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Transcription:

. q {y j } I( ( L y j =Δy j = u j = C l ε j l = C(L ε j, {ε j } i.i.d.(,i q ( l= y O p ( {u j } q {C l } A l C l <, C = C( =., A i l= A q C l = [tr(c l C l] /2 C V(ε j = I q A i C(L L ( B-N Δ y j =[C +(C(L C] ε j = Cε j +Δ C(L ε j (2 C(L = l= C l L l, Cl = C k k=l+ C(L ε j C l C k = k C k <. l= l= k=l+ k= (2 j y j = C ε l + y + C(L ε j C(L ε (3 l= j {y j } 2 2 y j y j CZC, Z = j= W (t W (t dt (4

{W (t} 2 2 q Brown (4 Z 2 C y j C α C = α ( (2 α Δ α y j = α Cε j +Δα C(Lεj =Δα C(Lεj α y j = α C(Lεj I( y j α y j y j α α α ( ( β y 2j = β yj y j + v j, α =, y j =, v y j = α C(Lεj (5 2j I( C rank(c = r α C = α q r q r Engle-Granger (987 d I(d {y j } α y j I(b b<d d b {y j } CI(d, d b CI(,.2 C j Y j C j = α + βy j + u j I( u j α β α β u j I( 2 ( ( xj σ 2 z j = = ξ y j, {ξ j } i.i.d.(, Σ, Σ= x j σy 2 > (6 Σ 2 {x j } {y j } 2

{x j } {y j } r xy = j= (x j x(y j ȳ j= (x j x 2 j= (y j ȳ 2 2 I( 3 8 2 r xy W (t W 2 (t dt W 2 (t dt W, Wi (t =W i (t 2 2 (t dt W i (s ds Bm W (t W 2 (t I( (6 x j y j 2 y j = ˆβ x j +û j, y j =ˆα + ˆβ 2 x j +ˆv j (j =,, (7 ˆα ˆβ ˆβ 2 OLS (7 û j ˆv j x j y j 2 I( û j ˆv j I( 2 I( ˆβ = j= x j y j j= x 2 j R = σ y σ x W (t W 2 (t dt W 2 (t dt (8 ˆβ 2 = j= (x j x(y j ȳ j= (x j x 2 R 2 = σ y σ x W (t W 2 (t dt W 2 (t dt (9 2 R R 2 k μ (k μ 2 (k anaka (993 3 μ ( =, μ (2 =.897 σ2 y, μ σx 2 (3 =, μ (4 = 4.9539 μ 2 (2 μ 2 ( =, μ 2 (2 =.3965 σ2 y, μ σx 2 2 (3 =, μ 2 (4 = 4.838 μ 2 2 (2 - R / μ (2 R 2 / μ 2 (2 N(, R μ (4/μ 2 (2 3.9539 R 2.838 3

- (7 t DW Granger-Newbold (974 Phillips (986 (a ˆα = O p (, ˆβ = O p (, ˆβ2 = O p ( (b t O p ( (c O p ( (d 2 = O p ( 2 (e DW O p (/ (b (c t (d (e (d (e DW / Σ 2 I(.3 ( ( y j = β uj Σ Σ x j + v j, Δx j = u j, ξ j = i.i.d.(, Σ, Σ= 2 ( v j Σ 2 Σ 22 y j x j p β p ξ j Σ Σ p p, Σ 2 p, Σ 2 p, Σ 22 Σ 2 x j v k j k β 3 ( y = Xβ + v, ΔX = X X = U ( y =(y,,y, X =(x,, x, X =(, x,, x v =(v,,v, U =(u,, u 4

