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1 40, 2, Tests on Long Memory Time Series Haruhisa Nishino This paper surveys and explains testing problems on long memory time series. The first testing is a test where a null hypothesis is that a time series is short-memory stationary and alternatives is that it is long-memory. The paper explains a change point test and a two sample test for comparison of long memory parameters. It compares the performance of these tests with computer simulations and applies the tests to the realized volatility data of the Nikkei index. : 1. Box=Jenkins ARIMA (long-memory) ARIMA Granger and Joyeux (1980) Hosking (1981) (fractional difference) nishino@le.chiba-u.ac.jp.

2 ARMA ARFIMA (fractional ARIMA) ARIMA ARFIMA Ding et al. (1993) 2 2 (realized volatility) (2007) Beran (1994), Robinson (2003), Doukhan et al. (2003), Teyssiere and Kirman (2007), Palma (2007) (2003). ARFIMA(p, d, q) fractional Brownian motion 3 3 R/S KPSS V/S Lobato-Robinson (long-memory) (fractional difference) {y t ; t = 1, 2,..., } γ(k)

3 149 γ(k) = Cov(y t, y t+k ), k = 0, 1, 2,.... f(λ) γ(k) = π π e ikλ f(λ)dλ. ρ(k) = γ(k), k = 0, 1, 2,... γ(0) f(λ) Cλ 2d, as λ +0. (2.1) C (long memory time series) k C γ(k) Ck 2d 1, where 0 < d < 1/2, (2.2) hyperbolic (short-memory) γ(k) <, (2.3) k= 2.1 B By t = y t 1 1 B Box=Jenkins Granger and Joyeux (1980) Hosking (1981) (fractional difference) 1 B d d (1 B) d y t = ε t, 1/2 < d < 1/2, (2.4)

4 {ε t } W N(0, σ 2 ) (1 B) d = j=0 Γ(j + d) Γ(j + 1)Γ(d) Bj. (2.5) Γ(x) Γ(x) = 0 z x 1 e z dz, x Γ(x) = (x 1)! Hosking (1981) (2.4) d < 1/2 {y t } MA y t = j=0 1/2 < d {y t } Γ(j + d) Γ(j + 1)Γ(d) ε t j. (2.6) ARMA(p, q) φ(b)(1 B) d (y t µ) = θ(b)ε t, (2.7) ARFIMA(p, d, q) 0 < d < 1/2 (2.7) f(λ) = σ2 θ(e iλ ) 2π φ(e iλ ) 1 eiλ 2d = σ2 θ(e iλ ) 2π φ(e iλ ) 2 sin(λ 2 ) 2d Cλ 2d, as λ +0, (2.8) (2.1) 0 < d < 1/2, lim λ 0 f(λ) = 1/2 < d < 0 lim λ 0 f(λ) = d d Dickey and Fuller (1979) AR AR(1) (1 φb)y t = ε t, t = 1, 2,..., (2.9) φ = 1 φ < 1 (1 B) d y t = ε t, t = 1, 2,..., (2.10)

5 151 (2.9) φ = 1 (2.10) d = 1 Hosking (1981) d < 1/2 (2.10) d 1/2 1/2 d < 1 I(d) {y t } d = d 1 1/2 d < 0 I(d ) {z t } {ε t } (1 B) d z t = ε t I(d) {y t } y t = t j=1 z t, (t = 1, 2,...) {z t } I(1) {y t } {ε t } y t = t j=1 ε t, (t = 1, 2,...) (2.10) I(d) d (i) d = 0, (short-memory process) (ii) 0 < d < 1/2, (stationary long-memory process). (iii) 1/2 d < 1, (nonstationary fractionally integrated process) (iv) d = 1,, nonstationary process I(1) I(0) (2.9) {ε t } 0 < d < 1/2 I(d) Sowell (1990) fractional Brownian motion Sowell (1990) fractional Brownian motion(fbm) 2.3 Marinucci and Robinson (1999) Type I fbm (Marinucci and Robinson (1999 p. 120) ) Marinucci and Robinson (1999) fractional Brownian motion Type II Kwiatkowski et al. (1992) Saikkonen and Luukkonen (1993) I(0) I(1). Kwiatkowski et al. (1992) KPSS d 1/2 d tapering d Hurvich and Ray (1995), Velasco (1999a, b), Velasco and Robinson (2000) Shimotsu and Phillips (2005), Abadir

