Working Paper Series No March 2012 日本の商品先物市場におけるボラティリティの長期記憶性に関する分析三井秀俊 Research Institute of Economic Science College of Economics, Nihon Un

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1 Working Paper Series No March 2012 日本の商品先物市場におけるボラティリティの長期記憶性に関する分析三井秀俊 Research Institute of Economic Science College of Economics, Nihon University

2 2012 3, FIGARCH, FIEGARCH.,,, ( ),., Student-t,, skewed-student t.,,.,. 1,, (commodity).,,,., 1)., (ETF)., 2)., (volatility).,, mitsui.hidetoshi@nihon-u.ac.jp 1), Gorton and Rouwenhorst (2006). 2) (2010),,.,.,, (2004). 1

3 3).,,.,. (1996),,,,,,,.,,. (2000),,,,, 2.,,. (2000), Schwartz (1997) Ornstein-Uhlenbeck (Convenience yield )..,. (2000),, Bollerslev (1986) GARCH Glosten et al. (1993) GJR., GARCH, GJR 4) (2006),,,,, Granger.,,,,, Granger,. (2008), 3), Engle (1982) ARCH (Autoregressive Conditional Heteroskedasticity), Bollerslev (1986) GARCH (Generalized ARCH), Nelson (1991) EGARCH (Exponential GARCH)., Tsay (2010). 4) (2010), GJR WTI (West Texas Intermediate).. 2

4 , Russell and Engle (2005) ACM-ACD 5) 6).,.,,., (long memory). (short memory) (2000) GARCH GJR.,,.,.,. Baillie et al. (1996) FIGARCH (Fractionally Integrated GARCH) Bollerslev and Mikkelsen (1996) FIEGARCH (Fractionally Integrated Exponential GARCH)., (Nikkei Commodity Futures Index; NCF Index), (TG Index), (Nikkei-TOCOM Commodity Index; NTC Index), ( ).,.,., ( ).,. (1),,, ( ),,,. (2). (3).. 2, FIGARCH FIEGARCH,. 3, FIGARCH FIEGARCH. 4,. 5) Autoregressive Conditional Multinomial-Autoregressive Conditional Duration. 6), (2005), (2008). 3

5 2 2.1 FIGARCH FIEGARCH t R t. Index t t t R t R t = Index t Index t 1 Index t (2.1), S t t t R t, R t. R t = S t S t 1 S t (2.2) R t = μ + λσ t + ɛ t, (2.3) ɛ t = σ t z t, σ t > 0, (2.4) z t i.i.d., E[z t ]=0,Var[z t ]=1. (2.5), (2.3) μ λ (risk premium),ɛ t i.i.d. (independent and identically distributed) E[ ] Var[ ], 7), Baillie et al.(1996) FIGARCH Bollerslev and Mikkelsen (1996) FIEGARCH. FIGARCH(p, d, q), σt 2 8). { σt 2 = ω [1 β(l)] [1 β(l)] 1 φ(l)(1 L) d} ɛ 2 t. (2.6) 7), (2003), (2007). 8),. X σt 2 = ω + ψ il i ɛ 2 t = ω + ψ(l)ɛ 2 t, 0 d 1., ω = ω [1 β(l)] 1, ψ(l) =1 [1 β(l)] 1 φ(l)(1 L) d. i=1 4

6 , β(l) =β 1 L + β 2 L β p L p, φ(l) =[1 α(l) β(l)](1 L) 1, α(l) = α 1 L + α 2 L α p L p., L (Lag operater), L i y t = y t 1, (i =0, 1...). (1 L) d,. (1 L) d = k=0 =1+ Γ(d +1) Γ(k + 1)Γ(d k +1) Lk k=1 d(d 1) (d k +1) ( L k ). (2.7) k!, Γ( ) (gamma function) 9). (1 L) d d. 0<d<1, σt 2., 0<d<0.5, 0.5 d<1. d =1, σt 2. d =0, Bollerslev (1986) GARCH (p, q) 10)., FIGARCH(1,d,1). { σt 2 = ω [1 β 1 (L)] [1 β 1 (L)] 1 φ 1 (L)(1 L) d} ɛ 2 t. (2.8) FIGARCH (1,d,1) σ 2 11). ω>0, β 1 φ 1 2 d ( 3, d φ 1 1 d ) β 1 (φ 1 β 1 + d). (2.9) 2, φ 1 =0 FIGARCH (1,d,0). FIEGARCH (p, d, q), σt 2. ln (σ 2 t )=ω + φ(l) 1 (1 L) d [1 + α(l)]g(z t 1 ), (2.10) g(z t 1 )=θz t 1 + γ[ z t 1 E z t 1 ] (2.11) g(z t 1 )= { 9). (θ + γ) z t 1 γe( z t 1 ), if z t 1 > 0, ( θ + γ) z t 1 γe( z t 1 ), if z t 1 < 0. Γ(ν) = Z 0 x ν 1 e x dx, for ν > 0. 10) GARCH (p, q), σ 2 t. σ 2 t = ω + qx i=1 α iɛ 2 t i + px j=1 β jσ 2 t j. L GARCH(p, q),. σ 2 t = ω + α(l)ɛ 2 t + β(l)σ 2 t. 11), Baillie et al. (1996). 5