3 ˆβ OLS = (X X X y = β +(X X X v ˆβ 2SLS = (X P X X P y = β +(X P X X P v ˆβ ML = (X M X X M y = β +(X M X X M v P = X (X X X, M = I ΔX(ΔX ΔX ΔX ˆβ OLS LSE ˆβ 2SLS 2 LSE X X ˆX = P X y ˆX ˆβ ML β MLE 4 3 Σ =HH H ( /2 Σ H = Σ 2 Σ /2 Σ /2 22, Σ 22 =Σ 22 Σ 2 Σ Σ 2 ( ( xj uj z j = = z j + = z j + ξ j, V(ξ j =Σ=HH w j v j 5 2 j= z j z j = 2 ( xj x j x j w j w j= j x j wj 2 H W (t W (t dt H W (t =(W (t,w 2(t (p + Brown W (t W 2 (t p Brown (, 2 j= x j x j = 2 X X Σ /2 W (t W (t dt Σ/2 = R (2 2 X P X R, 2 X MX R Ito 5 z j ξ j = j= ( xj u j x j v j w j= j u j w j v j (, 2 H W (t dw (t H j= x j v j = X v = j= x j v j + u j v j Q + Q 2 +Σ 2 j= 5

Q =Σ /2 W (t dw (tσ /2 Σ 2, Q 2 =Σ /2 X P v Q + Q 2, X Mv Q 2 W (t dw 2 (tσ /2 22 (3. ( β 3 ˆβ OLS ˆβ 2SLS ˆβ ML (ˆβ OLS β Y OLS = R (Q + Q 2 +Σ 2 (ˆβ 2SLS β Y 2SLS = R (Q + Q 2 (ˆβ ML β Y ML = R Q 2 R (2 Q Q 2 (3 ˆβ OLS ˆβ 2SLS R Q ˆβ ML Phillips 99 3 Σ 2 = x j v j p = P ( ( ˆβ OLS β x P (Y OLS x =P (S S = a 2 x W 2 (t dt ab a = σ, b = σ 2 σ, c = W (t dw (t ac σ 22 σ 2 2/σ, d = σ 2 W (t dw 2 (t d W = {W (t} S E(S W =a 2 x W 2 (t dt ab 2 W 2 ( + ab 2 d V(S W =a 2 c 2 W 2 (t dt S 6. φ (θ = E [ exp{iθe(s W θ 2 V(S W /2} ] { } [ { ( iθ = exp (ab 2d E exp a 2 iθ x + c2 iθ 2 2 { } [ iθ = exp (ab 2d cos ν + abiθ sin ν 2 ν 6 W 2 abiθ (t dt 2 W 2 ( ] /2, ν = a 2 iθ(2x + c 2 iθ }]

( ˆβ OLS β lim P ( ( ˆβ OLS β x = 2 + π θ Im (φ (θ dθ (4 ( ˆβ 2SLS β ( ˆβ ML β φ (θ d =, b = d = 3 2 anaka (993 E(Y OLS = σ 2 (a +2, E(YOLS 2 2σ =σ2 + σ2 2 (a 8σ 2 + a 2 +4 E(Y 2SLS = σ 2 ( a +2, E(Y 2σ 2SLS 2 =σ 2 + σ2 2 ( 6a 8σ 2 + a 2 +4 E(Y ML =, E(Y 2 ML =σ2 a = u du =5.5629, a 2 = cosh u σ 2 = a (σ 22 σ 2 2 /σ σ u 3 cosh u du = 35.6625, (5 σ 2 Y OLS Y 2SLS σ 2 Y OLS Y 2SLS Y ML σ 2 = 3 a σ 22 /σ ( ( y j = α + β uj Σ Σ x j + v j, Δx j = u j, ξ j = i.i.d.(, Σ, Σ= 2 (6 v j Σ 2 Σ 22 x j u j p y j v j Σ u j v j 3 β OLS = (X MX X My = β +(X MX X Mv β 2SLS = ( X M X X My = β +( X M X X Mv β ML = (X M 2 X X M 2 y = β +(X M 2 X X M 2 v M = I ee /, e =(,, : ( e X =(e,x e e X ( e X e X X X X 7