6 et al. (2007), Shao (2010) tapering local Whittle Whittle 2.3 fractinal Brownian motion fractinal Brownian motion {B(t); 0 t 1} Brownian motion B(t) = B(t) tb(1) { B(t); 0 t 1} Brownian bridge fractional Brownian motion {B d (t); 0 t 1} Brownian motion Marinucci and Robinson (1999) Type I fractional Brownian motion B d (t) = 1 [ t 0 ] (t s) d db(s) + (t s) d ( s) d db(s), A(d) 0 [ 1 where A(d) = 2d { (1 + r) d r d} 2 dr ] 1/2, B d (t) = B d (t) tb d (1) { B d (t); 0 t 1} fractional Brownian bridge fractional Brownian motion 3 (i) k= γ(k) <, (ii) BM (functional central limit theorem) [nt] 1 (y t E(y t )) σ L B(t), as n. n j=1 σl 2 = k= γ(k) = 2πf(0). (iii) 4 sup h r,s= κ(h, r, s) <, 4 κ(h, r, s) = E(y k y k+h y k+r y k+s ) {γ(h)γ(r s) + γ(r)γ(h s) + γ(s)γ(h r)}

7 153 3 (i) γ(k) ck 1 2d as k, c (ii) fractional BM (functional central limit theorem) [nt] n 1/2 d j=1 (y t E(y t )) c d B d (t), as n. c 2 d = c/{d(2d + 1)}. (iii) n κ(h, r, s) = O(n 2d ), r,s= n 3.. {y t } (1 B) d y t = u t, t = 1, 2,..., n (3.1) {u t } d = 0 I(0) I(d)(0 < d < 1/2) u t ARMA LM Breitung-Hassler u t R/S KPSS V/S 3 Lobato-Robinson 3.1 ARFIMA(p, d, q) d Robinson (1994) Tanaka (1999) d LM Tanaka (1999) Breitung and Hassler (2002) Tanaka (1999) Harris et al. (2008) Tanaka (1999) Tanaka (1999) LM Breitung and Hassler (2002)

8 Tanaka (1999) LM Tanaka (1999) LM d = 0 d > 0 Tanaka (1999) S T = n n 1 j=1 ˆρ j/j ˆω (3.2) ˆρ j {y t } j ˆω 2 Tanaka (1999) (50) ω 2 ARMA(p, q) u t u t = ε t ω 2 = π 2 /6 u t u t = φu t 1 + ε t 1 AR(1) ω 2 = π2 6 (1 φ2 ) φ 2 (log (1 φ)) 2 (3.3) (3.3) d = 0 AR(1) ˆφ ˆω 2 LM ARFIMA(p, d, q) d = 0 ARMA(p, q) d = 0 S T d > 0 S T Breitung and Hassler (2002) Breitung and Hassler (2002) Tanaka (1999) LM (1 B) d y t = ε t {ε t } d d = 0 d > 0 t 1 yt 1 = y t j /j, (3.4) j=1 y t = ρy t 1 + e t (3.5) ρ = 0 t ρ t ˆτ n = n t=2 y ty t 1 ˆσ 2 e n t=2 y2 t 1 (3.6) ˆσ e 2 (3.5) ˆτ n

9 155 AR(1) (1 φb)(1 B) d y t = ε t d AR(1) ˆφ, AR(1) ˆx t = y t ˆφy t 1 (3.4) t 1 x t 1 = ˆx t j /j, (3.7) j=1 ˆx t = ρx t 1 + ψy t 1 + e t (3.8) ρ t AR(p) Breitung and Hassler (2002) MA 3.2 R/S KPSS V/S 3 {u t } (long-run variance) R/S {y t } rescaled range (R/S) Lo (1991) R/S R/S Q n (q) = 1 k max (y j ȳ) min ŝ n,q 1 k n 1 k n ŝ 2 n,q = 1 n j=1 n (y t ȳ) t=1 k (y j ȳ), (3.9) j=1 q ω j (q)ˆγ(j), (3.10) {ω j } Bartlett ω j = 1 j/(q + 1) ˆγ(j) j=1 ˆγ(j) = 1 n j (y t ȳ)(y t+j ȳ), (3.11) n t=1 ŝ 2 n,q 2πf(0) = k= γ(k) (long-run variance) (2.1) ŝ 2 n,q Lo (1991) q Andrews (1991) q = [k n ], k n = ( ) ( 1/3 3n 2 ˆφ ) 2/3 2 1 ˆφ (3.12) 2

