21世紀の統計科学 <Vol. III>

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1 21 III HP, II 6 1 ( ) tanaka@stat.hit-u.ac.jp 141

2 1 20 Journal of the Royal Statistical Society, Series B M.S. Bartlett, M.H. Quenouille, H. Wold, A.M. Walker, P. Whittle, E.J. Hannan, T.W. Anderson 1970 G.E.P Box G.M. Jenkins 1970 Time Series Analysis: Forecasting and Control identification estimation diagnostic checking Akaike (1973) AIC (Akaike s Information Criterion: ) GARCH

3 {X t } t {y t } 2 (i) E(y t )=μ (ii) Cov(y t,y t+h )=γ( h ) γ(h) =0(h = 0) 0 y t = α j ε t j, {ε t } i.i.d.(0,σ 2 ), α 0 =1, j=0 αj 2 < (2.1) j=0 i.i.d.(0,σ 2 ) 0 σ 2 i.i.d. 0 (2.1) {α j } {y t } E(y t )=0, Cov(y t,y t+h )=σ 2 j=0 α j α j+ h (2.2) 143

4 (2.1) Brockwell-Davis (1991), Fuller (1996) i.i.d. s t Cov(y s,y t ) s t γ( s t ) s t ρ(h) = γ(h) γ(0) = γ(h) V(y t ) = ρ( h) (2.3) h ρ(h) ρ(h) h

5 {y t } S = h= Cov(y t,y t+h ) = h= γ(h) (2.4) S (2.1) {α j } j α j = O( λ j ), ( λ < 1) S < α j = O(j d 1 ), (0 <d<1/2) S 145

6 {y t } ARMA(p, q) y t = m + φ 1 y t φ p y t p + ε t θ 1 ε t 1 θ q ε t q (2.5) {ε t } i.i.d.(0,σ 2 ) ARMA(p, q) φ(l) =1 φ 1 L φ p L p, θ(l) =1 θ 1 L θ q L q φ(l) y t = m + θ(l) ε t (2.6) ARMA(p, q) AR φ(x) =0 1 MA θ(x) =0 θ(x) =0 1 2 Anderson (1971) (2.6) ARMA(p, q) {y t } μ μ = m φ(1) = m 1 φ 1 φ p (2.6) ARMA(p, q) y t = μ + φ 1 (L) θ(l) ε t = μ + α j ε t j (2.7) j=0 α j φ 1 (L) θ(l) L j (2.7) ARMA MA( ) ARMA(p, q) AR( ) MA θ(x) =0 1 (2.7) ε t = θ 1 (L) φ(l)(y t μ) = β j (y t j μ) j=0 β j θ 1 (L) φ(l) L j ε t y t,y t 1, 146

6) ARMA(p, q) AR φ(x) =0 1 MA θ(x) =0 θ(x) =0 1 2 Anderson (1971) (2.6) ARMA(p, q) {y t } μ μ = m φ(1) = m 1 φ 1 φ p (2.

7 ARMA(p, q) ARMA y t = ε t ε t 1 MA(1) ARMA MA MA 1 ARMA(2, 1) y t = y t y t 2 + ε t 0.5ε t L = L +0.64L ε 2 t (2.8) 5 ARMA(2, 1) σ 2 =1 AR ± i 0.48 = 0.8exp{±iπ/3} MA( ) (1 0.8L+ 0.64L 2 )(1 + α 1 L + α 2 L 2 + )=1 0.5L α 1 =0.3, α 2 = 0.4, α j =0.8α j α j 2 2 (2.8) ARMA(2, 1) 6 AR 6 re iω ω π/3 147

1 1 σ 2 =1 AR 2 0.4 ± i 0.48 = 0.8exp{±iπ/3} MA( ) (1 0.8L+ 0.

