chap9.dvi

9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j = α 1 + β 1 j + η j (j =4k +1 ) = α 2 + β 2 j + η j (j =4k +2 ) = α 3 + β 3 j + η j (j =4k +3 ) = α 4 + β 4 j + η j (j =4k ) η j = ρ 4 η j 4 + ε j α i β i (1) H : ρ S =1 vs. H 1 : ρ S < 1 S =1 4 T S N = T/S 1 (i) LBI R S1 = S Tj=T S+1 η 2 j Tj=1 ( η j η j S ) 2 (ii) LBIU R S2 = S2 T Tj=1 η 2 j Tj=1 ( η j η j S ) 2 1

(iii) DF R S3 = N(ˆρ S 1), ˆρ S = T / ˆη j S ˆη j T ˆη j S 2 j=s+1 j=s+1 (iv) DW R S4 = S2 T Tj=1 ˆη 2 j Tj=S+1 (ˆη j ˆη j S ) 2 ˆη (1) OLS η GLS ρ S =1 (c/n) =1 (cs/t ) (1) α i β i 3 Tanaka (1996) 9.1 (1) α i = β i =4 ρ S =1 (cs/t ) R S1 1 e 2c χ 2 (S), R S2 2c S i=1 Z 2 i (t) dt R S3 Si=1 Z i(t) dz i (t) Si=1, R S4 Z2 i (t) dt S i=1 Z 2 i (t) dt Z i (t) (i =1,,S) O-U t Z i (t) =e ct e cs dw i (s) W i (t) (i =1,,S) Bm 9.2 (1) β i =4 ρ S = 1 (cs/t ) R S1 1 e 2c χ 2 (S), R S2 2c S i=1 Z 2 i (t) dt R S3 Si=1 Z i (t) dz i (t) Si=1, R Z i 2 S4 (t) dt S i=1 Z 2 i (t) dt Z i (t) (i =1,,S) O-U Z i (t) =Z i (t) Z i (s) ds 9.3 (1) α i β i R S1 3 ρ S =1 (cs/t ) R S2 S i=1 (Z i (t) tz i (1)) 2 dt, R S3 Si=1 Si=1 Z i dz i (t) Z i 2 (t) dt 2

R S4 S i=1 Z 2 i (t) dt Z i (t) (i =1,,S) O-U Z i (t) =Z i (t) (4 6s)Z i (s) ds t (12s 6)Z i (s) ds S =1 S 2 9-1 5% LBI S =1 β i = DF LBIU S =1 DF DW S =1 9-1 POI S =1 3 H : ρ S =1 vs. H 1 : ρ S = ρ S (θ) =1 θ N =1 θs T θ R T (θ) =T Tt=1 () ( η t η () ) 2 t S Tt=1 (1) ( η t ρ S (θ) η (1) ) 2 t S Tt=1 () ( η t η () ) 2 (2) t S H Tanaka (1996) η () t η (1) t H H 1 GLS 9.4 (1) ρ S =1 (c/n) =1 (cs/t ) R T (θ) R(c, θ) R(c, θ) α i = β i = β i = S ] R(c, θ) = [θ 2 Zi 2 (t) dt +2θ Z i (t) dz i (t) i=1 α i β i S [ R(c, θ) = θ 2 i=1 Z 2 i + θ +1 θ2 3δ Z2 i (1) δ 2(θ +1) 1 (t) dt Z i (1) tz i (t) dt δ ( ) 2 ] tz i (t) dt + θs 3

δ =(θ 2 +3θ +3)/3 α P (R(c, c) x c,α ) x c,α R(,c) 1α% POI P (R(c, θ) x θ,α ) x θ,α R(,θ) 1α% 9-2 5% θ POI α i β i POI S =4 θ =2 S =12 θ =1 5% α i β i S =4 θ =6 S =12 θ =4 5% 9-1 β i = S =4 LBI DF POI POI θ =2 c =2 LBI S =1 8 1 DF 9-2 9-1 2 AR Dickey-Hasza-Fuller (1984) (26) 9.1.2 (1 re iθ L)(1 re iθ L) y j = u j y j =2r cos θy j 1 r 2 y j 2 + u j (3) r <r 1 θ <θ<π u j u j = α k ε j k, k= k=1 k α k <, {ε j } i.i.d.(,σ 2 ) y 1 = y = (3) 2π/θ e iθ /r, e iθ /r r =1 e ±iθ θ H : r =1 vs. H 1 : r<1 (4) (3) y j = φ 1 y j 1 + φ 2 y j 2 + u j, y 1 = y = (j =1,,T) (5) φ 1 =2r cos θ, φ 2 = r 2 4

