = X y. Its variace is Var[b] = σ (X X) - X ΩX(X X) - = σ Λ K, which meas that OLS ad GLS are idetical i this case. 6. Write b = β + (X X) - X ε ad ˆβ = β + (X Ω - X) - X Ω - ε. The covariace matrix is E[(b - β)( ˆβ - β) ] = E[(X X) - X εε Ω - X(X Ω - X) - ] = (X X) - X (σ Ω)Ω - X(X Ω - X) - = σ (X Ω - X) -. For part (b), e = Mε as always, so E[ee ] = σ MΩM. No further simplificatio is possible for the geeral case. For part (c), ˆε = y - X ˆβ = y - X[β + (X Ω - X) - X Ω - ε] = Xβ + ε - X[β + (X Ω - X) - X Ω - ε] = [I - X(X Ω - X) - X Ω - ]ε. 45

49 Thus, E[ ˆε ˆε ] = [I - X(X Ω - X) - X Ω - ]E[εε ][I - X(X Ω - X) - X Ω - ] = [I - X(X Ω - X) - X Ω - ](σ Ω)[I - X(X Ω - X) - X Ω - ] = [σ Ω - σ X(X Ω - X) - X ][I - X(X Ω - X) - X Ω - ] = [σ Ω - σ X(X Ω - X) - X ][I - Ω - X(X Ω - X) - X ] = σ Ω- σ X(X Ω - X) - X - σ X(X Ω - X) - X + σ X(X Ω - )X) - X Ω - X(X Ω - X) - X = σ [Ω - X(X Ω - X) - X ] The GLS residual vector appears i the precedig part. As always, the OLS residual vector is e = Mε = [I - X(X X) - X ]ε. The covariace matrix is E[e ˆε ] = E[(I - X(X X) - X )εε (I - X(X Ω - X) - X Ω - ) ] = (I - X(X X) - X )(σ Ω)(I - Ω - X(X Ω - X) - X ) = σ Ω - σ X(X X) - X Ω - σ ΩΩ - X(X Ω - X) - X + σ X(X X) - X ΩΩ - X(X Ω - X) - X = σ Ω - σ X(X X) - X = σ MΩ. 7. The GLS estimator is ˆβ = (X Ω - X) - X - y = [Σ i x i x i /(β x i ) ] - [Σ i x i y i /(β x i ) ]. The log-likelihood for this model is ll = -Σ i l(β x i ) - Σ i y i /(β x i ). The likelihood equatios are ll/ β = -Σ i (/β x i )x i + Σ i [y i /(β x i ) ]x i = 0 or Σ i (x i y i /(β x i ) ) = Σ i x i /(β x i). Now, write Σ i x i /(β x i ) = Σ i x i x i β/(β x i ), so the likelihood equatios are equivalet to Σ i (x i y i /(β x i). ) = Σ i x i x i β/(β x i)., or X Ω - y = (X Ω - X)β. These are the ormal equatios for the GLS estimator, so the two estimators are the same. We should ote, the solutio is oly implicit, sice Ω is a fuctio of β. For aother more commo applicatio, see the discussio of the FIML estimator for simultaeous equatios models i Chapter The covariace matrix is ρ ρ ρ ρ ρ ρ σ Ω = σ ρ ρ ρ. ρ ρ ρ The matrix X is a colum of s, so the least squares estimator of μ is y. Isertig this Ω ito (0-5), we σ obtai Var[ y] = ( ρ + ρ). The limit of this expressio is ρσ, ot zero. Although ordiary least squares is ubiased, it is ot cosistet. For this model, X ΩX/ = + ρ( ), which does ot coverge. Usig Theorem 8. istead, X is a colum of s, so X X =, a scalar, which satisfies coditio. To fid the characteristic roots, multiply out the equatio Ωx = λx = (-ρ)ix + ρii x = λx. Sice i x = Σ i x i, cosider ay vector x whose elemets sum to zero. If so, the it s obvious that λ = ρ. There are - such roots. Fially, suppose that x = i. Pluggig this ito the equatio produces λ = - ρ + ρ. The characteristic roots of Ω are ( ρ) with multiplicity ad ( ρ + ρ), which violates coditio. 9. This is a heteroscedastic regressio model i which the matrix X is a colum of oes. The efficiet estimator is the GLS estimator, β = (X Ω - X) - X Ω - y = [Σ i y i /x i ] / [Σ i /x i ] = [Σ i (y i /x i )] / [Σ i (/x i )]. As always, the variace of the estimator is Var[ β ] = σ (X Ω - X) - = σ /[Σ i (/x i )]. The ordiary least squares estimator is (X X) - X y = y. The variace of y is σ (X X) - (X ΩX)(X X) - = (σ / )Σ i x i. To show that the variace of the OLS estimator is greater tha or equal to that of the GLS estimator, we must show that (σ / )Σ i x i > σ /Σ i (/x i ) or (/ )(Σ i x i )(Σ i (/x i )) > or Σ i Σ j (x i /x j ) >. The double sum cotais terms equal to oe. There remai (-)/ pairs of the form (x i /x j + x j /x i ). If it ca be show that each of these 46

50 sums is greater tha or equal to, the result is proved. Just let z i = x i. The, we require z i /z j + z j /z i - > 0. But, this is equivalet to (z i + z j - z i z j ) / z i z j > 0 or (z i - z j ) /z i z j > 0, which is certaily true if z i ad z j are positive. They are sice z i equals x i. This completes the proof. 0. Cosider, first, y. We saw earlier that Var[ y ] = (σ / )Σ i x i = (σ /)(/)Σ i x i. The expected value is E[ y ] = E[(/)Σ i y i ] = α. If the mea square of x coverges to somethig fiite, the y is cosistet for α. That is, if plim(/)σ i x i = q where q is some fiite umber, the, plim y = α. As such, it follows that s ad s * = (/(-))Σ i (y i - α) have the same probability limit. We cosider, therefore, plim s * = plim(/(-))σ i ε i. The expected value of s * is E[(/(-)) Σ i ε i ] = σ (/Σ i x i ). Oce agai, othig more ca be said without some assumptio about x i. Thus, we assume agai that the average square of x i coverges to a fiite, positive costat, q. Of course, the result is uchaged by divisio by (-) istead of, so lim E[s * ] = σ q. The variace of s * is Var[s * ] = Σ i Var[ε i ]/( - ). To characterize this, we will require the variaces of the squared disturbaces, which ivolves their fourth momets. But, if we assume that every fourth momet is fiite, the the precedig is (/(-) ) times the average of these fourth momets. If every fourth momet is fiite, the the term is domiated by the leadig (/(-) ) which coverges to zero. It follows that plim s * = σ q. Therefore, the covetioal estimator estimates Asy.Var[ y ]= σ q /. The appropriate variace of the least squares estimator is Var[ y ]= (σ / )Σ i x i, which is, of course, precisely what we have bee aalyzig above. It follows that the covetioal estimator of the variace of the OLS estimator i this model is a appropriate estimator of the true variace of the least squares estimator. This follows from the fact that the regressor i the model, i, is urelated to the source of heteroscedasticity, as discussed i the text.. The sample momets are obtaied usig, for example, S xx = x x - x ad so o. For the two samples, we obtai y x S xx S yy S xy Sample Sample The parameter estimates are computed directly usig the results of Chapter 6. Itercept Slope R s Sample /3 4/9 (500/9)/48 = 3.47 Sample - 4/3 6/30 (400/9)/48 = The pooled momets based o 00 observatios are X X = , X y = , y y = The coefficiet vector based o these data is [a,b] = [0,]. This might have bee predicted sice the two X X matrices are idetical. OLS which igores the heteroscedasticity would simply average the estimates. The sum of squared residuals would be e e = y y - b X y = = 700, so the estimate of σ is s = 700/98 = 7.4. Note that the earlier values obtaied were 3.47 ad 9.7, so the pooled estimate is betwee the two, oce agai, as might be expected. The asymptotic covariace matrix of these estimates is s (X X) = To test the equality of the variaces, we ca use the Goldfeld ad Quadt test. Uder the ull hypothesis of equal variaces, the ratio F = [e e /( - )]/[e e /( - )] (or vice versa for the subscripts) is the ratio of two idepedet chi-squared variables each divided by their respective degrees of freedom. Although it might seem so from the discussio i the text (ad the literature) there is othig i the test which requires that the coefficiet vectors be assumed equal across groups. Sice for our data, the secod sample has the larger residual variace, we refer F[48,48] = s /s = 9.7 / 3.47 =.8 to the F table. The critical value for 95% sigificace is.6, so the hypothesis of equal variaces is rejected. The method of Example 8.5 ca be applied to this groupwise heteroscedastic model. The two step estimator is β = [(/s )X X + (/s )X X ] - [(/s )X y + (/s )X y ]. The X X matrices are the same i 47

51 this problem, so this simplifies to β = [(/s + /s )X X] - [(/s )X y + (/s )X y ]. The estimator is, therefore = ?=======================================================? Applicatio 8.?======================================================= a. The ordiary least squares regressio of Y o a costat, X, ad X produces the followig results: Sum of squared residuals R Stadard error of regressio Variable Coefficiet Stadard Error t-ratio Oe X X b. Covariace Matrix White s Corrected Matrix c. To apply White's test, we first obtai the residuals from the regressio of Y o a costat, X, ad X. The, we regress the squares of these residuals o a costat, X, X, X, X, ad X X. The R i this regressio is.7896, so the chi-squared statistic is = The critical value from the table of chi-squared with 5 degrees of freedom is.08, so we would coclude that there is evidece of heteroscedasticity. d. Lagrage multiplier test. Regress;Lhs=y;rhs=oe,x,x ; Res=e ; het $ create ; lmi=e*e/(sumsqdev/) - $ Name ; x=oe,x,x $ Calc ; list ;.5*xss(x,lmi)$ The result was reported with the regressio, Br./Paga LM Chi-sq [ ] (prob) = 7.78 (.0000) e. Two step estimator read;obs=50;var=;ames=y;byva $ read;obs=50;var=;ames=x;byva $ read;obs=50;var=;ames=x;byva $ Regress;Lhs=y;rhs=oe,x,x ; Res=e $ Ordiary least squares regressio Model was estimated May, 007 at 08:33:0PM LHS=Y Mea = Stadard deviatio = WTS=oe Number of observs. = 50 Model size Parameters = 3 48

52 Degrees of freedom = 47 Residuals Sum of squares = 9.98 Stadard error of e = Fit R-squared = E-0 Adjusted R-squared = E-0 Model test F[, 47] (prob) =.93 (.4033) Diagostic Log likelihood = Restricted(b=0) = Chi-sq [ ] (prob) =.93 (.3806) Ifo criter. LogAmemiya Prd. Crt. = Akaike Ifo. Criter. = Autocorrel Durbi-Watso Stat. = Rho = cor[e,e(-)] = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat X X Create ; e = e*e $ Create ; loge = log(e) $ Regress ; lhs = loge ; Rhs = oe,x,x ; keep=vi $ Create ; vi = /exp(vi) $ Regress ; Lhs = y ; rhs = oe,x,x ; wts = vi $ Ordiary least squares regressio Model was estimated May, 007 at 08:33:0PM LHS=Y Mea = Stadard deviatio = WTS=VI Number of observs. = 50 Model size Parameters = 3 Degrees of freedom = 47 Residuals Sum of squares = Stadard error of e = Fit R-squared =.693 Adjusted R-squared =.78657E-0 Model test F[, 47] (prob) = 3.09 (.0548) Diagostic Log likelihood = Restricted(b=0) = Chi-sq [ ] (prob) = 6.8 (.0456) Ifo criter. LogAmemiya Prd. Crt. = Akaike Ifo. Criter. = Autocorrel Durbi-Watso Stat. = Rho = cor[e,e(-)] = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat X X

53 Applicatios?=======================================================? Applicatio 8. Gasolie Cosumptio?=======================================================? Reame variable for coveiece Create ; y=lgaspcar $? RHS of ew regressio Namelist ; x = oe,licomep,lrpmg,lcarpcap $? Base regressio. Is cars per capita sigificat? Regress ; Lhs = y ; Rhs = x $ Ordiary least squares regressio LHS=Y Mea = Stadard deviatio = WTS=oe Number of observs. = 34 Model size Parameters = 4 Degrees of freedom = 338 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 3, 338] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LINCOMEP LRPMG LCARPCAP Calc ; r0 = rsqrd $ Namelist ; Ctry=c,c3,c4,c5,c6,c7,c8,c9,c0,c,c,c3,c4,c5,c6,c7,c8$ Regress;lhs=y;rhs=x,ctry ; Res = e $ Ordiary least squares regressio LHS=Y Mea = Stadard deviatio = WTS=oe Number of observs. = 34 Model size Parameters = Degrees of freedom = 3 Residuals Sum of squares = Stadard error of e = E-0 Fit R-squared = Adjusted R-squared = Model test F[ 0, 3] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LINCOMEP LRPMG LCARPCAP C C C C C C C C C C C

54 C C C C C C Calc ; r = rsqrd $ Calc ; list ; Fstat = ((r - r0)/7) / ((-r)/(-4-7)) $ Calc ; list ; Fc =ftb(.95,7,(-4-7)) $ Listed Calculator Results FSTAT = FC = Plot ; lhs = coutry ; rhs = e ; Bars = 0 ;Title=Plot of OLS Residuals by Coutry $ Regress;lhs=y;rhs=x,ctry ; Het $ Ordiary least squares regressio LHS=Y Mea = Stadard deviatio = WTS=oe Number of observs. = 34 Model size Parameters = Degrees of freedom = 3 Residuals Sum of squares = Stadard error of e = E-0 Fit R-squared = Adjusted R-squared = Model test F[ 0, 3] (prob) = (.0000) White heteroscedasticity robust covariace matrix Br./Paga LM Chi-sq [ 0] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LINCOMEP LRPMG LCARPCAP C C C C C C C

55 C C C C C C C C C C Create ; e = e*e $ Regress ; Lhs = e ; Rhs = oe,ctry $ Calc ; List ; White = *rsqrd ; ctb(.95,7) $ Listed Calculator Results WHITE = Result = Calc ; s = e'e/ $ Matrix ; sg = {/9} * ctry'e ; sg = /s * sg ; g = sg - ; List ; lmstat = {9/}*g'g $ Matrix LMSTAT has rows ad colums Name ; All = c,ctry $ Matrix ; vg = /9*all'e $ Create ; wt = /vg(coutry) $ Regress ; Lhs = y ; rhs = x,ctry;wts=wt $ Ordiary least squares regressio LHS=Y Mea = Stadard deviatio = WTS=WT Number of observs. = 34 Model size Parameters = Degrees of freedom = 3 Residuals Sum of squares = Stadard error of e =.48779E-0 Fit R-squared = Adjusted R-squared = Model test F[ 0, 3] (prob) =89.9 (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LINCOMEP LRPMG LCARPCAP C C C C C C C C C C C C C C C C

56 C ?=======================================================? Applicatio 8.3 Iterative estimator?======================================================= create ; logc = log(c) ; logq=log(q) ; logq=logq^ ; logp=log(pf) $ Name ; x = oe,logq,logq,logp $ Regress ; lhs = logc ; rhs = x ; Res = e $ Matrix ; b0=b $ Procedure$ Create ; e = e*e ; le = e/(sumsqdev/)- $ (MLE)?le = log(e) $ (Iterative two step) Regress ; quiet ; lhs=le ; rhs=oe,lf ; keep = si $ Create ; wi = /exp(si) $ Regress ; lhs = logc ; rhs = x ; wts=wi ; res=e $ Matrix ; db = b-b0 ; b0 = b $ Calc ; list ; db = db'db $ Edproc $ Exec ; = 0 $ These are the two step estimators from Example 8.4 Ordiary least squares regressio LHS=LOGC Mea =.9005 Stadard deviatio =.944 WTS=WI Number of observs. = 90 Model size Parameters = 4 Degrees of freedom = 86 Residuals Sum of squares =.889 Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 3, 86] (prob) =96.37 (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGQ LOGQ LOGP These are the maximum likelihood estimates Ordiary least squares regressio Residuals Sum of squares = Stadard error of e =.594 Fit R-squared =.9890 Adjusted R-squared = Model test F[ 3, 86] (prob) =68.35 (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGQ LOGQ LOGP

57 Chapter 9 Models for Pael Data. The pooled least squares estimator is y = x, e e = (.95595) ( ) The fixed effects regressio ca be computed just by icludig the three dummy variables sice the sample sizes are quite small. The results are y = i -.836i +.66i x e e = (.05079) The F statistic for testig the hypothesis that the costat terms are all the same is F[6,] = [( )/]/[79.83/6] = 6.8. The critical value from the F table is 9.458, so the hypothesis is ot rejected. I order to estimate the radom effects model, we eed some additioal parameter estimates. The group meas are y x Group Group Group I the group meas regressio usig these three observatios, we obtai y i. = x i. with e ** e ** = There is oly oe degree of freedom, so this is the cadidate for estimatio of σ ε /T + σ u. I the least squares dummy variable (fixed effects) regressio, we have a estimate of σ ε of 79.83/6 = Therefore, our estimate of σ u is σ u =.9747/ /0 = Obviously, this wo't do. Before abadoig the radom effects model, we cosider a alterative cosistet estimator of the costat ad slope, the pooled ordiary least squares estimator. Usig the group meas above, we fid 3 Σ i= [ y i. - ( ) x i. ] = Oe ought to proceed with some cautio at this poit, but it is difficult to place much faith i the group meas regressio with but a sigle degree of freedom, so this is probably a preferable estimator i ay evet. (The true model uderlyig these data -- usig a radom umber geerator -- has a slope, β of.000 ad a true costat of zero. Of course, this would ot be kow to the aalyst i a real world situatio.) Cotiuig, we ow use σ u = /0 = 3.67 as the estimator. (The true value of ρ = σ u /(σ u +σ ε ) is.5.) This leads to θ = - [ / /(0(3.67) ) / ] =.754. Fially, the FGLS estimator computed accordig to (6-48) is y = -.345(.786) (.08998)x. For the LM test, we retur to the pooled ordiary least squares regressio. The ecessary quatities are e e = , Σ t e t = , Σ t e t = , Σ t e 3t = Therefore, LM = {[3(0)]/[(9)]}{[(-.5534) + (3.784) + (4.838) ]/ } = The statistic has oe degree of freedom. The critical value from the chi-squared distributio is 3.84, so the hypothesis of o radom effect is rejected. Fially, for the Hausma test, we compare the FGLS ad least squares dummy variable estimators. The statistic is χ = [( ) ]/[( ) - (.05060) ] = This is relatively small ad argues (oce agai) i favor of the radom effects model. 54