8 2 ARMA {y t } (2.4) S γ(h) Fourier f(ω) = 1 2π h= γ(h) e ihω ( π ω π) (2.9) f(ω) f(ω) {y t } 2π f(ω) [0,π] ω 2π/ω π

9 {γ(h)} (2.9) f(ω) f(ω) (2.9) e ihω [ π, π] ω γ(h) 1 γ(h) f(ω) γ(h) = π π f(ω) e ihω dω (2.10) 1 1 f(ω) (2.10) γ(0) ω f(ω) ω {u t } σ 2 f(ω) =σ 2 /(2π) y t = μ + α j ε t j, {ε t } i.i.d.(0,σ 2 ), j=0 αj 2 < (2.11) j=0 f y (ω) 2 f y (ω) = α j e ijω j=0 f ε (ω) = σ2 2π α(e iω ) 2 (2.12) α(x) = j=0 α j x j ARMA(p, q) (2.8) ARMA(2, 1) f y (ω) = σ2 2π 1 0.5e iω e iω +0.64e 2iω 2 (2.13) 149

10 3 σ 2 =1 ω = π/3 6 3 ARMA pi/2 pi ARMA Taniguchi- Kakizawa (2000) (2.10) (2006) 150

11 2.4 ARMA ARMA {y t } μ T y(t )=(y T,y T 1, ) h y T +h ŷ T +h (a) e T +h = y T +h ŷ T +h 0 (b) V(e T +h ) 2 y(t )=(y T,y T 1, ) y T +h y T +h V(e T +h ) y T +h =E(y T +h y(t )), V(e T +h) =E [ {y T +h E(y T +h y(t ))} 2] ARMA 0 ARMA(p, q) y T +h y T +h y T +h = φ 1 y T +h φ p y T +h p + ε T +h θ 1 ε T +h 1 θ q ε T +h q y j = y j (j T ), ε j = { εj (j T ) 0 (j>t) {ε t }. h MA( ) ARMA(p, q) y T +h = α j ε T +h j, α 0 = 1 (2.14) j=0 MA( ) y T +h = j=h α j ε T +h j, e T +h = h 1 j=0 α j ε T +h j, h 1 V(e T +h) =σ 2 αj 2 j=0 151

T +h 1 + + φ p y T +h p + ε T +h θ 1 ε T +h 1 θ q ε T +h q y j = y j (j T ), ε j = { εj (j T ) 0 (j>t) {ε t }.

12 V(y t ) 2.5 ARMA ARMA AR p MA q ARMA(p, q) ARMA(p, q) φ(l) y t = θ(l)ε t, {ε t } i.i.d.(0,σ 2 ) (2.15) 0 0 φ =(φ 1,,φ p ), θ =(θ 1,,θ q ), σ 2 =V(ε t ) T φ θ 2 { } T T 2 φ(l) f(φ, θ) = ε 2 yt t = (2.16) θ(l) t=1 t=1 MA θ(l) MA ε t φ MA ε t NLSE ( 2 ) (2.16) Box-Jenkins-Reinsel (1994) 152

(0,σ 2 ) (2.15) 0 0 φ =(φ 1,,φ p ), θ =(θ 1,,θ q ), σ 2 =V(ε t ) 2 2.5.1 2 T φ θ 2 { } T T 2 φ(l) f(φ, θ) = ε 2 yt t = (2.

13 σ 2 φ θ NLSE ARMA(p, q) y =(y 1,,y T ) N(0,σ 2 Σ) Σ T φ θ L(φ, θ,σ 2 )= T 2 log(2πσ2 ) 1 2 log Σ 1 2σ 2 y Σ 1 y (2.17) φ θ (2.17) σ 2 y Σ 1 y/t σ 2 l(φ, θ) = T 2 log(y Σ 1 y) 1 log Σ (2.18) 2 φ θ MLE ( ) σ 2 ˆσ 2 = y ˆΣ 1 y/t ˆΣ Σ φ θ MLE MLE T Σ Σ 1 Box-Jenkins-Reinsel (1994) Brockwell-Davis (1991) MLE NLSE 3 ARMA β =(φ, θ ) NLSE MLE T (ˆβ β) N(0, Ω 1 ) ˆβ β MLE NLSE Ω β Fisher Ω= 1 ( E(ut u t ) E(u tv t ) ) σ 2 E(v t u t ) E(v tv t ) u t =(u t,,u t p+1 ), v t =(v t,,v t q+1 ) {u t }, {v t } AR(p) φ(l)u t = ε t AR(q) θ(l)v t = ε t

18) 2 φ θ MLE ( ) σ 2 ˆσ 2 = y ˆΣ 1 y/t ˆΣ Σ φ θ MLE MLE T Σ Σ 1 Box-Jenkins-Reinsel (1994) Brockwell-Davis (1991) MLE NLSE 3 ARMA β =(φ,

