3. SEEMINGLY UNRELATED REGRESSIONS (SUR)

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1 3. SEEMINGLY UNRELATED REGRESSIONS (SUR) [] Examples Demad for some commodities: y Nike,t = x Nike,t β Nike + ε Nike,t y Reebok,t = x Reebok,t β Reebok + ε Reebok,t ; where y Nike,t is the quatity demaded for Nike seakers, x Nike,t is a k Nike vector of regressors such as the uit price of Nike seakers, prices of other seakers, icome..., ad t idexes time. Grufeld s ivestmet data: I: gross ivestmet ($millio) F: market value of firm at the ed of previous year. C: value of firm s capital at the ed of previous year. I it = β i + β i F it + β 3i C it + ε it, where i = GM, CH (Chrysler), GE, etc. Notice that although the same regressors are used for each i, the values of the regressors are differet across differet i. CAPM (Capital Asset Pricig Model) r it - r ft : excess retur o security i over a risk-free security. r mt -r ft : excess market retur. r it -r ft = α i + β i (r mt -r ft ) + ε it. Notice that the values of regressors are the same for every security. SUR-

2 VAR (Vector Autoregressios) g CPI,t = growth rate of CPI (iflatio rate) g GDP,t = growth rate of ormial GDP g CPI,t = α CPI + β CPI g CPI,t- + γ CPI g GDP,t- + ε CPI,t g GDP,t = α GDP + β GDP g CPI,t- + γ GDP g GDP,t- + ε GDP,t Notice that the values of regressors are the same. Cobb-Douglas Cost fuctio system Cost fuctio is a fuctio of output ad iput prices. Assume M iputs (labor, capital, lad, material, eergy, etc). Cobb-Douglas Cost fuctio: lc = α + β ly +Σ β l( P ) + ε. (CD.) M y j= j j c Share fuctios (s j = p j x j /C, where x j = quatity of iput j used) s lc = = β + ε, j =,..., M. (CD.) l P j j j j M M M Observe that Σ s = 0. This imples that Σ β = 0 ad Σ ε = 0. j= j j= j j= j Traslog Cost fuctio System Assume three iputs (j =,, ad 3 idex iputs, capital (k), labor (l), ad fuel (f), respectively. SUR-

3 Traslog productio fuctio: lc = α +Σ j= β jl Pj +.5Σ j= Σi= δ jilog Pjlog Pi +Σ γ l P ly + θ l Y +.5 θ (l Y) + ε, (T.) 3 j= y, j j y yy c where δ ji = δ ij. (See Greee Ch. 4.) This a geeralizatio of Cobb-Douglas cost fuctio. If the productio fuctio is homothetic, γ y, = γ y, = γ y,3 = 0. If the productio fuctio is Cobb-Douglas, δ ji = 0, γ y,j = 0, θ yy = 0, for all j ad i. If the productio fuctio is Cobb-Douglas ad costat returs to scale, δ ji = 0, γ y,j = 0, θ yy = 0, for all j ad i; ad θ y =. Share fuctios: lc s = = β +Σ δ l P + γ l Y + ε ; 3 i= i i y, l P lc s = = β +Σ δ l P + γ l Y + ε ; 3 i= i i y, l P lc s = = β +Σ δ l P + γ l Y + ε i= 3 i i y,3 3 l P3 Note that s + s + s 3 =. Thus, Σ β = ; Σ Σ δ = 0; Σ γ = 0; Σ ε = j= j j= i ji j= y, j j= j (T.) SUR-3

4 [] Basic Model Model: y = X β + ε ; y = X β + ε ; : y = X β + ε, where y i = T, X i = T k i, β i = k i, ε i = (ε i,..., ε it ) = T, ad i =,...,. Assumptios o the model: ) SIC hold. (I fact, WIC are sufficiet). ) cov(ε it,ε jt ) = E(ε it ε jt ) = σ ij for ay t, i ad j. cov(ε it,ε it ) = var(ε it ) = σ ii. 3) cov(ε it,ε js ) = E(ε it ε js ) = 0 for ay t s. 4) The ε it are ormally distributed. (For simplicity. Not required) Implicatio of ozero σ ij : A uobservable macro shock at time t ca ifluece all of the y it. SUR-4