58 . There is o effect o the coefficiets of the other variables. For the dummy variable coefficiets, with the full set of dummy variables, each coefficiet is y i * = mea residual for the ith group i the regressio of y o the xs omittig the dummy variables. (We use the partitioed regressio results of Chapter 6.) If a overall costat term ad - dummy variables (say the last -) are used, istead, the coefficiet o the ith dummy variable is simply y i * - y * while the costat term is still y * For a full proof of these results, see the solutio to Exercise 5 of Chapter 8 earlier i this book. i= i i i= i i 3. (a) The pooled OLS estimator will be b= Σ X X Σ X y where X i ad y i have T i observatios. It remais true that y i = X i β + ε i + u i i, where Var[ε i + u i i X i ] = Var[w i X i ] = σ ε I + σ u ii ad, maitaiig the assumptios, both ε i ad u i are ucorrelated with X i. Substitutig the expressio for y i ito that of b ad collectig terms, we have i= i i i= i i b= β + Σ X X Σ X w. Ubiasedess follows immediately as log as E[w i X i ] equals zero, which it does by assumptio. Cosistecy, as metioed i Sectio 9.3., is covered i the discussio of Chapter 4. We would eed for the matrix Q = Σ XX to coverge to a matrix of costats, or ot to degeerate to a matrix of zeros. The i= T i i i requiremets for the large sample behavior of the vector i the secod set of brackets is quite the same as i our earlier discussios of cosistecy. The vector (/ ) Σ i= Xw i i = (/ ) Σi= v i has mea zero. We would require the coditios of the Lideberg-Feller versio of the cetral theorem to apply, which could be expected. (b) We seek to establish cosistecy, ot ubiasedess. As such, we will igore the degrees of freedom correctio, -K, i (9-37). Use (T-) as the deomiator. Thus, the questio is whether T Σi= Σt= ( eit ei. ) plim = σε T ( ) If so, the the estimator i (9-37) will be cosistet. Usig (9-33) ad e it - ei = yi xb i ai, it follows that eit ei =εit εi ( xit xi )( b β ). Summig the squares i (9-37), we fid that the estimator i (9-37) T Σi= Σt= ( eit ei. ) T = σ ˆ ( i) + ( b β) ( )( ) ( ) i= i t it i it i T ( ) = x x x x = T b β T - ( b β) ( )( it i εit εi. ) i= x x t= T The secod term will coverge to zero as the ceter matrix coverges to a costat Q ad the vectors coverge to zero as b coverges to β. (We use the Slutsky theorem.) The third term will coverge to zero as both the leadig vector coverges to zero ad the covariace vector betwee the regressors ad the disturbaces coverges to zero. That leaves the first term, which is the average of the estimators i (9-34). The terms i the average are idepedet. Each has expected value exactly equal to σ ε. So, if each estimator has fiite variace, the the average will coverge to its expectatio. Appedix D discusses various differet coditios uderwhich a sample average will coverge to its expectatio. For example, fiite fouth momet of ε it would be sufficiet here (though weaker coditios would also suffice). Note that this derivatio follows through for ay cosistet estimator of β, ot just for b. 4. To fid plim(/)lm = plim [T/((T-))]{[Σ i (Σ t e it ) ]/[Σ i Σ t e it ] - } we ca cocetrate o the sums iside the curled brackets. First, Σ i (Σ t e it ) = T {(/)Σ i [(/T)Σ t e it ] } ad Σ i Σ t e it = T(/(T))Σ i Σ t e it. The ratio equals [Σ i (Σ t e it ) ]/[Σ i Σ t e it ] = T{(/)Σ i [(/T)Σ t e it ] }/{(/(T))Σ i Σ t e it }. Usig the argumet used i Exercise 8 to establish cosistecy of the variace estimator, the limitig behavior of this statistic is the same as that which is computed usig the true disturbaces sice the OLS coefficiet estimator is cosistet. Usig the true disturbaces, the umerator may be writte (/)Σ i [(/T)Σ t ε it ] = (/)Σ i ε i. Sice E[ ε i. ] = 0, 55

59 plim(/)σ i ε i. = Var[ ε i. ] = σ ε T + σ u The deomiator is simply the usual variace estimator, so plim(/(t))σ i Σ t ε it = Var[ε it ] = σ ε + σ u Therefore, isertig these results i the expressio for LM, we fid that plim (/)LM = [T/((T-))]{[T(σ ε T + σ u )]/[σ ε + σ u ] - }. Uder the ull hypothesis that σ u = 0, this equals 0. By expadig the ier term the collectig terms, we fid that uder the alterative hypothesis that σ u is ot equal to 0, plim (/)LM = [T(T-)/][ σ u /(σ ε +σ u )]. Withi group i, Corr [ε it,ε is ] = ρ = σ u /(σ u + σ ε ) so plim (/)LM = [T(T-)/](ρ ). It is worth otig what is obtaied if we do ot divide the LM statistic by at the outset. Uder the ull hypothesis, the limitig distributio of LM is chi-squared with oe degree of freedom. This is a radom variable with mea ad variace, so the statistic, itself, does ot coverge to a costat; it coverges to a radom variable. Uder the alterative, the LM statistic has mea ad variace of order (as we see above) ad hece, explodes. It is this latter attribute which makes the test a cosistet oe. As the sample size icreases, the power of the LM test must go to. 5. The ordiary least squares regressio results are R =.9803, e e = 46.76, 40 observatios Variable Coefficiet Stadard Error X X Costat Period Period Period Period Period Period Period Period Period Group Group Group Estimated covariace matrix for the slopes: β β β β For testig the hypotheses that the sets of dummy variable coefficiets are zero, we will require the sums of squared residuals from the restrictios. These are Regressio Sum of squares All variables icluded Period variables omitted Group variables omitted Period ad group variables omitted The F statistics are therefore, () F[9,5] = [( )/9]/[46.76/5] = 3.5 () F[3,5] = [( )/3]/[46.76/5] =.639 (3) F[,5] = [( )/]/[46.76/5] = 6.3 The critical values for the three distributios are.83,.99, ad.65, respectively. All sample statistics are larger tha the table value, so all of the hypotheses are rejected. 6. The covariace matrix would be 56

60 i =, t = i =, t = i =, t = i =, t = i =, t = σε + σu + σv σu σv 0 i =, t = σu σε + σu + σv 0 σv i =, t = σv 0 σε + σu + σv σu i =, t = 0 σv σu σε + σu + σv 7. The two separate regressios are as follows: Sample Sample b = x y/x x 4/5 =.8 6/0 =.6 e e = y y - bx y 0-4(4/5) = 84/5 0-6(6/0) = 64/0 R = - e e/y y - (84/5)/0 =.6 - (64/0)/0 =.36 s = e e/(-) (84/5)/9 =.884 (64/0)/9 = Est.Var[b] = s /x x.884/5 = /0 = To carry out a Lagrage multiplier test of the hypothesis of equal variaces, we require the separate ad commo variace estimators based o the restricted slope estimator. This, i tur, is the pooled least squares estimator. For the combied sample, we obtai b = [x y + x y ]/[x x + x x ] = (4 + 6) / (5 + 0) = /3. The, the variace estimators are based o this estimate. For the hypothesized commo variace, e e = (y y + y y ) - b(x y + x y ) = (0 + 0) - (/3)(4 + 6) = 70/3, so the estimate of the commo variace is e e/40 = (70/3)/40 = Note that the divisor is 40, ot 39, because we are comptutig maximum likelihood estimators. The idividual estimators are e e /0 = (y y - b(x y ) + b (x x ))/0 = (0 - (/3)4 + (/3) 5)/0 = ad e e /0 = (y y - b(x y ) + b (x x ))/0 = (0 - (/3)6 + (/3) 0)/0 =.3. The LM statistic is give i Example 6.3, LM = (T/)[(s /s - ) + (s /s - ) ] = 0[(.84444/ ) + (.3/ ) ] = This has oe degree of freedom for the sigle restrictio. The critical value from the chi-squared table is 3.84, so we would reject the hypothesis. I order to compute a two step GLS estimate, we ca use either the origial variace estimates based o the separate least squares estimates or those obtaied above i doig the LM test. Sice both pairs are cosistet, both FGLS estimators will have all of the desirable asymptotic properties. For our estimator, we used σ = e j e j /T from the origial regressios. Thus, σ =.84 ad σ =.3. The GLS estimator is β = [(/ σ )x y + (/ σ )x y ]/[ (/ σ )x x + (/ σ )x x ] = [4/ /.3]/[5/ /.3] =.63. The estimated samplig variace is /[ (/ σ )x x + (/ σ )x x ] = This implies a asymptotic stadard error of (.0688) = To test the hypothesis that β =, we would refer z = (.63 - ) /.6395 = -.45 to a stadard ormal table. This is reasoably large, ad at the usual sigificace levels, would lead to rejectio of the hypothesis. The Wald test is based o the urestricted variace estimates. Usig b =.63, the variace estimators are σ = [y y - b(x y ) + b (x x )]/0 = ad σ = [y y - b(x y ) + b (x x )]/0 =.305 while the pooled estimator would be σ = [y y - b(x y) + b (x x)]/40 = The statistic is give at the ed of Example 6.3, W = (T/)[( σ / σ - ) + ( σ / σ - ) ] = 0[( / ) + ( / ) ] = We reach the same coclusio as before. To compute the maximum likelihood estimators, we begi our iteratios from the two separate ordiary least squares estimates of b which produce estimates σ =.84 ad σ =.3. The iteratios are Iteratio σ σ β

61 coverged Now, to compute the likelihood ratio statistic for a likelihood ratio test of the hypothesis of equal variaces, we refer χ = 40l l l to the chi-squared table. (Uder the ull hypothesis, the pooled least squares estimator is maximum likelihood.) Thus, χ = 4.564, which is roughly equal to the LM statistic ad leads oce agai to rejectio of the ull hypothesis. Fially, we allow for cross sectioal correlatio of the disturbaces. Our iitial estimate of b is the pooled least squares estimator, /3. The estimates of the two variaces are ad.3 as before while the cross sectioal covariace estimate is e e /0 = [y y - b(x y + x y ) + b (x x )]/0 = Before proceedig, we ote, the estimated squared correlatio of the two disturbaces is r =.4444 / [(.84444)(.3)] / =.77, which is ot particularly large. The LM test statistic give i (6-4) is.533, which is well uder the critical value of Thus, we would ot reject the hypothesis of zero cross sectio correlatio. Noetheless, we proceed. The estimator is show i (6-6). The two step FGLS ad iterated maximum likelihood estimates appear below. Iteratio σ σ σ β coverged Because the correlatio is relatively low, the effect o the previous estimate is relatively mior. 8. If all of the regressor matrices are the same, the estimator i (8-35) reduces to β = (X X) - Σ i = {(/σ i )/[Σ = (/σ j j )]}X y i = Σ i= w i b i a weighted average of the ordiary least squares estimators, b i = (X X) - X y i with weights w i = (/σ i )/[Σ = (/σ j j )]. If it were ecessary to estimate the weights, a simple two step estimator could be based o idividual variace estimators. Either of s i = e i e i /T based o separate least squares regressios (with differet estimators of β) or based o residuals computed from a commo pooled ordiary least squares slope estimator could be used. 9. The various least squares estimators of the parameters are Sample Sample Sample 3 Pooled a (9.658) (0.46) (7.38) b (.438) (.4756) (.3590) e e (464.88) (73.560) (7.40) ( ) (Values of e e i paretheses above are based o the pooled slope estimator.) The FGLS estimator ad its estimated asymptotic covariace matrix are b =, Est.Asy.Var[b] = Note that the FGLS estimator of the slope is closer to the of sample 3 (the highest of the three OLS estimates). This is to be expected sice the third group has the smallest residual variace. The LM test statistic is based o the pooled regressio, LM = (0/){[(464.88/0)/( /30) - ] +...} =

62 To compute the Wald statistic, we require the urestricted regressio. The parameter estimates are give above. The sums of squares are , , ad for i =,, ad 3, respectively. For the commo estimate of σ, we use the total sum of squared GLS residuals, The, W = (0/){[(396.6/30)/( /0) - ] +...} = 5.. The Wald statistic is far larger tha the LM statistic. Sice there are two restrictios, at sigificace levels of 95% or 99% with critical values of 5.99 or 9., the two tests lead to differet coclusios. The likelihood ratio statistic based o the FGLS estimates is χ = 30l(396.6/30) - 0l( /0)... = 6.4 which is betwee the previous two ad betwee the 95% ad 99% critical values. Applicatios As usual, the applicatios below require ecoometric software. The computatios ca be doe with ay moder software package, so o specific program is recommeded. --> read $ Last observatio read from data file was 00 Ed of data listig i edit widow was reached --> REGRESS ; Lhs = I ; Rhs = F,C,oe $ Ordiary least squares regressio LHS=I Mea = Stadard deviatio = WTS=oe Number of observs. = 00 Model size Parameters = 3 Degrees of freedom = 97 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[, 97] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat > CALC ; R0=Rsqrd $ --> REGRESS ; Lhs = I ; Rhs = F,C,oe ; Cluster = 0 $ Ordiary least squares regressio LHS=I Mea = Stadard deviatio = WTS=oe Number of observs. = 00 Model size Parameters = 3 Degrees of freedom = 97 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[, 97] (prob) = (.0000) Covariace matrix for the model is adjusted for data clusterig. Sample of 00 observatios cotaied 0 clusters defied by 0 observatios (fixed umber) i each cluster. Sample of 00 observatios cotaied strata defied by 00 observatios (fixed umber) i each stratum

63 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat The stadard errors icrease substatially. This is at least suggestive that there is correlatio across observatios withi the groups. A formal test would be based o oe of the pael models below. Whe the radom effects model is fit by maximum likelihood, for example, the log likelihood fuctio is The log likelihood fuctio for the pooled model is Thus, the correlatio is highly sigificat. The Lagrage multiplier statistic reported below is 798.6, which is far larger tha the critical value of Oce agai, these results do suggest withi groups correlatio. --> REGRESS ; Lhs = I ; Rhs = F,C,oe ; Pael ; Pds=0 ; Fixed $ Least Squares with Group Dummy Variables Ordiary least squares regressio LHS=I Mea = Stadard deviatio = WTS=oe Number of observs. = 00 Model size Parameters = Degrees of freedom = 88 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[, 88] (prob) = (.0000) Pael:Groups Empty 0, Valid data 0 Smallest 0, Largest 0 Average group size 0.00 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Test Statistics for the Classical Model Model Log-Likelihood Sum of Squares R-squared () Costat term oly D () Group effects oly D (3) X - variables oly D (4) X ad group effects D Hypothesis Tests Likelihood Ratio Test F Tests Chi-squared d.f. Prob. F um. deom. P value () vs () (3) vs () (4) vs () (4) vs () (4) vs (3) > CALC ; R = Rsqrd $ --> MATRIX ; bf = b(:) ; vf = varb(:,:) $ --> CALC ; List ; Fstat=((R-R0)/9)/((-R)/(--0)) ; FC=Ftb(.95,9,(--0)) $ Listed Calculator Results FSTAT =

64 FC = The F statistic of 49.8 is far larger tha the critical value, so the hypothesis of equal costat terms is rejected. --> REGRESS ; Lhs = I ; Rhs = F,C,oe ; Pael ; Pds=0 ; Radom $ Radom Effects Model: v(i,t) = e(i,t) + u(i) Estimates: Var[e] =.78446D+04 Var[u] =.6849D+04 Corr[v(i,t),v(i,s)] = Lagrage Multiplier Test vs. Model (3) = ( df, prob value = ) (High values of LM favor FEM/REM over CR model.) Sum of Squares.8409D+07 R-squared D Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat The LM statistic, as oted earlier, is very large, so the hypothesis of o effects is rejected. --> MATRIX ; br = b(:) ; vr = varb(:,:) $ --> MATRIX ; db = bf-br ; vdb = vf-vr ; List ; Hausma=db'<vdb>db $ > CALC ; List ; Ctb(.95,) $ Listed Calculator Results Result = The Hausma statistic is quite small, which suggests that the radom effects approach is cosistet with the data. 6