( e M 2 = I (e, ΔX e e ΔX ( e ΔX e ΔX ΔX ΔX β OLS (6 LSE β 2SLS 2 LSE X X X y X β ML β MLE 7 3.2 (6 β 3 β OLS β 2SLS β ML ( β OLS β Z OLS = R ( Q + Q 2 +Σ 2 ( β 2SLS β Z 2SLS = R ( Q + Q 2 ( β ML β Z ML = R Q2 Q =Σ /2 W (t dw (tσ /2 Σ 2, Q2 =Σ /2 W (t dw 2 (tσ /2 22 R =Σ /2 W (t W (t dt Σ/2, W (t =W (t W (s ds p = P ( ( β OLS β x P (Z OLS x =P (S 2 S 2 = a 2 x W 2 (t dt ab W (t dw (t ac S 2 φ 2 (θ =exp W (t dw 2 (t d { }[ iθ 2a 2 b 2 θ 2 ( (ab 2d ( cos ν+ a2 b 2 θ 2 sin ν 2 ν 4 ν 2 ν ] /2 6 ν = a 2 iθ(2x + c 2 iθ Z OLS (4 Z 2SLS Z ML φ 2 (θ d = b = d = 3 2 E(Z OLS = σ 2 b, E(ZOLS 2 2σ =σ2 2 + σ2 2 (4b 8σ 2 8b 2 + b 3 E(Z 2SLS = E(Z OLS, E(Z 2 2SLS =E(Z 2 OLS 8

E(Z ML =, E(Z 2 ML =σ2 2 b = b 3 = u 3/2 du =.7583, b 2 = sinh u u(cosh u du =2.645 (7 sinh 3/2 u u 7/2 sinh u du = 372.3572, σ 2 2 = b (σ 22 σ 2 2 /σ σ (8 β 2SLS β OLS β ML -2 Σ =I 2 ( ˆβ β/σ ( β β/σ 2 N(, 3.576 95% 5.349.826 95% 3.772-2.4 ( y j = β x j + g (L ε j, Δx j =Φ (L ε j, {ε j } i.i.d.(,i q (9 x j (q g(l ε j q Φ(L q (q Φ( q g (Lε j Φ (Lε j β q ( Δx Δz j = j y j β = x j {z j } ( Φ (L g ε (L j =Ψ(L ε j (2 ( Φ Ω=Ψ(Ψ ( = (Φ( Φ ( (g( Ω Ω g (Φ( g = 2 (g( Ω 2 Ω 22 Ω Ω ( /2 Ω Ω=HH, H = Ω 2 Ω /2 Ω /2, Ω 22 =Ω 22 Ω 2 Ω Ω 2 22 2 z j z j H j= W (t W (t dt H 9

{W (t} q Brown Ito 8 z j Δz j = j= z j (Ψ(Lε j H j= W (t dw (t dt H + Λ (2 Λ= k= E ( Ψ(Lε j (Ψ(Lε j+k ( Λ Λ = 2 Λ 2 Λ 22 β OLS ˆβ OLS = x j x j j= j= x j y j = β + x j x j j= j= x j g (Lε j.3 (9 β OLS ˆβ OLS (ˆβ OLS β Y OLS = R (Q + Q 2 +Λ 2 Q =Ω /2 W (t dw (tω /2 Ω 2, Q 2 =Ω /2 W (t dw 2 (tω /2 22 R =Ω /2 W (t W (t dt Ω /2, Λ 2 = k= E(Φ (Lε j g (Lε j+k. q =2 2 E(Y OLS = (2λ 2 ω 2 a +2ω 2 2ω (22 E ( YOLS 2 ω 2 = 2 + 4ω 22 3ω2 2 /ω +2ω 2 λ 2 /ω 2 a 2ω 2 + (2λ 2 ω 2 2 a 4ω 8ω 2 2 (23 a a 2 (5 (9 2SLS ML Phillps-Hansen (99 OLS R Q R Λ 2 (2 (, j= x j Δx jω Ω 2 Q +Λ Ω Ω 2