14 ARMA T ARMA(p, q) φ(l)y t = θ(l)ε t e t = y t p j=1 ˆφ j y t j + q j=1 ˆθ j e t j (t =1,,T) (2.19) h r h = T h t=1 e t e t+h Tt=1 e 2 t (h =1,,T 1) (2.20) ARMA(p, q) Trh N(0, 1) h 5% r h [ , ] T T 0 ±2/ T m p + q 2 Q = T m h=1 r 2 h, Q = T (T +2) m h=1 1 T h r2 h (2.21) ARMA(p, q) m p q χ 2 Q Box-Pierce Q Ljung-Box Q χ 2 154

20) ARMA(p, q) Trh N(0, 1) h 5% r h [ 1.96 1, 1.

15 Kullback-Leibler Akaike (1973) AIC ARMA(p, q) AIC AIC(p, q) = 2 +2(p + q + 1) (2.22) p q ARMA(p, q) AIC (2.22) 1 2 p + q AIC Schwarz (1978) SBC(p, q) = 2 +(p + q +1)logT (2.23) SBC 2 AIC 2 SBC SBC AIC 3 ARIMA ARFIMA d d I(d) I Integrated I(d) φ(l)(1 L) d y t = θ(l)ε t, {ε t } i.i.d.(0,σ 2 ) (3.24) d φ(l) =1 φ 1 L φ p L p, θ(l) =1 θ 1 L θ q L q 2 φ(x) =0,θ(x) =0 1 {y t } ARIMA(p, d, q) ARIMA I Integrated ARIMA ARIMA(0, 1, 0) y t = y t 1 + ε t = ε ε t, y 0 = 0 (3.25) 155

23) SBC 2 AIC 2 SBC SBC AIC 3 ARIMA ARFIMA d d I(d) I Integrated I(d) φ(l)(1 L) d y t = θ(l)ε t, {ε t } i.i.d.(0,σ 2 ) (3.

16 AR 1 y t = O p ( t) ARIMA(p, d, q) y t = O p (t d 1/2 ) ARIMA d ARMA (3.24) ARIMA(p, d, q) d ARFIMA(p, d, q) F Fractional d <1/2 d> 1/2 0 <d<1/2 (Hosking (1981)) 4 ARFIMA(p, d, q) 0 <d<1/2 ARFIMA(p, d, q) ARMA f(ω) = σ2 2π θ(e iω ) 2 1 e iω 2d φ(e iω ) 2, γ(h) = π π f(ω) e ihω dω (3.26) ARFIMA(0,d,0) γ(h) =σ 2 Γ(1 2d)Γ(h + d) Γ(d)Γ(1 d)γ(h d +1) (h >0) (3.27) (3.26) ω 0 f(ω) =O(ω 2d ) (3.27) ARFIMA(0,d,0) h γ(h) =O(h 2d 1 ) ARFIMA(p, d, q) Hosking (1981) ARFIMA(0, 0.45, 0) σ 2 =1 4 ARMA 156

ARFIMA(p, d, q) ARMA f(ω) = σ2 2π θ(e iω ) 2 1 e iω 2d φ(e iω ) 2, γ(h) = π π f(ω) e ihω dω (3.

17 ARFIMA ARMA Hosking (1996) 4 μ ARFIMA(p, d, q) T ȳ T 1/2 d (ȳ μ) = 1 ( ) T (y T d+1/2 t μ) N 0,σ 2 (d) θ2 (1) φ 2 (1) σ 2 (d) = lim T V ( 1 T d+1/2 t=1 ) T (1 L) d ε t = t=1 σ 2 Γ(1 2d) (1 + 2d)Γ(1+d)Γ(1 d) ARMA(p, q) d =0 157