10 q k n ˆφ 1 q KPSS n Q n U MRS = max B(t) min B(t), (3.13) 0 t 1 0 t 1 { B(t); 0 t 1} Brownian bridge B(t) = B(t) tb(1) Brownian motion B(t) U MRS (3.13) U MRS Lo (1991) (3.9) F MRS (x) = (1 4k 2 x 2 )e 2k2 x 2, (3.14) k=1 U MRS Lo (1991, II) ( q n ) d 1 n Q n sup B d (t) inf B d (t), 0 t 1 0 t 1 { B d (t)} B d (t) = B d (t) tb d (1) fractional Brownian bridge KPSS Kwiatkowski et al. (1992) KPSS Saikkonen and Luukkonen (1993) I(0) I(1). Lee and Schmidt (1996) Lee and Amsler (1997) d = 0 (d = 1) (0 < d < 1/2) (1/2 d < 1) Lee and Schmidt (1996) Lee and Amsler (1997) ξ = 0 KPSS T n = 1 ŝ 2 n,qn 2 ( n k 2 (y t ȳ)), (3.15) k=1 t=1 T n U KP SS = 1 0 B(t) 2 dt, (3.16)

11 157 { B(t); 0 t 1} Brownian bridge Anderson and Darling (1952) (4.35) U KP SS Anderson and Darling (1952, Table 1) Anderson and Darling (1952) U KP SS U KP SS = k=1 Y 2 k π 2 k 2 N(0, 1) {Y k } U KP SS φ KP SS (θ) = ( sin ) 1/2 2iθ 2iθ (3.17) Tanaka (1996, Figure 1.2) (3.17) Tanaka (1996, Table 1.2) U KP SS Giraitis et al. (2003) ( q n ) 2d Tn 1 0 B d (t) 2 dt, { B d (t)} fractional Brownian bridge KPSS (long-run variance) Hobijn et al. (2004) KPSS Bartlett Quadratic Spectral 2 Bartlett ω j = 1 j/(q + 1) (3.10) Quadratic Spectral ω j = [ ] 25 sin (6π(j/m)/5) 12π 2 (j/q) 2 cos (6π(j/m)/5) 6π(j/m)/5 (3.10) 2 Andrews (1991) Newey and West (1994) q Quadratic Spectral 2 Hobijn et al. (2004, Table 3) q V/S Giraitis et al. (2003) KPSS V/S rescaled variance

12 M n (q) = 1 ŝ 2 n,qn 2 ( n k ) 2 ( (y t ȳ) 1 n n k=1 t=1 k=1 t=1 ) 2 k (y t ȳ), (3.18) S k = k t=1 (y t ȳ) M n S k ŝ2 n,q n M n = 1 n 1 n n j=1 (S j S ) 2 ŝ 2, n,q M n U V S = 1 0 ( 1 B(t) 2 dt 0 B(t)dt) 2, (3.19) { B(t); 0 t 1} Brownian bridge Watson (1961) (22) U V S F V S (x) = 1 + 2( 1) k e 2k2 π 2x, (3.20) k=1 U V S Kolmogorov- Smirnov F KS (x) F V S (x) = F KS (π x) Kolmogorov-Smirnov Smirnov (1948) U V S U V S Watson (1961) (16) U V S = k=1 Y 2 k + Z2 k 4π 2 k 2 N(0, 1) ({Y k }, {Z k }) Watson (1961) U V S φ V S (θ) = ( sin ) 1/2 iθ/2 iθ/2 (3.21) Giraitis et al. (2003) ( q n ) 2d Mn 1 0 ( 1 B d (t) 2 dt 0 B d (t)dt) 2,

13 Lobato-Robinson Lobato and Robinson (1998) d = 0 I(0) I(d) d Robinson (1995) Whittle Shao and Wu (2007) R/S KPSS V/S 3 Whittle Whittle d n 1 < m < n/2 m λ j = 2πj/n, j = 1,..., m, R(d) = log 1 m m I(λ) = 1 n 2 y 2πn t e itλ, (3.22) j=1 λ 2d t=1 j I(λ j ) 2d m m log(λ j ), (3.23) R(d) d Whittle Whittle R(d) R(d) d = 0 LM LM LM = t 2 m ν j = log j m 1 m j=1 log j, Ĉ0 = m 1 m j=1 I(λ j), Ĉ1 = m 1 m j=1 ν ji(λ j ) Shao and Wu (2007) (8) j=1 1/2 Ĉ1 t m = m, (3.24) Ĉ t 2 m χ2 1 t m n Whittle ˆd loc d = 0 ˆd loc = t m /2 d > 0 100α% t m α d = 0 d > 0 t m m 1 m + m5 (log m) 2 n 4 0, as n,