7 ,. θ<0,,., ω, β, α, θ, γ. d =0, Nelson (1991) EGARCH (p, q) 12)., FIGARCH (p, d, q) FIEGARCH (p, d, q) p, q, p =1, q =0 13), FIGARCH (1,d,0), FIEGARCH (1,d,0). FIGARCH (1,d,0),. { σt 2 = ω [1 β 1(L)] [1 β 1 (L)] 1 (1 L) d} ɛ 2 t. (2.12), FIEGARCH (1,d,0). ln (σ 2 t )=ω +[1 β 1 (L)] 1 (1 L) d g(z t 1 ), (2.13) g(z t 1 )=θz t 1 + γ[ z t 1 E z t 1 ].. 2.2, Mandelbrot (1963), Fama (1965) (normal distribution) (fat tail) 14). ARCH,.,, z t, 12) EGARCH (p, q), σ 2 t. ln(σ 2 t )=ω + px β j ln(σ 2 t j)+ qx j=1 i=1 α i [θz t i + γ ( z t i E( z t i ))]. L EGARCH(p, q),. ln (σt 2 )=ω +[1 β(l)] 1 [1 + α(l)]g(z t 1), g(z t 1) =θz t 1 + γ[ z t 1 E z t 1 ]. 13), (2005, 2006), ( ) (2008), ( )(2009)., FIEGARCH(1,d,0). 14) (1998),,,, 13.,. 6

8 (standard normal distribution), Student-t (standardized Student-t distribution), (GED: Generalized Error Distribution), skewed-student t (standardized skewed Student-t distribution) 15). (i) Student-t : Student-t f (t) (z t ; ν). Γ((ν +1)/2) f (t) (z t ; ν) = Γ(ν/2) π(ν 2) ( ) (ν+1)/2 1+ z2 t, ν > 2. (2.14) ν 2, ν (degree of freedom). Student-t 0, ν>4 3 16)., ν. (ii) GED: GED f (GED) (z t ν). f (GED) (z t ν) = ν exp ( 1 2 z t/λ ν ν), λ ν 2 (1+ 1 ν ) Γ(1/ν) ν > 0, (2.17) λ ν = Γ(1/ν)2 ( 2/ν) Γ(3/ν), ν. ν =2 z t. ν<2 17), ν>2 18). (iii) skewed-student t : skewed-student t f (skt) (z t ν, ξ). f (skt) (z t ν, ξ) = Γ((ν +1)/2) Γ(ν/2) π(ν 2) ( 2s )(1+ (sz t + m) 2 ) (ν+1)/2 ξ 2It, ν > 2. ξ +1/ξ ν 2 (2.18) 15) ARCH, Bollerslev et al. (1994) t (generalized t dstribution) Michelfelder (2005) skewed GED (SGED)., Knight and Stachell (eds.) (2001), Rachev (ed.) (2003). 16) ν Student-t K t, K t 3(ν 2) = ν 4 (2.15) =3+ 6 ν 4, ν > 4 (2.16)., 3. 17) ν =1 z t, double exponential distribution, Laplace distribution. 18) ν = z t, ( 3, 3) (uniform distribution). 7