ˆβ FM = x j x j j= j= ( x j yj Δx j ˆΩ ˆΩ 2 (ˆΛ2 ˆΛ ˆΩ ˆΩ 2 ˆΛ, ˆΛ 2 ˆΩ ˆΩ 2 Park-Phillips (988 (2 (2 Λ = E(Δx j Δx j+k, Λ 2 = E(Δx j (y j+k β Δx j+k k= k= ˆΛ = ( l k k Δx j Δx j+k l + ˆΛ 2 = k= ( l k= k l + j= k j= Δx j (y j+k ˆβ OLS Δx j+k l (2 {Δz j } f(ω = 2π Ψ(eiω Ψ (e iω ( 2πf( = Ψ(Ψ Ω Ω ( = Ω = 2 Ω 2 Ω 22 f( ˆf( = [ ˆΓ + 2π ( l k ] l + (ˆΓ k + ˆΓ k k= Ω Ω 2 ˆΓ k = ( k Δẑ j Δẑ j+k, Δẑ Δx j j = y j ˆβ OLS x j j= (ˆβFM β Y ML = R Q 2 FM FM fully modified q =2 Y ML 2 (22 (23 ω 2 = λ 2 =,ω 22 = ω 22 E(Y ML =, E(Y 2 ML =ω 22 ω a

y j = α + β x j + g (L ε j, Δx j =Φ (L ε j, {ε j } i.i.d.(,i q (24 β 2 β OLS = β FM = (x j x(x j x (x j x(y j ȳ j= j= (x j x(x j x j= j= (x j x ( y j Δx j Ω Ω 2 ( Λ2 Λ Ω Ω ] 2 Λ, Λ 2 Ω Ω 2.4 (24 β OLS β OLS FM β FM ( β OLS β Z OLS = R ( Q + Q 2 +Λ 2 ( βfm β Z ML = R Q2 Q =Ω /2 W (t dw (tω /2 Ω 2, Q2 =Ω /2 W (t dw 2 (tω /2 22 R =Ω /2 W (t W (t dt Ω /2, W (t =W (t W (s ds q =2 2 E(Z OLS = 2λ 2 ω 2 b, E(Z ML = 2ω E ( ZOLS 2 ω 2 = 2 (b 2ω 2 2b 2 + ω 22 b + (2λ 2 ω 2 2 ω 8ω 2 E ( ZML 2 ω 22 = b ω b,b 2,b 3 (7, (8 Phillips-Hansen (99 OLS FM ( uj y j = βx j + u 2j, Δx j = u j, u j = = ξ j +Θξ j =Ψ ε j +Ψ ε j (25 u 2j b 3 2

{ξ j } i.i.d.(, Σ, {ε j } i.i.d.(,i 2 Θ= (.6 θ, Σ=.4.3 ( Ψ = δ δ 2 ( δ δ ( δθ +.6 θ δ 2, Ψ =.3δ.4.3 δ 2 ( ( Δxj φ ( (L +(δθ +.6L θ δ2 L Δz j = = y j βx j g ε (L j = δ +(.3δ.4L δ 2 ε ( +.3L j ( φ (φ( φ ( (g( ω ω Ω= g (φ( g = 2 (g( ω 2 ω 22 ω = (δθ +.6 2 + θ 2 ( δ 2 ω 2 = (δθ +.6(.3δ.4 +.3θ( δ 2 ω 22 = (.3δ.4 2 +.69( δ 2 ( ˆβ OLS β ( ˆβ FM β -3 δ =.8, θ = OLS FM OLS (22 (23 2.36 2.4 FM,.5 OLS FM -3.5 Engle-Granger (987 Phillips-Ouliaris (99 Engle-Granger (987 q I( I( y j q I( x j ˆη j = y j ˆβ x j 2 Δˆη j ˆη j Δˆη j Δˆη j = ˆδˆη j + p ˆφ k ˆη j k +ˆv j (26 k= H : δ = H : δ< ˆδ t 3

