経済時系列分析と単位根検定 : これまでの発展と今後の Title 展望 Author(s) 黒住, 英司 Citation Issue 27-12 Date Type Technical Report Text Version publisher URL http://hdl.handle.net/186/15111 Right Hitotsubashi University Repository
Hi-Stat Discussion Paper Series No.228 経済時系列分析と単位根検定 : これまでの発展と今後の展望 黒住英司 December 27 Hitotsubashi University Research Unit for Statistical Analysis in Social Sciences A 21st-Century COE Program Institute of Economic Research Hitotsubashi University Kunitachi, Tokyo, 186-863 Japan http://hi-stat.ier.hit-u.ac.jp/
27 9 3 ADF
1. 3 AR ADF ADF 2 ADF 1
3 4 2. 2.1. y t y t = μ z t + x t, φ(l)x t = u t (t =1, 2,,T). (1) z t u t i.i.d.(,σ 2 ) φ(l) L p φ(l) =1 φ 1 L φ 2 L 2 φ p L p x,x 1,,x 1 p T x t y t μ z t p (AR(p)) z t z t z t =1( ) z t =[1,t] ( ) y t μ z t ( x t = y t μ z t ) ( ) y t (1) x t φ(z) = 1 H : φ(1) = v.s. H 1 : φ(1) > (2) φ(1) > φ(1) < φ() = 1 (, 1) ( ) 2
I(1) (an integrated process of order 1) ( ) I() μ z t = μ (1) y t = μ + x t, φ(l)x t = u t φ(l) y t y t = c + φ 1 y t 1 + + φ p y t p + u t (3) c = φ(1)μ Δy t = c + ρy t 1 + ψ 1 Δy t 1 + + ψ p 1 Δy t p+1 + u t. (4) Δ =1 L p ρ = φ i 1, i=1 p ψ i = φ j (i =1, 2,,p 1) j=i+1 ρ = φ(1) c = φ(1)μ (2) (4) H : ρ = c = v.s. H 1 : ρ< (5) ( ) : Δy t = ψ 1 Δy t 1 + + ψ p 1 Δy t p+1 + u t, ( ) : y t = c + φ 1 y t 1 + + φ p y t p + u t, y t y t = μ + μ 1 t + x t, φ(l)x t = u t 3
y t y t = c + c 1 t + φ 1 y t 1 + + φ p y t p + u t (6) c = φ(1)μ φ (1)μ 1 c 1 = φ(1)μ 1 (3) (4) Δy t = c + c 1 t + ρy t 1 + ψ 1 Δy t 1 + + ψ p 1 Δy t p+1 + u t (7) (2) (7) H : ρ = c 1 = v.s. H 1 : ρ< (8) ( ) : Δy t = c + ψ 1 Δy t 1 + + ψ p 1 Δy t p+1 + u t, ( ) : y t = c + c 1 t + φ 1 y t 1 + + φ p y t p + u t, y t Dickey and Fuller (1981) (5) (8) c c 1 ρ = H : ρ = v.s. H 1 : ρ< (9) 2.2. ADF (9) H ρ 4
(4) Δy t y t 1 Δy t 1,, Δy t p+1 y t 1 ˆρ μ (7) Δy t y t 1 Δy t 1,, Δy t p+1 y t 1 ˆρ τ Dickey and Fuller (1979) Said and Dickey (1984) Phillips (1987) T ˆρ μ d 1 2 (B2 (1) 1) B(1) 1 B(s)ds 1 B2 (s)ds ( 1, Tˆρ τ B(s)ds)2 d 1 2 (B2 (1) 1) + A D. (1) B(r) r 1 ( 1 A =12 sb(s)ds 1 1 )( 1 B(s)ds B(s)ds 1 ) 1 2 2 B(1) B(1) B(s)ds, 1 ( 1 2 1 1 ( 1 ) 2 D = B 2 (s)ds 12 sb(s)ds) +12 B(s)ds sb(s)ds 4 B(s)ds T ˆρ μ T ˆρ τ ρ t (4) (7) ρ t t ADF μ t ADF τ t ADF μ d 1 2 (B2 (1) 1) B(1) 1 B(s)ds 1 B2 (s)ds (, t ADF τ 1 B(s)ds)2 d 1 2 (B2 (1) 1) + A D (11) (4) (7) 2 T ˆρ μ T ˆρ τ t ADF μ t ADF τ ADF(augmented Dickey-Fuller) p =1 AR(1) DF(Dickey-Fuller) t ADF t 2.3. ADF-GLS ADF ADF Elliott, Rothenberg 5
and Stock (1996, ERS ) ADF-GLS ADF-GLS POI(point optimal invariant) POI H 1 H l : ρ = v.s. H l 1 : ρ = ρ(θ) = θ T (θ>). α 1 ρ = ρ = θ /T θ ρ(θ) y =[y 1,y 2,,y T ] f(y; θ) Neyman-Pearson L(θ; α) f(y; θ) f(y;) k α ρ(θ) k α P θ =(L(θ; α) k α )=α ϕ(θ, θ ; α) P θ (L(θ; α) k α ) θ θ ϕ(θ, θ ; α) θ θ ρ(θ) POI POI ρ(θ) ρ(θ ) POI ρ(θ) θ θ = θ ϕ(θ,θ ; α) 6
