Simultaneous equation models with spatially autocorrelated error components

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Beatrice Reeves
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1 MPRA Munich Personal RePEc Archive Simultaneous equation models with spatially autocorrelated error components Claude Marius AMBA OYON and Taoufiki Mbratana University of Yaounde II, University of Yaounde II October 2017 Online at MPRA Paper No , posted 10 November :03 UTC

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5 l y l = Y l α l + X l β l + υ l = Z l δ l + υ l l = 1,..., L y l=1,...,l T N 1 Y l T N (M l 1) l X l T N K l Z l = [Y l, X l ] δ l = (α l, β l) Z l ρ l W l υ l + ϵ l = (I ρ l W l ) 1 ϵ l υ l = Λ l ϵ l = ρ l W l ϵ l + ϵ l = (I + ρ l W l )ϵ l i = 1,..., N t = 1,..., T N T W l = I T W ln I T T W ln N N 1 l L W ln ρ l < 1 ϵ l T N 1 ϵ l ϵ l = Z η η l + ξ l Z η = ι T I N η l = ( η 1l... η Nl ) ξ l = ( ξ 11l ξ 12l... η T Nl ) ι T T 1 η l ξ l ( ηl ξ l ) (η q ξ q [ ] ) σ 2 = ηlq I N 0 0 σ 2 ξ lq I T N M l l M l +M l = L

6 Ω ϵlq = (ϵ l ϵ q) l q Ω ϵlq = σ 2 η lq (J T I N ) + σ 2 ξ lq I T N for l, q = 1,..., L 1 Ω ϵlq = σh 2 lq Q h for l, q = 1,..., L h=0 Q h = B h I N, B h,{h=0,1} T B 0 = E T B 1 = J T JT = ι T ι T /T E T = I T J T σ 2 0 lq = σ 2 ξ lq σ 2 1 lq = T σ 2 η lq + σ 2 ξ lq Q 0 Q 1 (M) M Q 0 + Q 1 = I T N, (Q 0 ) = (T 1)N (Q 1 ) = N Ω υlq l q Ω υlq = Λ l (ϵ l ϵ q)λ q = Λ l (σ 2 0 lq Q 0 + σ 2 1 lq Q 1 )Λ q = σ 2 0 lq Λ l Q 0 Λ q + σ 2 1 lq Λ l Q 1 Λ q = σ 2 0 lq Q 0 Λ l Λ q + σ 2 1 lq Q 1 Λ l Λ q = σ 2 0 lq Q 0 Λ lq + σ 2 1 lq Q 1 Λ lq Λ lq = Λ l Λ q Λ l l W l Q h = (I T W ln )(B h I N ) = (I T B h ) (W ln I N ) = (B h I T ) (I N W ln ) = Q h W l ρ l < 1

7 Λ l Q h = (I ρ l W l ) 1 Q h = (I + ρ l W l + ρ 2 l Wl 2 + ρ 3 l Wl )Q h = (Q h + ρ l W l Q h + ρ 2 l Wl 2 Q h + ρ 3 l Wl 3 Q h +... ) = Q h (I + ρ l W l + ρ 2 l Wl 2 + ρ 3 l Wl ) = Q h (I ρ l W l ) 1 = Q h Λ l y 1 = Z 1 δ 1 + υ 1 = Z 1 δ 1 + Λ 1 ϵ 1 Q h y (h) 1 = Q h y 1 Z (h) 1 = QZ 1 ϵ (h) 1 = Q h ϵ 1 y (h) 1 = Z (h) 1 δ 1 + Λ 1 ϵ (h) 1 (Λϵ (h) 1 ) = Λ 1 Q h (ϵ 1 ) = 0 ( Λ 1 ϵ (h) ) 1 = (Qh υ 1 ) = σh 2 Q 11 hλ 11 X h = Q h X X hy (h) 1 = X hz (h) 1 δ 1 + X hλ 1 ϵ (h) 1 ( X hλ 1 ϵ (h) ) 1 = σ 2 X h11 hλ 11 X h