r =1 φ =(φ 1,φ 2 ) LSE ˆφ =(ˆφ 1, ˆφ 2 ) φ φ φ =(2cosθ, 1) u j i.i.d. Chan-Wei (1988) 9.5 (5) u j i.i.d.(,σ 2 ) φ LSE ˆφ T (ˆφ φ ) (Z 1,Z 2 ) (6) Z 1 = 2 W ( ) (t)p (θ) dw (t) cos θ sin θ W, P(θ) = (t)w (t) dt sin θ cos θ Z 2 = 2 W (t) dw (t) W (t)w (t) dt W (t) =(W 1 (t),w 2 (t)) 2 Bm f(θ) u j θ Z 2 S T = T ( ˆφ 2 +1) θ S T φ 2 = r 2 ˆφ 2 S T S T 9-3 Z 2 4 9-3 (5) u j Tanaka (28 9.6 (5) u j γ(j) f(ω) φ LSE ˆφ T (ˆφ φ ) ( Z1, Z2 ) (7) Z 1 = 2 [ πf(θ) W (t)p (θ) dw (t)+sinθ j=1 γ(j)cos(j 1)θ ] πf(θ) W (t)w (t) dt Z 2 = 2 [ W (t) dw (t)+1 γ()/(2πf()) ] W (t)w (t) dt S T = T ( ˆφ 2 +1) Z 2 u j λ = γ()/(2πf()) λ 9.5 Z 2 5

9.2 MA AR MA AR MA MA(1) y j = ε j αε j 1, {ε j } i.i.d.(,σ 2 ) (8) y j H : α =1 vs. H 1 : α<1 y j y j =(1 L)x j H (1 L)x j = (1 L)ε j 1 L AR H 1 AR MA AR MA AR (8) ε j y =(y 1,,y T ) α σ 2 L(α, σ 2 )= T 2 log(2πσ2 ) 1 2 Ω(α) = 1+α 2 α α 1+α 2 α α 1+α 2 log Ω(α) 1 2σ 2 y Ω 1 (α) y (9) α MLE ˆα ˆα MA(1) ˆα AR T (ˆα 1) Tanaka-Satchell (1989), Davis-Dunsmuir (1996) α l(α) l(α) =L(α, ˆσ 2 (1)) = T 2 log(2π y Ω 1 (α) y/t ) 1 2 log Ω(α) T 2 (1) ˆσ 2 (α) =y Ω 1 (α) y/t dl(1) dα = T 2 y Ω 1 ΩΩ 1 y y Ω 1 y 1 2 tr(ω 1 Ω) =, Ω=Ω(1) 6

α 1 α =1 ( d 2 ) lim P (ˆα = 1) = lim P l(α) T T dα 2 < ( α=1 Zn 2 = P n 2 π < 1 ) =.6574, {Z 2 n } NID(, 1) 6 n=1 Tanaka (1996) T (ˆα 1).6574 α 1 H 2 S T 1 = 1 T y Ω 2 y y Ω 1 y H AR LBIU 5 S T 1 α =1 (c/t ) Tanaka (199), (1996) 9.7 (11) LBIU S T 1 α =1 (c/t ) [ S T 1 S 1 = K1 (s, t)+c 2 K 1(2) (s, t) ] dw (s) dw (t) [ ] D 1 = n 2 π + c2 Z 2 2 n 4 π 4 n, {Z n} NID(, 1) (12) n=1 K 1 (s, t) =min(s, t) st, K 1(2) (s, t) = K 1 (s, u) K 1 (u, t) du (11) K 1(2) K 1 5 7 S T 1 S 1 (12) 5.7 9.8 9.7 S 1 [ ] 1/2 E(e iθs 1 sin μ sin ν )= (13) μ ν μ = iθ + θ 2 +2ic 2 θ, ν = iθ θ 2 +2ic 2 θ MA(1) AR y = Xβ + η, E(η) =, V(η) =σ 2 Ω(α) (14) 7