65 . create ; logc=log(cost/pfuel) ; logp=log(pmtl/pfuel) ; logp=log(peqpt/pfuel) ; logp3=log(plabor/pfuel) ; logp4=log(pprop/pfuel) ; logp5=log(kprice/pfuel) ; logq=log(output) ; logq=.5*logq^ $ Namelist ; cd = logp,logp,logp3,logp4,logp5 $ create ; p=.5* logp^ ; p=.5* logp^ ; p33=.5* logp3^ ; p44=.5* logp4^ ; p55=.5* logp5^ ; p=logp*logp ; p3=logp*logp3 ; p4=logp*logp4 ; p5=logp*logp5 ; p3=logp*logp3 ; p4=logp*logp4 ; p5=logp*logp5 ; p34=logp3*logp4 ; p35=logp3*logp5 ; p45=logp4*logp5 $ Namelist ; tl = p,p,p3,p4,p5,p,p3,p4,p5,p33,p34,p35,p44,p45,p55$ Namelist ; z = loadfctr,stage,poits $ regress;lhs=logc;rhs=oe,logq,logq,cd,z $ Ordiary least squares regressio LHS=LOGC Mea = Stadard deviatio = WTS=oe Number of observs. = 56 Model size Parameters = Degrees of freedom = 45 Residuals Sum of squares = Stadard error of e =.004 Fit R-squared = Adjusted R-squared = Model test F[ 0, 45] (prob) =407.3 (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X Costat LOGQ LOGQ LOGP LOGP LOGP LOGP LOGP LOADFCTR STAGE D POINTS ?? Turs out the traslog model caot be computed with the firm? dummy variables. I'll use the Cobb Douglas form.? regress;lhs=logc;rhs= oe,logq,logq,cd ; pael ; pds=ti $ OLS Without Group Dummy Variables Ordiary least squares regressio LHS=LOGC Mea = Stadard deviatio = WTS=oe Number of observs. = 56 6

66 Model size Parameters = 8 Degrees of freedom = 48 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 7, 48] (prob) = (.0000) Pael Data Aalysis of LOGC [ONE way] Ucoditioal ANOVA (No regressors) Source Variatio Deg. Free. Mea Square Betwee Residual E-0 Total Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X LOGQ LOGQ LOGP LOGP LOGP LOGP LOGP Costat Least Squares with Group Dummy Variables Ordiary least squares regressio LHS=LOGC Mea = Stadard deviatio = WTS=oe Number of observs. = 56 Model size Parameters = 3 Degrees of freedom = 4 Residuals Sum of squares = Stadard error of e =.64689E-0 Fit R-squared = Adjusted R-squared = Model test F[ 3, 4] (prob) =6.94 (.0000) Pael:Groups Empty 0, Valid data 5 Smallest, Largest 5 Average group size 0.4 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X LOGQ LOGQ LOGP LOGP LOGP LOGP LOGP Test Statistics for the Classical Model Model Log-Likelihood Sum of Squares R-squared () Costat term oly D () Group effects oly D (3) X - variables oly D (4) X ad group effects D Hypothesis Tests Likelihood Ratio Test F Tests 63

67 Chi-squared d.f. Prob. F um. deom. P value () vs () (3) vs () (4) vs () (4) vs () (4) vs (3) Radom Effects Model: v(i,t) = e(i,t) + u(i) Estimates: Var[e] =.48468D-0 Var[u] =.70D-0 Corr[v(i,t),v(i,s)] =.7533 Lagrage Multiplier Test vs. Model (3) = ( df, prob value = ) (High values of LM favor FEM/REM over CR model.) Baltagi-Li form of LM Statistic = Fixed vs. Radom Effects (Hausma) = ( 7 df, prob value =.00000) (High (low) values of H favor FEM (REM).) Sum of Squares.64877D+0 R-squared D Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X LOGQ LOGQ LOGP LOGP LOGP LOGP LOGP Costat regress;lhs=logc;rhs=z,oe,logq,logq,cd ; pael ; pds=ti $ OLS Without Group Dummy Variables Ordiary least squares regressio LHS=LOGC Mea = Stadard deviatio = WTS=oe Number of observs. = 56 Model size Parameters = Degrees of freedom = 45 Residuals Sum of squares = Stadard error of e =.004 Fit R-squared = Adjusted R-squared = Model test F[ 0, 45] (prob) =407.3 (.0000) Pael Data Aalysis of LOGC [ONE way] Ucoditioal ANOVA (No regressors) Source Variatio Deg. Free. Mea Square Betwee Residual E-0 Total Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X LOADFCTR STAGE D POINTS LOGQ LOGQ LOGP LOGP LOGP

68 LOGP LOGP Costat Least Squares with Group Dummy Variables Ordiary least squares regressio LHS=LOGC Mea = Stadard deviatio = WTS=oe Number of observs. = 56 Model size Parameters = 35 Degrees of freedom = Residuals Sum of squares = Stadard error of e =.5965E-0 Fit R-squared = Adjusted R-squared = Model test F[ 34, ] (prob) = (.0000) Pael:Groups Empty 0, Valid data 5 Smallest, Largest 5 Average group size 0.4 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X LOADFCTR STAGE D POINTS LOGQ LOGQ LOGP LOGP LOGP LOGP LOGP Test Statistics for the Classical Model Model Log-Likelihood Sum of Squares R-squared () Costat term oly D () Group effects oly D (3) X - variables oly D (4) X ad group effects D Hypothesis Tests Likelihood Ratio Test F Tests Chi-squared d.f. Prob. F um. deom. P value () vs () (3) vs () (4) vs () (4) vs () (4) vs (3) Radom Effects Model: v(i,t) = e(i,t) + u(i) Estimates: Var[e] = D-0 Var[u] = D-0 Corr[v(i,t),v(i,s)] =.706 Lagrage Multiplier Test vs. Model (3) = ( df, prob value = ) (High values of LM favor FEM/REM over CR model.) Baltagi-Li form of LM Statistic = 70.0 Fixed vs. Radom Effects (Hausma) = (0 df, prob value = ) (High (low) values of H favor FEM (REM).) Sum of Squares.45094D+0 65

69 R-squared.9848D Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X LOADFCTR STAGE D POINTS LOGQ LOGQ LOGP LOGP LOGP LOGP LOGP Costat matrix ; List ; bz=b(:3);vz=varb(:3,:3) ; wald = bz'<vz>bz $ Matrix WALD has rows ad colums

70 Chapter 0 Systems of Regressio Equatios y. The model ca be writte as y i = i + ε μ. Therefore, the OLS estimator is ε m = (i i + i i) - (i y + i y ) = ( y + y ) / ( + ) = ( y + y )/ =.5. The samplig variace would be Var[m] = (/) {Var[ y ] + Var[ y ] + Cov[( y, y )]}. We would estimate the parts with Est.Var[ y ] = s / = ((50-00() )/99)/00 =.005 Est.Var[ y ] = s / = ((550-00() )/99)/00 =.05 Est.Cov[ y, y ] = s / = ((60-00()())/99)/00 =.006 Combiig terms, Est.Var[m] = The GLS estimator would be [(σ + σ )i y + (σ + σ )i y ]/[(σ + σ )i i + (σ + σ )i i] = w y + (-w) y where w = (σ + σ ) / (σ + σ + σ ). Deotig Σ = σ σ σ σ, Σ - σ σ = σσ σ σ σ. The weight simplifies a bit as the determiat appears i both the deomiator ad the umerator. Thus, w = (σ - σ ) / (σ + σ - σ ). For our sample data, the two step estimator would be based o the variaces computed above ad s =.505, s =.55, s =.606. The, w =.50. The FGLS estimate is.5() + ( -.5)() =.875. The samplig variace of this estimator is w Var[ y ] + ( - w) Var[ y ] + w( - w)cov[ y, y ] =.0050 as compared to.0079 for the OLS estimator.. The model is y = y y = Xβ + ε = i 0 0 x β + ε β, σ Ω = ε σ σ I σ I σ I I. The geeralized least squares estimator is β = [ X' Ω X] X' Ω y = σ σ ii ' i'x σ iy ' + σ iy ' σ i'x σ x'x σ xy ' + σ xy ' σ σ x σ y + σ y = σ x σ sxx σ sx + σ sx where s xx = x x/, s x = x y /, s x = x y / ad σ ij = the ijth elemet of the Σ -. To obtai the explicit form, ote, first, that all terms σ ij are of the form σ ji /(σ σ - σ ) But, the deomiator i these ratios will be cacelled as it appears i both the iverse matrix ad i the vector. Therefore, i terms of the origial parameters, (after cacellig ), we obtai β σ = σ σ x σ s x σ y σ y σ s + σ s xx x x = σ σ s ( σ x) xx σ s σ x xx σ x σ y σ y σ σ s + σ s The two elemets are β = [σ s xx (σ y - σ y ) - σ x (σ s x -σ s x )]/[σ σ s xx - (σ x ) ] The asymptotic covariace matrix is β = [σ x (σ y - σ y ) - σ (σ s x - σ s x )]/[σ σ s xx - (σ x ) ] x x. 67

71 [X Ω - X] - σ σ x = = σ x σ sxx σ σ σ σ σ σ x σ sxx x The OLS estimator is b = (X X) - y X y =. The samplig variace is x'y / x'x (X X) - X ΩX(X X) - = 0 0 s σ σ ad reduce to (/). This leaves Var[b] = x σ x σ s 0 s xx xx xx 0 σ / σ x/( s ) xx. σ x/( sxx ) σ /( sxx ). The s are carried outside the product Usig the results above, the OLS coefficiets are b = y = 50/50 = 3 ad b = x y /x x = 50/00 = /. The estimators of the disturbace (co-)variaces are s = Σ i (y i - y ) / = (500-50(3))/50 = s = Σ i (y i - b x i ) / = (90 - (/)50)/50 =.3 s = Σ i (y i - y )(y i - b x i ) / = [y y - y y - b x y + b y x ]/ = (40-50(3)() - (/) (/)(3)()/50 =. Therefore, we estimate the asymptotic covariace matrix of the OLS estimates as / 50. ( )[ 50( 90)] Est.Var[b] =.()[ 50 / 90] 3. / 90 = To compute the FGLS estimates, we use our results from part a. The ecessary statistics for the computatio are s =, s =.3, s =., s xx = 00/50 =, x = 00/50 =, y = 50/50 = 3, y = 50/50 = s x = 60/50 =., s x = 50/50 = The, β = {()[.3(3) -.()] -.()[.(.) - ()]}/{(.3) - [.()] } = 3.57 β = {()[.3(3) -.()] -.3[.(.) - ()]}/{(.3) - [.()] } =.0 The estimate of the asymptotic covariace matrix is (/50)[(.3) - (.) ]/{(.3) - [.()] ( ). ( ) }.() 3. = Notice that the estimated variace of the FGLS estimator of the parameter of the first equatio is larger. The result for the true GLS estimator based o kow values of the disturbace variaces ad covariace does ot guaratee that the estimated variaces will be smaller i a fiite sample. However, the estimated variace of the secod parameter is cosiderably smaller tha that for the OLS estimate. Fially, to test the hypothesis that β = we use the z-statistic (asymptotically distributed as stadard ormal), z = (.0 - ) / ( ) =.3. The hypothesis caot be rejected. 3. The ordiary least squares estimates of the parameters are b = x y /x x = 4/5 =.8 ad b = x y /x x = 6/0 =.6 The, the variaces ad covariace of the disturbaces are s = (y y - b x y )/ = (0 -.8(4))/0 =.84 s = (y y - b x y )/ = (0 -.6(6))/0 =.3 s = (y y - b x y - b x y + b b x x )/ = (6 -.6(3) -.8(3) +.8(.6)())/0 =.46 68

72 We will require S = = s. The, the FGLS estimator is s s β = β s x ' x s x ' x s x ' y + s x ' y. Isertig the values give i the problem produces s x' x s x' x s x' y + s x' y the FGLS estimates, β = , β =.5474 with estimated asymptotic covariace matrix equal to the iverse matrix show above, Est.Var β = To test the hypothesis, we use the t statistic, t = ( )/[ ( )] = which is quite small. We would ot reject the hypothesis. To compute the maximum likelihood estimates, we would begi with the OLS estimates of σ, σ, ad σ. The, we iterate betwee the followig calculatios () Compute the matrix, S - () Compute the matrix [X (S - I)X] = s s x' x x' x s x' x s x' x [X (S - I)y] = s s x' y + x' y s x' y + s x' y (3) Compute the coefficiet vector β = [X (S - I)X] - [X (S - I)y] Compare this estimate to the previous oe. If they are similar eough, exit the iteratios. (4) Recompute S usig s ij = y i y j - β i x i y j - β j x j y i + β i β j x i x j, i,j =,. (5) Go back to step () ad cotiue. Our iteratios produce the two slope estimates : : : : : : : coverged. At covergece, we fid the estimate of the asymptotic covariace matrix of the estimates as [XN(S - I)X] = ad S = To use the likelihood ratio method to test the hypothesis, we will require the restricted maximum likelihood estimate. Uder the hypothesis,the model is the oe i Sectio 5... The restricted estimate is give i (5-) ad the equatios which follow. To obtai them, we make a small modificatio i our algorithm above. We replace step (3) with (3') β = [s x y + s x y + s (x y + x y )]/[s x x + s x x + s x x ]. Step 4 is the computed usig this commo estimate for both β ad β. The iteratios produce : : : : : : coverged. 69

73 At this estimate, the estimate of Σ is The likelihood ratio statistic is give i (5-56). Usig our ucostraied ad costraied estimates, we fid W u =.4774 ad W r = The statistic is λ = 0(l l.4774) =.03. This is far below the critical value of 3.84, so oce agai, we do ot reject the hypothesis. 4. The GLS estimator is β σ X' X σ X' X σ X' y + σ X' y = σ X' X σ X' X σ X' y + σ X' y The matrix to be iverted equals [Σ - X X] -. But, [Σ - X X] - = Σ (X X) -. (See (-76).) Therefore, β = σ - σ - (X'X) (X'X) σ X'y + σ X'y - - σ (X'X) σ (X'X) σ X'y + σ X'y We ow make the replacemets X y = (X X)b ad X y = (X X)b. After multiplyig out the product, we fid that β σσ b + σσ b + σσ b + σσ b σσ σσ )b σσ σσ )b = σσ b + σσ b + σσ b + σσ b = ( + + ( + ( σσ + σσ )b + ( σσ + σσ ) b The four scalar terms i the matrix product are the correspodig elemets of ΣΣ - = I. Therefore, β b =. b 5. The algebraic result is a little tedious, but straightforward. The GLS estimator which is computed is β σ σ σ σ = β x ' x x ' x x ' y + x ' y. σ x' x σ x' x σ x' y + σ x' y It helps at this poit to make some simplifyig substitutios. The elemets i the iverse matrix, σ ij, are all equal to elemets of the origial matrix divided by the determiat. But, the determiat appears i the leadig matrix, which is iverted ad i the trailig vector (which is ot). Therefore, the determiat will cacel out. Makig the substitutios, β σ σ σ = x ' x x ' x x ' y σ x ' y σ σ σ σ β x x x x x y +. Now, ' ' ' x ' y we are cocered with probability limits. We divide every elemet of the matrix to be iverted by, the because of the iversio, divide the vector o the right by as well. Suppose, for simplicity, that lim x i x j / = q ij, i,j =,,3. The, plim β σ σ σ = q q x ' y / σ x ' y / σ σ plim σ σ β q q x ' y / + x ' y / The, we will use plim (/)x y = β q + plim (/)x Nε = β q plim (/)x y = β q + β 3 q 3 plim (/)x y = β q plim (/)x y = β q + β 3 q 3. Therefore, after multiplyig out all the terms, plim β σ σ βσ β βσ = q q q σ q q σ σ βσ βσ βσ β q q q + q + q The iverse matrix is σ σ q q ( σ q ) σ σ q σ q σ q q, so with Δ = (σ F q q - (F q ) ) 70

74 βˆ σq σq βσ q βσq β3σq3 plim = ˆ q q q q 3 q. Takig the first coefficiet β Δ σ σ β σ + β σ + β σ 3 separately ad collectig terms, plimβ = β [σ σ q q -(σ q ) ]/Δ + β [σ q σ q + σ q σ q ]/Δ + β 3 [σ q σ q 3 + σ q σ q 3 ]/Δ The first term i brackets equals Δ while the secod equals 0. That leaves plim β = β - β 3 [σ σ (q q 3 - q q 3 )]/Δ which is ot equal to β. There are two special cases worthy of ote, though. The right had side does equal β if either () σ = 0; the regressios are actually urelated, or () q = q 3 = 0; the regressors i the two equatios are ucorrelated. The secod of these is similar to our fidig for omitted variables i the classical regressio model. 6. The model is y y θ 0 i x 0 = α 0 i + ε β. The GLS estimator of the full coefficiet vector, θ, is ε α x σ y y σ σ + σ = x x'x x x'y x'y ( ) σ σ x σ y + σ y. Let q xx equal x x/, q x = x y / ad, q x = x y /. The s i the iverse ad i the vector cacel. Also, as suggested, we assume that x = 0. As i the previous exercise, we replace elemets of the iverse with elemets from the origial matrix ad cacel the determiat which multiplies the matrix (after iversio) ad divides the vector. Thus, θ σ 0 σ σ y σ y = 0 σqxx 0 σqx σqx. The iverse of the matrix is straightforward. Proceedig σ σ σ + σ 0 y y σ directly, we obtai θ σqxx 0 σσq = 0 σ 0 σ σ σqxx ( σσ σ ) σσqxx 0 σqxx It remais oly to multiply the matrices ad collect terms. The result is α = y, α = y, β = [(q x /q xx ) - (σ σ )(q x /q xx )] = b - γb. xx σ y σ y σqx σqx. σ + y σ y 7. Oce agai, othig is lost by assumig that x = 0. Now, the OLS estimators are a = y, a = y, a 3 = y 3, b = x y /x x. The vector of residuals is e i = y i - y - bx i e i = y i - y e i3 = y i3 - y 3 Now, if y i + y i3 = at every observatio, the (/)Σ i (y i + y i3 ) = y + y 3 = as well. Therefore, by just addig the two equatios, we see that e i + e i3 = 0 for every observatio. Let e i be the 3 vector of residuals. The, e i c = 0, where c = [0,,]. The sample covariace matrix of the residuals is S = [(/)Σ i e i e i ]. The, Sc = [(/)Σ i e i e i ]c = [(/)Σ i e i e i c] = [(/)Σ i e i 0] = 0, which meas, by defiitio, that S is sigular. We ca proceed simply by droppig the third equatio. The addig up coditio implies that α 3 = - α. So, we ca treat the first two equatios as a seemigly urelated regressio model ad estimate a 3 usig the estimate of α. 7