8 [ ˆδ (h) ( ) ] 1,S2SLS = Z (h) 1 X h σ 2 h11 X hλ 1 1 [ 11 X h X h Z (h) ( ) ] 1 Z (h) 1 X h σ 2 h11 X hλ 1 11 X h X h y (h) 1 = ( Z (h) 1 P h Z (h) ) 1 ( (h) 1 Z 1 P h y (h) ) 1 P h = X h (X hλ 11 X h ) 1 X h ρ 1 ˆσ 2 h 11 = ( ) y 1 Z1 (h) ( ) ˆδ 1,S2SLS Qh y 1 Z1 (h) ˆδ 1,S2SLS (Q h ) y1 = Λ 1 y 1 Z1 = Λ 1 Z 1 y 1 Z 1 Q h σh 2 11 ρ 1 δ 1 h = 0 h = 1 ρ 1 = 0 Λ 11 = I T N P h P Xh ρ 1 ˆδ 1,2SLS σh 2 11 (ˆδ1,S2SLS ) ( = σ 2 (h) h11 Z 1 P h Z (h) ) 1 1 δ 1 δ 1 ( X 0 y (0) ) 1 X 1y (1) = 1 ( X 0 Z (0) ) 1 X 1Z (1) δ ( X 0 υ (0) ) 1 X 1υ (1) 1 ρ 1 ˆδ 1,SEC2SLS ˆδ 1,SEC2SLS = Z (0) 1 P 0 Z (0) 1 ˆσ Z (1) 1 P 1 Z (1) 1 1 Z (0) 1 P 0 y (0) 1 ˆσ ˆσ Z (1) 1 P 1 y (1) 1 ˆσ

9 P h = X h (X hλ 11 X h ) 1 X h X h = Q h X y (h) 1 = Q h y 1 Z (h) 1 = QZ 1 h = 0 h = 1 ˆσ ˆσ ( X 0 υ (0) ) 1 X 1υ (1) = ( ) σh 2 lq X hλ 11 X h 1 ρ l ρ l ρ l a ϵ = W a ϵ, 0 ϵ = W 0 ϵ = ϵ 1 ϵ = W ϵ 2 ϵ = W 2 ϵ = ϵ Q h (ϵ 1 ϵ 1) = Q h Ω ϵ11 = σh 2 σ Q h = 11 Q 0 h = 0 σ Q 1 h = 1 ( ā ) ϵ 1 Q h b ϵ1 = ϵ 1W 1... W 1 Q h W 1... W 1 ϵ }{{}}{{} 1 a b = ( ) ϵ 1(W 1) a Q h W1 b ϵ 1 = ( (W 1) a W b 1 Q h Ω ϵ11 ) = σ 2 h 11 (B h ) ( (W 1N) a W b 1N ) (W 0 1N) = (I N ) = N (W 1N ) = 0

10 (ϵ 1Q h ϵ 1 ) = σ 2 h 11 (Q h ) ( ϵ 1Q h ϵ 1 ) = σ 2 h 11 (B h ) (W 1N W 1N) ( ϵ 1Q h ϵ 1 ) = σ 2 h 11 (B h ) (W 1N ) = 0 T 2 ϵ 1Q h ϵ 1 /(Q h ) ϵ 1Q h ϵ 1 /(Q h ) ϵ 1Q h ϵ 1 /(Q h ) = σ 2 h 11 1 (W 1N W 1N)/N 0 ρ 1 σ σ ϵ 1 ϵ 1Q h ϵ 1 /(Q h ) σ 2 h 11 υ 1 = (I ρ 1 W 1 ) 1 ϵ 1 = ϵ 1 = υ 1 ρ 1 ῡ 1 ϵ 1 = ῡ 1 ρ 1 ῡ 1 ϵ 1 ϵ 1 υ 1 ῡ 1 ῡ 1 ϵ 1Q h ϵ 1 = υ 1Q h υ 1 ρ 1 (υ 1Q h ῡ 1 + ῡ 1Q h υ 1 + ρ 2 1ῡ 1Q h ῡ 1 ϵ 1Q h ϵ 1 = ῡ (ῡ ) 1Q h ῡ 1 ρ 1 1 Q h ῡ 1 + ῡ 1Q h ῡ 1 + ρ 2 1 ῡ 1Q h ῡ 1 ϵ 1Q h ϵ 1 = ῡ 1Q (ῡ ) h υ 1 ρ 1 1 Q h ῡ 1 + ῡ 1Q h υ 1 + ρ 2 1 ῡ 1Q h ῡ 1 h = 0, 1 ρ 1 σh 2 11 Γ 1h ρ 1 ρ 2 1 Θ 1h = 0 σ 2 h 11