5 Form of E(ε i ε j ) ε... i εε i j εε i j εε i jt ε ε iε j εiε j... εiε i jt E( εε i j ) = E ( εj εj... εjt) = E : : : : ε εitε j εitε j... ε it itε jt σ ij σ ij... 0 = = σ ijit : : : σ ij Matrix represetatio of the model y = X β + ε y = X β + ε : y = X β + ε y X 0 T k... 0 T k β ε y 0 T k X... 0 T k β ε = + : : : : : : y 0T k 0 T k... X β ε T k T T k, where k = Σ i k i. y = X β + ε SUR-5

6 Digressio to Kroecker Products Let A = [a ij ] m ad B = [b ij ] p q. The two matrices do ot have to be of the same dimesios. The, A ab ab... a B a B a B... a B B = : : : a B a B... a B m m m mp q Example: 3 A= ; B = A B = = 4 4 Facts: Let A, B ad C be coformable matrices. The, (A+B) C = (A C) + (B C); (A B)(C D) = AC BD, if AC ad BD ca be defied; (A B) - = A - B -, if both A ad B are ivertible; (A B) = (A B ). Ed of Digressio SUR-6

7 Covariace matrix of ε i the system y = X β + ε Let Σ = [σ ij ]. (Note that σ ij = σ ji ) εε εε... εε εε εε... εε Cov( ε ) = E( ε ε ) = E : : : εε εε... εε σit σit... σ IT σ I σ I... σ I : : : σit σit... σit T T T = =Σ T Ω I If we let ε t = (ε t,..., ε t ), the ε t are iid N(0, Σ). SUR-7

8 [3] OLS ad GLS Two possible OLS OLS o idividual i: β i = ( X X ) X y, for i =,...,. OLS o y = X β + ε : i i i i β β β = = : β ( X X ) X y. Propositio : β j = β, for i =,,...,. j <Proof> Ad X X X X... 0 X X = : : : : : : X X X X X X... 0 = : : : X X. SUR-8

9 ( X X) ( X X)... 0 ( X X ) = ; : : : ( X X) Thus, = X y X y X y. : X y SUR-9

10 β β β = = ( X X ) X y : β ( X X) X y 0 ( X X)... 0 X y = : : : : ( X X) X y ( X X) X y β ( X X) X y β = = : : ( X ) X X y β Implicatio: Note that Cov(ε ) = Σ I T σ ε I T. So, i geeral, the OLS o y = X β + ε would ot be efficiet, ad so are idividual OLS estimators. You ca use idividual OLS estimators β i ad Cov( β ) = σ ( X X ). But they would be iefficiet. i ii j j SUR-0

11 Propositio : Let Ω - = (Σ I T ) - = Ω = ( Σ I ) =Σ I. The, the (ifeasible) GLS T T estimator β = asymptotically efficiet <Proof> Obvious. ( X Ω X ) X Ω y is ubiased, efficiet, cosistet ad Structure of the GLS estimator Deote Σ - = [σ ij ]. The, σ X X σ X X... σ X X σ X X σ X X... σ X X X Ω X = ; : : : σ X X σ X X... σ X X j Σ j σ X y = j j Σ j σ X y = j X Ω y =. : Σ j= X y j (Justify this by yourself.) Efficiecy gai of GLS over OLS: SUR-

12 Digressio Suppose A A A = A A ; A A A = A where A ad A are ivertible square matrices. The, A = (A -A (A ) - A ) - = (A ) - + (A ) - A (A -A (A ) - A ) - A (A ) - A = - (A ) - A (A -A (A ) - A ) - = - (A -A (A ) - A ) - A (A ) - A = - (A -A (A ) - A ) - A (A ) - = - (A ) - A (A -A (A ) - A ) - A = (A -A (A ) - A ) - = (A ) - + (A ) - A (A -A (A ) - A ) - A (A ) - Ed of Digressio A Digressio (Review): Let B = [b ij ] x be a symmetric matrix, ad c = [c,..., c ]. The, the scalar c Bc is called a quadratic form of B. If c Bc > (<) 0 for ay ozero vector c, B is called positive (egative) defiite. If c Bc ( ) 0 for ay ozero c, B is called positive (egative) semidefiite. SUR-