75 Applicatios. By addig the share equatios vertically, we fid the restrictios β + β + β 3 = δ + δ + δ 3 = 0 δ + δ + δ 3 = 0 δ 3 + δ 3 + δ 33 = 0 γ y + γ y + γ y3 = 0. Note that the addig up coditio also implies ε + ε + ε 3 = 0. We will elimiate the third share equatio. The restrictios imply β 3 = - β - β δ 3 = - δ - δ δ 3 = - δ - δ δ 33 = - δ 3 - δ 3 = δ + δ + δ γ y3 = - γ y - γ y. By isertig these i the three share equatios, we fid S = β + δ lp + δ lp - δ lp 3 - δ lp 3 + γ y ly + ε = β + δ l(p /p 3 ) + δ l(p /p 3 ) + γ y ly + ε S = β + δ lp + δ lp - δ lp 3 - δ lp 3 + γ y ly + ε = β + δ l(p /p 3 ) + δ l(p /p 3 ) + γ y ly + ε S 3 = - β - β - δ lp - δ lp - δ lp - δ lp + δ lp 3 + δ lp 3 + δ lp 3 + δ lp 3 - γ y lp 3 - γ y lp 3 - ε - ε = - S - S For the cost fuctio, makig the substitutios for β 3, δ 3, δ 3, δ 33, ad γ y3 produces lc = α + β (lp - lp 3 ) + β (lp - lp 3 ) + δ ((l p )/ - lp lp 3 + (l p 3 )/) + δ ((l p )/ - lp lp 3 + (l p 3 )/) + δ (lp lp - lp lp 3 - lp lp 3 + (l p 3 )) + γ y ly(lp - lp 3 ) + γ y ly(lp - lp 3 ) + β y ly + β yy (l Y)/ + ε c = α + β l(p /p 3 ) + β l(p /p 3 ) + δ (l (p /p 3 ))/ + δ (l (p /p 3 ))/ + δ l(p /p 3 )l(p /p 3 ) + γ y lyl(p /p 3 ) + γ y lyl(p /p 3 ) + β y ly + β yy (l Y)/ + ε c The system of three equatios (cost ad two shares) ca be estimated as discussed i the text. Ivariace is achieved by usig a maximum likelihood estimator. The five parameters elimiated by the restrictios ca be estimated after the others are obtaied just by usig the restrictios. The restrictios are liear, so the stadard errors are also striaghtforward to obtai. The least squares estimates are show below. Estimated stadard errors appear i paretheses. Variable Cost Fuctio Capital Share Labor Share Oe 5.3 (45.9) (.4697).7 (.408) l(p k /p f ) -.74 (0.4).380 (.045).0033 (.0534) l(p l /p f ) 3.39 (.8).0065 (.059).068 (.054) l (p k /p f )/ (4.604) (.0098) -.07 (.0050) l (p l /p f )/ 8.6 (5.59) l(p k /p f )l(p l /p f ) (4.684) ly.674 (.997) l Y/, (.033) lyl(p k /p f ) -.33 (.65) lyl(p l /p f ).0863 (.98) The estimates do ot eve come close to satisfyig the cross equatio restrictios. The parameters i the cost fuctio are extremely large, owig primarily to rather severe multicolliearity amog the price terms. The results of estimatio of the system by direct maximum likelihood are show. The covergece criterio is the value of Belsley (discussed ear the ed of Sectio 5.5). The value α show below is g H - g where g is the gradiet ad H is the Hessia of the log-likelihood. Iteratio 0, F= , l*s*= , α=

76 Iteratio, F= , l*s*= , α= Iteratio, F= , l*s*= , α= Iteratio 3, F=47.68, l*s*= , α=.0004 Residual covariace matrix Cost Capital Labor Cost Capital Labor Coefficiet Estimate Std. Error α β k β l δ kk δ ll δ kl β y β yy γ yk γ yl β f δ kf δ lf δ ff.6767 γ yf The meas of the variables are: Y = 353.8, p k = 69.35, p l =.039, p f = 6.4. The three factor shares computed at these meas are S k =.48, S l =.0865, S f = (The sample meas are.4,.0954, ad.4936.) The matrix of elasticities computed accordig to (5-7) is k l f.05 k Σ = l f (Two of the three diagoals have the `wrog' sig. This may be due to the very small sample size. The cross elasticities however do coform to what oe might expect, the primary oe beig the evidet substitutio betwee capital ad fuel. To test the hypothesis that γ yi = 0, we reestimate the model without the iteractio terms betwee ly ad the prices i the cost fuctio ad without ly i the factor share equatios. The iteratios for this restricted model are show below. Iter.= 0, F= , log S = , α=.93 Iter.=, F=3.75, log S = , α= Iter.=, F=36.340, log S = , α= Iter.= 3, F=4.349, log S = , α= Iter.= 4, F=4.559, log S = , α=.0636 Coverged achieved Sice we are iterested oly i the test statistic, we have ot listed the parameter estimates. The test statistic give i (7-6) is λ = T(l S r - l S u ) = 0( ( )) = There are two restrictios sice oly two of the three parameters are free. The critical value from the chi-squared table is 5.99, so we would reject the hypothesis. 73

77 ?===========================================? Applicatio 0.?===========================================? a. Separate regressios ad aggregatio test.? This saves the residuals to be used later. CALC ; SS=0 $ MATRIX ; EOLS = Iit(0,0,0) $ PROCEDURE $ Iclude ; ew ; Firm = compay $ REGRESS ; Lhs = I ; Rhs = F,C,oe ; Res = e$ CALC ; SS=SS + Sumsqdev $ MATRIX ; EOLS(*,compay) = e $ ENDPROC $ EXECUTE ; Compay=,0 $ SAMPLE ; -00 $ Residuals Sum of squares = Stadard error of e = Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X 74

78 F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared =.954 Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = 0.3 Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat Residuals Sum of squares = Stadard error of e = Fit R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat

79 Ordiary least squares regressio LHS=I Mea = Stadard deviatio = WTS=oe Number of observs. = 00 Model size Parameters = 3 Degrees of freedom = 97 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[, 97] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat ? b. Aggregatio test REGRESS ; LHS = I ; RHS = F,C,oe $ CALC ; SS0=Sumsqdev $ CALC ; List ; Fstat = ((SS0 - SS)/(9*3)) / (SS0/(-0*3)) ; FC = Ftb(.95,7,70) $ Listed Calculator Results FSTAT = FC =.55534? c. SUR model NAMELIST ; X=F,C,oe $ NAMELIST ; X=F,C,oe $ NAMELIST ; X3=F3,C3,oe $ NAMELIST ; X4=F4,C4,oe $ NAMELIST ; X5=F5,C5,oe $ NAMELIST ; X6=F6,C6,oe $ NAMELIST ; X7=F7,C7,oe $ NAMELIST ; X8=F8,C8,oe $ NAMELIST ; X9=F9,C9,oe $ NAMELIST ; X0=F0,C0,oe $ NAMELIST ; Y=I,I,I3,I4,I5,I6,I7,I8,I9,I0 $ SAMPLE ; - 0 $ SURE ; Lhs = Y ; Eq=X;Eq=X;Eq3=X3;Eq4=X4;Eq5=X6;Eq6=X6 ; Eq7=X7;Eq8=X8;Eq9=X9;Eq0=X0 ; Maxit=0 ; OLS $ Criterio fuctio for GLS is log-likelihood. Iteratio 0, GLS = Iteratio, GLS = Estimates for equatio: I Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat

80 Estimates for equatio: I3 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I4 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I5 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I6 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I7 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I8 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I9 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat Estimates for equatio: I0 77

81 Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X F C Costat ? c. Aggregatio test accordig to (0-5) MATRIX ; Z=Iit(3,3,0) ; J=Ide(3); L=-*J $ MATRIX ; R=[j,z,z,z,z,z,z,z,z,l / z,j,z,z,z,z,z,z,z,l / z,z,j,z,z,z,z,z,z,l / z,z,z,j,z,z,z,z,z,l / z,z,z,z,j,z,z,z,z,l / z,z,z,z,z,j,z,z,z,l / z,z,z,z,z,z,j,z,z,l / z,z,z,z,z,z,z,j,z,l / z,z,z,z,z,z,z,z,j,l ] ; d = R*b ; Vd = R*Varb*R' ; list ; AggF = /7 * d'<vd>d $ Matrix AGGF has rows ad colums CALC ; List ; Ftb(.95,7,(00-0*3)) $ Listed Calculator Results Result =.55534? d. Usig separate OLS regressios, compute LM statistic? OLS residuals were saved i matrix EOLS earlier. MATRIX ; VEOLS = /0*EOLS'EOLS ; VI = Diag(VEOLS) ; SDI = ISQR(VI) ; ROLS = SDI*VEOLS*SDI ; RR = ROLS' *ROLS $ CALC ; List ; LMStat = (0/)*(Trc(RR)-0) ; Ctb(.95, (9*0/))$ Listed Calculator Results LMSTAT = Result = ? Costraied Sur model with oe coefficiet vector.? This is the ucostraied model i (0-9)-(0-) SAMPLE ; - 00 $ REGRESS; Lhs = I ; Rhs = F,C,oe $ Ordiary least squares regressio LHS=I Mea = Stadard deviatio = WTS=oe Number of observs. = 00 Model size Parameters = 3 Degrees of freedom = 97 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[, 97] (prob) = (.0000) Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat TSCS ; Lhs = I ; Rhs = F,C,oe ; Pds=0 ; Model=S,R0 $ 78

82 Groupwise Regressio Models Estimator = Step GLS Groupwise Het. ad Correlated (S) Noautocorrelated disturbaces (R0) Test statistics agaist the correlatio Deg.Fr. = 45 C*(.95) = 6.66 C*(.99) = Test statistics agaist the correlatio Likelihood ratio statistic = Log-likelihood fuctio = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] F C Costat CREATE ; WI = (SDI(firm,firm))^ $ REGRESS; Lhs = I ; Rhs = F,C,oe ; Wts = WI $ Ordiary least squares regressio LHS=I Mea = Stadard deviatio = WTS=WI Number of observs. = 00 Model size Parameters = 3 Degrees of freedom = 97 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X F C Costat

83 Chapter Noliear Regressio Models Exercises. We caot simply take logs of both sides of the equatio as the disturbace is additive rather tha multiplicative. So, we must treat the model as a oliear regressio. The liearized equatio is 0 β β 0 0 β y α x 0 + x 0 ( α α ) + α (log x) x 0 ( β β 0 ) where α 0 ad β 0 are the expasio poit. For give values of α 0 ad β 0, the estimatig equatio would be 0 β 0 β 0 β β 0 β α + α x + α x x = α( x ) + β( α x x ) 0 0 (log ) 0 0 (log ) 0 0 β β 0 or y+ α (log x) x = α( x ) + β( α x x ) 0 0 (log ) β0 + ε *. y x + ε * Estimates of α ad β are obtaied by applyig ordiary least squares to this equatio. The process is repeated with the ew estimates i the role of α 0 ad β 0. The iteratio could be cotiued util covergece. Startig values are always a problem. If oe has o particular values i mid, oe cadidate would be α 0 = y ad β 0 = 0 or β 0 = ad α 0 either x y/x x or y / x. Alteratively, oe could search directly for the α ad β to miimize the sum of squares, S(α,β) = Σ i (y i - αx β ) = Σ i ε i. The first order coditios for miimizatio are S(α,β)/ α = -Σ i (y i - αx β )x β = 0 ad S(α,β)/ β = -Σ i (y i - αx β )α(lx)x β = 0. Methods for solvig oliear equatios such as these are discussed i Appedix E... The proof ca be doe by mathematical iductio. For coveiece, deote the ith derivative by f i. The first derivative appears i Equatio (0-34). Just by pluggig i i=, it is clear that f satisfies the relatioship. Now, use the chai rule to differetiate f, f = (-/λ )[x λ (lx) - x (λ) ] + (/λ)[(lx)x λ (lx) - f ] Collect terms to yield f = (-/λ)f + (/λ)[x λ (lx) - f ] = (/λ)[x λ (lx) - f ]. So, the relatioship holds for i = 0,, ad. We ow assume that it holds for i = K-, ad show that if so, it also holds for i = K. This will complete the proof. Thus, assume f K- = (/λ)[x λ (lx) K- - (K-)f K- ] Differetiate this to give f K = (-/λ)f K- + (/λ)[(lx)x λ (lx) K- - (K-)f K- ]. Collect terms to give Now, we take the limitig value lim λ 0 f i = lim λ 0 [x λ (lx) i - if i- ]/λ. Use L'Hospital's rule oce agai. lim λ 0 f i = lim λ 0 d{[x λ (lx) i - if i- ]/dλ}/lim λ 0 dλ/dλ. The, lim λ 0 f i = lim λ 0 {[x λ (lx) i+ - if i ]} f K = (/λ)[x λ (lx) K - Kf K- ], which completes the proof for the geeral case. Just collect terms, (i+)lim λ 0 f i = lim λ 0 [x λ (lx) i+ ] or lim λ 0 f i = lim λ 0 [x λ (lx) i+ ]/(i+) = (lx) i+ /(i+). 80

84 Applicatios. First, the two simple regressios produce Liear Log-liear Costat (73.4) (.368) Labor (.039) (.60) Capital (.4) (.08535) R Stadard Error I the regressio of Y o, K, L, ad the predicted values from the logliear equatio mius the predictios from the liear equatio, the coefficiet o α is with a estimated stadard error of 335. Sice this is ot sigificatly differet from zero, this evidece favors the liear model. I the regressio of ly o, lk, ll ad the predictios from the liear model mius the expoet of the predictios from the logliear model, the estimate of α is with a stadard error of Therefore, this cotradicts the precedig result ad favors the logliear model. A alterative approach is to fit the Box-Cox model i the fashio of Exercise 4. The maximum likelihood estimate of λ is about -., which is much closer to the log-liear model tha the loear oe. The log-likelihoods are at the MLE, at λ=0 ad at λ =. Thus, the hypothesis that λ = 0 (the log-liear model) would ot be rejected but the hypothesis that λ = (the liear model) would be rejected usig the Box-Cox model as a framework.. The search for the miimum sum of squares produced the followig results: λ e e

85 The sum of squared residuals is miimized at λ = At this value, the regressio results are as follows: Parameter Estimate OLS Std.Error Correct Std.Error α β k β l λ Estimated Asymptotic Covariace Matrix α β k β l λ α β k β l λ The output elasticities for this fuctio evaluated at the sample meas are ly/ lk = β k K λ = (.783) =.695 ly/ ll = β l L λ = ( ) = The estimates foud for Zeller ad Revakar's model were.54 ad.88, respectively, so these are quite similar. For the simple log-liear model, the correspodig values are.790 ad The Wald test is based o the urestricted model. The statistic is the square of the usual t-ratio, W = (-.3 /.077) = The critical value from the chi-squared distributio is 3.84, so the hypothesis that λ = 0 ca be rejected. The likelihood ratio statistic is based o both models. The sum of squared residuals for both urestricted ad restricted models is give above. The log-likelihood is ll = -(/)[ + l(π) + l(e e/)], so the likelihood ratio statistic is LR = [l(e e/) λ=0 - l(e e/) λ=-.38] = l[(e e λ=0) / (e e λ=-.38) = 5l(.7843/.54369) = Fially, to compute the Lagrage Multiplier statistic, we regress the residuals from the log-liear regressio o a costat, lk, ll, ad (/)(b k l K + b l l L) where the coefficiets are those from the log-liear model (.7898 ad.973). The R i this regressio is.300, so the Lagrage multiplier statistic is LM = R = 5(.300) = All three statistics suggest the same coclusio, the hypothesis should be rejected. 4. Istead of miimizig the sum of squared deviatios, we ow maximize the cocetrated log-likelihood fuctio, ll = -(/)l(+l(π)) + (λ - )Σ i ly i - (/)l(ε ε/). The search for the maximum of ll produced the results o the ext page The log-likelihood is maximized at λ =.4. At this value, the regressio results are as follows: Parameter Estimate OLS Std.Error Correct Std.Error α β k β l λ σ Estimated Asymptotic Covariace Matrix α β k β l λ σ α.54 β k β l λ σ

86 λ ll The output elasticities for this fuctio evaluated at the sample meas, K =.75905, L = , Y = , are ly/ lk = b k (K/Y) λ =.674 ly/ ll = b l (L/Y) λ =.907. These are quite similar to the estimates give above. The sum of the two output elasticities for the states give i the example i the text are give below for the model estimated with ad without trasformig the depedet variable. Note that the first of these makes the model look much more similar to the Cobb Douglas model for which this sum is costat. State Full Box-Cox Model lq o left had side Florida Louisiaa Califoria Marylad Ohio Michiga Oce agai, we are iterested i testig the hypothesis that λ = 0. The Wald test statistic is W = (.3 /.48) =.455. We would ow ot reject the hypothesis that λ = 0. This is a surprisig outcome. The likelihood ratio statistic is based o both models. The sum of squared residuals for the restricted model is give above. The sum of the logs of the outputs is , so the restricted log-likelihood is ll 0 = (0-)(9.9336) - (5/)[ + l(π) + l(.78403/5)] = The likelihood ratio statistic is -[ ( )] = Oce agai, the statistic is small. Fially, to compute the Lagrage multiplier statistic, we ow use the method described i Example.8. The result is LM =.56. All of these suggest that the log-liear model is ot a sigificat restrictio o the Box-Cox model. This rather peculiar outcome would appear to arise because of the rather substatial reductio i the log-likelihood fuctio which occurs whe the depedet variable is trasformed alog with the right had side. This is ot a cotradictio because the model with oly the right had side trasformed is ot a parametric restrictio o the model with both sides trasformed. Some further evidece is give i the ext exercise. 83