2 0 5 10 15 20 ARFIMA ARMA Hosking (1996) 4 μ ARFIMA(p, d, q) T ȳ T

18 ARFIMA d 1/4 Hosking (1996) ARIMA ARFIMA 1 L m SARIMA, SARFIMA S Seasonal Box-Jenkins-Reinsel (1994) Journal of Econometrics (1996), Vol AR(1) y t = ρy t 1 + ε t Δ y t = δy t 1 + ε t (Δ = 1 L, δ = ρ 1) (4.28) y 0 =0,{ε t } i.i.d.(0,σ 2 ) H 0 : ρ =1(δ =0) H 1 : ρ<1(δ<0) ρ δ LSE H 0 T δ LSE ˆδ H 1 : ρ =1 c/t (c ) (Phillips-Perron (1988)) T ˆδ = 1 Tσ 2 T t=2 y t 1 Δy t / 1 T 2 σ 2 T yt 1 2 t= Y (t) dy (t) Y 2 (t) dt {Y (t)} [0,1] Ornstein-Uhlenbeck (O-U) (4.29) dy (t) = cy (t) dt + dw (t), Y(0) = 0 Y (t) =e ct t 0 e cs dw (s) {W (t)} [0,1] (4.29) (Nabeya-Tanaka (1990)) 158

(0,σ 2 ) H 0 : ρ =1(δ =0) H 1 : ρ<1(δ<0) ρ δ LSE H 0 T δ LSE ˆδ H 1 : ρ =1 c/t (c ) (Phillips-Perron (1988)) T ˆδ = 1 Tσ 2 T t=2 y t 1 Δy t / 1 T 2

19 Fuller (1996), Tanaka (1996) ARMA AR MA AR MA MA(1) y t = ε t αε t 1, {ε t } i.i.d.(0,σ 2 ) (4.30) y t H 0 : α =1vs.H 1 : α<1 {y t } y t =(1 L)x t H 0 (1 L)x t =(1 L)ε t 1 L AR H 1 AR MA AR (4.30) {ε t } S T = 1 T y Ω 2 y y Ω 1 y (4.31) H 0 (LBIU) Ω y H 0 S T α =1 (c/t ) Tanaka (1996), (2006) S T n=1 [ 1 n 2 π 2 + c2 n 4 π 4 ] Z 2 n, {Z n } NID(0, 1) (4.32) MA Elliott- Rothenberg-Stock (1996) (2006) 159

20 5 1 Perron (1989) T B D t (T B )= { 0 (t TB ) 1 (t>t B ) (5.33) y t = α 0 + α 1 D t (T B )+η t, η t = ρη t 1 + ε t, {ε t } i.i.d.(0,σ 2 ) (5.34) ρ Perron (1989) Zivot-Andrews (1992), Vogelsang-Perron (1998) 1 Tong (1983) threshold ( ) AR y t = φ (1) 1 y t φ (1) p 1 + ε (1) t φ (2) 1 y t φ (2) p 2 + ε (2) t (x t <a ) (x t a a x t y t Hamilton (1989) 0 1 S t P (S t =1 S t 1 =1)=p, P (S t =0 S t 1 =1)=1 p 160

34) ρ Perron (1989) Zivot-Andrews (1992), Vogelsang-Perron (1998) 1 Tong (1983) threshold ( ) AR y t = φ

21 P (S t =0 S t 1 =0)=q, P(S t =1 S t 1 =0)=1 q ARCH GARCH EM MCMC 6 2 {x t } x =(x 1,x 2,,x T ) T T =2 J J x DWT Discrete Wavelet Transform w 1 w = W x, w =. w J v J, W = W 1. W J V J (6.35) W W j T/2 j j V J W J 1/ T w w j j T/2 j 1 T/2 j w J w v J 161

22 J V J v J = V J x = T x x x 3 (a) (b) (c) ARFIMA (a) (b) (c) (b) 5 1 ARFIMA(0, 0.45,0) T =

23 (c) ARFIMA ARFIMA Percival-Walden (2000) 7 VARMA m (p, q) y t =Φ 1 y t 1 + +Φ p y t p + ε t Θ 1 ε t 1 Θ q ε t q (7.36) {ε t } i.i.d.(0, Σ) y t ε t m Φ k,θ l,σ m m Φ(L) =I m Φ 1 L Φ p Θ(L) =I m Θ 1 L Θ q L q Φ(L) y t =Θ(L) ε t (7.37) Φ(x) =0 1 VMA m ( ) VARMA MA ARMA VAR VAR (1988), Hamilton (1994) VAR {y t } I(1) Δ y t Δ y t = C j ε t j = C(L) ε t (7.38) j=0 = [C(1) + (C(L) C(1))] ε t = C(1) ε t +Δ C(L) ε t C(L) { C(L)ε t } (7.37) (7.38) Φ(L)Δy t =Δε t =Φ(L) C(L) ε t 163