9 , I t = { 1 if z t m s 1 if z t < m s (2.19)., ν. ξ,., m = Γ((ν +1)/2) ( ν 2 ξ 1 ), (2.20) πγ(ν/2) ξ ( s = ξ + 1 ) ξ 1 m 2 (2.21). ξ =1,, ln(ξ) =0 Student-t. ξ>1,, ln(ξ) > 0., ξ<1,, ln(ξ) < 0. z t, Student-t, GED, skewed-student t (2.3) z t 19). z t i.i.d.n (0, 1) (2.22) z t i.i.d.t (0, 1,ν) (2.23) z t i.i.d.ged (0, 1,ν) (2.24) z t i.i.d.skt (0, 1,ν,ξ) (2.25) 2.3 Θ, FIGARCH (1,d,0) Θ=(μ, λ, ω, d, β 1 ), Student-t, GED ν Θ=(μ, λ, ω, d, β 1, ν), skewed-student t ξ Θ=(μ, λ, ω, d, β 1, ν, ξ)., FIGARCH(1,d,0), FIGARCH (1,d,0) θ γ 20).. L(Θ) = f(r 1,R 2,,R T Θ) T ( ) 1 ɛt = f. (2.26) σ t t=1 19), Bauwens and Laurent (2005). 20) Θ=(μ, λ, ω, d, β 1 θ, γ), Student-t, GED Θ=(μ, λ, ω, d, β 1,θ,γ,ν), skewed-student t Θ=(μ, λ, ω, d, β 1,θ,γν,ξ). σ t 8

10 ,, T T ln L(Θ) = ln(σ t )+ ln f t=1 t=1 ( ɛt σ t ) (2.27).,, Student-t, GED, skewed-student t ln L (n), ln L (t), ln L (GED), ln L (skt). ln L (n) = 1 T 2 t=1 { ( ν +1 ln L (t) = T ln Γ 2 ln L (GED) = T t=1 [ ln(2π) + ln(σ 2 t )+zt 2 ], (2.28) [ ( ) ν ln 1 λ ν 2 ) ( ν ) ln Γ 1 } 2 2 ln[π(ν 2)] 1 2 ν z t λ ν { ( ) ν +1 ( ν ) ln L (skt) = T ln Γ ln Γ T { ln(σt 2 )+(ν +1)ln 2 t=1 ( 1+ 1 ) ln (2) ln Γ ν T t=1 ( )] [ln(σ 2t ) + (1 + ν)ln 1+ z2 t, ν 2 ( ) 1 12 ] ν ln(σ2t ), ( ) } 1 2 ln[π(ν 2)] + ln 2 ξ ln(s) ξ [1+ (sz t + m) 2 ν 2 (2.29) (2.30) ξ 2It ]}. (2.31) FIEGARCH (1,d,0), (2.13), Student-t, GED, skewed-student t E( z t ) (n),e( z t ) (t),e( z t ) (GED), E( z t ) (skt). E( z t ) (n) = 2/π, (2.32) E( z t ) (t) = 2Γ((1 + ν)/2) ν 2, πγ(ν/2) (2.33) E( z t ) (GED) =2 (1/ν) λ ν Γ(2/ν) Γ(1/ν), (2.34) E( z t ) (skt) = 4ξ2 Γ((1 + ν)/2) ν 2. (2.35) ξ +1/ξ πγ(ν/2), G@RCH 4.2 OxMetrix 21). 21), Doornik (2006), Laurent and Peters (2006), (2007), (2010), Xekalaki and Degiannakis (2010). 9

11 2.4,, 2.1 FIGARCH (1,d,0) FIEGARCH (1,d,0).,, 2.2,, Student-t, GED, skewed-student t FIGARCH(1,d,0)-n (2.3) (2.5), (2.12), (2.22) 22). 2. FIGARCH(1,d,0)-t (2.3) (2.5), (2.12), (2.23). 3. FIGARCH(1,d,0)-GED (2.3) (2.5), (2.12), (2.24). 4. FIGARCH(1,d,0)-skt (2.3) (2.5), (2.12), (2.25). 5. FIEGARCH(1,d,0)-n (2.3) (2.5), (2.11), (2.13), (2.22) 23). 6. FIEGARCH(1,d,0)-t (2.3) (2.5), (2.11), (2.13), (2.23). 7. FIEGARCH(1,d,0)-GED (2.3) (2.5), (2.11), (2.13), (2.24). 8. FIEGARCH(1,d,0)-skt (2.3) (2.5), (2.11), (2.13), (2.25). -n, -t, -GED, -skt,,, Student-t, GED, skewed- Student t 3 3.1,, 24), 25) 22), FIGARCH(1,d,0)-n,. R t = μ + λσ t + ɛ t, ɛ t = σ tz t, σ t > 0, z t i.i.d.n (0, 1), σ 2 t = ω [1 β 1(L)] 1 + n 1 [1 β 1(L)] 1 (1 L) do ɛ 2 t. 23), FIEGARCH(1,d,0)-n,. R t = μ + λσ t + ɛ t, ɛ t = σ tz t, σ t > 0, z t i.i.d.n (0, 1), ln(σt 2 )=ω +[1 β 1(L)] 1 (1 L) d g(z t 1), g(z t 1) =θz t 1 + γ[ z t 1 E z t 1 ]. 24),,,, ( / ),, ( / ), Non-GMO,,,,, ( / )., web site. 25),, ( ), ( ),, ( ),,,,,,,., 10