η 2 LBIU S T = 1 T η Ω 2 η η Ω 1 η, η = My, M = IT X(X Ω 1 X) 1 X Ω 1 (15) 6 X 2 1 1 1 1 2 X = = e, X = =(e, d) 1... 1 1 T MA(1) MA(1) S T Tanaka (1996) 9.9 (14) (15) MA S T LBIU α =1 (c/t ) [ S T S = K(s, t)+c 2 K (2) (s, t) ] dw (s) dw (t) K (2) K E(e iθs )= [ D ( iθ + θ 2 +2ic 2 θ ) D ( iθ θ 2 +2ic 2 θ )] 1/2 K(s, t) D(λ) X 2 i) X = e K(s, t) =min(s, t) st 3st(1 s)(1 t), D(λ) = 12 λ 2 ( 2 λ sin λ 2cos λ ) ii) X =(e, d) K(s, t) =min(s, t) st 2st(1 s)(1 t)(4 5s 5t +1st) D(λ) = 864 λ 4 ( 2+ λ 3 λ ( 2 λ 12 ) sin λ ( 2 2λ 3 ) cos λ ) D(λ) K(s, t) FD Fredholm 5 9-2 MA(1) 3 LBIU MA(1) MA(1) 1 MA(1) 9-3 3 AR Tanaka (1996) 8

9-2 9-3 MA i.i.d. y j = a + bj+ u j αu j 1, u j = φ k ε j k, k= k φ k <, k=1 (15) S T Tanaka (199), (1996) k= φ k (16) 9.1 (15) S T (16) α =1 (c/t ) S T σ2 L σ 2 S [ K(s, t)+c 2 K (2) (s, t) ] dw (s) dw (t) ( ) σl 2 = 2 σ2 φ k, σs 2 = σ2 φ 2 k k= k= σ 2 L σ2 S u j σ 2 L σ2 S ũ =(ũ 1,, ũ T ) = H ( I T X(X Ω 1 X) 1 X Ω 1) y = H η H Ω 1 Cholesky Ω 1 = H H H = 1 2 1 6 2 6 1 T (T +1) 2 T (T +1) T T (T +1) σ S 2 = 1 T ũ 2 j T = 1 j=1 T η Ω 1 η, σ L 2 = σ2 S + 2 T ( l 1 k ) T l +1 k=1 j=k+1 ũ j ũ j k l o(t 1/4 ) (16) S T = σ2 S S σ L 2 T = 1 η Ω 2 η T 2 σ L 2 [ K(s, t)+c 2 K (2) (s, t) ] dw (s) dw (t) 9

S T MA 9.3 MA AR y j = x j β + γ j + ε j, γ j = γ j 1 + ξ j, γ = (17) y j x j β {ε j } {ξ j } i.i.d.(,σ 2 ε ) i.i.d.(,σ 2 ξ ) σ2 ε σ2 ξ H : δ = σ2 ξ σ 2 ε = vs. H 1 : δ> (18) H γ j y j H 1 AR δ MLE ˆδ H ˆδ MA(1) MLE T 2 ˆδ LM δ, σ 2 ε, β (17) y j = x j β + ε j + ξ 1 + + ξ j ε j ξ j y =(y 1,,y T ) y = X β + C ξ + ε N ( X β,σ 2 ε (I T + δcc ) ), (19) X =(x 1,, x T ), ξ =(ξ 1,,ξ T ), ε =(ε 1,,ε T ) C (i, j) i j 1 L(δ, σ 2 ε, β) = T 2 log(2πσ2 ε ) 1 2 log I T + δcc 1 (y X β) (I 2 σε 2 T + δcc ) 1 (y X β) (2) L(δ, σε 2, β) T (T +1) = + T δ H 4 2 (y X ˆβ) CC (y X ˆβ) (y X ˆβ) (y X ˆβ) ˆβ =(X X) 1 X y (19) β OLS S T = 1 T (y X ˆβ) CC (y X ˆβ) (y X ˆβ) (y X ˆβ) (21) 1

