ᰬঠᮦᦤ࠼᷆фᰬঠ᫇⁗ශᾸ2015 ᒤ 07 ᴸ 22 ᰕ 2016.08.12(Spatial Econometrics) (Spatial Effect) (Spatio-time Models)§1 ᓅᤦ(Spatial Econometrics) (Spatial Effect)(Anselin 1988, Anselin 1999) (Spatial Dependence) (Spatial Correlation), (Spatial Heterogeneity) Dependence) (Cross sectional data)(Crosssectional( Paelinck and Klaassen 1979, Cliff and Ord 1981, Upton and Fingleton 1985, (Spatial Interaction) Anselin 1988, Haining 1990, Anselin and Florax 1995) (Anselin and Bera 1998, Anselin 2001, Anselin 2002, Florax and Van DerVlist 2003, Anselin et al. 2004) (Spatio-temporal Data Analysis) (Spatial-temporal models) 2 3 4 5 61§2§2.1yit = xit Ø + ≤it yit ≤it i y = XØ + ≤ y = (y1 , y2 , ..., yT ) X NT £ K00(1) K £1 t (2) y NT £ 1 Ø K £1iytxitityi , i = 1, 2, ...T ≤ NT £ 1N £1(Spatial Ordering)E [≤it ≤jt ] 6= 0, 8i 6= j(Stability) (Homogeneity)§2.2 W j (Linkage) i j wij = 0 1s wijN £Nwij wij i j 0 1 wiji (Network) wij = 1ijs wij = wij /Pj(subjectivity) Cliff and Ord 1981( Weights)(ad hoc) ( 17-19 ) Ng (Lag operator) L (3) Anselin 1988( ) ) (Block (Case 1991, Case 1992, Lee 2002)1/Ng ° 1Lyt = yt°1 2, j wijziPjwij zj z W NT £ NT ziiW y = (IT ≠ WN )y W X = (IT ≠ WN )X W ≤ = (IT ≠ WN )≤ §2.3 (Spatio-time Lag Model) y = Ω(IT ≠ WN )y + X Ø + ≤ Ω Ω W (4)(Endogenous) i i i X ≤ y = [IT ≠ (IN ° ΩWN )°1 ]X Ø + [IT ≠ (IN ° ΩWN )°1 ]≤ t2 2 yt = Xt Ø + ΩWN Xt Ø + Ω2 WN Xt Ø + ... + ≤t + ΩWN ≤t + Ω2 WN ≤t ...jijj (Spatial Multiplier)ij (Anselin 2003)i y(5)(6)ytXtXt ≤tOLS (Spatial Filter) : [IT ≠ (IN ° ΩWN )]y = X Ø + ≤ (Detrend) Ω=1 §2.4 t T ¿N N £ (N ° 1)/2 N £ (N ° 1)/2 W IN ° ΩWN Ω=1 (7)3§2.4.1 (Direct Representation) (Geostatistics) 8i 6= j, t = 1, 2, ..., T : E [≤it ≤jt ] = æ 2 f (ø , dij ) ø dij i j ( ) f (Durbin 1988) æ (Isotropy) (8) (Cressie,1993)(Distance Decay Function)§2.4.2 (Spatial Error Process) (Spatial Autoregressive) (Spatial Moving Average) ≤t = µWN ≤t + ut µ ut N £12 E [ut ut ] = æu 2 Ωt,N = æu (BN BN )°10 0SAR (9)BN = IN ° µWN : (10)2 