t t 9 (26 p = ˆη j 2 I( x j y j ˆδ = ˆη j Δˆη j / ˆη j 2, ˆη j = y j ˆβx j, ˆβ = x j y j / x 2 j j= j= ˆδ t ˆδ ˆσ2 / ˆη 2 j Q(t dq(t κ Q2 (t dt ˆσ 2 = (Δˆη j ˆδˆη j 2 / Q(t =W 2 (t W (tw 2 (t dt W 2 (t dt W (t, κ =+ ( W (tw 2 (t dt 2 ( W 2 (t dt 2 (W (t,w 2 (t 2 Brown 8 4 Δy j = δy j + ε j, {ε j } i.i.d.(,σ 2 t δ σ2 / y 2 j W (t dw (t W 2 (t dt δ = y j Δy j / y 2 j σ2 = (Δy j δy j 2 / I( 8 Engle-Granger (987 Davidson-MacKinnon (993 Zivot-Wang (23 Phillips-Ouliaris (99 ε j y j = β x j + α C L + g (Lε j, Δx j =Φ (Lε j, {ε j } i.i.d.(,i q (27 β x j (q α ε j q q I( Δz j = C(Lε j = Cε j +Δ C(Lε j, C = C( (28 z j =(x j,y j (27 H :rank(c =q vs. H :rank(c =q H H Phillips-Ouliaris (99 4

i y j x j OLS ˆη j = y j ˆβ x j ii ˆη j Ẑ ρ = (ˆρ ˆσ 2 L ˆσ2 S 2 ˆη 2 j / 2 (29 ˆρ = ˆη j ˆη j / ˆη 2 j, ˆσ2 S = (ˆη j ˆρˆη j 2 ˆσ 2 L = σ2 S + 2 ( l k= k l + k (ˆη j ˆρˆη j (ˆη j+k ˆρˆη j+k l iii Ẑρ Phillips-Ouliaris (99 H Engle-Granger (987 H W (t = ( ( W (t A, C = W 2 (t A 2 ( ( K (A K = K 2 = A /2 K 3 A 2A (A A /2 (A 2A 2 A 2A (A A A A 2 /2 W (t q Bm W (t : (q W 2 (t : C K q q CC = KK (ˆρ = / ˆη j (ˆη j ˆη j ˆη 2 2 j = ˆα z j Δz j ˆα / ˆα 2 z j z j ˆα ˆα =( ˆβ, ˆα, z j Δz j, z 2 j z j (( X,,X 2,X 3 X = (K W (tw (t dt K (K 5 W (t(k 2 W (t+k 3 W 2 (t dt

X 2 = K W (t dw (t K + E ( C(Lε t (C(Lε t+j j= X 3 = K W (tw (t dt K ( ( X, X X 2 (ˆρ ( = ( X, X X 3 Q(t dq(t + R (3 Q2 (t dt Q(t =W 2 (t ( W 2 (tw (t dt W (tw (t dt W (t ( X, j= E ( C(Lε t (C(Lε t+j ( X R = ( ( X, X X 3 (3 R = (28 C(L =C {Q(t} {W 2 (t} {W (t} Bm (3 R (29 Phillips (988 R ( ˆσ L 2 ˆσ S 2 lk= k k l+ j= Δz j Δz j+k ˆα 2 2 ˆη 2 j = ˆα 2 ˆη 2 j + o p ( l H Ẑ ρ Q(t dq(t Q2 (t dt (3 H H (27 y j = β x j + g (Lε j (32 Ẑρ = O p ( ˆσ Ẑρ L 2 =ˆρ ˆσ2 S 2 ˆη j / 2 ˆη j = y j ˆβ x j = g (Lε j (ˆβ β x j = O p ( + O p ( /2 6