S T LBI S T X 3 i) X ii) X = e iii) X =(e, d) e =(1,, 1), d =(1, 2,,T) δ = c 2 /T 2 Nabeya-Tanaka (1988), Tanaka (1996) 9.11 (19) (18) (21) S T LBI X 3 δ = c 2 /T 2 S T S = [ K(s, t)+c 2 K (2) (s, t) ] dw (r) dw (s) E(e iθs )= [ D ( iθ + θ 2 +2ic 2 θ ) D ( iθ θ 2 +2ic 2 θ )] 1/2 K (2) K D(λ) K FD D K X i) X K(s, t) =1 max(s, t), D(λ) =cos λ ii) X = e K(s, t) =min(s, t) st, D(λ) = sin λ λ iii) X =(e, d) K(s, t) =min(s, t) st 3st(1 s)(1 t), D(λ) = 12 λ 2 ( 2 λ sin λ 2cos λ ) MA(1) 9.11 ii) MA(1) 1 iii) MA(1) 1 2 1 MA(1) 1 Δy j = b + ξ j +Δε j (22) Δy t MA(1) 9.11 i) MA(1) LBI ii) iii) 11

y j = x j β + γ j + u j, γ j = γ j 1 + ξ j, γ = (23) u j (16) (21) S T δ = c 2 /T 2 9.1 u j σs 2 σ2 L ˆσ2 S ˆσ2 L ˆσ 2 S = 1 T T û 2 j, j=1 ˆσ2 L =ˆσ2 S + 2 T ( l 1 k ) T l +1 k=1 j=k+1 û j û j k û j H ŜT =ˆσ 2 S S T /(T ˆσ 2 L ) Kwiatkowski et al. (1992) KPSS 9.4 Quah (1994) DF (17) y ij = x ij β i + γ ij + ε ij, γ ij = γ i,j 1 + ξ ij, γ i = (i =1,,N; j =1,,T) (24) i j y ij x ij β i {ε ij } {ξ ij } i.i.d.(,σ 2 ε ) i.i.d.(,σ2 ξ ) σ 2 ε σ2 ξ (24) Hadri (2) H : δ = σ2 ξ σ 2 ε = vs. H 1 : δ> (25) LBI (24) y = Xβ +(I N C)ξ + ε N ( Xβ, σ 2 ε(i N (I T + δcc )) ) (26) Kronecker C (19) y 1 y i1 x i1 y =., y i =., X =diag(x 1,,X N ), X i =. y N β 1 β =. β N, ξ = y it ξ 1. ξ N, ξ i = ξ i1. ξ it, ε = L(δ, σ 2 ε, β) = NT 2 log(2πσ2 ε) 1 2 log I N (I T + δcc ) ε 1. ε N, ε i = x it 1 (y X β) ( I 2 σε 2 N (I T + δcc ) 1) (y X β) (27) 12 ε i1. ε it

δ H 7 L(δ, σε, 2 β) NT(T +1) = + NT (y X ˆβ) CC (y X ˆβ) δ H 4 2 (y X ˆβ) (y X ˆβ) ˆβ =(X X) 1 X y (26) β OLS S NT = 1 T (y X ˆβ) CC (y X ˆβ) (y X ˆβ) (y X ˆβ) = 1 Ni=1 1 N 1 N (y T 2 i X i ˆβi ) CC (y i X i ˆβi ) Ni=1 1 (y X ˆβ) T (y X ˆβ) S NT LBI S NT X =diag(x 1,,X N ) 3 i) X ii) X i = e iii) X i =(e, d) e =(1,, 1), d =(1, 2,,T) 3 S (m) NT (m =1, 2, 3) δ = S (m) NT T N N T Phillips-Moon (1999) N T (28) S (m) NT S (m) N = 1 N N i=1 K (m) (s, t) dw i (s)dw i (t), (m =1, 2, 3) (29) W 1 (t),,w N (t) Bm K (m) (s, t) X 3 9.11 8 E(S (1) N )= 1 2, V(S(1) N )= 1 3, E(S(2) N )= 1 6, V(S(2) N )= 1 45, E(S(3) N )= 1 15, V(S(3) N )= 11 63 S (m) N 9.12 (24) (25) (28) S NT LBI X 3 T (29) S (m) N (m =1, 2, 3) S(m) N N N(S (m) N E(S (m) N )) V(S (m) N ) N(, 1) (3) H 1 : δ = c 2 /T 2 9 ε ij 13