13 Let B be a symmetric ad square matrix give by: Defie the pricipal miors by: B b b... b b b... b = : : : b b... b. b b b b b B b ; B ; B b b b ;... 3 = = 3 = 3 b b b3 b3 b33 B is positive defiite iff B, B,..., B are all positive. B is egative defiite iff B < 0, B > 0, B 3 < 0,.... Ed of Digressio Digressio 3 (Review): Let θ ad θ be two p ubiased estimators. Let c = [c,..., c p ] be ay ozero vector. The, θ is said to be efficiet relative to θ iff var( c θ) var( c θ). ccov ( θ) c ccov ( θ) c 0. c [ Cov( θ) Cov( θ)] c 0. [ Cov( θ) Cov( θ)] is positive semidefiite. SUR-3

14 If θ is more efficiet tha θ, var( θ ) var( j θ j ), for ay j =,..., p. But, the reverse is ot true. Ed of Digressio 3 Retur to Efficiecy gai of GLS over OLS: Cosider the cases of two equatios: y = X β + ε y = X β + ε, with σ σ ; Σ= σ σ σ σ Σ =. σ σ Σ must be positive defiite; that is, σ > 0 ad σ σ -(σ ) > 0. β σ X X σ X X Cov( β ) = Cov = ( X Ω X ) = β σ X X σ X X A A A A = ( say) A A = A A Cov( β ) = A ; Cov( β ) = A. SUR-4

15 Usig the fact that A = (A - A (A ) - A ) -, Cov( β ) = [σ X X - {(σ ) /σ }X X (X X ) - X X ] - = [σ X X - {(σ ) /σ }X X + {(σ ) /σ }X X - {(σ ) /σ }X X (X X ) - X X ] - = [{σ -(σ ) /σ }X X + {(σ )/σ }X M(X )X ] -, where M(X ) = I T - X (X X ) - X = [(/σ )X X + {(σ ) /σ }X M(X )X ] - = σ [X X + {σ (σ ) /σ }X M(X )X ] - = σ [X X + {(σ ) /(σ σ -(σ ) )}X M(X )X ] - = σ [X X + m X M(X )X ] - σ, where m = σ σ σ (m 0). Note that Cov( β) = σ( X X). Thus, β is more efficiet tha β, because: [ Cov( β)] [ Cov( β)] = m XM ( X ) X σ is psd. Cov( β ) Cov( β ) is psd. β is more efficiet. Similarly, we ca show that β is more efficiet tha β. SUR-5

16 There are three possible cases i which β is as efficiet as β (m X M(X )X = 0): ) σ = 0 m = 0. ) X = X X M(X )X = X M(X )X = 0. 3) X = [X,W] X M(X )X = X 0 = 0. For ) ad ), β ad β are equally efficiet. But for 3), β is still more efficiet tha β. Three Cases i which OLS = GLS: Case I: σ ij = 0 for ay i j. Σ = diag(σ,..., σ ) Σ - = diag(/σ,..., /σ ). σ X X σ X X... 0 X Ω X = ; : : : σ X X σ X y σ X y X Ω y =. : σ X y SUR-6

17 Thus, β = ( X Ω X ) X Ω y X X σ X y σ 0 σ X X... 0 σ X y = : : : : σ X X σ X y ( X X) σ σ X y σ X y ( X X) = σ : : : ( X X σ ) σ ( X X) X y ( X X) X y = = β : ( X X) X y : X y Case II: X = X =... = X. This is the case where the values of regressors are the same for all equatios. X X 0 X... 0X 0 X X X... 0X X = = = I X : : : : : : X 0X 0 X... X SUR-7

18 The, β = [X Ω - X ] - X Ω - y = [(I X) (Σ - I T )(I X)] - (I X) (Σ - I T )y = [Σ - X X] - (Σ - X )y = (Σ (X X) - )(Σ - X )y = (I (X X) - X )y = ( XX ) X y ( X X) Xy 0 ( )... 0 y XX X ( X X) Xy = : : : : : ( XX) X y ( X X) Xy = β Case III: X, X,..., X m X m+, X m+,..., X β i = β i for i =,,..., m. But β j are still more efficiet tha β j for j = m+,...,. SUR-8