87 5. --> lsq ; lhs = y ; labels = b,b ; fc=b*( - /sqr(+*b*x)) ; start = 500,.000 ;output=$ Begi NLSQ iteratios. Liearized regressio. Iteratio= ; Sum of squares= ; Gradiet= Iteratio= ; Sum of squares= ; Gradiet= Iteratio= 3; Sum of squares= ; Gradiet= Iteratio= 4; Sum of squares= ; Gradiet= Iteratio= 5; Sum of squares= ; Gradiet= Iteratio= 6; Sum of squares= ; Gradiet= Iteratio= 7; Sum of squares= ; Gradiet= Iteratio= 8; Sum of squares= ; Gradiet= Iteratio= 9; Sum of squares= ; Gradiet= Iteratio= 0; Sum of squares= ; Gradiet= Iteratio= ; Sum of squares= ; Gradiet= Iteratio= ; Sum of squares= ; Gradiet= Iteratio= 3; Sum of squares= ; Gradiet=.3365 Iteratio= 4; Sum of squares= e-0; Gradiet=.96769E-05 Iteratio= 5; Sum of squares= e-0; Gradiet= E-3 Iteratio= 6; Sum of squares= e-0; Gradiet= E-8 Iteratio= 7; Sum of squares= e-0; Gradiet= E-4 Covergece achieved Noliear least squares regressio LHS=Y Mea = Stadard deviatio =.8065 WTS=oe Number of observs. = 4 Model size Parameters = Degrees of freedom = Residuals Sum of squares = E-0 Stadard error of e = E-0 Fit R-squared = Not usig OLS or o costat. Rsqd & F may be < Variable Coefficiet Stadard Error b/st.er. P[ Z >z] B B D > lsq ; lhs = y ; labels = b,b ; fc=b*( - /sqr(+*b*x)) ; start = 600,.000 ;output=$ Begi NLSQ iteratios. Liearized regressio. Iteratio= ; Sum of squares= ; Gradiet= Iteratio= ; Sum of squares= ; Gradiet= Iteratio= 3; Sum of squares= e-0; Gradiet= E-06 Iteratio= 4; Sum of squares= e-0; Gradiet= E-3 Iteratio= 5; Sum of squares= e-0; Gradiet= E-8 Iteratio= 6; Sum of squares= e-0; Gradiet= E-3 Covergece achieved Noliear least squares regressio LHS=Y Mea = Stadard deviatio =.8065 Residuals Sum of squares = E-0 Stadard error of e = E-0 Fit R-squared = Adjusted R-squared = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] B B D

88 Chapter Istrumetal Variables Estimatio Exercises. There is o eed for a separate proof differet from the usual for OLS. Formally, however, it follows from the results at (-4) that XX X ε b = β + The, XX X ε b plim b = QXXγ ad XX X ε ( b plim b) = QXXγ The large sample distributio of this statistic will be the same as the large sample of the statistic with X X/ replaced with its probablity limit, which is Q XX. Thus, X ε ( b plim b) QXX γ To deduce the large sample behavior of this statistic, we ca ivoke the results from chapter 4. The oly chage here is the ozero mea (probability limit) of the vector i brackets. [See (-3).] Thus, the same proof applies. The cosistecy, asymptotic ormality ad asymptotic covariace matrix equal to Asy.Var[b] = σ ε (X X) -. A logical solutio to this oe is simple. For y ad x*, Cov (y,x*)/[var(y)var(x*)] = β (σ * ) /[(β σ * +σ ε )(σ * )] Cov (y,x) /[Var(y)Var(x)] = Cov[βx*+ε,x*+u] / [Var(y)Var(x)] = {Cov[y,x*] +Cov[y,u]} / [Var(y)Var(x)]. The secod term is zero, sice y=βx*+ε which is ucorrelated with u. Thus, Cov (y,x) /[Var(y)Var(x)] = Cov[y,x*] / [Var(y)Var(x)]. The umerator is the same. The deomiator is larger, sice [Var(y)Var(x)] = Var[y](Var[x*] + Var[u]), so the squared correlatio must be smaller. If both variables are measured with errors, the we are comparig Cov (y*,x*)/{var[y*]var[x*]} to Cov (y,x)/{var[y]var[x]}. The umerator is the covariace of (βx* + ε + v) with (x* + u), so the umerator of the fractio is still β (σ * ). The deomiator is still obviously larger, so the same result holds whe both variables are measured with error. 3. We work off (-6), usig repeatedly the result Σ uu = (σ u j)(σ u j) where j has a i the first positio ad 0 i the remaiig K-. From (-6), plim b = β - [Q* + Σ uu ] - Σ uu β. The vector is Σ uu β equals [σ u β,0,...,0]. The iverse matrix is [Q* + Σ uu ] - = ( ) Q* Q* ( )( ) * uj u + ( σuj) ( Q* ) ( σuj) ( ) σ σ j ( Q ) 85

89 This ca be simplified sice the quadratic form i the deomiator just picks off the, diagoal elemet. Thus, +σ [Q* + Σ uu ] - = ( Q* ) ( Q* ) ( σ j)( σ j) ( Q *) The q * u [Q* + Σ uu ] - Σ uu β= ( *) ( *) ( σ )( σ ) ( *) u u Q Q ( u )( u ) * u u q +σ Q j j σ j σ j β u = ( Q *) ( σuj)( σu j) β - * ( *) ( u )( u ) ( *) q Q σ j σ j Q ( σuj)( σu j) β +σu = ( j σ * *) σuq Q u β - * ( Q* ) j σβ u +σuq * = ( Q *) σuq j * σ u β +σuq =( Q *) j * q σ u β +σu =( Q *) σβ u j * q +σu Fially, ( Q *) j equals the first colum of ( Q *) = [q*, q*,...,q* k ]. Therefore, the first elemet, give by (-7a) is For (-7b), plim b = β - plim b = β - σβ u * +σ q u q* σuq = β +σuq σβ +σ u q * u q* k * * 4. To obtai the result, ote first: plim b = β + Q XX - γ Asy.Var[b] = (σ /)Q XX - Asy.Var[b sls ] = (σ /)Q ZX - Q ZZ Q XZ -. 86

90 The mea squared error of the OLS estimator is the variace plus the squared bias, M(b β) = (σ /)Q - XX + Q - - XX γγ Q XX the mea squared error of the SLS estimator equals its variace. For OLS to be more precise the SLS, we would have to have (σ /)Q - XX + Q - XX γγ Q - XX << (σ /)Q - ZX Q ZZ Q - XZ. For coveiece, let δ = Q - XX γ so M(b β) = (σ /)Q - XX + δδ. If the mea squared error matrix of the OLS estimator is smaller tha that of the SLS estimator, the its iverse is larger. Use (A-66) to do the iversio. The result would be [(σ /)Q - XX + δδ ] - >> [(σ /)Q - ZX Q ZZ Q - XZ ] - Now, use A-66 [(σ /)Q - XX + δδ ] - = (/σ ) Q XX - + δ ( / σ ) QXXδ (/σ ) Q XX δδ (/σ ) Q XX Reisert δ = Q - XX γ ad the right had side above reduces to (/σ ) Q XX ( / σ ) γ Q XX γ (/σ ) γγ Therefore, if the mea squared error matrix of OLS is smaller, the (/σ ) Q XX ( / σ ) γ Q XX γ (/σ ) γγ >> (/σ )Q XZ Q - ZZ Q ZX Collect the terms, ad this implies (/σ )[ Q XX - Q XZ Q - ZZ Q ZX ] >> - + ( / σ ) γ Q XX γ (/σ ) γγ divide both sides by (/σ ), Q XX - Q XZ Q - ( / σ ) ZZ Q ZX >> - + ( / σ ) γ Q XX γ γγ ad divide umerator ad deomiator of the fractio by /σ Q XX - Q XZ Q - ZZ Q ZX >> - ( σ / ) + γ γ γγ Q XX which is the desired result. Is it possible? It is possible, sice Q XX - Q XZ Q - ZZ Q ZX = plim (/)[X X - X Z(Z Z) - Z X] = plim (/) X M Z X which is a positive defiite matrix. SIce γ varies idepedetly of Z ad X, certaily there is some cofiguratio of the data ad parameters for which this is the case. The result is that it is, ideed, possible for OLS to be more precise, i the mea squared error sese, tha SLS. 5. The matrices are X = [i,x] ad Z = [i,z]. For the OLS estimators, we kow from chapter that a = y bx ad b = Cov[x,y]/var[x]. For the IV estimator, (Z X) - Z y, we obtai the result i detail. Give the forms, Σxi x x x y ( ZX ) =, ( ), z x = = i x ZX = = ( x x) Zy Σ y where subscript idicates the mea of the observatios for which z equals, ad is the umber of observatios. Multiplyig the matrix times the vector ad cacellig terms produces the solutios a IV = a IV x y x y ad b y = = y x IV x x x 87

91 Applicatio a. The statemet of the problem is actually a bit optimistic. GIve the way it is stated, it would imply that the exogeous variables i the demad equatio would be, i priciple, (Ed, Uio, Fem) which are also i the supply equatio, plus the remaider, (Exp, Exp, Occ, Id, South, SMSA, Blk). The problem is that the model as stated would ot be idetified the supply equatio would, but the demad equatio would ot be. The way out would be to assume that at least oe of (Ed, Uio, Fem) does ot appear i the demad equatio. Sice surely educatio would, that leaves oe or both of Uio ad Fem. We will assume both of them are omitted. So, our equatio is lwage it = α + α Ed it + α 3 Exp it + α 4 Exp it + α 5 Occ it + α 6 Id it + α 7 South it + α 8 SMSA it + α 9 Blk it + γ Wks it + u it. NAMELIST ; X = oe,ed,exp,expsq,occ,id,south,smsa,blk,wks $ NAMELIST ; Z = oe,ed,exp,expsq,occ,id,south,smsa,blk,uio,fem $ Regress ; Lhs = lwage ; Rhs = X $ SLS ; Lhs = lwage ; Rhs = X ; Ist = Z $ REGRESS ; Lhs = Wks ; Rhs = Z ; cls:b(0)=0,b()=0$ Ordiary least squares regressio LHS=LWAGE Mea = Stadard deviatio =.465 WTS=oe Number of observs. = 465 Model size Parameters = 0 Degrees of freedom = 455 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[ 9, 455] (prob) = 4.74 (.0000) Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat ED EXP EXPSQ D OCC IND SOUTH SMSA BLK WKS Two stage least squares regressio LHS=LWAGE Mea = Stadard deviatio =.465 WTS=oe Number of observs. = 465 Model size Parameters = 0 Degrees of freedom = 455 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared =.3777 Model test F[ 9, 455] (prob) = 6.50 (.0000) Istrumetal Variables: ONE ED EXP EXPSQ OCC IND SOUTH SMSA BLK UNION FEM Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat ED

92 EXP EXPSQ D OCC IND SOUTH SMSA BLK WKS This is the test of relevace of the istrumetal variables. I the regressio of WKS o the full set of exogeous variables, we test the hypothesis that the coefficiets o the istrumets, UNION ad FEM are joitly zero. The results show that the hypothesis is rejected. We coclude that the istrumets are relevat. Liearly restricted regressio Ordiary least squares regressio LHS=WKS Mea = Stadard deviatio = WTS=oe Number of observs. = 465 Model size Parameters = 9 Degrees of freedom = 456 Residuals Sum of squares = Stadard error of e = Fit R-squared = E-0 Adjusted R-squared =.69705E-0 Model test F[ 8, 456] (prob) = 4.6 (.0000) Restricts. F[, 454] (prob) = (.0000) Not usig OLS or o costat. Rsqd & F may be < 0. Note, with restrictios imposed, Rsqd may be < 0. Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat ED EXP EXPSQ OCC IND SOUTH SMSA BLK UNION D-5.855D FEM (Fixed Parameter)... 89

93 Chapter 3 Simultaeous Equatios Models. (a) Sice othig is excluded from either equatio ad there are o other restrictios, either equatio passes the order coditio for idetificatio. () We use (3-) ad the equatios which follow it. For the first equatio, [A 3,A 5 ] = β, a scalar which has rak M- = uless β = 0. For the secod, [A 3,A 5 ] = β 3. Thus, both equatios are idetified. () This restrictio does ot restrict the first equatio, so it remais uidetified. The secod equatio is ow idetified, as [A 3,A 5 ] = [β,β ] has rak if either of the two ceofficiets are ozero. (3) If γ equals 0, the model becomes partially recursive. The first equatio becomes a regressio which ca be estimated by ordiary least squares. However, the secod equatio cotiues to fail the order coditio. To see the problem, cosider that eve with the restrictio, ay liear combiatio of the two equatios has the same variables as the origial secod eqatio. (4) We kow from above that if β 3 = 0, the secod equatio is idetifiable. If it is, the γ is idetified. We may treat it as kow. As such, γ is kow. By regressig y - γ y o the xs, we would obtai estimates of the remaiig parameters, so these restrictios idetify the model. It is istructive to aalyze this from the stadpoit of false structures as doe i the text. A false structure which icorporates γ λ f f the kow restrictios would be β β. If the false structure is to obey the restrictios, f f β β β3 0 the f - γ f =, f - γ f =, f - γf = f - γ f, β 3 f = 0. It follows the that f = 0 so f =. The, f - γf = -γ or f = (f - )γ so that f - γ (f - ) =. This ca oly hold for all values of γ if f = ad, the, f = 0. Therefore, F = I which establishes idetificatio. (5) If β 3 = 0, the first equatio is idetified by the usual rak ad order coditios. Cosider, the, the off-diagoal elemet of Σ = Γ ΩΓ. Ω is idetified sice it is the reduced form covariace matrix. The off-diagoal elemet is σ = ω + ω - (γ + γ )ω = 0. Sice γ is zero, γ = ω /(ω + ω ). With γ kow, the remaiig parameters are estimable by least squares regressio of (y - γ y ) o the xs. Therefore, the restrictios idetify the model. (6) Sice this is oly a sigle restrictio, it will ot likely idetify the etire model. Cosider agai the false structure. The restrictios implied by the theory are f - γ f =, f - γ f =, β f + β f = β f + β f. The three restrictios o four ukow elemets of F do ot serve to pi dow ay of them. This restrictio does ot eve partially idetify the model. (7) The last four restrictios remove x ad x 3 from the model. The remaiig model is ot idetified by the usual rak ad order coditios. From part (5), we see that the first restrictio implies σ = ω + ω - (γ + γ )ω = 0. But, with either γ or γ specified, this does ot idetify either parameter. (8) The first equatio is idetified by the covetioal rak ad order coditios. The secod equatio fails the order coditio. But, the restrictio σ = 0 provides the ecessary additioal iformatio eeded to idetify the model. For simplicity, write the model with the restrictios imposed as y = γ y + ε ad y = γ y + βx + ε. The reduced form is y = π x + v ad y = π x + v where π = γ β/δ ad π = β/δ with Δ = ( - γ γ ), ad v = (ε + γ ε )/Δ ad v = (ε + γ ε )/Δ. The reduced form variaces ad covariaces are ω = (γ σ + σ )/Δ, ω = (γ σ + σ )/Δ, ω = (γ σ + γ σ )/Δ. All reduced form parameters are estimable directly by usig least squares, so the reduced form is idetified i all cases. Now, γ = π /π. σ is the residual variace i the euqatio (y - γ y ) = ε, so σ must be estimable (idetified) if γ is. Now, with a bit of maipulatio, we fid that γ ω - ω = -σ /Δ. Therefore, with σ ad 90

94 γ "kow" (idetified), the oly remaiig ukow is γ, which is therefore idetified. With γ ad γ i had, β may be deduced from π. With γ ad β i had, σ is the residual variace i the equatio (y - βx - γ y ) = ε, which is directly estimable, therefore, idetified.. Followig the method i Example 3.6, for idetificatio of the ivestmet equatio, we require that the ( ) ( ) ( 3) ( 4) ( 5) ( 6) ( 7) ( 8) ( 9) α3 0 0 α γ γ matrix 3 γ have rak 5. Colums (), (4), (6), (7), ad (8) each have oe elemet i a differet row, so they are liearly idepedet. Therefore, the matrix has rak five. For ( ) ( ) ( 3) ( 4) ( 5) ( 6) ( 7) ( 8) ( 9) ( 0) 0 α 0 α α 0 0 β β β the third equatio, the required matrix is 3. Colums (4), (6), (7), (9), ad (0) are liearly idepedet. 3. We fid [A 3,A 5 ] for each equatio. () () (3) (4) γ3 γ34 γ 0 γ4 γ γ 0 β β β 3 4 4, [ 0 β43 β44],, β3 β3 β33 0 β β β β5 0 β β Idetificatio requires that the rak of each matrix be M- = 3. The secod is obviously ot idetified. I (), oe of the three colums ca be writte as a liear combiatio of the other two, so it has rak 3. (Although the secod ad last colums have ozero elemets i the same positios, for the matrix to have short rak, we would require that the third colum be a multiple of the secod, sice the first caot appear i the liear combiatio which is to replicate the secod colum.) By the same logic, (3) ad (4) are idetified. 4. Obtai the reduced form for the model i Exercise uder each of the assumptios made i parts (a) ad (b), (b6), ad (b9). (). The model is y = γ y + β x + β x + β 3 x 3 + ε y = γ y + β x + β x + β 3 x 3 + ε. β β γ Therefore, Γ = ad B = ad Σ is urestricted. The reduced form is γ 0 β β3 0 β + γβ γβ + β Π= γβ β ad γγ β3 γβ3 9