12 ( ) (daily data)., ,.,,,,,,,, 26), 1 1., ( ),,,,,, 6. 27)., ( ),.,,, 28). ( ), (1 )., ( )., ( ) NEEDS-FinancialQuest, web site. 2 FIGARCH FIEGARCH., ( 1 ) 29). R t,(2.1), (2.2) ( 2 )., , (Descriptive statistics),,,,,, 30) 1. web site. 26). 27) 2011, (30.00%), (22.75%), (2.37%), (30.00%), (9.47%), (5.41%)., web site. 28) 2011, (27.34%), (0.82%), (9.22%), (0.66%), (18.73%), (8.78%), (29.85%), (4.6%)., web site. 29), P cgive ( ). P cgive, Doornik and Hendry (2001), (2006). 30),, Jarque and Bera (1987). Jarque - Bera JB, JB = ˆ skew 2 T 6 + ( kurt ˆ 3) 2 T 24 χ 2 (2)., skew, ˆ kurt ˆ, T. JB =0, JB., Jarque and Bera (1987). 11

13 [ 1, 2] [ 1],.,,, ( ),.,,,, ( ) 3.,,., 3., (density) (normal approximation).,, N(s =1.32), , N(0.061, ). 2,., ADF (Augmented Dickey - Fuller)., (trend) 2 31). ADF ,,,, ( ) 1 ADF(1), 5 ADF(5)., Mackinnon (1991).,. [ 3] [ 2] 3.2, (1) 31), ADF(n). Δx t = α + bx t 1 + nx i=1 Δx t = α + μt + bx t 1 + γ iδx t 1 + u t nx i=1 γ iδx t 1 + u t ADF, Dickey and Fuller (1981). ( ), ( ). 12

14 (i) FIGARCH (1,d,0) : μ, ω,, Student-t, GED, skewed-student t. λ,. λ,. d, 0.951, 0.717, 0.740, 0.717,., d 0.5 d<1, σ 2,. β 1,. ν, t, skewed-student t ν>4., GED ν<2.,., ln(ξ) 0.109,. (ii) FIEGARCH (1,d,0) : μ, ω,, GED, Student-t, skewed-student t. λ, FIGARCH(1,d,0). d, 0.718, 0.875, 0.827, 0.758,.,d 0.5 d<1, FIEGARCH(1,d,0) σ 2,. β 1,. θ,. ν, Student-t, skewed-student t ν>4., GED ν<2.,., ln(ξ) 0.017,.,,, ( ). (2) 13

15 FIGARCH (1,d,0) FIEGARCH (1,d,0) λ,. d, FIGARCH(1,d,0), 0.310, 0.321, 0.311, 0.318, FIEGARCH (1,d,0) 0.425, 0.398, 0.402, , d 0 <d<0.5, σ 2,. β 1,. FIEGARCH (1,d,0) θ,. ν. (3) λ,. d, FIGARCH (1,d,0), 0.783, 0.796, 0.774, 0.830, FIEGARCH(1,d,0) 0.821, 0.832, 0.887, , d 0.5 d<1, σ 2,. β 1,. FIEGARCH(1,d,0) θ,. ν. (4) ( ) λ,. d, FIGARCH (1,d,0), 0.387, 0.413, 0.395, 0.402, FIEGARCH (1,d,0), 0.288, 0.268, 0.410, d 0 <d<0.5 ( ) σ 2,., β 1,, FIEGARCH (1,d,0) θ,. ν., FIGARCH (1,d,0) FIEGARCH (1,d,0) Ljung - Box Q 32). 3 6 Q(20) Q 2 (20), 20 (ˆɛˆσ 1 ) 2 Ljung - Box Q 32) Ljung - Box,. Q LB = T (T +2) nx i=1 r 2 i T i 14