ˆη 2 j γ(, ˆη j ˆη j γ(, γ(h =E(g (Lε j g (Lε j+h ˆρ γ(/γ( = ρ ˆη j ˆρˆη j = g (Lε j ˆρg (Lε j (ˆβ β (x j ˆρx j ˆσ 2 S = (ˆη j ˆρˆη j 2 E [ (g (Lε j ρg (Lε j 2] = γ2 ( γ 2 ( γ( {g (Lε j ρg (Lε j } f(ω ˆσ 2 L 2πf( = (g ( ρg ((g( ρg( = (γ( γ(2 g (g( γ 2 ( Ẑρ γ( γ( [ (γ( γ( 2 g (g( γ2 ( γ 2 ] ( 2γ( γ 2 ( γ( (γ( γ(2 = 2γ 2 ( ( + g (g( γ( Ẑρ Phillips-Ouliaris (99 t Ẑ t = /2 2ˆσ ˆη 2 L 2 j Ẑ ρ (33 H ( ˆη 2 j, ˆσ 2 2 L K3 2 Q 2 (t dt, K3 2 S S ( S = ( W (tw 2(t dt W (tw (t dt, ˆσ L 2 ( l ˆα Δz j Δz j +2 k k Δz j Δz j+k ˆα k= l + ( ( ( X, KK X =( X K + K 2,K K X 3 + K 2 K 3 = K3 2 S S 7

H Ẑ t Q(t dq(t S S Q2 (t dt (34 H Ẑ t = /2 ˆη 2 ˆσ L 2 j Ẑρ γ 3 ( (γ( γ( 2 g (g( ( γ( γ( = 2 γ(g (g( ( ( (γ( γ(2 2γ 2 ( + g (g( γ( + g (g( γ( H t ˆη j Bm Bm Bm (. OPIX 8 5 OPIX 8-6 -4-6 OPIX 3-4 -5-6 (26 t (33 Ẑt - (A (B (C 3 Ẑt l l =4 l =8 2 Zivot-Wang (23 5% OPIX -4-5 -6 - (.2 8 5 8-8 -7-8 -9 3 (. Ẑt l =5 l = -2 5% % -7-8 -9-2 8

.6 q y j VAR(p y j = B y j + + B p y j p + ε j, {ε j } i.i.d.(, Σ (35 B(L B(L y j = ε j, B(L =I q B L B p L p (36 y j I( B(x = = q rank(b( = q r B( = B( B( r q B( r B( B(L L B(L =( L B(L B( B( r (35 VAR(p Δy j = C k ε j k = C(L ε j (37 k= (36 (37 B(LΔy j =Δε j = B(L C(L ε j B(L C(L =ΔI q =( L I q B( C( = (38 β C( = β ( (38 B( rank(b( Johansen (995 r VAR(p (36 B(L B(L =B( L + B(L B( L = B( L +ΔΓ(L Γ(L =I q Γ L Γ p L p, p Γ j = B i i=j+ VAR(p Δy j = γα y j +Γ Δ y j + +Γ p Δ y j p+ + ε j (39 9

γα = B( γ α q r r r Q γα = γqq α = γ α = B( γ α (39 ECM: Error Correction Model {α y t } I( r r ARIMA α α γ (.3 {y t } 2 VAR( I( ECM y j = B y j + ε j Δy j = (I 2 B y j + ε j = γα y j + ε j y j 2 I( (I 2 B α y j I( α =(α,α 2 ECM ( ( Δyj Δy j = = γα γ (α y Δy j + ε j = y,j + α 2 y 2,j +ε j 2j γ 2 (α y,j + α 2 y 2,j +ε 2j y j α y,j + α 2 y 2,j γ 2 α α =( α, y 2,j αy,j (39 θ ε j θ MLE Δy,, Δy L(θ = q 2 log(2π 2 log Σ (z j γα z j Γz 2j Σ (z j γα z j Γz 2j (4 2 j= z j =Δy j, z j = y j, z 2j =(Δy j,, Δy j p+, Γ=(Γ,, Γ p L(θ S ab = û aj û bj, z aj = ˆΦ a z 2j + û aj, (a, b =, j= û aj z aj z 2j 2