24 Φ(L) C(L) =ΔI m I m m Φ(1) C(1) = 0 (7.39) (7.38) Φ(1) (7.39) Φ(1) y t =Φ(1) C(L) ε t {y t } {y t } cointegration Φ(1) Johansen (1995) r VAR(p) (7.37) Φ(L) Φ(L) =Φ(1)L +Φ(L) Φ(1) L =Φ(1)L +ΔΓ(L) Γ(L) =I q Γ 1 L 1 Γ p 1 L p 1, p Γ j = Φ i i=j+1 VAR(p) Δy t = γα y t 1 +Γ 1 Δ y t 1 + +Γ p 1 Δ y t p+1 + ε t (7.40) γα = Φ(1) γ α q r r (7.40) ECM Error Correction Model: 1 {α y t 1 } I(1) r r ARIMA 4 1 Hamilton (1994), Johansen (1995) 164

25 8 ARIMA ARFIMA SARIMA, SARFIMA S Seasonal SARIMA Box-Jenkins-Reinsel (1994) SARFIMA Journal of Econometrics (1996), Vol. 73 (2006) Arellano-Bond (1991), Blundell-Bond (1998) Anderson, T. W. (1971). The Statistical Analysis of Time Series, Wiley, New York. Arellano, M. and Bond, S. (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations, Review of Economic Studies, 58, Beran, J. (1994). Statistics for Long-Memory Processes, Chapman & Hall, New York. Blundell, R. and Bond, S. (1998). Initial conditions and moment restrictions in dynamic panel data models, Journal of Econometrics, 87, Box, G. E. P. and Jenkins, G. M. (1970). Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco. Box, G. E. P., Jenkins, G. M., and Reinsel, G. C. (1994). Time Series Analysis: Forecasting and Control, 3rd Edition, Holden-Day, San Francisco. Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 3rd Edition, Springer, New York. Elliott, G., Rothenberg, T. J., and Stock, J. H. (1996). Efficient tests for an autoregressive unit root, Econometrica, 64,

26 Fuller, W. A. (1996). Introduction to Statistical Time Series, 2nd Edition, Wiley, New York. Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, Hamilton, J. D. (1994). Time Series Analysis, Princeton University Press, Princeton. Hosking, J. R. M. (1981). Fractional differencing, Biometrika, 68, Hosking, J. R. M. (1996). Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series, Journal of Econometrics, 73, Johansen, S. (1995). Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press, Oxford. (2006). X-12-ARIMA CIRJE No. R-5 Nabeya, S. and Tanaka, K. (1990). A general approach to the limiting distribution for estimators in time series regression with nonstable autoregressive errors, Econometrica, 58, Percival, D. B. and Walden, A. T. (2000). Wavelet Methods for Time Series Analysis, Cambridge University Press, Cambridge. Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis, Econometrica, 57, Phillips, P. C. B. and Perron, P. (1988). Testing for a unit root in time series regression, Biometrika, 75, Schwarz, G. (1978). Estimating the dimension of a model, Annals of Statistics, 6, Tanaka, K. (1996). Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Wiley, New York. (2006).. Taniguchi, M. and Kakizawa, Y. (2000). Asymptotic Theory of Statistical Inference for Time Series, Springer, New York. 166

27 Tong, H. (1983). Threshold Models in Non-Linear Time Series Analysis, Springer, New York. Vogelsang, T. J. and Perron, P. (1998). Additional tests for a unit root allowing for a break in the trend function at an unknown time, International Economic Review, 39, (1988).. Zivot, E. and Andrews, D. W. (1992). Further evidence on the great crash, the oil price shock, and the unit root hypothesis, Journal of Business and Economic Statistics, 10,

seminar0220a.dvi

seminar0220a.dvi 1 Hi-Stat 2 16 2 20 16:30-18:00 2 2 217 1 COE 4 COE RA E-MAIL: ged0104@srv.cc.hit-u.ac.jp 2004 2 25 S-PLUS S-PLUS S-PLUS S-code 2 [8] [8] [8] 1 2 ARFIMA(p, d, q) FI(d) φ(l)(1 L) d x t = θ(l)ε t ({ε t }