1: POI ϕ(θ,θ ; α) θ POI 1 θ = θ1 POI θ1 θ2 POI POI King (1983) Tanaka (1996) POI ϕ(θ, θ ; α) 5% θ POI θ ERS POI θ =7 θ =13.5 POI ERS POI ADF ADF 7
ADF-GLS (1) y t z t (2) y qd t { { y qd y1 : t =1 t = y t ay t 1 : t 2, z1 : t =1 zqd t = z t az t 1 : t 2 a =1 7/T a =1 13.5/T z qd t ˆμ qd (3) y t x qd t = y t ˆμ qdz t (4) x qd t ADF (t ) Δx qd t OLS ρ t t GLS μ ) = ρx qd t 1 + ψ 1Δx qd t 1 + + ψ p 1Δx qd t p+1 + e t ( ) t GLS τ ( ERS t GLS μ t GLS μ d t GLS τ 1 2 (B2 (1) 1) 2 B2 (s)ds, t GLS τ d 1 2 (V 2 (1,θ) 1) 2 V 2 (s, θ)ds. V (r, θ) =B(r) r{λb(1) + 3(1 λ) 1 sb(s)ds} λ =(1 θ)/(1 θ + θ2 /3) θ =7 θ =13.5 t GLS μ ADF-GLS t GLS τ 2 ADF ADF-GLS ADF-GLS ADF ADF-GLS 8
1 1.8.8.6.6 ADF ADF-GLS ADF ADF-GLS.4.4.2.2 5 1 15 2 25 c (i) 5 1 15 2 25 c (ii) 2: ADF ADF-GLS 2.4. Box and Jenkins (1971) p q (ARMA(p, q)) y t = μ z t + x t, φ(l)x t = ϑ(l)u t. p φ(l) 1 ϑ(l) q φ(l) ϑ(l) φ(l) (ARIMA(p 1, 1,q)) (MA) ARIMA ARIMA Hall (1989) Pantula and Hall (1991) Ahn (1993) ARIMA MA ϑ(l) 1 ϑ(l) AR( ) φ (L)x t = u t, φ (L) φ(l)ϑ 1 (L) = φ i L i, (12) i= 9
AR( ) AR(p) ADF Berk (1974) Said and Dickey (1984) Ng and Perron (1995) p T ARIMA y t = μ z t + x t, (1 φl)x t = w t, w t. (13) Wold w t w t = ψ i u t i, ψ =1, i=1 ψi 2 < i=1 w t MA( ) Brillinger(1981) MA w t (12) AR( ) (13) AR( ) AR (4) (7) ADF ADF-GLS Ng and Perron (1995) (1) T Schwartz (1989) { ( ) } T 1/4 p =int 4 1 p =int { 12 ( ) } T 1/4 1 (2) AIC BIC p (3) p p p = p φ p t AR( p) φ p t AR( p 1) AR( p 1) 1
φ p 1 t AR( p 2) p Ng and Perron (1995) 3 Ng and Perron (21) MA Ng and Perron (21) AIC BIC (MIC) MIC =lnˆσ p 2 + C T (τ T (p)+p 1). T p ˆρ û p,t (4) (7) OLS ˆσ 2 p = 1 T p T û 2 p,t, t= p+1 τ T (p) = ˆρ2 T t= p+1 ( x qd t 1 )2 ˆσ 2 p AIC MAIC C T =2 BIC MBIC C T = ln(t p) Ng and Perron (21) MA MAIC 2.5. 2.4 ADF-GLS ADF Müller and Elliott (23) ADF-GLS ADF 11
AR(1) y t = μ z t + x t, x t = φx t 1 + u t. (14) φ =1 t x t = x + u i i=1 x z t x t x t = ρ t x + ρ t i u i i=1 ρ t x Müller and Elliott (23) x F (x ) N(,kσ 2 ) ϕ (φ,k) ϕ (φ,k) = arg ϕ max P (ϕ rejects φ = φ,x ) df (x ) ϕ ϕ (φ,k) k = ϕ(φ, ) x = k Müller and Elliott (23) ADF ADF-GLS ADF-GLS k = ADF k AR(1) (14) T = 1 3 ( 3(i-a)) ADF-GLS ADF 5 ( 3(i-b))ADF-GLS ADF ( 3(ii)) 12
1 ADF (x=) ADF-GLS (x=) 1 ADF (x=5) ADF-GLS (x=5).8.8.6.6.4.4.2.2.65.7.75.8.85.9.95 1 phi.65.7.75.8.85.9.95 1 phi (i-a) (x =) (i-b) (x =5) 1 ADF (x=) ADF-GLS (x=) 1 ADF (x=5) ADF-GLS (x=5).8.8.6.6.4.4.2.2.65.7.75.8.85.9.95 1 phi.65.7.75.8.85.9.95 1 phi (ii-a) (x =) (ii-b) (x =5) 3: ρ t Elliott and Müller (26) ˆQ m = q m + q m 1 ( ŷm T ) 2 + q m 3 ŷ m ŷm T T + q m 4 Tt=1 (ŷm t 1 ) 2 T 2. m = μ ŷ μ t =(y t T s= y s /T )/ˆω ˆω 2 x t AR(1) 13