α = γ = L(θ L = q 2 log(2π 2 log S q 2 (4 r α L(θ ˆΣ(r = S S α(α S α α S = S α (S S S S α α S α Johansen (995, Hatanaka (996 ( S S α α S α = S S α α S α α S S S α = α S α S S α(α S α α S (42 α S α = I r S r j= ( ˆλj ˆλ ˆλ r S S S ˆV = S ˆV ˆΛ, ˆΛ =diag (ˆλ,, ˆλ r ˆV =(ˆv,, ˆv r α (42 ˆα = ˆV =(ˆv,, ˆv r, ˆα S ˆα = I r rank(α =r (4 L r = q 2 log(2π 2 log S 2 r j= log ( ˆλ j q 2 (43 (39 γα Anderson (95.7 VAR(p Δy j = γα y j +Γ Δ y j + +Γ p Δ y j p+ + ε j (44 y j q I( 2

H :rank(α =r<q vs. H :rank(α =q γα q y j r q = ε j L(θ (43 J = 2(L r L q = q i=r+ log ( ˆλ i H J Johansen (988 ( ( J tr dw (t W (t W (t W (t dt W (t dw (t (45 W (t q r Bm Johansen (995 able 5. Johansen (995 H :rank(α =r<q vs. H :rank(α =r + ( ˆλ r+ H (45 Osterwald-Lenum (992.8 (35 VAR(p y j = μ + μ j + B y j + + B p y j p + ε j, {ε j } i.i.d.(, Σ (46 Δy j = C(L (μ + μ j + ε j =C( μ + C(L(μ j + ε j = μ + μ j + γα y j +Γ Δ y j + +Γ p Δ y j p+ + ε j (47 (35 (46 μ μ 22

α γ α α =,γ γ = q r q (q r α γ Johansen (99 (47 C( C( = α ( γ B( α γ (48 B( B-N B(L =B( + ( L B(L (47 t j = μ + μ t 5 (a t j = Δy j = γα y j +Γ Δ y j + +Γ p Δ y j p+ + ε j (b t j = μ, γ μ = μ C(μ = (47 μ γ μ = γ μ α y j μ Δy j = γ(α y j + μ +Γ Δ y j + +Γ p Δ y j p+ + ε j (c t j = μ, γ μ μ Δy j = μ + γα y j +Γ Δ y j + +Γ p Δ y j p+ + ε j (d t j = μ + μ j, γ μ = μ 2 μ = γ μ α y j Δy j = μ + γ(α y j + μ j+γ Δ y j + +Γ p Δ y j p+ + ε j (e t j = μ + μ j, γ μ 2 (47 5 (a - (e (b - (e (q r (q r ( R = dw (t F (t F (t F (t dt F (t dw (t (49 23

W (t q r Bm {F (t} q r q r + tr(r R (b - (e F (t (b : F (t q r + F j (t =W j (t (j =,,q r, F q r+ (t = Johansen (995 able 5.2 (c : F (t q r F j (t =W j (t W j (s ds (j =,,q r, F q r (t =t 2 Johansen (995 able 5.3 (d : F (t q r + F j (t =W j (t W j (s ds (j =,,q r, F q r+ (t =t 2 Johansen (995 able 5.4 (e : F (t q r F j (t =W j (t a j b j t (j =,,q r, F q r (t =t 2 a bt a j b j W j (t dt = a j + b j tdt tw j (t dt = a j tdt+ b j t 2 dt a b t 2 dt = a + b t 3 dt = a tdt tdt+ b t 2 dt a = /6, b = Johansen (995 able 5.5 24

(.4 2-3 2 8 5 5 (.2 9 3 y j =(y j,y 2j,y 3j - 3 VAR AIC VAR (4 (b (c (b : -3 H(r H : rank(α =r Zivot-Wang (23 3 α γ ˆα =(, 3.27,.84, ˆγ =(.5,.75,.7-3 (c : -4 (b (b (b 3 α γ ˆα =(, 3.25,.82, ˆγ =(.53,.73,.5 (b -4 Johansen-Schaumburg (999 Jeganathan (999 I(d Hubrich et al. (2 25