9.5 2 1 1 Perron (1989) T B f(t) = { α (t T B ) α + α 1 (t>t B ) (31) D t (T B )= { (t TB ) 1 (t>t B ) (32) f(t) =α + α 1 D t (T B ) (33) y t = α + α 1 D t (T B )+η t, η t = ρη t 1 + ε t, {ε t } i.i.d.(,σ 2 ) (34) y t = a + bd t 1 (T B )+α 1 I t (T B +1)+(ρ 1) y t 1 + ε t (35) a = α (1 ρ), b = α 1 (1 ρ) H : ρ =1 I t (T B +1) I t (T B +1)= { (t TB +1) 1 (t = T B +1) (36) ρ OLS ˆρ ˆρ T B T λ ( <λ<1) ( TB lim T T ) = λ ( <λ<1) λ DF T (ˆρ 1) 1 t T T B +2 t T 14

1 1 yt 1 1 y t yt 1 y t 1 yt 1 y t T (ˆρ 1) = T 1 1 yt 1 1 1 y t 1 yt 1 y t 1 y 2 t 1 1 1 λ W(1) 1 λ 1 λ W(1) W (λ) 1 W (r) dr λ W (r) dr W (r) dw (r) 1 λ 1 W (r) dr λ 1 λ 1 λ W (r) dr 1 W (r) dr λ W (r) dr W 2 (r) dr (37) Perron (1989) Zivot-Andrews (1992), Vogelsang-Perron (1998) 15

9 1. S y j = ρy j S + ε j, {ε j } NID(,σ 2 ) H : ρ =1 L S = S Tj=T S+1 y 2 j Tj=1 (y j y j S ) 2 H LBI L S 2. 1 Tj=1 R S = S2 yj 2 T 2 Tj 1 (y j y j S ) 2 ρ =1 3. 1 4. W (t) 2 Bm Z = 2 W (t) dw (t) W (t)w (t) dt 5. MA(1) y j = ε j αε j 1, {ε j } NID(,σ 2 ) H : α =1 LBIU 6. y = Xβ + η, η N(,σ 2 Ω(α)) σ 2 Ω(α) 5 MA(1) H : α =1 LBIU 7. y = Xβ +(I N C)ξ + ε N ( Xβ, σ 2 ε (I N (I T + δcc )) ) (38) Kronecker C 1 T T δ H : δ = 16

8. R R = [min(s, t) st 3st(1 s)(1 t)] dw (s)dw (t) 9. 7 H : δ = LBI δ = c 2 /T 2 LBI 17

9-1 5%% S =4 S =12 5% c =1 5 1 5% c =.5 1 5 α j = β j = LBI.711 19.9 87. 99.3 5.226 23.6 56.1 1. LBIU.641 17.9 98.2 1. 3.223 19.5 47.5 1. DF 2.268 2. 99.2 1..965 23.3 57.7 1. DW.641 17.9 98.2 1. 3.223 19.5 47.5 1. β j = LBI.711 19.9 87. 99.3 5.226 23.6 56.1 1. LBIU.641 17.9 98.2 1. 3.223 19.5 47.5 1. DF 6.961 13.1 66.7 99.6 4.929 14.6 28.1 99.4 DW.31 15.3 8.4 99.9 1.271 17.3 37.7 1. LBIU.31 5.99 34.2 89.6 1.271 5.53 7.21 8.4 DF 13.67 5.74 23.8 76.7 1.36 5.39 6.55 56.8 DW.156 5.8 27. 82.9.588 5.42 6.7 65.2 18

9-2 POI % 5% S =4 S =12 c = 1 2 5 1 c =.5 1 2 5 α j = β j = β j = 2.84 49.98 99.19 1. 23.85 58.3 97.89 1. θ = 1 2.84 49.16 97.8 1. 23.73 58.3 97.63 1. θ =1.5 2.77 49.83 98.14 1. 23.5 58.1 97.84 1. θ = 2 2.65 49.98 98.65 1. 23.25 57.65 97.89 1. 5.99 9.1 35.48 94.53 5.53 7.21 15.46 83.4 θ = 4 5.98 8.99 35.45 93.74 5.53 7.19 15.4 82.98 θ = 5 5.98 8.98 35.48 94.8 5.52 7.17 15.34 83.4 θ = 6 5.97 8.96 35.46 94.29 5.52 7.16 15.28 83. 19

9-3 T ( ˆφ 2 +1) =1.664, =3.46.1.5.1.5.9.95.99 2.956 2.28 1.539.775 6.1 8.389 14.115 2

9-1 1..8.6.4.2. 5 1 21

9-2 3 MA 3 2 1..5.1.15.2 22

9-3 3 MA 1..8.6.4.2. 2 4 6 23