11 Γ 1h = 2γ1 1h γ2 1h 1 γ 2γ3 1h ) γ4 1h γ5 1h 1h Θ 1h 0 = γ 1h 2 γ 1h 3 0 γ1 1h ( γ 1h 2 + γ6 1h γ 1h i γ 1h 1 = (ῡ 1Q h υ 1 ) (Q h ), γ 1h 4 = (ῡ 1Q h ῡ 1 ) (Q h ), γ1h 2 = (ῡ 1Q h ῡ 1 ) (Q h ), γ1h γ1h 3 = (ῡ 1Q h ῡ 1 ) (Q h ) 5 = (W 1NW 1N ), γ6 1h = (ῡ 1Q h υ 1 ) N (Q h ), γ1h 0 = (υ 1Q h υ 1 ) (Q h ) υ 1 = (I + ρ 1 W 1 )ϵ 1 = υ 1 = ϵ 1 + ρ 1 ϵ 1 ῡ 1 = ϵ 1 + ρ 1 ϵ 1 υ 1Q h υ 1 = ϵ 1Q h ϵ 1 + ρ 1 (ϵ 1Q h ϵ 1 + ϵ 1Q h ϵ 1 ) + ρ 2 1 ϵ 1Q h ϵ 1 ῡ 1Q h ῡ 1 = ϵ 1Q h ϵ 1 + ρ 1 ( ϵ 1 Q h ϵ 1 + ϵ 1Q h ϵ 1 ) + ρ 2 1 ϵ 1Q h ϵ 1 ῡ 1Q h υ 1 = ϵ 1Q h ϵ 1 + ρ 1 ( ϵ 1 Q h ϵ 1 + ϵ 1Q h ϵ 1 ) + ρ 2 1 ϵ 1Q h ϵ 1 (υ 1Q h υ 1 ) = σ 2 h 11 (B h ) { N + ρ 1 [ (W 1N ) + (W 1N )] + ρ 2 1 (W 1NW 1N ) } (ῡ 1Q h ῡ 1 ) = σh 2 11 (B h ) { [ ( ) ( )] (W 1NW 1N ) + ρ 1 W 1N W1N 2 + (W 1N ) 2 W 1N + ρ 2 1 ( (W 1N) 2 W1N 2 (ῡ 1Q h υ 1 ) = σh 2 11 (B h ) { [ (W 1N ) + ρ 1 (W 1N W 1N ) + ( )] W1N 2 + ρ 2 1 ( )} (W 1N) 2 W 1N )}

12 ( ) [ υ 1 Q h υ 1 = σ 2h ρ 2 1 (Q h ) (ῡ ) [ 1 Q h ῡ 1 (W = σ 2 1N W 1N ) h (Q h ) 11 N (ῡ ) 1 Q h υ 1 (Q h ) (W 1NW ] 1N ) N = σ 2 h 11 ρ 1 [ (W 1N W 1N ) N + 2ρ 1 (W 1NW 2 1N) N + (W 2 1N) N + ρ 2 ((W 1N) 2 W 2 ] 1N) 1 N ] + ρ 1 ((W 1N) 2 W 1N ) N 3 3 Γ 1h 3 1 Θ 1h γ 1 1 = (W 1NW 1N ) N, γ 1 2 = (W 1NW 2 1N) N, γ3 1 = (W 1N) 2, γ4 1 = ((W 1N) 2 W1N) 2 N N σ 2 Γ 1h h 11 ρ 1 σh 2 11 Θ 1h = 0 ρ 2 1σ 2 h γ 1 Γ 1h 1 = γ1 1 2γ2 1 γ4 1, Θ 1h = 0 (γ1 1 + γ3) 1 γ2 1 1 (Q h ) υ 1Q h υ 1 ῡ 1Q h ῡ 1 ῡ 1Q h υ 1 δ 1 υ 1 = y 1 Z 1 δ 1 g 1h Θ 1h G 1h Γ 1h υ 1 ῡ 1 ῡ 1 G 1h ψ h g 1h = e(ρ 1, σ 2 h 11 ) ψ ψ h SAR h = ( ) ρ 1 ρ 2 1 σh 2 = 11 ψ SMA h = ( ) σh 2 11 ρ 1 σh 2 11 ρ 2 1σh 2 11 e(ρ 1, σh 2 11 ) ρ 1 σh 2 11

13 σ ρ 1 σ σ ρ 1 ρ 1 ( ρ 1, σ 2 h 11 ) = arg min{e(ρ 1, σ 2 h 11 ) e(ρ 1, σ 2 h 11 )} ρ 1 ˆδ 1,SEC2SLS ˆδ 1,SEC2SLS = Z (0) 1 P 0 Z (0) 1 ˆσ Z (1) 1 P 1 Z (1) 1 1 Z (0) 1 P 0 y (0) 1 ˆσ ˆσ Z (1) 1 P 1 y (1) 1 ˆσ ˆρ 1 σξ 2 11 σ P h = X h (X hλ 11 X h ) 1 X h X h = Q h X y (h) 1 = Q h y 1 Z (h) 1 = QZ 1 h = 0 h = 1 ρ 1 σ 2 ξ 11 σ ˆρ 1 ˆσ 2 ξ 11 ˆσ υ l υ q L l = 1,..., L L y = Zδ + υ y = ( y 1 y L) Z = (Zl ) δ = ( δ 1 δ L) υ = ( υ 1 υ L) Z l = [ Y l X l ]