. q y j y j = y j + C(Lε j, y =, {ε j } i.i.d.(,i q 2 y j y j C( j= W (t W (t dt C ( 2. ( ( xj σ 2 z j = = ξ y j, {ξ j } i.i.d.(, Σ, Σ= x j σy 2 > x y r xy r xy W (t W 2 (t dt W 2 (t dt W, Wi (t =W i (t 2 2 (t dt W i (s ds 3. R R 2 W (t W 2 (t Bm R = W (t W 2 (t dt W, R 2 2 = (t dt W (t W 2 (t dt, Wi (t =W W 2 i (t (t dt W i (s ds 4. β MLE y j = β x j + v j, Δx j = u j, ξ j = 5. z j = ( xj w j = z j + ( uj v j ( uj v j ( Σ Σ NID(, Σ, Σ= 2 Σ 2 Σ 22 = z j + ξ j, {ξ j } i.i.d.(,hh 2 2 j= z j z j = 2 ( xj x j x j w j w j= j x j wj 2 H W (t W (t dt H z j ξ j = j= ( xj u j x j v j w j= j u j w j v j H W (t dw (t H 26

6. 2 S S 2 W (t W 2 (t Bm W i (t Bm S = a 2 x W 2 (t dt ab W (t dw (t ac W (t dw 2 (t d S 2 = a 2 x W 2 (t dt ab W (t dw (t ac W (t dw 2 (t d 7. β MLE y j = α + β x j + v j, Δx j = u j, ξ j = 8. q z j ( Δx Δz j = j y j β = x j Ω=Ψ(Ψ ( = HH, Λ= ( uj v j ( Σ Σ NID(, Σ, Σ= 2 Σ 2 Σ 22 ( Φ (L g ε (L j =Ψ(L ε j, {ε j } i.i.d.(,i q k= E ( Ψ(Lε j (Ψ(Lε j+k z j Δz j = j= z j (Ψ(Lε j H j= W (t dw (t dt H +Λ 9. 2 2 x j y j ˆβ = x j y j / x 2 j, j= j= ˆη j = y j ˆβx j ˆδ = ˆη j Δˆη j / ˆη 2 j, ˆσ 2 = ( Δˆηj ˆδˆη 2 j tˆδ = ˆδ ˆσ2 / ˆη 2 j 27

- OPIX t Ẑt (l =4 Ẑ t (l = 8 % 5% % (A.92.77.83 2.47 2.79 3.4 (B 2.92 3.23 3.24 3.8 3.39 3.99 (C 3.3 3.2 3.2 3.56 3.86 4.47-2 t Ẑt (l =5 Ẑ t (l = % 5% % (A 3. 4.3 4.4 2.47 2.77 3.37 (B 3.5 5.49 5.66 3.6 3.36 3.94 (C 3.49 6.3 6.33 3.53 3.82 4.39 28

-3 : (b 5% % 5% % H( 53.26 34.9 4.7 38.25 22. 26.8 H( 5. 9.96 24.6 2.79 5.67 2.2 H(2 2.22 9.24 2.97 2.22 9.24 2.97-4 : (c 5% % 5% % H( 52.5 29.68 35.65 38.6 2.97 25.52 H( 3.99 5.4 2.4 2.78 4.7 8.63 H(2.2 3.76 6.65.2 3.76 6.65 29

-.5.4.3.2.. 3 2 2 3 3

-2.7.6.5.4.3.2.. 3 2 2 3 3

-3 OLS FM δ =.8, θ=.7.6.5.4.3.2.. 3 2 2 3 32

-4 OPIX (A.4.3.2....2.3.4.5.6 2 4 6 8 2 33

-5 OPIX (B.5..5..5..5.2.25.3.35 2 4 6 8 2 34

-6 OPIX (C.2....2.3.4 2 4 6 8 2 35

-7 (A 5 4 3 2 2 5 5 2 25 36

-8 (B 5 4 3 2 2 5 5 2 25 37

-9 (C 5 4 3 2 2 5 5 2 25 38

- 2 4 2 8 6 4 2 5 5 2 25 39

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