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1 1 Hi-Stat :30-18: COE 4 COE RA ged0104@srv.cc.hit-u.ac.jp S-PLUS S-PLUS S-PLUS S-code 2 [8] [8] [8] 1

2 2 ARFIMA(p, d, q) FI(d) φ(l)(1 L) d x t = θ(l)ε t ({ε t } IID(0,σ 2 )). (1) MA( ) AR( ) x t = ψ j ε t j, π j x t j = ε t, ψ 0 = π 0 =1, (2) ARMA f(λ) {x t } T t=1 x Var( T x) 2πf(0), T (x µ) d N(0, 2πf(0)), as T, (3) x = O p (1/ T ) 1 ARFIMA O p (1/ T ) ARFIMA(p, d, q) x = O p (T d 1/2 ) d (0, 1/2) O p (1/ T ) [1] [3] 1.2 ARMA ARFIMA(p, d, q) σ 2 d, φ 1,φ 2,...,φ p,θ 1,θ 2,...,θ q β =(d, φ 1,φ 2,...,φ p,θ 1,θ 2,...,θ q ) ARFIMA(p, d, q) {x t } x n =(x 1,...,x n ) L n (β, σ 2 ) L n (β, σ 2 1 )= (2π) Γ n(β, σ 2 ) 1/2 exp ( 12 ) x nγ n/2 n (β, σ 2 ) 1 x n. (4) Γ n (β, σ 2 ) x n β, σ 2 2

3 L n(β,σ 2 ) Γ n(β,σ 2 ) n 500 S-PLUS Whittle (2) ψ j β ψ j (β) g(λ; β) g(λ; β) = ψ j (β)e ijλ I n (λ) I n (λ) = n l=1 x te itλ 2 2πn f(λ) U n (β, σ 2 )= 1 2 log σ2 1 2πI n (λ j ) 2σ 2 ng(λ j ; β) j (5) β, σ 2 Whittle log L n (β, σ 2 ) (5) 1 2 log Γ n (β, σ 2 ) /2 x n Γ n(β, σ 2 ) 1 x n / CSS (2) π j β π j (β) t 1 ε t (β) = π j (β)x t j CSS S(β, σ 2 ) S n (β, σ 2 )= n 2 log 2π n 2 log σ2 1 2σ 2 ε t (β) 2 (6) t=1 β, σ 2 CSS Q n (β) = ε t (β) 2 t=1 β β ε t ( β) = t 1 π j( β)x t j σ 2 σ 2 = 1 n ε t ( β) 2 t=1 β Q n (β) 0 t 0 x t =0 3

4 x {x t x} {x t} 1/ n FI(d) 3 1/ n 0 6/π 2 Whittle CSS d 1/2 d [8] p.154- ARIMA 2 S-PLUS S-PLUS S-PLUS arima.fracdiff.sim arima.fracdiff arima.fracdiff.var arima.fracdiff [5] 10 HP [1] S-PLUS ARMA(p, q) [2] arima.fracdiff S-PLUS [1] Whittle Gauss-Newton arima.fracdiff arima.fracdiff.var Godfrey COE 1 (1 0.2L)(1 L) 0.3 x t =(1 0.4L)ε t,