19 [4] Feasible GLS Estimator The GLS estimator defied above is ot feasible i the sese that it depeds o the ukow covariace matrix Σ. Feasible GLS estimator is a GLS estimator obtaied by replacig Σ by a cosistet estimate of it. Feasible GLS estimators caot be said to be ubiased. But they are cosistet ad asymptotically equivalet to the ifeasible couterpart. () Two-Step Feasible GLS Let T sij = ( yi Xiβ )( i yj X jβ j) = Σ t= etietj ; ad S = [s ij ]. T T It ca be show that plim s ij = σ ij. The two-step feasible GLS estimator is the give by: where Ω= S I. T β = ( X Ω X ) X Ω y, () Iterative Feasible GLS. Usig the two-step GLS estimator, recompute S. Usig this recomputed S, recompute the feasible GLS estimator. Repeat this procedure, util the value of FGLS does ot chage. SUR-9

20 (3) Facts: The two-step ad iterative feasible GLS are asymptotically equivalet to the MLE uder the ormality assumptio about the ε s. I fact, iterative feasible GLS = MLE, umerically. SUR-0

21 [4] MLE of β ad Σ Cosider a simple regressio model: y = Xβ + ε; ε ~ N(0 T,Ω). l T = -(T/)l(π) - (/)l[det(ω)] - (/)(y-xβ) Ω - (y-xβ). The log-likelihood fuctio of the SUR model: y = X β + ε ; ε ~ N(0 T,Σ I T ). l = -(T/)l(π) - (/)l[det(σ I T )] - (/)(y -X β ) (Σ - I T )(y -X β ). Let A ad B are p p ad q q matrices. The, det(a B) = [det(a)] q [det(b)] p (Theil, p. 305). (y -X β ) (Σ - I T )(y -X β ) = Σ i Σ j σ ij (y i -X i β i ) (y j -X j β j ). l = -(T/)l(π) + (T/)l[det(Σ - )] - (/)(y -X β ) (Σ - I T )(y -X β ) = -(T/)l(π) + (T/)l[det(Σ - )] - (/)Σ i Σ j σ ij (y i -X i β i ) (y j -X j β j ) = -(T/)l(π) + (T/)l[det(Σ - )] - (/)Σ i σ ii (y i -X i β i ) (y i -X i β i ) -Σ i Σ j<i σ ij (y i -X i β i ) (y j -X j β j ) l[det( A )] = a ij a ij for A = [a ij ] ad A - = [a ij ]. For a symmetric A, l[det( A )] a ii = a ii ; ad l[det( A )] a ij = a ij for j i. SUR-

22 Maximize l w.r.t. β ad σ ii ad σ ij to get MLE of β ad Σ. Let A be a symmetric p p matrix ad; x be a p vector. The, x Ax = Ax. x FOC: l = X ( Σ IT )( y X β ) = 0; β l T = σii ( yi X ii iβi )( yj X jβ j ) = 0 ; σ l = Tσij ( yi Xiβi )( yj X jβ j ) = 0. ij σ MLE estimators, β ad Σ solves: β = [ X ( Σ I ) X ] X ( Σ I ) y ; σ ij = ( yi X iβ i )( yj X jβ j ) T. T T SUR-

23 [5] Testig Hypotheses Let β be a feasible GLS estimator; S be a cosistet estimator of Σ; ad C = Cov( β ) = [X (S - I T )X ] -. () Testig liear hypotheses: H o : Rβ = r, where R ad r are kow matrices with m rows. Uder H 0, W = ( Rβ r)[ RC R ] ( Rβ r) χ ( m). T Example : l(q coke,t ) = β coke, + β coke, l(p coke,t ) + β coke,3 l(p pep,t ) + ε coke,t ; l(q pep,t ) = β pep, + β pep, l(p coke,t ) + β pep,3 l(p pepe,t ) + ε pep,t. βcoke, β coke, βcoke,3 β = β. pep, β pep, β pep,3 H 0 : Ow-price elasticities are the same. H 0 : β coke, - β pep,3 = 0 R = (0,,0,0,0,-), r = 0. SUR-3