ΣN T = æ u [IT ≠ (BN BN )°1 ]0(11) WN (BN BN )°1 WN (Global)0WBN SAR 8t = 1, 2, ..., T : ≤t = ∞ WN ut + ut(12)2 Ωt,N = E [≤t ≤t ] = !u [IN + ∞ (WN + WN ) + ∞ 2 WN WN ]000(13)2 ΣN T = æ u (IT ≠ [IN + ∞ (WN + WN ) + ∞ 2 WN WN ])00(14) SARW SMA §2.4.3[IN + ∞ (WN + WN ) + ∞ 2 WN WN ] (Local)00(Spatial Error Components,SEC) son 1995, Anselin and Moreno 2003)KelejianRobinson(Kelijian and Robin-≤t = WN √t + ªt 4(15)ªtN £1√tE [√it ªjt ] = 0 8i, j, tN £108t02 2 Ωt,N = E [≤t ≤t ] = æ√ WN WN + æª IN(16)2 2 ΣN T = æxi IN T + æpsi (IT ≠ WN WN )0(17) SMASEC§2.4.4 (Two Way Error Component) ≤it = µi + ∏t + uit µi2 æµ ∏t 2 æ∏ uit 2 æu(18)µi∏t≤t = µ + ∏t ∂N + ut µ, ut N £1 ∏t ∂N0(19)1N £102 2 2 E [≤t ≤t ] = æµ IN + æ∏ ∂N ∂N + æu IN(20)≤ = (∂T ≠ IN )µ + (IT ≠ ∂N )∏ + u(21)2 2 2 ΣN T = æ µ (∂T ∂T ≠ IN ) + æ∏ (IT ≠ ∂N ∂N ) + æu IN T00(22)Andrews(2005) ≤it = ±i ft + uit ±i i j2 E [≤it ≤jt ] = ±i ±j æf(23)ft(Loading) uit (24)≤it = ±i ft + uit ± f m ft uit Ωij = ±i ±j 0 0 1 / (1 + ±i ±i ) 2 (1 + ±j ±j )1/200(25)(26)5§3§3.1 (Spatial SUR Model)(Anselin 1988 ≤t t = 1, 2, ..., T yt = Xt Øt + ≤t E [≤t ≤s ] = æts IN , s 6= t æts t s y = XØ + ≤ E [≤≤ ] = ΣT ≠ IN0 010)(27)(28)(29) (30)yt = Ωt WN yt + Xt Øt + ≤t 0 1(31)B B 0 X=B B ··· @ 0 0X10 X2 ··· 0··· ···C 0 C C ··· ··· C A · · · XT ··· ··· 0 10B B 0 RT = B B ··· @ 0Ω1C 0 C C ··· ··· ··· C A 0 · · · ΩT Ω2 (32)0y = (RT ≠ WN )y + X Ø + ≤ :Ω1 = Ω2 = ... = ΩT = Ωy = Ω(IT ≠ WN )y + X Ø + ≤(33)≤t = µt WN ≤t + ut(34)µ1 = µ2 = ... = µT = µ6§3.2 ≤it = µi + ∫it2 µi ª IID(0, æµ ) 2 ∫it ª IID(0, æ∫ )(35) µi ∫it (36)≤t = µ + ∫t µ N ≠1 ∫t = µWN ∫t + ut(37)2 2 ΣN T = E [≤≤ ] = æµ (∂T ∂T ≠ IN ) + æu [IT ≠ (BN BN )°1 ]000(38)BN = IN ° µWN(39) Kapoor2003 : ≤ = µ(IT ≠ WN )≤ + ∫ ∫ = (∂T ≠ IN )µ + u0 0 0(40)°1 °1 2 2 ΣN T = E [≤≤ ] = (IT ≠ BN )[æµ (∂T ∂T ≠ IN ) + æu IN T ](IT ≠ BN )(41)§3.3 (Space Recursive Model) yt = ∞ WN yt°1 + Xt Ø + ≤t ∞ WN yt°1 N £1 t°1 WN yt°1 WN Xt°1 Xt°1 WN Xt°1 (43) : (42) WN Xt2 yt = ∞ 2 WN yt°2 + Xt Ø + ∞ WN Xt°1 Ø + ≤t + ∞ WN ≤t°1yt = ¡yt°1 + ∞ WN yt°1 + Xt Ø + ≤t WN Xt 2004) WN Xt°1 (Time-space Simultaneous Models) yt = ¡yt°1 + ΩWN yt + Xt Ø + ≤t 7(44)(Giacomini, Granger(45)Ωyt = (IN ° ΩWN )°1 [¡yt°1 + Xt Ø + ≤t ] yt°1 : yt = (IN ° ΩWN )°1 [¡[(IN ° ΩWN )°1 (¡yt°2 + Xt°1 Ø + ≤t°1 )] + Xt Ø + ≤t ](46)(47)yt= ¡2 (IN ° ΩWN )°2 yt°2+(IN ° ΩWN )°1 Xt Ø + ¡(IN ° ΩWN )°2 Xt°1 Ø +(IN ° ΩWN )°1 ≤t + ¡(IN ° ΩWN )°2 ≤t°1X (Time-space Dynamic Models) yt = ¡yt°1 + ΩWN yt + ∞ WN yt°1 + Xt Ø + ≤t §3.4 Cliff 1975 Cliff Ord 1973 Marten Oeppen 1975 (48)zi (t) = p ¡kl q µklk=1 l=0p X ∏k X¡kl L(l) zi (t ° k ) ° ∏k k ≤i (t)k=1 l=0q X mk Xµkl L(l) ≤i (t ° k ) + ≤i (t) mk k(49)E [≤i (t)] = 0 ( æ 2 i = j, s = 0 E [≤i (t)≤j (t + s)] = 0 STARMA(p∏1 ,∏2 ,...∏p , qm1 ,m2 ,...mq )p X ∏k X q X mk X(Pfeifer and Deutsch 1980)z (t) = ≤(t)k=1 l=0¡kl W (l) z (t ° k ) °k=1 l=0µkl W (l) ≤(t ° k ) + ≤(t)(50)E [≤(t)] = 0 ( æ 2 IN 0 E [≤(t)≤(t + s) ] = 0 STARMA xu (Stationary)s=0 z (t) (51)∑ ∏ p X ∏k X p l p°k det xu I ° ¡kl W xu =0k=1 l=08|xu | < 1xu ∑ ∏ q X mk X q l q °k det xu I ° µkl W xu =0k=1 l=0(52) z (t)|xu | < 1 STARMA(0, qm1 ,m2 ,...mq )(Invertible)STARMA(p∏1 ,∏2 ,...∏p , 0)§4§4.1: y = ΩW y + ≤ E [≤≤ ] = æ 2 IN y = (I ° ΩW )°1 ≤ E [W y ≤ ] = E [W (I ° ΩW )°1 ≤≤ ] = æ 2 W (I ° ΩW )°1 6= 00 0 0(53)(54)(55)§4.2 (Ord 1975, Mardia and Marshall 1984, Anselin 1988, Cressie 1993, Anselin and Bera 1998)(4) L = ln |IT ≠ (IN ° ΩWN )| ° ≤ = y ° Ω(IT ≠ WN )y ° X Ø |IT ≠ (IN ° ΩWN )| L = T ln |IN ° ΩWN | ° NT 1 0 2 ln æ≤ ° 2≤ ≤ 2 2æ≤ Jacobi NT 1 0 2 ln æ≤ ° 2≤ ≤ 2 2æ ≤ (56) Jacobi(57)≤ = (∂T ≠ IN )µ + u0 0 1 1 0 2 2 2 2 ln |æµ (∂T ∂T ≠ IN ) + æu IN T | ° ≤ [æµ (∂T ∂T ≠ IN ) + æu IN T ]°1 ≤ 2 2(58)L = T ln |IN ° ΩWN | °(59)9(27) L= Xtln |IN ° Ωt WN | °N 1 0 1 ln |ΣT | ° ≤ (Σ° T ≠ IN )≤ 2 2(60) Anselin(1988)≤ = [IN T ° (RT ≠ WN )]y ° X Ø(Magnus 1978) ≤ ª N (0, Σ) L= (9)-(11) L=° ≤ = y ° XØ0 NT 1 0 2 ln æu + T ln |BN | ° 2 ≤ [IT ≠ (BN