5 2 NA ARFIMA(1,d,1) 3 #1# generate a fractionally-differenced ARIMA(1,d,1) model given initial values ts.sim <- arima.fracdiff.sim(model = list(d =.3, ar =.2, ma =.4), n = 3000) #2# estimate the parameters in an ARIMA(1,d,1) model for the simulated series fd.out <- arima.fracdiff(ts.sim, model = list(ar = NA, ma = NA)) #3# modify the covariance estimate by changing the finite-difference interval arima.fracdiff.var(ts.sim, fd.out, h =.0001) # arima.fracdiff d 1/2 "d.range" # d 1/2 FI(d) # 100 exact # "M" ARIMA(p,d,q) # exact # # arima.fracdiff > # ########## d=-0.40 ########### # arima.fracdiff d 0 # FI(-0.40) 200 FI(-0.40) > FIdm040 <- FIdm040.gen(500)[301:500] # arima.fracdiff exact > arima.fracdiff(fidm040, model = list(ar = NULL, ma = NULL), d.range=c(-1,.5), M=200)$model$d [1] # arima.fracdiff > arima.fracdiff(fidm040, model = list(ar = NULL, ma = NULL), d.range=c(-1,.5))$model$d [1] > > ########## d=1.40 ########### # d=1.40 # FI(1.40) 200 FI(1.40) > FId140 <- FId140.gen(500)[301:500] > FId140diff <- diff(fid140) # # arima.fracdiff exact > arima.fracdiff(fid140diff, model = list(ar = NULL, ma = NULL), d.range=c(-1,.5), M=200)$model$d [1] # arima.fracdiff > arima.fracdiff(fid140diff, model = list(ar = NULL, ma = NULL), d.range=c(-1,.5))$model$d [1] # > CSSE.FId.nocat(1.40,FId140) [1] # > CSSE.FId.nocat(0.40,FId140diff) [1] d = I(1) > FId100.gen <-function(n){cumsum(rnorm(5*n+100))[(5*n n + 1):(5*n+100)]} > FId100 <- FId100.gen(200) > FId100diff <- diff(fid100) > CSSE.FId.nocat(1.00,FId100) [1] FI(1.40) FId140.gen [1] 100 5

6 > CSSE.FId.nocat(0.00,FId100diff) [1] > arima.fracdiff(fid100diff, model = list(ar = NULL, ma = NULL), d.range=c(-1,.5))$model$d [1] > arima.fracdiff(fid100diff, model = list(ar = NULL, ma = NULL), d.range=c(-1,.5), M=200)$model$d [1] ( 1) H 0 : I(0) vs H 1 : I(1), ( 2) H 0 : I(1) vs H 1 : I(0) (7) d =1 d =0 2 d (0, 1) ( 3) H 0 : d =0 vs H 1 : d 0 (8) LM [7] 3 LM LM LM f(λ) f(λ) =Cλ 2d, (λ d <1/2 C ) (9) Whittle λ j ( π, π) I j R(d) ( ) R(d) 2 /( LM s = m 2 ) R(d) d d=0 d 2 (10) d=0 ( ) k LM s = m(ĉ 1 /Ĉ 0 ) 2 where Ĉ k = 1 m log j 1 m log i I j,k=0, 1; λ j = 2πj m m n i=1 (11) LM m n 1 χ 2 χ 2 ARFIMA f(λ) Cλ 2d, (as λ 0), 6

7 3.2 LM ARFIMA(p, d, q) CSS L n (β, σ 2 ) S n (β, σ 2 ) 3 CSS S n (β, σ 2 ) ARMA γ S n (d, γ, σ 2 ) ( LM p = 1 S n (d, γ, σ 2 ) n d d=0,γ=bγ,σ2 =bσ 2 ) / ( 2 S n (d, γ, σ 2 ) d 2 LM p where ρ(j) = 2) 1/2 (12) d=0,γ=bγ,σ2 =bσ LM p = n 1 / 1 n j ρ(j) ω, (13) n j t=1 ε t ε t+j t 1 n, ε t = ε t ( β) = π j ( β)x t j (14) t=1 ε2 t ω d CSS d n( d d) d N(0,ω 2 ), as n. ω [7] LM p ( 3 ) H 0 : d =0 vs H 1 : d>0 ( d<0) (15) p = q =0FI(d) (3) [7] LM p = n 1 / 1 π n j ρ(j) 2, where ρ(j) = 6 l=1 n j t=1 ε t ε t+j n, ε t = x t, (16) t=1 ε2 t LM p (3) Godfrey [4] ( i 1 ) ε i l X = n, l ε i j φ(l) ε i j θ(l) 1 p q (17) X n (1 + p + q) (ε 1 (β),ε 2 (β),...,ε n (β)) β d =0 ARMA ε t =0fort 0 Î 1 = n σ 2( X X ) 1 where σ 2 = 1 n ε 2 t. (18) β Î 1 (1, 1) ω t=1 7