14 m Lobato and Robinson (1998) m = 0.06n 4/5 m = 1.2n 4/5 ( ) 4/5 3n m opt = ˆφ 4π (1 ˆφ) 2 ˆφ {y t } AR(1) y t 1 1 m m opt Lobato and Savin (1998) 2/5 3.4 Tanaka (1999) LM Breitung-Hassler, R/S KPSS V/S Lobato-Robinson 6 ARFIMA(0, d, 0) ARFIMA(1, d, 0)(φ = 0.2, 0.5, 0.8), ARFIMA(0, d, 1)(θ = 0.8) 5, d = 0 d = 0.1, 0.2, 0.3 n = 500, %, 1% 1000 Tanaka (1999) LM LM Breitung and Hassler (2002) BH R/S (mr/s) KPSS V/S q KPSS V/S Bartlett Quadratic Spectral q Hobijn et al. (2004, 3) Bartlett n 0 = [8(n/100) 1/4 ] Quadratic Spectral n 0 = [8(n/100) 2/25 ] BT, QS R/S KPSS V/S Bartlett Quadratic Spectral (3.12) Lo (1991) Bartlett Lo Lobato and Robinson (1998) L-R Lobato-Robinson m opt Ox (Doornik (2008)) 1 ARFIMA(0, d, 0) ARFIMA(0, d, 0) R/S LM Breitung-Hassler (BH) Lobato-Robinson 3 R/S KPSS V/S R/S KPSS V/S 3 Lo (1991) q R/S

15 161 1 ARFIMA(0, d, 0) LM BH mr/s KPSS V/S L-R n 5%test n = 500 n = %test n = 500 n = 1000 q Lo BT QS BT QS BT QS m opt d V/S KPSS (Lobato-Robinson ) ARFIMA(1, d, 0) AR ARFIMA(0, d, 1) MA AR φ = 0.2, 0.5, 0.8 MA θ = 0.5 q m opt ARFIMA(0, d, 0) AR AR 3 φ = 0.2, 0.5, φ = 0.2 ARFIMA(0, d, 0) LM Lobato-Robinson Breitung-Hassler (BH) 3 φ = 0.5 KPSS V/S 4 φ = 0.8

16 ARFIMA(1, d, 0)(φ = 0.2) LM BH mr/s KPSS V/S L-R n 5%test n = 500 n = %test n = 500 n = 1000 q Lo BT QS BT QS BT QS m opt d LM AR Lobato-Robinson LM 1 R/S 5 MA(1)(θ = 0.8) Breitung-Hassler 5 LM LM MA(1) Lobato-Robinson

17 163 3 ARFIMA(1, d, 0)(φ = 0.5) LM BH mr/s KPSS V/S L-R n 5%test n = 500 n = %test n = 500 n = 1000 q Lo BT QS BT QS BT QS m opt d LM AR MA Breitung-Hassler 1 Lobato-Robinson MA(1) φ φ = 0.2 LM φ φ q Bartlett Quadratic Spectral (long-run variance) KPSS V/S Bartlett Quadratic Spectral KPSS V/S

18 ARFIMA(1, d, 0)(φ = 0.8) LM BH mr/s KPSS V/S L-R n 5%test n = 500 n = %test n = 500 n = 1000 q Lo BT QS BT QS BT QS m opt d (realized volatility) (2007) 5, RV t

19 165 5 ARFIMA(0, d, 1)(θ = 0.5) LM mr/s KPSS V/S L-R n 5%test n = 500 n = %test n = 500 n = 1000 q Lo BT QS BT QS BT QS m opt d RV t = c c = n ri,t, 2 (3.25) i=1 T t=1 (R t R) 2 T n t=1 i=1 r2 i,t (3.26) R t R r i,t t 5 c (3.26) c (, Hansen and Lunde (2005), (2007) ). 1 1 (2007) (2007)