BN )]≤ 2 2æu1 1 0 ln |Σ| ° ≤ Σ°1 ≤ 2 2(61)(62) : (63)2 2 ¥ = æµ /æuØ ˆ = [X 0 (IT ≠ B 0 BN )X ]°1 X 0 (IT ≠ B 0 BN )y Ø N N (35)-(38)2 ΣN T = æ u ΨN TΨN T = ∂T ∂T ≠ ¥ IN + [IT ≠ (BN BN )°1 ] ΨN T |ΨN T | = |(BN BN )°1 + (T ¥ )IN ||BN |°2(T °1)1 Ψ° NT0 0 ∂T ∂T ∂T ∂T = ≠ [(BN BN )°1 + (T ¥ )IN ]°1 + (IT ° ) ≠ (BN BN ) T T 0 0 000(64)(65)(66)L = °0 NT 1 2 ln æu + (T ° 1) ln |BN | ° ln |(BN BN )°1 + (T ¥ )IN | 2 2 ∑ 0 ∏ ∑ ∏ 0 0 0 1 0 ∂T ∂T 1 0 ∂T ∂T ° 2≤ ≠ [(BN BN )°1 + (T ¥ )IN ]°1 ≤ ° 2 ≤ (IT ° ) ≠ (BN BN ) ≤ 2æu T 2æ u T≤ = y ° XØ §4.3(Anselin 1988, Anselin 1990, Kelejian and Prucha 1998, Kelejian and Prucha 1999, Conley 1999)XWX 10KelejianRobinson(1993) Kelejian Prucha(1998) Lee(2003) (4) (I T≠W N)X (I≠W N y) X (32) Z t=[W N y t,X t] ∞t=[Ωt,Ø0t]0Z=0BB B B@Z10 00Z2 0············00···Z T1CC C C AH t=[X t,W N X t]:H=0BB B B@H10 00H2 0············00···H T1CC C C Aˆ∞=∑Z0H[H0(ˆΣT≠I N)H]°1H0Z∏°1Z0H[H0(ˆΣT≠I N)H]°1H0y(67) ˆΣT≠I NVar[ˆ∞]=∑Z0H[H0(ˆΣT≠I N)H]°1H0Z∏°1(68) (3SLS) : (2SLS) ˆΣ (67) (Kelejian and Robinson1993,Kelejian and Prucha1998)(63) (9) µ µ (63) ØKelejian Prucha(1999)≤=µ(I T≠W N)≤+uu uªIID[0,æ2uI NT]E[1NTu0u]=æ2uE[1NTu0(I T≠W0N)(I T≠W N)u]=1Næu Tr(W0N W N) E[1NTu0(I T≠W N)u]=0u=≤°µ(I T≠W N)≤ ≤ µ,µ2,æ2u µ (63) ا5(33)-(34) H0:Ω1=Ω2=...=ΩT=0,µ1=µ2=...=µT=0(LM Test)(Anselin2001b)LM (Anselin 1988b) (Anselin1988,Baltagi 2003) Moran ILM Burridge(1980) Moran I (2)LM1=[e0W N e/(e0e/N)]2Tr(W2N+W0NW N)(69)(2) (2)LM1=[e0(I T≠W N)e/(e0e/NT)]2Tr[(I T≠W2N)+(I T≠W0N W N)](70)¬2(1) (I T≠W N) W N Anselin(1988) Ω=0 LM (4) AnselinLM2=[e0(I T≠W N)y/(e0e/NT)]2[(Wˆy)0M(Wˆy)/ˆæ2]+T Tr(W2N+W0NW N)(71)Wˆy=(I T≠W N)XØ M=I NT°X(X0X)°1X0 ¬2(1)(27)-(30) H0:µ1=...=µT Anselin(1988b) LMLM3=∂0T(ˆΣTØE0W N E)J°1(ˆΣ°1TØE0W N E)0∂T(72) Ø HadamardJ=[Tr(W2N)]I T+[Tr(W0N W N)](ˆΣ°1T؈ΣT)¬2(T)§6(1) (2) (3)Andrews,DonaldW.K.(2005).Cross-section regression with common shocks.Econometrica,73:1551 1585. 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