95 Ω = (Γ - ) Σ(Γ - ) = ( γγ ) σ + γ σ + γσ γ σ + γ σ + ( γ + γ ) σ γ σ + ( γ + γ ) σ + σ γ σ + γσ + γ σ (6) The model is y = β x + β x + β 3 x 3 + ε y = γ y + β x + β x + β 3 x 3 + ε The first equatio is already a reduced form. Substitutig it ito the secod provides the secod reduced form. β β + γβ The coefficiet matrix is P= β β γ β, Γ - = so Ω = (Γ - ) Σ(Γ - + γ σ γσ ) = 0 β3 β3 + γβ3 γσ γ σ + σ (9) The model is y = γ y + ε y = γ y + β x + ε The, Π = -BΓ - σ + γ σ γσ + γ σ = [β γ /(-γ γ ) β /(-γ γ )] ad Ω =. γσ γσ γ + σ + σ The relevat submatrices are X X =, X y =, X y = , y y = 0, y y = 0, y y = 6, X Z = 6, X Z = Z Z =, Z Z =, Z Z =, Z y =, Z y =, Z y =, Z y = The two OLS coefficiet vectors are d = (X X) - X y = [.43904, ] d = (X X) - X y = [.9306,.3847,.9746]. The two stage least squares estimators are δ = [Z X(X X) - X Z ] - [Z X(X X) - X y ] = [.36886,.5787]. δ = [Z X(X X) - X Z ] - [Z X(X X) - X y ] = [ ,.36788,.09375]. σ = (y y - y Z δ + δ Z Z δ ) / 5 =.60397, σ = The estimated asymptotic covariace matrices are Est.Var[ δ ] = σ [Z X(X X) - X Z ] = Est.Var[Est.Var[ δ ]] = The three stage least squares estimate is 9

96 σ [ Z' XXX ( ' ) XZ ' ] σ [ Z' XXX ( ' ) XZ ' ] σ [ Z' XXX ( ' ) XZ ' ] σ [ Z' XXX ( ' ) XZ ' ] σ σ σ [ Z' XXX ( ' ) Xy ' ] + [ Z' XXX ( ' ) Xy ' ] [ Z' X( X' X) X' y ] + [ Z ' X( X' X) X' Z] σ = [.36887, ,.4706, ,.6894]. The estimated stadard errors are the square roots of the diagoal elemets of the iverse matrix, [.4637,.4466,.366,.76,.68], compared to the SLS values, [.4637,.4466,.3639,.74,.08]. To compute the limited iformatio maximum likelihood estimator, we require the matrix of sums of squares ad cross products of residuals of the regressios of y ad y o x ad o x, x, ad x 3. These are W 0 = Y Y - Y x (x x ) x Y =, W = Y Y - Y X(X X) X Y = The two characteristic roots of (W ) - W 0 are.5357 ad We carry the smaller oe ito the k-class computatio [see, for example, Theil (97) or Judge, et al (985)]; δ ( ) (. 553) 3676 k = = Fially, the two estimates of the reduced form are (OLS) P = ad (SLS) Π = = For the model y = γ y + β x + β x + ε y = γ y + β 3 x 3 + β 4 x 4 + ε show that there are two restrictios o the reduced form coefficiets. Describe a procedure for estimatig the model while icorporatig the restrictios. β 0 γ The structure is [y y ] β 0 [ x x x x ] = [ε ε ]. γ 0 β3 0 β4 or y Γ + x B = ε. The reduced form coefficiet matrix is β γβ π π Π = -BΓ - = β γ β π π = The two restrictios are π /π = π /π ad γγ γβ 3 β3 π3 π3 γβ 4 β4 π4 π4 π 3 /π 3 = π 4 /π 4. If we write the reduced form as y = π x + π x + π 3 x 3 + π 4 x 4 + v y = π x + π x + π 3 x 3 + π 4 x 4 + v. We could treat the system as a oliear seemigly urelated regressios model. Oe possible way to hadle the restrictios is to elimiate two parameters directly by makig the substitutios π = π π /π ad π 3 = π 3 π 4 /π 4. 93

97 The pair of equatios would be y = π x + π x + (π 3 π 4 /π 4 )x 3 + π 4 x 4 + v y = (π π /π )x + π x + π 3 x 3 + π 4 x 4 + v. This oliear system could ow be estimated by oliear GLS. The fuctio to be miimized would be Σ i v i σ = + v i σ + v i v i σ = tr(σ - W). Needless to say, this would be quite ivolved. 7. We would require that all three characteristic roots have modulus less tha oe. A ituitive guess that the diagoal elemet greater tha oe would preclude this would be correct. The roots are the solutios to. 899 λ det λ 0 = 0. Expadig this produces -( λ)( λ)( λ) λ ( λ).899 = 0. There is o eed to go ay further. It is obvious that λ =.087 is a solutio, so there is at least oe characteristic root larger tha. The system is ustable. 8. Prove plim Y j ε/t = ω j - Ω jj γ j. Cosistet with the partitioig y = [y j Y j Y * i ], partitio Ω ito ω jj ω j ω * j Ω = ω j Ω jj Ω j ω * j Ω * j * Ω j ad, as i the equatio precedig (3-8), partitio the jth colum of Γ as Γ j = γ. Sice the full set of 0 reduced form disturbaces is V = EΓ -, it follows that E = VΓ. I particular, the jth colum of E is ε j = VΓ j. I the reduced form, ow referrig to (5-8), Y j = XΠ j + V j, where Π j is the M j colums of Π correspodig to the icluded edogeous variables ad V j is the T M j matrix of their reduced form disturbaces. Sice X is ucorrelated with all colums of E, we have plim Y j ε j /T = plim V j Γ j /T = [ω j Ω jj Ω j * ] γ = ω j - Ω jj γ j as required Prove that a uderidetified equatio caot be estimated by two stage least squares. If the equatio fails the order coditio, the the umber of excluded exogeous variables is less tha the umber of icluded edogeous. The matrix of istrumetal variables to be used for two stage least squares is of the form Z = [XA,X j ], where XA is M j liear combiatio of all K colums i X ad X j is K j colums of X. I total, K = K * j + K j. If the equatio fails the order coditio, the K * j < M j, so Z is M j + K j colums which are liear combiatios of K = K * j + K j < M j + K j. Therefore, Z caot have full colum rak. I order to compute the two stage least squares estimator, we require ( Z Z ) -, which caot be computed. 94

98 Applicatio?=========================================================? Applicatio 3. - Simultaeous Equatios?=========================================================? Read the data? For coveiece, reame the variables so they correspod? to the example i the text. sample ; - 04 $ create ; ct=realcos$ create ; it=realivs$ create ; gt=realgovt$ create ; rt=tbilrate $? Impose (artifically) the addig up coditio o total demad. create ; yt=ct+it+gt $ create ; ct=ct[-] $ create ; yt = yt[-] $ create ; dyt = yt - yt $ sample ; -04 $ ames ; xt = oe,gt,rt,ct,yt$? Estimate equatios by sls ad save coefficiets with? the ames used i the example. sls ; lhs = ct ; rhs=oe,yt,ct ; ist = xt $ Two stage least squares regressio LHS=CT Mea = Stadard deviatio = WTS=oe Number of observs. = 03 Model size Parameters = 3 Degrees of freedom = 00 Residuals Sum of squares = Stadard error of e = Fit R-squared = Adjusted R-squared = Model test F[, 00] (prob) =******* (.0000) Istrumetal Variables: ONE GT RT CT YT Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat YT CT calc ; a0=b() ; a=b() ; a=b(3) $ sls ; lhs = it ; rhs=oe,rt,dyt ; ist = xt $ Two stage least squares regressio LHS=IT Mea = Stadard deviatio = WTS=oe Number of observs. = 03 Model size Parameters = 3 Degrees of freedom = 00 Residuals Sum of squares =.77447E+08 Stadard error of e = 6.63 Fit R-squared = Adjusted R-squared = Istrumetal Variables: ONE GT RT CT YT Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat RT DYT calc ; b0=b() ; b=b() ; b=b(3) $ 95

99 ?? Create the coefficiets of the reduced form. We oly eed the parts? for the dyamics. These are i the secod half of the example. calc ; a=-a-b $?? Costruct the matrix that govers the dyamics of the system. Note that? the I equatio is static. It is a fuctio of y(t-) ad c(t-) but ot? of I(t-). This is the DELTA() submatrix i (3-4). The domiat? root is the largest rood of DELTA(). calc ; list ; C=(-b)/a ; C=-a*b/a ; C=a/a ; C=-b/a $ matrix ; C = [c,c / c,c] $ Listed Calculator Results C = C = C = C = Matrix ; list ; roots = cxrt(c)$ Calc ; list ; domroot = sqr(roots(,)^ + roots(,)^)$ --> Matrix ; list ; roots = cxrt(c)$ Matrix ROOTS has rows ad colums > Calc ; list ; domroot = sqr(roots(,)^ + roots(,)^)$ Listed Calculator Results DOMROOT =.09596? The largest root is larger tha o i absolute value. The system is ustable. 3sls ; lhs = ct,it ; eq=oe,yt,ct ; eq=oe,rt,dyt ; ist=xt ; maxit=0 $ Estimates for equatio: CT IstVar/GLS least squares regressio LHS=CT Mea = Residuals Sum of squares = Stadard error of e = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat YT CT Estimates for equatio: IT IstVar/GLS least squares regressio LHS=IT Mea = Residuals Sum of squares = E+08 Stadard error of e = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Costat RT DYT

100 Chapter 4 Estimatio Frameworks i Ecoometrics Exercise. A fully parametric model/estimator provides cosistet, efficiet, ad comparatively precise results. The semiparametric model/estimator, by compariso, is relatively less precise i geeral terms. But, the payoff to this imprecisio is that the semiparametric formulatio is more likely to be robust to failures of the assumptios of the parametric model. Cosider, for example, the biary probit model of Chapter, which makes a strog assumptio of ormality ad homoscedasticity. If the assumptios are correct, the probit estimator is the most efficiet use of the data. However, if the ormality assumptio or the homoscedasticity assumptio are icorrect, the the probit estimator becomes icosistet i a ukow fashio. Lewbel s semiparametric estimator for the biary choice model, i cotrast, is ot very precise i compariso to the probit model. But, it will remai cosistet if the ormality assumptio is violated, ad it is eve robust to certai kids of heteroscedasticity. Applicatios. Usig the gasolie market data i Appedix Table F., use the partially liear regressio method i Sectio to fit a equatio of the form l(g/pop) = β l(icome) + β lp ew cars + β 3 lp used cars + g(lp gasolie ) + ε crea;gp=lg;ip=ly;cp=lpc;upp=lpuc;pgp=lpg$ sort;lhs=pgp;rhs=gp,ip,cp,upp$ crea;dgp=.809*gp -.5*gp[-] -.309*gp[-]$ crea;dip=.809*ip -.5*ip[-] -.309*ip[-]$ crea;dc=.809*cp -.5*cp[-]-.309*cp[-]$ crea;duc=.809*upp -.5*upp[-]-.309*upp[-]$ samp;3-36$ regr;lhs=dgp;rhs=dip,dc,duc;res=e$ Ordiary least squares regressio Weightig variable = oe Dep. var. = DGP Mea= E-0, S.D.= E-0 Model size: Observatios = 34, Parameters = 3, Deg.Fr.= 3 Residuals: Sum of squares= e-0, Std.Dev.=.089 Fit: R-squared=.79947, Adjusted R-squared = Model test: F[, 3] = 6.80, Prob value = Diagostic: Log-L = , Restricted(b=0) Log-L = LogAmemiyaPrCrt.= , Akaike Ifo. Crt.= -4.7 Model does ot cotai ONE. R-squared ad F ca be egative! Autocorrel: Durbi-Watso Statistic =.34659, Rho = Variable Coefficiet Stadard Error t-ratio P[ T >t] Mea of X DIP E-0 DNC E E-0 DUC E E E-0 --> matr;varpl={+/(*)}*varb$ --> matr;stat(b,varpl)$

101 Number of observatios i curret sample = 34 Number of parameters computed here = 3 Number of degrees of freedom = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] B_ B_ E B_ E E Noparametric Regressio for G Observatios = 36 Poits plotted = 36 Badwidth = Statistics for abscissa values---- Mea =.366 Stadard Deviatio =.5735 Miimum = Maximum = Kerel Fuctio = Logistic Cross val. M.S.E. = Results matrix = KERNEL E[y xi] G Noparametric Regressio for G E[y xi] PG 3. A. Usig the probit model ad the Klei ad Spady semiparametric models, the two sets of coefficiet estimates are somewhat similar Biomial Probit Model Maximum Likelihood Estimates Model estimated: Jul 3, 00 at 05:6:40PM. Depedet variable P Weightig variable Noe Number of observatios 60 Iteratios completed 5 98

102 Log likelihood fuctio Restricted log likelihood Chi squared Degrees of freedom 5 Prob[ChiSqd > value] = Hosmer-Lemeshow chi-squared = P-value= with deg.fr. = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Idex fuctio for probability Z E E Z E E Z E Z E E Z E Costat Seimparametric Biary Choice Model Maximum Likelihood Estimates Model estimated: Jul 3, 00 at :0:4PM. Depedet variable P Weightig variable Noe Number of observatios 60 Iteratios completed 3 Log likelihood fuctio Restricted log likelihood Chi squared Degrees of freedom 4 Prob[ChiSqd > value] = Hosmer-Lemeshow chi-squared = P-value= with deg.fr. = 8 Logistic kerel f. Badwidth = Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Characteristics i umerator of Prob[Y = ] Z E E Z E Z Z Z Costat (Fixed Parameter)... 99

103 The probit model produces a set of margial effects, as discussed i the text. These caot be computed for the Klei ad Spady estimator Partial derivatives of E[y] = F[*] with respect to the vector of characteristics. They are computed at the meas of the Xs. Observatios used for meas are All Obs Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Idex fuctio for probability Z E E Z E E Z E E Z E E Z E E Costat These are the various fit measures for the probit model Fit Measures for Biomial Choice Model Probit model for variable P Proportios P0= P= N = 60 N0= 45 N= 50 LogL = LogL0 = Estrella = -(L/L0)^(-L0/) = Efro McFadde Be./Lerma Cramer Veall/Zim. Rsqrd_ML Iformatio Akaike I.C. Schwarz I.C. Criteria Frequecies of actual & predicted outcomes Predicted outcome has maximum probability. Threshold value for predictig Y= =.5000 Predicted Actual 0 Total Total These are the fit measures for the probabilities computed for the Klei ad Spady model. The probit model fits better by all measures computed Fit Measures for Biomial Choice Model Observed = P Fitted = KSPROBS Proportios P0= P= N = 60 N0= 45 N= 50 LogL = LogL0 = Estrella = -(L/L0)^(-L0/) = Efro McFadde Be./Lerma Cramer Veall/Zim. Rsqrd_ML

104 The first figure below plots the probit probabilities agaist the Klei ad Spady probabilities. The models are obviously similar, though there is substatial differece i the fitted values PROBITS KSPROBS Fially, these two figures plot the predicted probabilities from the two models agaist the respective idex fuctios, b x. Note that the two plots are based o differet coefficiet vectors, so it is ot possible to merge the two figures. 0

105 Chapter 5 Miimum Distace Estimatio ad The Geeralized Method of Momets Exercises. The elemets of J are b m b m 5/ 3/ = m ( 3/) m = m 3 3 b b b m m m 3 = m ( ) m = 0 = m b 0 m = Usig the formula give for the momets, we obtai, μ = σ, μ 3 = 0, μ 4 = 3σ 4. Isert these i the derivatives above to obtai 0 σ 3 0 J =. 6σ 0 σ 4 Sice the rows of J are orthogoal, we kow that the off diagoal term i JVJ will be zero, which simplifies thigs a bit. Takig the parts directly, we ca see that the asymptotic variace of b will be σ-6 Asy.Var[m 3 ], which will be Asy.Var[ b ] = σ-6 (μ 6 - μ 3 + 9μ 3-3μ μ 4-3μ μ 4 ). The parts eeded, usig the geeral result give earlier, are μ 6 = 5σ 6, μ 3 = 0, μ = σ, μ 4 = 3σ 4. Isertig these i the paretheses ad multiplyig it out ad collectig terms produces the upper left elemet of JVJ equal to 6, which is the desired result. The lower right elemet will be Asy.Var[b ] = 36σ -4 Asy.Var[m ] + σ -8 Asy.Var[m 4 ] - (6)σ -6 Asy.Cov[m,m 4 ]. The eeded parts are Asy.Var[m ] = σ 4 Asy.Var[m 4 ] = μ 8 - μ 4 = 05σ 8 - (3σ 4 ) Asy.Cov[m,m 4 ] = μ 6 - μ μ 4 = 5σ 6 - σ (3σ 4 ). Isertig these parts i the expasio, multiplyig it out ad collectig terms produces the lower right elemet equal to 4, as expected. 4. The ecessary data are give i Examples 5.5. The two momets are m =3.78 ad m. = Based o the theoretical results m = P/λ ad m = P(P+)/λ, the solutios are P = μ /(μ - μ ) ad λ = μ /(μ - μ ). Usig the sample momets produces estimates P =.0568 ad λ = The matrix of derivatives is μ / / '/ P μ '/ λ λ P λ G = = =. μ '/ '/ ( )/ ( )/ 3, ,0.08 P μ λ P+ λ P P+ λ The covariace matrix for the momets is give i Example 8.7; Φ= ,

