Simultaneous equation models with spatially autocorrelated error components

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 https://mpra.ub.uni-muenchen.de/82395/ MPRA Paper No. 82395, posted 10 November 2017 07:03 UTC

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

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

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 3 +... )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 3 +... ) = 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

[ 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 1 + 1 ( 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 0 2 11 + Z (1) 1 P 1 Z (1) 1 1 Z (0) 1 P 0 y (0) 1 ˆσ 1 2 11 ˆσ 0 2 11 + Z (1) 1 P 1 y (1) 1 ˆσ 1 2 11

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 2 0 11 ˆσ 2 1 11 ( 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 σ0 2 11 Q h = 11 Q 0 h = 0 σ 1 2 11 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

(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 2 0 11 σ 2 1 11 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

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 )}

( ) [ υ 1 Q h υ 1 = σ 2h11 1 + 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 11 1 0 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

σ1 2 11 ρ 1 σ0 2 11 σ1 2 11 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 0 2 11 + Z (1) 1 P 1 Z (1) 1 1 Z (0) 1 P 0 y (0) 1 ˆσ 1 2 11 ˆσ 0 2 11 + Z (1) 1 P 1 y (1) 1 ˆσ 1 2 11 ˆρ 1 σξ 2 11 σ1 2 11 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 2 1 11 ˆρ 1 ˆσ 2 ξ 11 ˆσ 2 1 11 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 ]

(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

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

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

Γ = ( ) ( ) 1 0.5 2 1.5 0 0 Λ = 4 1 0 0 3 1.8 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 ( ) ( ) 16 8 4 2 Ω η = Ω 8 16 ξ = 2 4 ( ) ( ) 12 6 8 4 Ω η = Ω 6 12 ξ = 4 8 ( ) ( ) 8 4 12 6 Ω η = Ω ξ = Ω η = 4 8 ( ) 4 2 2 4 Ω ξ = 6 12 ( ) 16 8 8 16 ( ) 20 10 Ω ϵ = 10 20 W 1 W 3 W 7 W 9 W J J W J N = 25 T = (7, 10, 15)

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 0.75 0.25 ˆα k k α k 1/2

NOMAD 0.125 0.100 0.075 0.07 0.06 0.05 2sls ec 2sls ec 3sls sec 2sls sec 3sls 0.050 0.025 0.04 NOMAD 0.07 0.06 0.05 ec 2sls ec 3sls sec 2sls sec 3sls 0.0475 0.0450 0.0425 0.0400 0.04 0.0375 0.03 2.5 5.0 7.5 Neighbour 0.0350 1 2 3 4 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 1/2 1.35 k=1 ˆα k 2 bias IQ RMSE

(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

0.07 2sls NOMAD 0.08 0.06 0.06 0.05 0.04 ec 2sls ec 3sls sec 2sls sec 3sls 0.04 0.03 0.055 0.0350 0.050 ec 2sls 0.0325 NOMAD 0.045 0.040 ec 3sls sec 2sls sec 3sls 0.0300 0.0275 0.035 0.0250 0.030 0.5 0.0 0.5 Rho 0.0225 8 10 12 14 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 )

0.0330 0.040 0.0327 NOMAD 0.036 sec 2sls sec 3sls 0.0324 0.0321 sec 2sls sec 3sls 0.032 1 2 3 4 Sigma 0.0318 2.5 5.0 7.5 W 0.0325 0.0330 0.0300 0.0325 NOMAD 0.0275 0.0250 0.0225 sec 2sls sec 3sls 0.0320 0.0315 sec 2sls sec 3sls 8 10 12 14 Time 0.0310 0.5 0.0 0.5 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

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

0.3 J=1 J=3 J=7 J=9 0.16 RMSE 0.2 0.12 V1 V2 V3 0.1 0.08 V4 0.5 0.0 0.5 Rho 0.5 0.0 0.5 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

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 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 W N = (ω ij ) ω ij = 1 i j ω ij = 0 W N ϵ l ϵ l = Z η η l + ξ l

6 Health Expenditure (ln) 5 4 3 2 6 7 8 9 10 Income (ln)

ρ l

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.14 0.13 0.11 0.1 0.09 0.07 0.06 0.04 0.03 0.01 0 0.33 0.3 0.27 0.23 0.2 0.17 0.13 0.1 0.07 0.03 0 0.11 0.1 0.09 0.08 0.07 0.06 0.04 0.03 0.02 0.01 0

α 1 β 11 β 12 α 2 β 21 β 22

ρ l α 1 β 11 β 12 α 2 β 21 β 22