(2π ) q μ = 1, qμ 1 = 4.95, qμ 2 =9.5, qμ 3 =.9, qμ 4 = 1 m = τ ŷ τ t ŷμ t 1 t q τ = 15, q τ 1 = 7.127, q τ 2 =12.166, q τ 3 = 3.31, q τ 4 = 225 Harvey and Leybourne (25) ADF ADF-GLS Harvey and Leybourne (26) ADF Elliott and Müller (26) ˆQ m Müller and Elliott (23) 2.5. ADF ADF-GLS Phillips and Perron (1988) (13) DF w t DF Phillips-Perron ADF Perron and NG (1996) OLS Pantula, Gonzalez-Farias and Fuller (1994) 14
AR AR WS (weighted symmetric) So and Shin (1999) y t 1 Cauchy Shin and So (21) ADF Wald Dickey and Fuller (1981) Schmidt and Phillips (1992) ADF Müller and Elliott (23) ADF 3. 3.1. Nelson and Plosser (1982) 4 common stock prices ( ) 1929 ( ) Perron (1989) Perron (1989) ( ) A: y t = μ a + μb DU t + μ 1 t + x t, x t = φx t 1 + w t, 15
5 4.5 4 3.5 3 2.5 2 1.5 1 187 188 189 19 191 192 193 194 195 196 197 4: B: y t = μ + μ a 1 t + μb 1 DT t + x t, x t = φx t 1 + w t, C: y t = μ a + μb DU t + μ a 1 t + μb 1 DT t + x t, x t = φx t 1 + w t, w t T B DU t = { : t T B 1 : t>t B,DT t = { : t T B t T B : t>t B,DT t = { : t T B t : t>t B φ =1 φ <1 A B C TB +1 y t AO (additive outlier) y t IO (innovational outlier) AO A B C y t 16
{ A: yt = c a + dd(t B ) t + y t 1 + w t, A: y t = c a + cb DU t + c 1 t + w t, { B: yt = c + c 1 DU t + y t 1 + w t, B: y t = c + c a 1 t + cb 1 DT t + w t, { C: yt = c + dd(tb ) t + C 1 DU t + y t 1 + w t, C: y t = c a + cb DU t + c a 1 t + cb 1 DT t + w t, D(TB ) t t = TB +1 1 Perron (1989) y t Perron (1989) T (λ = T B /T ) Park and Sung (1994) A C Perron (199) Perron and Vogelsang (1992b) 3.2. (1) Perron (1989) Christiano (1992) Banerjee, Lumsdaine and Stock (1992) Zivot and Andrews (1992) Banerjee, Lumsdaine and Stock (1992) Zivot and Andrews (1992) T B y t ADF t ρ (λ) λ = T B /T λ Λ 17
λ Λ t ρ (λ) inf λ Λ t ρ Λ Zivot and Andrews (1992) Λ=[2/T, (T 1)/T ] Perron (1997) c b c b 1 t t ˆλ c t ρ (ˆλ c ) Amsler and Lee (1995) Perron and Vogelsang (1992a) 2.1 Perron (1989) Perron (1989) 3.3. (2) Leybourne, Mills and Newbold (1998) ( ) ADF Perron (1989) 18
Perron (1989) Leybourne, Mills and Newbold (1998) Vogelsang and Perron (1998) Hatanaka and Yamada (1999) Zivot and Andews (1992) Vogelsang and Perron (1998) Hatanaka and Yamada (1999) Zivot and Andrews (1992) ˆλ t ρ (ˆλ) Harvey, Leybourne and Newbold (21) Lee and Strazicich (21), Carrion-i-Silvestre and Sansó (26) Perron and Rodríguez (23) ADF-GLS Liu and Rodrǵuez (26) Saikkonen and Lütkepohl (22) 3.4. Perron (1989) 2 2 1 Lumsdaine and Papell (1997) 2 3.3 Kim, Leybourne and Newbold (2) 2 19
1 Hatanaka and Yamada (1999) Lee and Strazicich (23) 2 4. 3 2 3 2 2 Ng and Perron (21) MAIC AR Elliott and Müller (23) Elliott and Müller (26) 2 ADF ADF ADF-GLS ADF-GLS Ng and Perron (21) MAIC Elliott and Müller (26) Harvey and Leybourne (25) 2
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