20 RV log(rv) ACF of log(rv) log(rv) of Nikkei225, 3 July June c = RV t log RV t log RV t 1 log RV t log RV t log RV t 6 log RV t Ljung-Box Ljung-Box 10 LB(10) = log RV t LM Breitung-Hassler, R/S KPSS V/S Lobato-Robinson 6 ARFIMA LM Breitung-Hassler AR(1) ARFIMA(1, d, 0) 7 5% 1% Breitung-Hassler, R/S 2 1% log RV t Breitung-Hassler R/S

21 167 6 (n = 487) LB(10) (n = 487) LM BH mr/s KPSS V/S L-R q Lo BT QS BT QS BT QS m opt % % Beran and Terrin (1996) Horvath and Shao (1999), Horvath (2001) Whittle Ling (2007) CSS (conditional sum of squares) Horvath (2001) Wald Yamaguchi (2010) Ling (2007) CSS Bai (1997) {y t, t = 1, 2,..., n} k = [nt](0 < t < 1) Horvath and Shao (1999), Horvath (2001) Whittle (σ 2, τ) S T (0, ) R r ARFIMA(p, d, q) r = p + q + 1 ARFIMA(1, d, 1)

22 τ = (d, φ, θ) Whittle (3.22) I(ω) f(ω τ) Q = 1 n 1 I(ω j ) 2n f(ω j=1 j τ) (4.1) τ ˆτ n(ˆτ τ) N (0, 4πW(τ) 1 ) W(τ) r r (i, j) w ij (τ) w ij (τ) = π π 2 1 f(ω τ) τ i τ j f(ω τ) dτ k = [nt](0 < t < 1) L 1 = 1 [nt] I(ω j ) 2n f(ω j=1 j τ) (4.2) ˆτ [nt],1 L 2 = 1 2n n 1 j=[nt]+1 I(ω j ) f(ω j τ) (4.3) ˆτ [nt],2 Ẑ n (t) = 1 { 1/2 n 1/2 t(1 t) (ˆτ [nt],1 ˆτ [nt],2 ) Ŵ(nt)(ˆτ [nt],1 ˆτ [nt],2 )} (4.4) 4π Ŵ(nt) k = [nt] Ŵ(k) = k n W(ˆτ k,1) + n k n W(ˆτ k,2) Ẑ n (t) M(t) (4.5) M(t) M(t) = ( B 1 j r (j) 2 (t))1/2 ( B (1) (t), B (2) (t),..., B (r) (t)) r Brownian bridge d M(t) = B(t) Brownian bridge B(t) = B(t) tb(1) E ε t 4+ρ, (ρ > 0) 4 Brownian bridge B(t) N (0, t(1 t))

23 169 Ling (2007) CSS 0 ARFIMA(p, d, q) (2.7) ε t = θ(b) 1 φ(b)(1 B) d y t y t 0 y t = 0 (t 0) ˆɛ t ˆɛ t = θ(b) 1 φ(b)(1 B) d y t, (4.6) y t = 0 (t 0) Conditional Sum of Squares L 1 = 1 2 k ˆɛ 2 t, L2 = 1 2 t=1 n t=k+1 ˆɛ 2 t, (4.7) τ 1 τ 2 ˆɛ 2 t/2 1 2 D t (τ) = τ ( 1 ) ( 2 ˆɛ2 t, P t (τ) = 2 τ 2 1 ) 2 ˆɛ2 t, τ 1 τ 2 ˆD t ( τ 1 ), ˆD t ( τ 2 ), ˆPt ( τ 1 ), ˆPt ( τ 2 ) ˆΣ n (k) = ˆΩ n (k) = k ˆP t ( τ 1 ) + t=1 n t=k+1 k ˆD t ( τ 1 ) ˆD t( τ 1 ) + t=1 ˆP t ( τ 2 ) n t=k+1 ˆD t ( τ 2 ) ˆD t( τ 2 ) Wald τ 1 τ 2 ˆΣ n (k) ˆΩ n (k) k(n k) W n (k) = n 2 ( τ 1 τ 2 ) [ˆΣn ] (k)ˆω n (k) 1 ˆΣn (k) ( τ 1 τ 2 ) (4.8) Ling (2007) (r < k < n r) r sup Ŵ n (r) = max W n(k) b n (r) r<k<n r a n (r) (4.9) a n (r) = b n (r)/(2 log log n), b n (r) = [2 log log n + (r log log log n)/ 2 log Γ(r/2)] 2 /(2 log log n) Γ( ) Ling (2007, Theorem 2.1) {ε t } near epoch dependence (NED) Darling-Erdös P [Ŵn(r) < x] exp ( 2e x/2 ) as n. (4.10)