16 ., 20 χ 2.,,, ( ) FIGARCH(1,d,0) FIEGARCH (1,d,0). Q(20) Q 2 (20), 10%., FIGARCH (1,d,0) FIEGARCH(1,d,0),., 4 FIGARCH(1,d,0)-skt (conditional variance)., 5 FIGARCH (1,d,0)-skt. [ 3 6] [ 4, 5] 4, FIGARCH (1,d,0), FIEGARCH (1,d,0).,,, ( ).. 1. FIGARCH (1,d,0), ( ),. 2. FIEGARCH (1,d,0), ( ),,,. 3.,,, ( ),, 33),, r i = T T i TX t=i+1 (ˆɛ 2 t ɛ)( ɛ t i ˆ 2 ɛ) TX t=1 (ˆɛ t 2 ɛ) 2, for ɛ =., 33), R t. R t = μ + λσ 2 + ɛ t. 1 T TX t=1 ɛ 2 t 15

17 . 4.,,, ( ). 5., Student-t, GED, skewed-student t.,. 1. ( ),.,. 2.,,,, FIGARCH, FIGARCH. 3. Fractionally Integrated ARCH, Tse (1998) FIAPARCH (Fractionally Integrated Asymmetric Power GARCH), Hwang (2001) ASYMM- FIGARCH (Asymmetric FIGARCH), Davidson (2004) HYGARCH (Hyperboric GARCH).,.,,, 34)., λ. 34) (2007), London Metal Exchnge. 16

18 [1] (2000),,, 4, 2, No.8, pp [2] (2007),,. [3] (2006),,, 10, 1, No.14, pp [4] (2004),,. [5] (2005), No [6] (2008),,, 11, 1, No.15, pp [7] ( ) (2009),,, 1. [8] ( ) (2008),, MTP, No.24 pp [9] (2007),,, 10, 1, No.14, pp [10] (1998), 1:,, 3, 2, No.6, pp [11] (2000),,, 4, 2, No.8, pp [12], D.F. J. A. ( [ ]) (2006), PcGive,. [13] (2007),, [14] (2010), G@RCH, 32, pp [15] (2008),, 11, 1, No.15, pp [16] (2010),,, 29, 2, pp [17] (2003),, [ ], ( 8),, pp [18] (1996),,, 2, 3, No.4, pp [19] (2000), 2,, 5, 1, No.9, pp [20] (2005), 225 FIEGARCH,, Vol.17, No.8, pp.1 4. [21] (2006), ARCH Realized Volatility,, 25, 2, pp

19 [22] Baillie, R. T., T. Bollerslev and H. O. Mikkelsen (1996), Fractionally Integrated Generalied Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 74, pp [23] Bauwens, L. and S. Laurent (2005), A New Class of Multivariate Skew Densities, with Application to GARCH Models, Journal of Business and Economic Statistics, 23, pp [24] Bollerslev, T. (1986), Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 31, pp [25] Bollerslev, T. and H. O. Mikkelsen (1996), Modeling and Pricing Long-Memory in Stock Market Volatility, Journal of Econometrics, 73, pp [26] Bollerslev, T., R. F. Engle and D. B. Nelson (1994), ARCH Models, in R. F. Engle and D. McFadden (eds.), Handbook of Econometrics, Vol.4, pp , North-Holland. [27] Davidson, J. (2004), Moment and Memory Properties of Linear Conditional Heteroskedasticity Models, and a New Model, Journal of Business and Economic Statistics, 22, pp [28] Dickey, D. A. and W. A. Fuller (1981), Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root, Econometrica, 49, pp [29] Ding, Z., C. W. J. Granger and R. F. Engle (1993), A Long Memory Property of Stock Market Returns and a New Model, Journal of Empirical Finance, 1, pp [30] Doornik, J. A. (2006), An Introduction to OxMetrics 4 - A Software System for Data Analysis and Forecasting, Timberlake Consultants Ltd. [31] Doornik, J. A. and D. F. Hendry (2001), Econometric Modelling Using PcGive 10 Volume III, Timberlake Consultants Ltd. [32] Engle, R. F. (1982), Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 50, pp [33] Engle, R. F. and T. Bollerslev (1986), Modeling the Persistence of Conditional Variances, Econometric Rewiews, 5, pp [34] Fama, E. (1965), The Behavior of Stock Prices, Journal of Business, 38, pp [35] Fernández, C. and M. F. J. Steel (1998), On Bayesian modeling of Fat Tails and Skewness, Journal of the American Statistical Association, 93, pp [36] Giot, P. and S. Laurent (2004), Modelling Daily Value-at-Risk Using Realized Volatility and ARCH Type Models, Journal of Empirical Finance, 11, pp [37] Glosten, L. R., R. Jagannathan and D. Runkle (1993), On the Relation between the Expected Value and the Volatility of Nominal Excess Returns on Stocks, Journal of Finance, 48, pp [38] Gorton, G. and K. G. Rouwenhorst (2006), Facts and Fantasies about Commodity Futures, Financial Analysts Journal 62, pp.47 68; [ ] (2006),, BP. [39] Hwang, Y. (2001), Asymmetric Long Memory GARCH in Exchange Return, Economics Letters, 73, pp.1 5. [40] Jarque, C. M. and A. K. Bera (1987), Test for Normality of Observations and Regression Residuals, International Statistical Review, 55, pp [41] Knight, J. and S. Stachell (eds.) (2001), Return Distributions in Finance, Butterworth- Heinemann. [42] Laurent, S. and J.-P. Peters (2006), Estimating and Forecasting ARCH Models Using G@RCH 4.2, Timberlake Consultants Ltd. 18

