03.Œk’ì

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1 HRS KG NG-HRS NG-KG AIC

2 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 n v n τ v n w n σ n αβ τ θ = αβ τ T r n = e α/ σ n w n logσ n =βlogσ n 1 + v n α = α /β β α

3 NelsonHarvey Ruiz and Shephard θ r n = e α σ n w n logr n =α +βlogσ n + logw n logr n α logσ n logw n χ 1 logw n 1 π exp e w w (5) N η, π η logw n

4 σ n σ n σ n 1 Kitagawa and Gersch log{ 1 (r m 1 + r m)} =logσ m + logu m logu m χ log u n exp {u e u } N ζ π ζ logu m logw n σ n

5 Kitagawa logw n logu m logw n θ σ n x n logr n x n logσ n y n logr n x n =βx n 1 + v n y n = x n + α + w n w n y n = logr n x n =βx n 1 + v n r n = e α/ e x n/ w n r n r n x n r n Ne α e x n

6 x n logσ n logσ µx n x n 1 µ βµ µ var x n τ (β ) x 0 Nτ β b 1 (b) (w n 1 τ, b) = τ P (b 1/ ) ( 1/ ) (w + τ ) n b (11) b <b< bbk k k t k dispersion Nagahara and Kitagawa v n j Y j y 1 y j n x n j n j nj n j n x n p x n y n p(y n ) = p(y n x n )p(x n )dx n

7 L(θ ) = p (Y N ) = p (Y N 1 )p (y N Y N 1 ) = = N p(y n ) n=1 p( y n ) N r(θ ) = log p( y n ) n=1 DFPBFGS θ θ^ NelsonHarvey, Ruiz and Shephard p (x n ) p (x n Y n ) p (x n ) N (x n n 1,V n n 1 ) p (x n Y n ) N (x n n,v n n ) x n n 1 =β x n n 1 V n n 1 =β V n 1 n 1 +τ K n = V n n 1 (V n n 1 +ξ ) 1 x n n = x n n 1 +K n (y n x n n 1 α)

8 V n n =(I K n ) V n n 1 ξ y n ε n = y n x n n 1 α s n = V n n 1 +ξ p ( y ) n Yn 1 = 1 π s n exp ε n s n (18) r(θ ) = N log π 1 N n = 1 logs n 1 N n = 1 ε (19) s n n p (x n ) = p (x n x n 1 )p (x n 1 )dx n 1 z n =βx n 1 = p (x n z n )p (z n )dz n = p (x n z n )p (z n )dz n p ( y n x n )p(x n ) p(x n Y n ) = p ( y n ) p ( y n ) p ( y n ) = p ( y n x n )p (x n )dx n

9 N r(θ) = log p ( y n ) n=1 p(x n ) p (x n Y n ) p(v n ) τ r n log σ n log σ n = β log σ n 1 +v n = β log σ n +β v n 1 +v n β β τ log σ n = log σ n 1 log u n w n w n 1 N v n N τ log (e v nw n +w n 1

10 τ τ η ζ AIC log r n Jacobian N J HRS = log exp {y n/} = log r n N n=1 n=1 AIC AIC = r(θˆ) +J HRS + 3 z n log { 1 (r m 1 + r m )} N/ N N/ J KG = log πe zn = log π + z n n=1 n=1 AIC AIC AIC Nr n HRSHarvey, Ruiz and Shephard 1994 NG-HRS KG NG-KG Direct NG-b b v n b

11 HRS KG α η ζ α HRSτ KG τ τ α KGNG-KG τ β τˆ β NG HG Direct AICNG-KG AIC HRS AIC NG-HRSNG-KG NGv n b b t AIC τ τ

12 αβ AIC β αβ AIC log σ n AICNG-HRS log σ n α log σ n log σ n α w r n e α σ n

13 log σ n α HRS HRS NG-HRS KG NG-HRS HRS KG NG-KG NG-HRSlog σ n α

15 KGNG-HRS NG- HRS KGNG-HRS NG-1.0 b n NG-HRS

16 α β τ N ˆα ˆβ τ α β τ HRS > KG> NG-HRS = NG-KG β HRS KGα β β τ α HRS KG β β β α β β NG-HRS NG-KGKGβ HRS τ τ τ τ τ τ τ τ τ

