01.Œk’ì/“²ﬁ¡*

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1 AIC AIC

2 y n r n = logy n = logy n logy n ARCHEngle r n = σ n w n logσ n 2 = α + β w n 2 () r n = σ n w n logσ n 2 = α + β logσ n 2 + v n (2) w n r n logr n 2 = logσ n 2 + logw n 2 logσ n 2 = α +β logσ n 2 + v n (3)

3 y n = T n + σ n ε n, ε n N(0,) (4) T n n ε n σ n y n σ n σ n σ 2 n Kitagawa and GerschT n k k T n = e n, e n N(0,τ 2 ) (5) σ 2 n logσ 2 n r r logσ 2 n = e 2n, e 2n N(0,τ 2 2 ) (6) e n e 2n τ 2 τ 2 2 k τ 2 rτ 2 2 k r y n = T n + p n +σ n ε n, ε n N(0,) (7) T n σ n ε n p n AR

4 p n = a j p n j + e 3n, e 3n N(0,τ 2 3 ) j= e 3n τ 2 3 k + +r x n = (T n,,t n k+ p n,, p n + logσ 2 2 n,, logσ n r+ ) Τ (9) T n p n logσ 2 n k + +rk + +rfk + +rg k k c c c 2 k k a a a F = 0 0, G = r r r c c2 Cr x n = Fx n + Gv n ()

5 v n v n = (e n,e 2n,e 3n ) Τ (2) Q=diag τ 2 τ 2 2 τ 2 3 T n p n logσ 2 n x n y n f y n = T n + p n +σ n ε n = f (x n, ε n ) (3) x n v n ε n σ n x n logσ 2 n σ n ε n y n j Y j ={y y j }n x n j nj n j n j n Y j x n T n p n σ 2 n σ n x n x n k + x n k k+ + x n k+ +

6 Kitagawa Kitagawa p(x n Y n ) p(x n Y n ) p(v n ) x n v n p(v n ) p(v n ) V () n V () n V () n V () n j= P n (x) = I (x;v n (j) ) x P(x) = p (t) dt I (x;a) 0 I(x;a) = x< a x a p(x n Y n ) P (j) n p(x n Y n ) F (j) n V n Kitagawa ( j F ) 0 p (x 0 Y 0 ) ( j V ) n p(v) ( j P ) n = f (F ( n, j ) ( j n ) α (j) ( j n = p( y n x n = P ) n ) ( j {P ) ( j n } {F ) n }

7 p(x 0 Y 0 ) N n = ( P j) n ( S j) n Kitagawa ( P j) ( n F j) n ( S j) n k + +rk +k + + T n L (j) (j) P n L logσ 2 ( j) n L { T () () n L,, T n L },{P() n L,, P() n L }{logσ2(),, n L logσ 2() n L } T n L (j) j L =n n N j = ( ) ( I x j ) σ nl j = j = ( j ) j I ( x; T nl ) I ( x; PnL) ( ; ) ( j ) T nl ( j ) P nl σ ( j) nl j = j = j = x n p(x n Y n ) y n p(y n Y n ) = p(y n x n )p(x n Y n )dx n p(x n Y n ) P () ( n P ) n

8 p( yn Yn ) p( yn xn) ( j ) I( x P dx n; n ) j = ( j ) = p Pn Yn j = ( ) n N Y N ={y y N } L(θ ) = p (Y N ) = p (Y N )p ( y N Y N ) = = N p (y n Y n ) n= p (y n Y n ) N r( θ ) = log p ( y n Y n ) n= N n= j= log p( y P ) n ( j ) n DFPBFGS θ θ τ 2 τ 2 2 τ 2 3 zn = xn θ

9 DECOMP y n = t n + w n t n =2t n t n 2 + v n w n v n σ 2 τ 2 σ 2 = τ 2 = AIC a cy n t n w n

11 AICAR w n f= DECOMPM 2 = AR y n = t n + p n + w n AR p n w n p n w n

13 k =r= τ 2 = τ 2 2 =AIC= AIC= σ n

14 y n = t n + σ n w n

15 T n = 2T n T n 2 + e n T n T n = T n T n T n = T n + (T n T n 2 ) + e n = T n + T n + e n T n = T n T n =(T n + T n + e n ) T n = T n + e n T n = T n + T n + e n T n = T n + e n Harvey Tn xn = Tn F =,, 0 G = e n4 T n = T n +δt n + e n + e n2 δ T n = δ T n + e n e n e n2 δ T n n T n T n T n

17 e n e n2 τ 2 τ 2 2 τ p(v τ 2 ) = π v 2 + τ 2 v Kitagawa and Matsuoto Pearson 2b (v τ 2 τ Γ (b) P, b) = Γ (b 2 ) Γ ( 2 ) (v + τ ) 2 2 b (32) b Kitagawa p(v) = ( α )φ (v) + αφ 2 (v) α φ φ 2 φ φ 2

18 φ 2 (v)u( d, 0) Pearson Nagahara AIC = AIC AIC = T n w n

19 y n = t n + p n + σ n w n

21 logs n 2 = logs n 2 - +b(dt n- ) + u n

22 β (x)x aβ (x) = cx c = 0 b x x x x y n = T n + p n + σ n w n T n = T n +δt n + v sn + v rn δt n = δt n + v sn logσ n 2 = logσ n 2 + β(δ T n ) + e 2n p n = a j p n j + e n j= AIC

24 logσ n 2 = logσ n 2 + u n u n ( β (δt n ))φ 0 (u) + β (δt n ) φ (u) β (x)δt n β (x) β (x) δt n logσ 2 n AIC

28 Akaike, H. et al., TIMSAC-78, Coputer Science Monographs, No., The Institute of Statistical Matheatics, 985. Harvey, A. C., Forcasting, Structural Tie Series Models and the Kalan Filter, Cabridge University Press, 989. and Shephard, N., Estiation of an Asyetric Stochastic Volatility Model of Asset Returns, Journal of Business & Econoic Statistics, 4 (4), 996, pp Jacquier, E., Polson, N and Rossi, P. E., Bayesian Analysis of Stochastic Volatility Models, with discussion, Journal of Business & Econoic Statistics, 2 (4), 994, pp Kitagawa, G., Non-Gaussian State-Space Modeling of Nonstationary Tie Series, (with discussion), Journal of the Aerican Statistical Matheatics, 82 (400), 987, pp , Monte Carlo Filter and Soother for Non-Gaussian Nonlinear State Space Models, Journal of Coputational and Graphical Statistics, 5 (), 996, pp.-25., Self-organizing State Space Model, Journal of the Aerican Statistical Association, 93 (444), 998, pp and Gersch, W., A Soothness Priors-State Space Modeling of Tie Series With Trend and Seasonality, Journal of the Aerican Statistical Association, 79(386), 984, pp and Matsuoto, N., Detection of Coseisic Effect Fro Underground Water Level, Journal of the Aerican Statistical Association, 9 (434), 996, pp Nagahara, Y., Non-Gaussian Distribution for Stock Returns and Related Stochastic Differential Equation, Financial Engineering and the Japanese Market, 3 (2), 996, pp and Kitagawa, G., Non-Gaussian Stochastic Volatility Model, Journal of Coputational Finance, 2 (2), 999, pp

03.Œk’ì

03.Œk’ì HRS KG NG-HRS NG-KG AIC Fama 1965 Mandelbrot Blattberg Gonedes t t Kariya, et. al. Nagahara ARCH EngleGARCH Bollerslev EGARCH Nelson GARCH Heynen, et. al. r n r n =σ n w n logσ n =α +βlogσ n 1 + v n w