8 / nr 2 = n ε X(X X) 1 X ε ε ε, where ε =( ε 1, ε 2,..., ε n ) (19) 1 χ 2 1 χ 2 ARMA 4 ARFIMA(p, d, q) 4.1 MA( ) {x t } x t = ψ j ε t j (20) {ε t } NID(0,σ 2 ), ψ2 j <, γ x (j) j γ x (j) β x (j) j 2d 1 (where d<0.5 and β x (j) <M for some constant M>0). (21) {x t } p t=1 x p+h ˆx (B) p+h ˆx (B) p+h =[x(p) ] Φ=Φ 1 x 1 + +Φ p x p (22) Φ 1, Φ 2,...,Φ p E[x p+h (Φ 1 x 1 + +Φ p x p )] 2 Φ 1, Φ 2,...,Φ p (23) x (p) =(x p,...,x 1 ), Φ=(Φ p,...,φ 1 ) =Σ 1 p γ (p) h, Σ p = E[x (p) [x (p) ] ]=[(γ x (i j))] p i,, γ (p) h = E[x p+h x (p) ]=(γ x (h),...,γ x (p + h 1)), (24) σp 2 (h) σp 2 (h) E[x p+h ˆx (B) p+h ]2 = γ x (0) + [γ (p) h ] Σ 1 p γ(p) h (25) Σ p [1] section 8.7 ). (23) h L 2 h 8

9 n + p + h 1 (x n h+2,...,x 0,x 1,...,x p ) n x p+h (x 1,...,x p ) Φ : y = XΦ+ξ (26) y p-vector y =(x n+p+1,...,x p ), X n p (i, j) x p h n+1+i j ξ n-vector ξ =(ξ n+p+1,...,ξ p ), ξ t Φ(L)x t, Φ(L) =1 Φ p L h Φ 1 L p+ (27) ξ t AR(p) ˆΦ ˆΦ =[X X] 1 X y (28) ˆx (B) p+h Φ (28) (28) p n (25) p R.J.Bhansali Bhansali HP J.Hidalgo [6] Φ 0 0 n p = n ˆx (B) n+h p p 10 ˆΦ p ˆΦ Φ ˆΦ p trade-off 4.2 AR( ) MA( ) : π j y t j = ε t, and y t = ψ j ε t j, t =0, ±1,..., (29) π2 j, ψ2 j <, π 0 = ψ 0 =1 y n,y n 1,... y n+h y n (h) y n (h) = c j (h)y n+1 j = ψ j ε n+h j, c j (h) = ψ i π j+h i 1 for j 1, (30) j=h 2 Bhansali.html i=0 9

10 [2] (A5.2.3) σ 2 y(h) [ ] 2 σy(h) 2 E y n+h y n (h) = E ψ j ε n+h j 2 = σ 2 ψj 2. (31) ARFIMA AR( ) MA( ) h n {x t } n t=1 n (1) ARFIMA(p, d, q) 0 ε t = x t =0,t 0 0 x n (h) (30) 0 y t x t x n (h) = c j (h)x n+1 j = n+ j=h (31) ψ j ε n+h j, c j (h) = [ ] 2 σx(h) 2 E x n+h x n (h) = E ψ j ε n+h j 2 i=0 ψ i π j+h i 1 for j 1. (32) = σ 2 ψj 2. (33) (32) c j (h) d ARMA 2 β c j (h, β) π j ψ j π j (β) ψ j (β) CSS β β CSS x n (h) x n (h) = c j (h, β)x n+1 j = n+ j=h { ε t } n t=1 ε t = t 1 π j( β)x t j. ψ j ( β) ε n+h j (34) bx n(h) n (33) σx(h) [1] Beran, J.: Statistics for Long-Memory Processes. Chapman and Hall, London, [2] Box, G.E.P. and Jenkins, G.M.: Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco, [3] Chung, C.-F. and Baillie, R.T.: Small sample bias in conditional sum-of-squares estimators of fractionally integrated ARMA models. Empirical Economics 18 (1993),

11 [4] Godfrey, L.G.: Misspecification Tests in Econometrics: The Lagrange Multiplier Principle and Other Approaches. Cambridge: Cambridge University Press, (1988). [5] Haslett, J. and Raftery, A.E.:. Space-time modelling with longmemory dependence: Assessing Ireland s wind power resource (with Discussion). J. of the Royal Stat. Soc., series C-Applied Statistics, (1989), 38:1?50. [6] Hidalgo, J. and Yajima, Y.: Prediction and signal extraction of strongly dependent processes in the frequency domain. Econometric Theory 18 (2002), [7] Tanaka, K.: The nonstationary fractional unit root. Econometric Theory 15 (1999), [8] :. 8 (2003),

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