14 (ρ l W l )υ + ϵ = (I ρ l W l ) 1 ϵ υ = Λϵ = (ρ l W l )ϵ + ϵ = (I + ρ l W l )ϵ Λ = (Λ l ) ϵ = ( ϵ 1 ϵ L) ϵ = (I L ι T I N )η + ξ = I L Z η η + ξ η = ( η 1 η L) ξ = ( ξ 1 ξ L) W l W l = W ρ 1 W 0 = (ρ l) W = ρ W 0 ρ L W Ω ϵ Ω ϵ = ( Ω ϵlq ) = ( σ 2 0lq ) Q0 + ( σ 2 1 lq ) Q1 = Σ 0 Q 0 + Σ 1 Q 1 Σ 1 = ( σ 2 1 lq ) Σ0 = ( σ 2 0 lq ) υ Ω υ = ΛΩ ϵ Λ = Λ (Σ 0 Q 0 ) Λ + Λ (Σ 1 Q 1 ) Λ Λ (Σ h Q h ) Λ = (Λ l ) ( ) σh 2 lq Q h (Λ l ) = ( ) σh 2 lq Λ l Q h Λ q = ( ) σh 2 lq Q h Λ lq Ω υ = (I L Q 0 ) Σ0 + (I L Q 1 ) Σ1 Σh = ( σ 2 h lq Λ lq ) IL Q h

15 y (h) = Z (h) δ + Λϵ (h) y (h) = (I L Q h )y Z (h) = (I L Q h )Z ϵ (h) = (I L Q h )ϵ (Λϵ (h) ) = Λ(I L Q h ) (ϵ) = 0 ( Λϵ (h)) = (I L Q h ) Σh x h = I L X h x hy (h) = x hz (h) δ + x hλϵ (h) ( x hλϵ (h)) = x h ( Λϵ (h)) x h = x h Σh x h Σh = ( σ 2 h lq Λ lq ) ˆδ (h) S3SLS = [ Z (h) x h (x h Σh x h ) 1 x hz (h)] 1 [ Z (h) x h (x h Σh x h ) 1 x hy (h)] = ( Z (h) P h Z (h)) 1 ( Z (h) P h y (h)) P h = x h (x h Σh x h ) 1 x h δ h = 0 (h) 1 ˆδ S3SLS δ δ ( x 0 y (0) ) ( x x y (1) = 0 Z (0) ) ( x x Z (1) δ + 0 υ (0) ) x υ (1) ˆδ SEC3SLS

16 ˆδ SEC3SLS = ( Z (0) P 0 Z (0) + Z (1) 1 P 1 Z (1) ) 1 ( 1 Z (0) P 0 y (0) + Z (1) 1 P 1 y (1) ) 1 P h = x h (x h Σh x h ) 1 x h y (h) = I M Q h y Z (h) = I M Q h Z h = 0 h = 1 ˆσ 2 h lq = ( y l Z l ) (h) ( ) ˆδ l,s2sls Qh y q Zq (h) ˆδ q,s2sls (Q h ) ρ l σh 2 lq ρ l σh 2 ll l ρ l Γy it + Λx it = υ it y it x it υ it W υ itl l = 1, 2 (I ρ l W ) 1 ϵ l υ itl = (I + ρ l W )ϵ l ϵ itl = η il + ξ itl Γ 2 2 Λ 2 4

17 Γ = ( ) ( ) Λ = X 11 X 12 X 21 X 22 x p,it = ζ p,i + z p,it p = 11, 12, 21, 22 ζ p,i iidu[ 10, 10] z p,it iidu[ 5, 5] 2(N + NT ) N (0, 1) 2N 2NT N (0, 1) Ω η Ω ξ ( ) ( ) Ω η = Ω 8 16 ξ = 2 4 ( ) ( ) Ω η = Ω 6 12 ξ = 4 8 ( ) ( ) Ω η = Ω ξ = Ω η = 4 8 ( ) Ω ξ = 6 12 ( ) ( ) Ω ϵ = W 1 W 3 W 7 W 9 W J J W J N = 25 T = (7, 10, 15)