24 Example : Assume that X,..., X cotai the same umber (k) of variables. H 0 : β = β =... = β. R Ik Ik 0 k k... 0k k 0k k 0 k k Ik Ik... 0k k 0 k k = : : : : : 0k k 0k k 0 k k... Ik I k ; r 0 k ( ) =. () Testig oliear hypotheses: H o : w(β ) = 0, where w is a m vecor of fuctios of β. Uder H 0, W = w( β )[ W( β ) CW( β )] w( β ) χ ( m), where T W ( β ) = w. β Example: l(q coke,t ) = β coke, + β coke, l(p coke,t ) + β coke,3 l(p pep,t ) + ε coke,t ; l(q pep,t ) = β pep, + β pep, l(p coke,t ) + β pep,3 l(p pep,t ) + ε pep,t. H 0 : β coke, β pep,3 = w( β ) = β β. coke, pep,3 W ( β ) = (0, β,0,0,0, β ). pep,3 coke, SUR-4

25 (3) Testig diagoality of Σ Digressio to LM test: Let θ be a ukow parameter vector (p ). H o : w(θ) = 0. Let θ R be the restricted MLE wihch max. l T subject to w(θ) = 0. lt Let st ( θ ) = ad I θ T lt( θo) ( θ ) = E θθ. The, LM = s ( θ )[ ( )] ( R I θ R s θ R). Ed of Digressio T T T Note that uder H o : Σ is diagoal, the restricted MLE of β i s = OLS of β i s; restricted MLE of σ ii = s ii = ( y X β )( y X β )/ T ; ad restricted MLE of σ ij = 0. i i i i i j Breusch ad Paga (979, Restud): Let r ij = s ij /(s ii s jj ) / (estimated correlatio coefficiet betwee e i ad e j ). LM T for H o = TΣ i Σ j<i r ij χ [(-)/]. Do ot eed to compute urestricted MLE. This statistic is obtaied uder the assumptio of ormal errors. Questio: Is this statistic still chi-squared eve if the errors are ot ormal? SUR-5

26 [6] Autocorrelatio: () AR() y it = x it β i + ε it ; ε it = ρ i ε i,t- + v it ; (v t,..., v t ) ~ N(0, Σ). Use OLS estimates of β i to estimate ρ i s as we did i ECN 55. Trasform each equatio usig Prais-Wiste or Cochrae-Orcutt methods: y ρ y = ρ x β + v ; i i i i i i ρ y = ( x ρ x ) β + v ; i i i i i i i i y it : ρ y = ( x ρ x ) β + v. i i, T it i i, T i it The, do SUR. () AR(p) Similar to the above procedures. (3) MA(q): Procedures are complicated. SUR-6

27 [6] Applicatio: Use taba5_a.wf (EVIEWS data set) from the CD attached to the textbook. Grufeld s ivestmet data: I: gross ivestmet ($millio) F: market value of firm at the ed of previous year. CS: value of firm s capital at the ed of previous year. I it = β i + β i F it + β 3i CS it + ε it, where i = GM (), CH (Chrysler, ), GE (3), WE (Westighouse, 4), ad US (U.S. Steel, 5). GLS Read the work file usig EVIEWS. Go to \objects\new Objects... Choose System ad click o the ok butto. The, a empty widow will pop up. Type the followigs o the widow: i = c()+c()f+c(3)cs i = c(4)+c(5)f+c(6)cs i3 = c(7)+c(8)f3+c(9)cs3 i4 = c(0)+c()f4+c()cs4 i5 = c(3)+c(4)f5+c(5)cs5 Click o proc\estimate. The, you will see the meu for etimatio of systems of equatios. Choose Seeigly Urelated Regressio. For Two-Step GLS, choose Iterate Coefs. For Iterative GLS, choose Sequetial. Do ot use Oe-Step Coefs or Simultaeous. SUR-7