24 Horvath and Shao (1999) Whittle ˆτ 1 ˆτ 2 Ẑn(t) max 1 k<n Ẑn(k/n) (4.10) sup 4.2 Lavancier et al. (2010) Giraitis et al. (2003) rescaled variance (V/S ) V/S ( n k ) 2 ( V = (y t ȳ) 1 n 2 k (y t ȳ)) (4.11) n k=1 t=1 k=1 t=1 ŝ 2 1q V/S M n (3.18) M n = V/ŝ 2 q V 1 /ŝ 2 1,q 2 V 2/ŝ 2 2,q T n T n = V 1/ŝ 2 1,q V 2 /ŝ 2 2,q + V 2/ŝ 2 2,q V 1 /ŝ 2, (4.12) 1,q Lavancier et al. (2010, Proposition 2.6) d 1 = d 2 = d U 1 U 2 U i = 1 0 T n T = U 1 U 2 + U 1 U 2, (4.13) ( 1 B i (t) 2 dt 0 B i (t)dt) 2, i = 1, 2, fractional Brownian bridge B i (t) fractional Brownian bridge 2 fractional Brownian motion (B d1 (t), B d2 (t)) 2 B d1 (t) B d2 (t) 0 d 1 d 2 T n p, (4.14) 2 1{y 1t } 2{y 2t } (long-run covariance) ŝ 2 12,q ỹ 1t = y 1t ŝ2 12,q ŝ 2 y 2t (4.15) 2,q

25 171 ỹ 1t (4.11) Ṽ1 Ṽ1 s 2 1,q = ŝ 2 1,q ŝ 2 12,q/ŝ 2 2,q Ṽ1, s 2 1,q, V 2, ŝ 2 2,q T n Ṽ 1 / s T 2 1,q n = V 2 /ŝ 2 2,q + V 2/ŝ 2 2,q, (4.16) Ṽ 1 / s 2 1,q Lavancier et al. (2010, Proposition 2.7) d 1 = d 2 T n T n (4.13) d 1 > d 2 T n d 1 < d 2 T n 1 2 T n Lavancier et al. (2010) 5% t 5% 3.7d d (4.17) V/S ARMA Lavancier et al. (2010) q Abadir et al. (2009) ARFIMA Horvath (2001) Ling (2007) Whittle ˆd CSS d Lavancier et al. (2010) (LPS) ARFIMA(0, d, 0) n = 500 n = 1000 t = 0.5 Lavancier et al. (2010) ARFIMA(0, d, 0) q = 0 d 1 = 0.1 d 2 d 2 = 0.1, d 2 = 0, 2, 0.3, d d π 2 /6 d

26 ARFIMA(0, d, 0)(d 1 = 0.1) n = 500 n = 1000 d 2 WH CSS LPS WH CSS LPS Whittle ˆd CSS d WH CSS Lavancier et al. (2010) LPS (LPS) 2 Whittle CSS ARFIMA(0, d, 0) (LPS) , 500 Lavancier et al. (2010) (LPS) Whittle CSS CSS ARFIMA(0, d, 0) 3.4 AR MA Ling (2007) Yamaguchi (2010) ARMA Lavancier et al. (2010) 5. I(0) I(d)(0 < d < 1/2) R/S KPSS V/S 3 Lobato-Robinson ARFIMA(0, d, 0) LM Lobato-Robinson R/S KPSS V/S 3 Shao and Wu (2007) I(d)

27 173 LM Lobato-Robinson KPSS V/S 2 Whittle CSS 2 V/S Lavancier et al. (2010) (2004) 2 2 (C)# Abadir, K. M., Distaso, W. and Giraitis, L. (2007). Nonstationarity-extended local Whittle estimation, J. Econom., 141, Abadir, K. M., Distaso, W. and Giraitis, L. (2009). Two estimators of the long-run variance: Beyond short memory, J. Econom., 150, Anderson, T. W. and Darling, D. A. (1952). Theory of certain Goodness of fit criteria based on stochastic processes, Ann. Math. Stat., 23, Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation, Econometrica, 59, Bai, J. (1997). Estimation of a change point in multiple regression models, Rev. Econ. Stat., 79,

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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 }