18 σ 2 η l q σ 2 ξ l q W N T ρ 1 ρ 2 α 1 β 11 β 12 α 2 β 21 β 22 W 3 W 3 W 3 W 3 W 1 W 3 W 7 W 9 W 3 W 3 W 3 W 3 W 3 W 3 W 3 W 3 RMSE (ˆα k ) = bias 2 (ˆα k ) + ( ) 2 IQ(ˆαk ) 1.35 IQ ˆα k k α k 1/2

19 NOMAD sls ec 2sls ec 3sls sec 2sls sec 3sls NOMAD ec 2sls ec 3sls sec 2sls sec 3sls Neighbour Sigma RMSE NOMAD(ˆα) = 1 RK K R k=1 r=1 ˆα k,r α k α k K R ˆα k,r k k r NORMSQD NORMSQD NORMSQD 1 (ˆα) = K [ K bias 2 (ˆα k ) + ( ) ] IQ(ˆα k ) 2 1/ k=1 ˆα k 2 bias IQ RMSE

20 (α 1, β 11, β 12, α 2, β 21, β 22 ) ( 0.5, 2, 1.5, 4, 3, 1.8) (ρ 1, ρ 2 ) ( 0.8, 0.3) W 3 Sigma

21 0.07 2sls NOMAD ec 2sls ec 3sls sec 2sls sec 3sls ec 2sls NOMAD ec 3sls sec 2sls sec 3sls Rho Time J J = 1 J = 3 J = 7 J = 9 (α 1, β 11, β 12, α 2, β 21, β 22 ) ( 0.5, 2, 1.5, 4, 3, 1.8) (ρ 1, ρ 2 ) ( 0.8, 0.3) V 1 W 1 W 3 W 7 W 9 W 1 W 1 W 9 T = 7 T = 10 T = 15 (α 1, β 11, β 12, α 2, β 21, β 22 ) (ρ 1, ρ 2 )

22 NOMAD sec 2sls sec 3sls sec 2sls sec 3sls Sigma W NOMAD sec 2sls sec 3sls sec 2sls sec 3sls Time Rho V 1 T ime (α 1, β 11, β 12, α 2, β 21, β 22 ) ( 0.5, 2, 1.5, 4, 3, 1.8) W 3 V 1 ρ 1 ρ 2 ρ l = 0.8 ρ l = 0.4 ρ l ρ l = 0.4 ρ l = 0.8 ρ ρ l ρ l = 0 ρ l

23 ˆρ 2SLS ˆρ 3SLS ρ l = 0.8 ρ l = 0.8 W J ˆρ 2SLS ˆρ 3SLS J = 3 ρ l ρ ρ 2SLS ρ 3SLS

24 0.3 J=1 J=3 J=7 J= RMSE V1 V2 V V Rho Rho y ti = β 01 + α 1 h ti + β 1 f ti + β 2 k ti + β 3 op ti + υ 1,ti h ti = β 02 + α 2 y ti + β 4 pub ti + β 5 old ti + β 6 young ti + υ 2,ti h ti y ti f k pub old young l = {1, 2} υ l = Λ l ϵ l = (I ρ l W ) 1 ϵ l = A 1 l ϵ l

25 Botswana Burkina Faso Burundi Cameroon RCA Chad Congo Rep Ivory coast Gabon Ghana Kenya Malawi Mozambique Namibia Nigeria Rwanda Senegal South Africa Tanzania Uganda Botswana Burkina Faso Burundi Cameroon RCA Chad Congo Rep Ivory coast Gabon Ghana Kenya Malawi Mozambique Namibia Nigeria Rwanda Senegal South Africa Tanzania Uganda W N = (ω ij ) ω ij = 1 i j ω ij = 0 W N ϵ l ϵ l = Z η η l + ξ l

27 6 Health Expenditure (ln) Income (ln)

28 ρ l

30 α 1 β 11 β 12 α 2 β 21 β 22

31 α 1 β 11 β 12 α 2 β 21 β 22

32 α 1 β 11 β 12 α 2 β 21 β 22

33 α 1 β 11 β 12 α 2 β 21 β 22

34 α 1 β 11 β 12 α 2 β 21 β 22

35 α 1 β 11 β 12 α 2 β 21 β 22

36 α 1 β 11 β 12 α 2 β 21 β 22

37 α 1 β 11 β 12 α 2 β 21 β 22

38 α 1 β 11 β 12 α 2 β 21 β 22

39 ρ l α 1 β 11 β 12 α 2 β 21 β 22

40 ρ l α 1 β 11 β 12 α 2 β 21 β 22

Chapter 8 - Appendixes

Chapter 8 - Appendixes Appendix 8.. Individual Preferences for Growth, Environment, and Income Distribution Funds to be invested in projects that Funds to be invested in projects to Funds to be invested