106 3. a. The log likelihood for samplig from the ormal distributio is logl = (-/)[logπ + logσ + (/σ )Σ i (x i - μ) ] write the summatio i the last term as Σx i + μ - μσ i x i. Thus, it is clear that the log likelihood is of the form for a expoetial family, ad the sufficiet statistics are the sum ad sum of squares of the observatios. b. The log of the desity for the Weibull distributio is logf(x) = logα + logβ + (β-)logx i - ασ i x i β. The log likelihood is foud by summig these fuctios. The third term does ot factor i the fashio eeded to produce a expoetial family. There are o sufficiet statistics for this distributio. c. The log of the desity for the mixture distributio is logf(x,y) = logθ - (β+θ)y i + x i logβ + x i logy i - log(x!) This is a expoetial family; the sufficiet statistics are Σ i y i ad Σ i x i.. 4. The questio is (deliberately) misleadig. We showed i Chapter 8 ad i this chapter that i the classical regressio model with heteroscedasticity, the OLS estimator is the GMM estimator. The asymptotic covariace matrix of the OLS estimator is give i Sectio 8.. The estimator of the asymptotic covariace matrices are s (X X) - for OLS ad the White estimator for GMM. 5. The GMM estimator would be chose to miimize the criterio q = m Wm where W is the weightig matrix ad m is the empirical momet, m = (/)Σ i (y i - Φ(x i β))x i For the first pass, we ll use W = I ad just miimize the sumof squares. This provides a iitial set of estimates that ca be used to compute the optimal weightig matrix. With this first roud estimate, we compute W = [(/ ) Σ i (y i - Φ(x i β)) x i x i ] - the retur to the optimizatio problem to fid the optimal estimator. The asymptotic covariace matrix is computed from the first order coditios for the optimizatio. The matrix of derivatives is G = m/ β = (/)Σ i -φ(x i β)x i x i The estimator of the asymptotic covariace matrix will be V = (/)[G WG] - 6. This is the compariso betwee (5-) ad (5-). The proof ca be doe by comparig the iverses of the two covariace matrices. Thus, if the claim is correct, the matrix i (5-) is larger tha that i (5- ), or its iverse is smaller. We ca igore the (/) as well. We require, the, that - G Φ G> GWG[GW Φ WG] GWG 7. Suppose i a sample of 500 observatios from a ormal distributio with mea μ ad stadard deviatio σ, you are told that 35% of the observatios are less tha. ad 55% of the observatios are less tha 3.6. Estimate μ ad σ. If 35% of the observatios are less tha., we would ifer that Φ[(. - μ)/σ] =.35, or (. - μ)/σ = μ = -.385σ. Likewise, Φ[(3.6 - μ)/σ] =.55, or (3.6 - μ)/σ = μ =.6σ. The joit solutio is μ = 3.30 ad σ = It might ot seem obvious, but we ca also derive asymptotic stadard errors for these estimates by costructig them as method of momets estimators. Observe, first, that the two estimates are based o momet estimators of the probabilities. Let x i deote oe of the 500 observatios draw from the ormal distributio. The, the two proportios are obtaied as follows: Let z i (.) = [x i <.] ad z i (3.6) = [x i < 3.6] be idicator fuctios. The, the proportio of 35% has bee obtaied as z (.) ad.55 is z (3.6). So, the two proportios are simply the meas of fuctios of the sample observatios. Each z i is a draw from a Beroulli distributio with success probability π(.) = Φ((.-μ)/σ) for z i (.) ad π(3.6) = Φ((3.6-μ)/σ) for z i (3.6). Therefore, E[ z (.)] = π(.), ad E[ z (3.6)] = π(3.6). The 03

107 variaces i each case are Var[ z (.)] = /[π(.)(-π(.))]. The covariace of the two sample meas is a bit trickier, but we ca deduce it from the results of radom samplig. Cov[ z (.), z (3.6)]] = / Cov[z i (.),z i (3.6)], ad, sice i radom samplig sample momets will coverge to their populatio i = couterparts, Cov[z i (.),z i (3.6)] = plim [{(/) z i(.)z i (3.6)} - π(.)π(3.6)]. But, z i (.)z i (3.6) must equal [z i (.)] which, i tur, equals z i (.). It follows, the, that Cov[z i (.),z i (3.6)] = π(.)[ - π(3.6)]. Therefore, the asymptotic covariace matrix for the two sample π(. )( π(. )) π(. )( π( 36. )) proportios is Asy. Var[ p(. ), p( 36. )] = Σ = (. )( ( 36. )) ( 36. )( ( 36. )). If we isert our π π π π.. sample estimates, we obtai Est. Asy. Var[ p(. ), p( 36. )] = S =. Now, ultimately, our estimates of μ ad σ are foud as fuctios of p(.) ad p(3.6), usig the method of momets. The momet equatios are. m. = z i (.) 0 i μ - Φ = =, σ Now, let Γ = 36. m3.6 = z i i (.) 36 μ - Φ = = σ m. / μ m. / σ ad let G be the sample estimate of Γ. The, the estimator of the m36. / μ m36. / σ asymptotic covariace matrix of (, ) is [GS - G ] -. The remaiig detail is the derivatives, which are just μ σ m. / μ = (/σ)φ((.-μ)/σ) ad m. / σ = (.-μ)/σ[ m. / σ] ad likewise for m 3.6. Isertig our sample estimates produces G =. Fially, multiplyig the matrices ad computig the ecessary iverses produces [GS - G ] =. The asymptotic distributio would be ormal, as usual. Based o these results, a 95% cofidece iterval for μ would be 3.30 ±.96(.078) =.6048 to

108 Chapter 6 Maximum Likelihood Estimatio Exercises. The desity of the maximum is [z/θ] - (/θ), 0 < z < θ. Therefore, the expected value is E[z] = θ θ 0 z dz = [θ + /(+)][/θ ] = θ/(+). The variace is foud likewise, E[z ] = z (z/) - (/θ)dz = θ /(+) so Var[z] = E[z ] - (E[z]) = θ /[( + ) (+)]. 0 Usig mea squared covergece we see that lim E[z] = θ ad lim Var[z] = 0, so that plim z = θ. x i = i. The log-likelihood is ll = -lθ - (/θ). The maximum likelihood estimator is obtaied as the solutio to ll/ θ = -/θ + (/θ ) x i = i = 0, or θ ˆ ML = (/) x i = i x i= i = x. The asymptotic variace of the MLE is {-E[ ll/ θ ]} - = {-E[/θ - (/θ 3 ) ]} -. To fid the expected value of this radom variable, we eed E[x i ] = θ. Therefore, the asymptotic variace is θ /. The asymptotic distributio is ormal with mea θ ad this variace. 3. The log-likelihood is ll = lθ - (β+θ) y i + lβ + i = x i = i xil y i= i - The first ad secod derivatives are ll/ θ = /θ- y i = i y i = i ll/ β = - + x i /β i = ll/ θ = -/θ ll/ β = - x /β i = i ll/ β θ = 0. i= l( x!) Therefore, the maximum likelihood estimators are θ ˆ ML = / y ad ˆβ = x / y ad the asymptotic covariace matrix is the iverse of E / θ 0. I order to complete the derivatio, we will require the 0 x i / β i = expected value of = E[x i ]. I order to obtai E[x i ], it is ecessary to obtai the margial x i = i ( β+ θ) y x distributio of x i, which is f(x) = θe ( βy) / x! dy = β ( θ/ x!) e y dy. This is β x (θ/x!) 0 x ( β+ θ) y x 0 times a gamma itegral. This is f(x) = β x (θ/x!)[γ(x+)]/(β+θ) x+. But, Γ(x+) = x!, so the expressio reduces to f(x) = [θ/(β+θ)][β/(β+θ)] x. Thus, x has a geometric distributio with parameter π = θ/(β+θ). (This is the distributio of the umber of tries util the first success of idepedet trials each with success probability -π. Fially, we require the expected value of x i, which is E[x] = [θ/(β+θ)] x = 0 / θ 0 θ / 0 covariace matrix is. = 0 ( β/ θ)/ β 0 βθ / x[β/(β+θ)] x = β/θ. The, the required asymptotic i 05

109 The maximum likelihood estimator of θ/(β+θ) is is θ /( β+θ ) = (/ y )/[ x / y + / y ] = /( + x ). Its asymptotic variace is obtaied usig the variace of a oliear fuctio V = [β/(β+θ)] (θ /) + [-θ/(β+θ)] (βθ/) = βθ /[(β+θ) 3 ]. The asymptotic variace could also be obtaied as [-/( + E[x]) ] Asy.Var[ x ].) For part (c), we just ote that γ = θ/(β+θ). For a sample of observatios o x, the log-likelihood would be ll = lγ + l(-γ) ll/dγ = /γ - x i = i /(-γ). x i = i A solutio is obtaied by first otig that at the solutio, (-γ)/γ = x = /γ -. The solutio for γ is, thus, ˆγ = / ( + x ).Of course, this is what we foud i part b., which makes sese. For part (d) f(y x) = ( β+ θ) y x x f ( x, y) = θ e ( β y) ( β + θ ) ( β + θ). f ( x) x! θ βx x+ λy x e y x dy 0 Cacellig terms ad gatherig the remaiig like terms leaves f(y x) = ( β+ θ)[( β+ θ) y] e / x! so the desity has the required form with λ = (β+θ). The itegral is {[ λ ]/!} x ( β+ θ) y. This itegral is a Gamma itegral which equals Γ(x+)/λ x+, which is the reciprocal of the leadig scalar, so the product is. The log-likelihood fuctio is ll = lλ - λ y i + lλ x i - l x i! i = x i = i x i = i i = y i = i ll/ λ = ( + )/λ -. ll/ λ = -( + )/λ. i = Therefore, the maximum likelihood estimator of λ is ( + x )/ y ad the asymptotic variace, coditioal o the xs is Asy.Var. λˆ = (λ /)/( + x ) Part (e.) We ca obtai f(y) by summig over x i the joit desity. First, we write the joit desity θy βy x θy βy x as f ( x, y) = θe e ( βy) / x!. The sum is, therefore, f ( y) = θe e ( βy) / x!. The sum is that of the probabilities for a Poisso distributio, so it equals. This produces the required result. The maximum likelihood estimator of θ ad its asymptotic variace are derived from ll = lθ - θ ll/ θ = /θ - y i = i y i = i ll/ θ = -/θ. Therefore, the maximum likelihood estimator is / y ad its asymptotic variace is θ /. Sice we foud f(y) by factorig f(x,y) ito f(y)f(x y) (apparetly, give our result), the aswer follows immediately. Just divide the expressio used i part e. by f(y). This is a Poisso distributio with parameter βy. The log-likelihood fuctio ad its first derivative are ll = -β y i + l x i + xi l yi - l x i! from which it follows that β= ˆ x / y. i = y i = i i = x i = i ll/ β = - + /β, 4. The log-likelihood ad its two first derivatives are logl = logα + logβ + (β-) log x i - α logl/ α = /α - β x i= i i = i = β x i= i x= 0 i = 06

110 logl/ β = /β + log x i - α (log xi) xi i = i= Sice the first likelihood equatio implies that at the maximum, ˆα = / β β x i= i, oe approach would be to sca over the rage of β ad compute the implied value of α. Two practical complicatios are the allowable rage of β ad the startig values to use for the search. The secod derivatives are ll/ α = -/α ll/ β = -/β β - α (log x ) x i = β i ll/ α β = - (log xi) xi. i= If we had estimates i had, the simplest way to estimate the expected values of the Hessia would be to evaluate the expressios above at the maximum likelihood estimates, the compute the egative iverse. First, sice the expected value of ll/ α is zero, it follows that E[x β i ] = /α. Now, β E[ ll/ β] = /β + E[ log x i ] - αe[ ]= 0 i = (log xi) x i= i as well. Divide by, ad use the fact that every term i a sum has the same expectatio to obtai /β + E[lx i ] - E[(lx i )x β i ]/E[x β i ] = 0. Now, multiply through by E[x β i ] to obtai E[x β i ] = E[(lx i )x β i ] - E[lx i ]E[x β i ] or /(αβ) = Cov[lx i,x β i ]. ~ 5. As suggested i the previous problem, we ca cocetrate the log-likelihood over α. From logl/ α = 0, β we fid that at the maximum, α = /[(/) x ]. Thus, we sca over differet values of β to seek the i= i value which maximizes logl as give above, where we substitute this expressio for each occurrece of α. Values of β ad the log-likelihood for a rage of values of β are listed ad show i the figure below. β logl The maximum occurs at β =.. The implied value of α is.79. The egative of the secod derivatives matrix at these values ad its iverse are I αβ = ad.,.. I αβ , = The Wald statistic for the hypothesis that β = is W = (. - ) / =.76. The critical value for a test of size.05 is 3.84, so we would ot reject the hypothesis. i 07

111 If β =, the ˆα = / = The distributio specializes to the geometric distributio i = xi if β =, so the restricted log-likelihood would be x i = i logl r = logα - α = (logα - ) at the MLE. logl r at α = is The likelihood ratio statistic is -logλ = ( ) =.36. Oce agai, this is a small value. To obtai the Lagrage multiplier statistic, we would compute log L/ α log L/ α β log L / α [ log L/ α log L/ β] log L/ α β log L/ β log L / β at the restricted estimates of α = ad β =. Makig the substitutios from above, at these values, we would have logl/ α = 0 logl/ β = + log x i - i= x x i i log = i = x logl/ α = x = logl/ β = - - x x i i(log i) = = x logl/ α β = x x = i = i log i The lower right elemet i the iverse matrix is The LM statistic is, therefore, (9.4003) = This is also well uder the critical value for the chi-squared distributio, so the hypothesis is ot rejected o the basis of ay of the three tests. 6. a. The full log likelihood is logl = Σ log f yx (y,x α,β). b. By factorig the desity, we obtai the equivalet logl = Σ[ log f y x (y x,α,β) + log f x (x α)] c. We ca solve the first order coditios i each case. From the margial distributio for x, Σ log f x (x α)/ α = 0 provides a solutio for α. From the joit distributio, factored ito the coditioal plus the margial, we have Σ[ log f y x (y x,α,β)/ α + log f x (x α)/ α = 0 Σ[ log f y x (y x,α,β)/ β = 0 d. The asymptotic variace obtaied from the first estimator would be the egative iverse of the expected secod derivative, Asy.Var[a] = {[-E[Σ log f x (x α)/ α ]} -. Deote this A αα -. Now, cosider the secod estimator for α ad β joitly. The egative of the expected Hessia is show below. Note that the A αα from the margial distributio appears there, as the margial distributio appears i the factored joit distributio. l L Bαα Bαβ A 0 A + B B αα αα αα αβ E = + = B B 0 0 B B α α βα ββ βα ββ β β The asymptotic covariace matrix for the joit estimator is the iverse of this matrix. To compare this to the asymptotic variace for the margial estimator of α, we eed the upper left elemet of this matrix. Usig the formula for the partitioed iverse, we fid that this upper left elemet i the iverse is [(A αα +B αα ) - (B αβ B ββ - B βα )] - = [A αα + (B αα - B αβ B ββ - B βα )] - which is smaller tha A αα as log as the secod term is positive. e. (Ufortuately, this is a error i the text.) I the precedig expressio, B αβ is the cross derivative. Eve if it is zero, the asymptotic variace from the joit estimator is still smaller, beig [A αα + B αα ] -. This makes sese. If α appears i the coditioal distributio, the there is additioal iformatio i the factored joit likelhood that is ot i the margial distributio, ad this produces the smaller asymptotic variace. 08

112 7. The log likelihood for the Poisso model is LogL = -λ + logλσ i y i - Σ i log y i! The expected value of / times this fuctio with respect to the true distributio is E[(/)logL] = -λ + logλ E 0 [ y ] E 0 (/)Σ i logy i! The first expectatio is λ 0. The secod expectatio ca be left implicit sice it will ot affect the solutio for λ - it is a fuctio of the true λ 0. Maximizig this fuctio with respect to λ produces the ecessary coditio E 0 (/)logl]/ λ = - + λ 0 /λ = 0 which has solutio λ = λ 0 which was to be show. 8. The log likelihood for a sample from the ormal distributio is LogL = -(/)logπ - (/)logσ /(σ ) Σ i (y i - μ). E 0 [(/)logl] = -(/)logπ - (/)logσ /(σ ) E 0 [(/) Σ i (y i - μ) ]. The expectatio term equals E 0 [(y i - μ) ] = E 0 [(y i - μ 0 ) ] + (μ 0 - μ) = σ 0 + (μ 0 - μ). Collectig terms, E 0 [(/)logl] = -(/)logπ - (/)logσ /(σ )[ σ 0 + (μ 0 - μ) ] To see where this is maximized, ote first that the term (μ 0 - μ) eters egatively as a quadratic, so the maximizig value of μ is obviously μ 0. Sice this term is the zero, we ca igore it, ad look for the σ that maximizes -(/)logπ - (/)logσ σ 0 /(σ ). The / is irrelevat as is the leadig costat, so we wish to miimize (after chagig sig) logσ + σ 0 /σ with respect to σ. Equatig the first derivative to zero produces /σ = σ 0 /(σ ) or σ = σ 0, which gives us the result. 9. The log likelihood for the classical ormal regressio model is LogL = Σ i -(/)[logπ + logσ + (/σ )(y i - x i β) ] If we reparameterize this i terms of η = /σ ad δ = β/σ, the after a bit of maipulatio, LogL = Σ i -(/)[logπ - logη + (ηy i - x i δ) ] The first order coditios for maximizig this with respect to η ad δ are logl/ η = /η - Σ i y i (ηy i - x i δ) = 0 logl/ δ = Σ i x i (ηy i - x i δ) = 0 Solve the secod equatio for δ, which produces δ = η (X X) - X y = η b. Isert this implicit solutio ito the first equatio to produce /η = Σ i y i (ηy i - ηx i b). By takig η outside the summatio ad multiplyig the etire expressio by η, we obtai = η Σ i y i (y i - x i b) or η = /[Σ i y i (y i - x i b)]. This is a aalytic solutio for η that is oly i terms of the data b is a sample statistic. Isertig the square root of this result ito the solutio for δ produces the secod result we eed. By pursuig this a bit further, you cashow that the solutio for η is just /e e from the origial least squares regressio, ad the solutio for δ is just b times this solutio for η. The secod derivatives matrix is 09