17 β β β β τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ NG-HRS NG-KGKG β HRS

18 AIC NG-HRS NG-KG NG-KGNG-KG KG β β β NG-HRS AIC

19 x n x min <x n <x max K x min = t 1 <t <...<t K = x max t = t i t i 1 t k tt k + t p n ( k ) f n ( k ) t + n = k t p n Yn 1 t k t p ( k ) ( x ) dx n t + n = k t t k t f ( k ) p ( x n Yn ) dx n (30) j k k = t + ( 1/) v ( j ) p (v) dv ( t 1/) (31) p(x n 1 ) p(z n ) p(x n 1 ) f n 1 (1),, f n 1 (K) a) g(k) = 0 k =1,..., K) b) j =1,..., K m = ( β t j t 1 )/ t g(m) = g(m) + f n 1 ( j) ( m+1 β t j / t) g(m+1) = g(m+1) + f n 1 ( j) (βt j / t m) p(z n ) p(x n ) j =1,..., K K p n ( j ) = g( j i )v (i) j= K p (x n ) p(x n Y n ) fn( j ) = P n( j ) r ( yn x n) f f n (k ) n( j ) = N i 1 f n(i ) (33) r (y n x n )

20 FORTRAN Blattberg, R. C. and Gonedes, N. J., A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices, Journal of Business, 47, 44-80, Bollerslev, T., Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 31, , Engle, Robert F., Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of U. K. Inflation, Econometrica, 50, , 198. Fama, E. F., The Behavior of Stock Market Prices, Journal of Business, 38, , Harvey, A. C., E. Ruiz and N. Shephard, Multivariate Stochastic Variance Model, Review of Economic Studies, 61, 47-64, and Shephard, N., Estimation of an Asymmetric Stochastic Volatility Model of Asset Returns, Journal of Business & Economic Statistics, Vol. 14, No. 4, , Heynen, R. C. and Kat, H. M., Volatility Prediction: A Comparison of the Stochastic Volatility, GARCH (1, 1), and EGARCH (1, 1) Models, The Journal of Derivatives, Winter, 50-65, Jacquier, E. and Polson, N and Rosi, P. E., Bayesian Analysis of Stochastic Volatility Models (with discussion), Journal of Business & Economic Statistics, Vol. 1, No. 4, , Kariya, T., Tsukuda, Y., Maru, J., Matsue, Y. and Omaki, K., An extensive analysis on the Japanese markets via S. Taylor's model, Financial Engineering and the Japanese Markets, Vol., No. 1, 15-86, Kitagawa, G., Non-Gaussian State-Space Modeling of Nonstationary Time Series (with discussion), Journal of the American Statistical Association, Vol. 8, No. 400, , and Gersch, W., A smoothness priors time varying AR coefficient modeling of nonstationary time series, IEEE Trans. on Automatic Control, AC-30, 48-56, and, Smoothness Priors Analysis of Time Series, Springer-Verlag, New York, Mandelbrot, B., The Variation of Certain Speculative Prices, Journal of Business, 36, , Nelson, D. B., Time Series Behaviour of Stock Market Volatility and Returns, Unpublished Ph.D. Dissertation, 1988., Conditional Heteroskedasticity in Asset Returns : A New Approach, Econometrica, 59, , Nagahara, Y., Non-Gaussian Distribution for Stock Returns and Related Stochastic Differential Equation, Financial Engineering and the Japanese Markets, Vol. 3, No., , and Kitagawa, G., Non-Gaussian Stochastic Volatility Model, Journal of Computational Finance, forthcoming, 1999.

01.Œk’ì/“²ﬁ¡*

01.Œk’ì/“²ﬁ¡* AIC AIC 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