28 <Two-Step GLS> Estimatio Results: System: SUR Estimatio Method: Seemigly Urelated Regressio (Marquardt) Sample: Icluded observatios: 0 Total system (balaced) observatios 00 Liear estimatio after oe-step weightig matrix Coefficiet Std. Error t-statistic Prob. C() C() C(3) C(4) C(5) C(6) C(7) C(8) C(9) C(0) C() C() C(3) C(4) C(5) Determiat residual covariace 6.8E+3 Equatio: I = C()+C()F+C(3)CS Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat Equatio: I = C(4)+C(5)F+C(6)CS Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat SUR-8

29 Equatio: I3 = C(7)+C(8)F3+C(9)CS3 Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat Equatio: I4 = C(0)+C()F4+C()CS4 Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var 9.09 S.E. of regressio Sum squared resid Durbi-Watso stat Equatio: I5 = C(3)+C(4)F5+C(5)CS5 Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat.0798 SUR-9

30 views/residuals/graphs I Residuals I Residuals I3 Residuals 0 I4 Residuals I5 Residuals SUR-30

31 views/residulas/correlatio matrix I I I3 I4 I5 I I I I I Testig H o : c() = c(4), c() = c(5), c(3) = c(6) views/wald Coefficiet Tests. Type: c() = c(4), c() = c(5), ad c(3) = c(6). Wald Test: System: SUR Null Hypothesis: C()=C(4) C()=C(5) C(3)=C(6) Chi-square Probability SUR-3

32 <Iterative GLS> System: SUR Estimatio Method: Iterative Seemigly Urelated Regressio (Marquardt) Date: 08/9/0 Time: 8:57 Sample: Icluded observatios: 0 Total system (balaced) observatios 00 Sequetial weightig matrix & coefficiet iteratio Covergece achieved after: 3 weight matrices, 4 total coef iteratios Coefficiet Std. Error t-statistic Prob. C() C() C(3) C(4) C(5) C(6) C(7) C(8) C(9) C(0) C() C() C(3) C(4) C(5) Determiat residual covariace 5.97E+3 Equatio: I = C()+C()F+C(3)CS Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat Equatio: I = C(4)+C(5)F+C(6)CS Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat.885 SUR-3

33 Equatio: I3 = C(7)+C(8)F3+C(9)CS3 Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat Equatio: I4 = C(0)+C()F4+C()CS4 Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var 9.09 S.E. of regressio Sum squared resid Durbi-Watso stat.4739 Equatio: I5 = C(3)+C(4)F5+C(5)CS5 Observatios: 0 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat SUR-33

34 <GLS with AR()> EVIEWS estimates β s ad ρ s joitly usig oliear GLS method. Type: i = c()+c()f+c(3)cs+[ar()=c(4)] i = c(5)+c(6)f+c(7)cs+[ar()=c(8)] i3 = c(9)+c(0)f3+c()cs3+[ar()=c()] i4 = c(3)+c(4)f4+c(5)cs4+[ar()=c(6)] i5 = c(7)+c(8)f5+c(9)cs5+[ar()=c(0)] The followig results are from two-step oliear GLS (Iterate Coefs). System: SUR Estimatio Method: Seemigly Urelated Regressio (Marquardt) Date: 08/9/0 Time: 9:00 Sample: Icluded observatios: 0 Total system (balaced) observatios 95 Iterate coefficiets after oe-step weightig matrix Covergece achieved after: weight matrix, 50 total coef iteratios Coefficiet Std. Error t-statistic Prob. C() C() C(3) C(4) C(5) C(6) C(7) C(8) C(9) C(0) C() C() C(3) C(4) C(5) C(6) C(7) C(8) C(9) C(0) SUR-34

35 Determiat residual covariace.6e+3 Equatio: I = C()+C()F+C(3)CS+[AR()=C(4)] Observatios: 9 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat Equatio: I = C(5)+C(6)F+C(7)CS+[AR()=C(8)] Observatios: 9 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat.7844 Equatio: I3 = C(9)+C(0)F3+C()CS3+[AR()=C()] Observatios: 9 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat.989 Equatio: I4 = C(3)+C(4)F4+C(5)CS4+[AR()=C(6)] Observatios: 9 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat Equatio: I5 = C(7)+C(8)F5+C(9)CS5+[AR()=C(0)] Observatios: 9 R-squared Mea depedet var Adjusted R-squared S.D. depedet var S.E. of regressio Sum squared resid Durbi-Watso stat SUR-35