113 logl/ η = -/η - Σ i y i logl/ δ δ = -Σ i x i x i logl/ δ η = Σ i x i y i. We ll obtai the expectatios coditioed o X. E[y i x i ] is x i β from the origial model, which equals x i δ/η. E[y i x i ] = /η (δ x i ) + /η. (The cross term has expectatio zero.) Summig over observatios ad collectig terms, we have, coditioed o X, E[ logl/ η X] = -/η - (/η )δ X Xδ E[ logl/ δ δ X] = -X X E[ logl/ δ η X] = (/η)x Xδ The egative iverse of the matrix of expected secod derivatives is XX ' (/ η) XX ' δ AsyVar. [ d, h] = (/ η) ' ' (/ η δ XX )[ + δxx ' δ (The off diagoal term does ot vaish here as it does i the origial parameterizatio.) 0. The first derivatives of the log likelihood fuctio are logl/ μ = -(/σ ) Σ i -(y i - μ). Equatig this to zero produces the vector of meas for the estimator of μ. The first derivative with respect to σ is logl/ σ = -M/(σ ) + /(σ 4 )Σ i (y i - μ) (y i - μ). Each term i the sum is Σ m (y im - μ m ). We already deduced that the estimators of μ m are the sample meas. Isertig these i the solutio for σ ad solvig the likelihood equatio produces the solutio give i the problem. The secod derivatives of the log likelihood are logl/ μ μ = (/σ )Σ i -I logl/ μ σ = (/σ 4 ) Σ i -(y i - μ) logl/ σ σ = M/(σ 4 ) - /σ 6 Σ i (y i - μ) (y i - μ) The expected value of the first term is (-/σ )I. The secod term has expectatio zero. Each term i the summatio i the third term has expectatio Mσ, so the summatio has expected value Mσ. Addig gives the expectatio for the third term of -M/(σ 4 ). Assemblig these i a block diagoal matrix, the takig the egative iverse produces the result give earlier. For the Wald test, the restrictio is H 0 : μ - μ 0 i = 0. The urestricted estimator of μ is x. The variace of x is give above, so the Wald statistic is simply ( x - μ 0 i ) Var[( x - μ 0 i )] - ( x - μ 0 i ). Isertig the covariace matrix give above produces the suggested statistic. 0

114 . The asymptotic variace of the MLE is, i fact, equal to the Cramer-Rao Lower Boud for the variace of a cosistet, asymptotically ormally distributed estimator, so this completes the argumet. I example 4.9, we proposed a regressio with a gamma distributed disturbace, where, y i = α + x i β + ε i f(ε i ) = [λ P /Γ(P)] ε i P- exp(-λε i ), ε i > 0, λ > 0, P >. (The fact that ε i is oegative will shift the costat term, as show i Example 4.9. The eed for the restrictio o P will emerge shortly.) It will be coveiet to assume the regressors are measured i deviatios from their meas, so Σ i x i = 0. The OLS estimator of β remais ubiased ad cosistet i this model, with variace Var[b X] = σ (X X) - where σ = Var[ε i X] = P/λ. [You ca show this by usig gamma itegrals to verify that E[ε i X] = P/λ ad E[ε i X] = P(P+)/λ. See B-39 ad (E-) i Sectio E.3. A useful device for obtaiig the variace is Γ(P) = (P-)Γ(P-).] We will ow show that i this model, there is a more efficiet cosistet estimator of β. (As we saw i Example 4.9, the costat term i this regressio will be biased because E[ε i X] = P/λ; a estimates α+p/λ. I what follows, we will focus o the slope estimators. The log likelihood fuctio is L L = Pl λ l Γ ( P) + ( P )l ε i= i λεi The likelihood equatios are ll/ α = Σ i [-(P-)/ε i + λ] = 0, ll/ β = Σ i [-(P-)/ε i + λ]x i = 0, ll/ λ = Σ i [P/λ - ε i ] = 0, ll/ P = Σ i [lλ - ψ(p) - ε i ] = 0. The fuctio ψ(p) = dlγ(p)/dp is defied i Sectio E.3.) To show that these expressios have expectatio zero, we use the gamma itegral oce agai to show that E[/ε i ] = λ/(p-). We used the result E[lε i ] = ψ(p)-λ i Example 5.5. So show that E[ ll/ β] = 0, we oly require E[/ε i ] = λ/(p-) because x i ad ε i are idepedet. The secod derivatives ad their expectatios are foud as follows: Usig the gamma itegral oce agai, we fid E[/ε i ] = λ /[(P-)(P-)]. Ad, recall that Σ i x i = 0. Thus, coditioed o X, we have -E[ ll/ α ] = E[Σ i (P-)(/ε i )] = λ /(P-), -E[ ll/ α β] = E[Σ i (P-)(/ε i )x i ] = 0, -E[ ll/ α λ] = E[Σ i (-)] = -, -E[ ll/ α P] = E[Σ i (/ε i )] = λ/(p-), -E[ ll/ β β ] = E[Σ i (P-)(/ε i )x i x i ] = Σ i [λ /(P-)]x i x i = [λ /(P-)](X X), -E[ ll/ λ β] = E[Σ i (-)x i ] = 0, -E[ ll/ P β] = E[Σ i (/ε i )x i ] = 0, -E[ ll/ λ ] = E[Σ i (P/λ )] = P/λ, -E[ ll/ λ P] = E[Σ i (/λ)] = /λ, -E[ ll/ P ] = E[Σ i ψ (P)] = ψ (P). Sice the expectatios of the cross partials witth respect to β ad the other parameters are all zero, it follows that the asymptotic covariace matrix for the MLE of β is simply Asy.Var[ ˆMLE β ] = {-E[ ll/ β β ]} - = [(P-)/λ ](X X) -. Recall, the asymptotic covariace matrix of the ordiary least squares estimator is

115 Asy.Var[b] = [P/λ ](X X) -. (Note that the MLE is ill defied if P is less tha.) Thus, the ratio of the variace of the MLE of ay elemet of β to that of the correspodig elemet of b is (P-)/P which is the result claimed i Example 4.9. Applicatios. a. For both probabilities, the symmetry implies that F(t) = F(-t). I either model, the, Prob(y=) = F(t) ad Prob(y=0) = F(t) = F(-t). These are combied i Prob(Y=y) = F[(y i -)t i ] where t i = x i β. Therefore, l L = Σ i l F[(y i -)x i β] f[(yi ) x iβ] b. ll/ β = i ( yi ) x = i= 0 F[(yi ) x iβ] where f[(y i -)x i β] is the desity fuctio. For the logit model, f = F(-F). So, for the logit model, ll/ β = { F[(yi ) x iβ]}(yi ) xi= 0 i= Evaluatig this expressio for y i = 0, we get simply F(x i β)x i. Whe y i =, the term is [- F(x i β)]x i. It follows that both cases are [y i - F(x i β)]x i, so the likelihood equatios for the logit model are ll/ β = [ y ( )] = 0. = i Λ xiβ xi i For the probit model, F[(y i -)x i β] = Φ[(y i -)x i β] ad f[(y i -)x i β] = φ[(y i -)x i β], which does ot simplify further, save for that the term y i iside may be dropped sice φ(t) = φ(-t). Therefore, ll/ β = i= φ[(yi ) x iβ] ( yi ) xi= 0 Φ[(y ) x β] i i c. For the logit model, the result is very simple. ll/ β β = Λ( x β)[ Λ( β)] x x. i= i i i For the probit model, the result is more complicated. We will use the result that dφ(t)/dt = -tφ(t). It follows, the, that d[φ(t)/φ(t)]/dt = [-φ(t)/φ(t)][t + φ(t)/φ(t)]. Usig this result directly, it follows that ll/ β β = φ[(y ) x β] φ[(y ) x β] x i= 0 i i i i (y ) ( ) i i y i= xβ+ i xi Φ[(yi ) x iβ] Φ[(yi ) x iβ] This actually simplifies somewhat because (y i -) = for both values of y i ad φ[(y i ) x iβ ] = φ( xi β )

116 d. Deote by H the actual secod derivatives matrix derived i the previous part. The, Newto s method is ˆ ˆ ˆ { } l L[ ˆ ( j)] ( j ) ( j) ( j) β β + = β H β βˆ( j ) where the terms o the right had side idicate first ad secod derivatives evaluated at the previous estimate of β. e. The method of scorig uses the expected Hessia istead of the actual Hessia i the iteratios. The methods are the same for the logit model, sice the Hessia does ot ivolve y i. The methods are differet for the probit model, sice the expected Hessia does ot equal the actual oe. For the logit model -[E(H)] - = { } Λ( xiβ)[ Λ( β)] xix i= i For the probit model, we eed first to obtai the expected value. Do obtai this, we take the expected value, with Prob(y=0) = - Φ ad Prob(y=) = Φ. The expected value of the ith term i the egative hessia is the expected value of the term, This is φ[(yi ) x iβ] φ[(yi ) x iβ] (yi ) x iβ+ x i i Φ[(yi ) x iβ] Φ[(yi ) x iβ] x φ[ x iβ] φ[ x iβ] Φ [ x iβ] x iβ+ i i [ ] [ ] x x + Φ x iβ Φ xiβ φ[ x iβ] φ[ x iβ] Φ[ x iβ] x iβ+ i i Φ[ x iβ] Φ[ x iβ] x x φ[ x iβ] φ[ x iβ] =φ[ x iβ] x iβ+ +xiβ+ xix Φ [ x i ] Φ[ x i β iβ] φ[ x iβ] φ[ x iβ] =φ[ x iβ ] + xix [ ] [ i Φ xiβ Φxiβ] = ( φ[ x iβ ]) + xix Φ [ x iβ] Φ[ x iβ] Φ [ x iβ] +Φ[ x iβ] = ( φ[ x iβ] ) x ix Φ [ x iβ] Φ[ x iβ] ( φ[ x iβ] ) = xx i [ Φ ( i Φ ( i xβ)] xβ) e.?====================================================? Applicatio 6.?==================================================== Namelist ; x = oe,age,educ,hsat,female,married $ LOGIT ; Lhs = Doctor ; Rhs = X $ Calc ; L = logl $

117 Biary Logit Model for Biary Choice Depedet variable DOCTOR Number of observatios 736 Log likelihood fuctio Number of parameters 6 Ifo. Criterio: AIC =.00 Ifo. Criterio: BIC =.0300 Restricted log likelihood Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Characteristics i umerator of Prob[Y = ] Costat AGE EDUC HSAT FEMALE MARRIED f. Matr ; bw = b(5:6) ; vw = varb(5:6,5:6) $ Matrix ; list ; WaldStat = bw'<vw>bw $ Calc ; list ; ctb(.95,) $ LOGIT ; Lhs = Doctor ; Rhs = Oe,age,educ,hsat $ Calc ; L0 = logl $ Calc ; List ; LRStat = *(l-l0) $ Matrix WALDSTAT has rows ad colums > Calc ; list ; ctb(.95,) $ Listed Calculator Results Result = > Calc ; L0 = logl $ --> Calc ; List ; LRStat = *(l-l0) $ Listed Calculator Results LRSTAT = Logit ; Lhs = Doctor ; Rhs = X ; Start = b,0,0 ; Maxit = 0 $ Biary Logit Model for Biary Choice Maximum Likelihood Estimates Model estimated: May 7, 007 at :49:4PM. Depedet variable DOCTOR Weightig variable Noe Number of observatios 736 Iteratios completed LM Stat. at start values LM statistic kept as scalar LMSTAT Log likelihood fuctio Number of parameters 6 Ifo. Criterio: AIC =.830 Fiite Sample: AIC =.830 Ifo. Criterio: BIC =.00 Ifo. Criterio:HQIC =.888 Restricted log likelihood McFadde Pseudo R-squared Chi squared Degrees of freedom 5 4

118 Prob[ChiSqd > value] = Hosmer-Lemeshow chi-squared = P-value=.0084 with deg.fr. = g. The restricted log likelihood give with the iitial results equals This is the log likelihood for a model that cotais oly a costat term. The log likelihood for the model is Twice the differece is about 3,00, which vastly exceeds the critical chi squared with 5 degrees of freedom. The hypothesis would be rejected.. We used LIMDEP to fit the cost frotier. The depedet variable is log(cost/pfuel). The regressors are a costat, log(pcapital/pfuel), log(plabor/pfuel), logq ad log Q. The Jodrow measure was the computed ad plotted agaist output. There does ot appear to be ay relatioship, though the weak relatioship such as it is, is ideed, egative Limited Depedet Variable Model - FRONTIER Depedet variable LCF Number of observatios 3 Log likelihood fuctio Variaces: Sigma-squared(v)=.085 Sigma-squared(u)=.033 Sigma(v) =.0884 Sigma(u) =.4944 Sigma = Sqr[(s^(u)+s^(v)]=.8488 Stochastic Cost Frotier, e=v+u Variable Coefficiet Stadard Error b/st.er. P[ Z >z] Mea of X Primary Idex Equatio for Model Costat LPK E E LPL E LQ E LQ E E Variace parameters for compoud error Lambda Sigma E

119 COSTEFF Q 6

120 Chapter 7 Simulatio Based Estimatio ad Iferece Exercises. Expoetial: The pdf is f(x) = θexp(-θx). The CDF is x Fx ( ) = θexp( θ tdt ) =θ exp( θx) exp( θ 0) = exp( θx). 0 θ θ We would draw observatios from the U(0,) populatio, say F i, ad equate these to F(x i ). Ivertig the fuctio, we fid that -F i = exp(-θx i ), or (/θ)l(-f i ) = x i. If x i has a expoetial desity, the the desity of y i = x P i is Weibull. If the survival fuctio is S(x) = λpexp[-(λx) p ], the we may equate radom draws from the uiform distributio, S i to this fuctio (a draw of S i is the same as a draw of F i = -S i ). Solvig for x i, we fid ls i = l(λp) (λx) p, so x i = (/λ)[l(λp) ls i ] /p.. We will eed a bivariate sample o x ad y to compute the radom variable, the average the draws o it. The precise method of usig a Gibbs sampler to draw this bivaraite sample is show i Example 8.5. Oce the bivariate sample of (x,y) is draw, a large umber of observatios o [x exp(y)+y exp(x)] is computed ad averaged. As oted there, the Gibbs sampler is ot much of a simplificatio for this particular problem. It is simple to draw a sample dircectly from a bivariate ormal distributio. Here is a program that does the simulatio ad plots the estimate of the fuctio Calc ; Ra(345) $ Sample ; -000$ Create ; xf=r(0,) ; yfb=r(0,) $ Matrix ; corr=iit(00,,0) ; fuctio=corr $ Calc ; i=0 $ Proc Calc ; i=i+ $ Matrix ; corr(i)=ro $ Matrix ; c=[/ro,] ; c=chol(c) $ Create ; yf = c(,)*xf + c(,)*yfb $ Create ; fr=xf^*exp(yf)+yf^*exp(xf) $ Calc ; ef = xbr(fr) ; ro=ro+.0 $ Matrix ; fuctio(i)=ef $ Edproc $ Calc ; ro=-.99 $ Execute; =00 $ Mplot ; Lhs = corr ; Rhs = Fuctio ; Fill ; Grid ; Edpoits = -, ; Title=E[x^*exp(y)+y^*exp(x) rho] $ 7

121 Applicatio?================================================================? Applicatio 7.. Mote Carlo Simulatio?================================================================? Set seed of RNG for replicability Calc ; Ra(3579) $? Sample size is 50. Geerate x(i) ad z(i) held fixed Sample ; - 50 $ Create ; xi = r(0,) ; zi = r(0,) $ Namelist ; X = oe,xi,zi ; X0 = oe,xi $? Momet Matrices Matrix ; XXiv = <X'X> ; X0X0iv = <X0'X0> $ Matrix ; Waldi = iit(000,,0) $ Matrix ; LMi = iit(000,,0) $?****************************************************************? Procedure studies the LM statistic?**************************************************************** Proc = LM (c) $? Three kids of disturbaces Create?; Eps = Rt(5)? Noormal distributio ; vi=exp(.*xi) ; eps = vi*r(0,)? Heteroscedasticity?;eps= R(0,)? Stadard ormal distributio ; y = 0 + xi + c*zi +eps $ Matrix ; b0 = X0X0iv*X0'y $ Create ; e0 = y - X0'b0 $ Matrix ; g = X'e0 $ Calc ; lmstat = qfr(g,xxiv)/(e0'e0/) ; i = i + $ Matrix ; Lmi (i) = lmstat $ EdProc $ 8