20 [43] MacKinnon, J. G. (1991), Critical Values for Cointegration Tests, in Engle, R. F. and C. W. J. Granger (eds.), Long-Run Economic Relationships, pp , Oxford University Press. [44] Mandelbrot, B. (1963), The Variation of Certain Speculative Prices, Journal of Business, 36, pp [45] Michelfelder, R. A. (2005), Volatility of Stock Returns: Emerging and Mature Markets, Managerial Finance, 31, pp [46] Nelson, D. B. (1991), Conditional Heteroskedasticity in Asset Returns: A New Approach, Econometrica, 59, pp [47] Rachev, S. T. (ed.) (2003), Handbook of Heavy Tailed Distributions in Finance, Elsevier. [48] Russell, J. R. and R. F. Engle (2005), A Discrete-State Continuous-Time Model of Financial Transactions Prices and Times: The Autoregressive Conditional Multinomial- Autoregressive Conditional Duration Model, Journal of Business and Economic Statistics, 23, pp [49] Schwartz, E. S.(1997), The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging, Journal of Finance, 52, pp [50] Tsay, R. S. (2010), Analysis of Financial Time Series, 3rd ed., John Wiley & Sons. [51] Tse, Y. K. (1998), The Conditional Heteroskedasticity of the Yen-Dollar Exchange Rate, Journal of Applied Econometrics, 193, pp [52] Xekalaki, E. and S. Degiannakis (2010), ARCH Models for Financial Applications, Wiley. Web Site [53] [54] 19

21 1: R t (%) , ( ) %. 20

22 2: (ADF ) Δx t = α + bx t γ i Δx t 1 + u t i=1 Δx t = α + μt + bx t γ i Δx t 1 + u t i=1 ( ), ( ). ADF b t ( ) ADF b t ( ) ADF b t ( ) ADF b t ( ) ADF b t ( ) ADF b t ( ) ADF b t ( ) ADF b t ( ) , 5% 2.86, 1% 3.44,, 5% 3.41, 1% %. 21

23 3: R t = μ + λσ t + ɛ t, ɛ t = σ t z t, σ t > 0, z t i.i.d., E[z t ]=0,Var[z t ]=1. { F IGARCH(1,d,0) : σt 2 = ω [1 β 1 (L)] [1 β 1 (L)] 1 (1 L) d} ɛ 2 t. F IEGARCH(1,d,0) : ln (σt 2 )=ω +[1 β 1 (L)] 1 (1 L) d g(z t 1 ), g(z t 1 )=θz t 1 + γ[ z t 1 E z t 1 ]. F IGARCH(1,d,0) F IEGARCH(1,d,0) n t GED skt n t GED skt μ (1.248) (1.977) (5.927) (2.030) (2.828) ( 0.121) (0.562) (1.789) λ (0.732) ( 0.206) ( 0.213) ( 0.762) ( 1.274) (1.109) ( 0.208) ( 0.762) ω (1.572) (2.245) (1.979) (2.055) (3.114) (3.053) (2.183) (2.059) d (6.589) (6.298) (5.930) (6.203) (4.174) (3.039) (4.779) (2.222) β (8.612) (5.980) (4.979) (5.917) (3.102) (2.493) (5.586) (4.029) θ ( 0.825) ( 0.299) ( 0.840) ( 0.135) γ (0.743) ( 0.728) (0.709) (0.132) ν (6.266) (10.14) (6.091) (7.363) (9.572) (4.838) ln(ξ) ( 3.046) ( 2.569) Log-lik Q(20) Q 2 (20) * 5%. t. 22

24 4: R t = μ + λσ t + ɛ t, ɛ t = σ t z t, σ t > 0, z t i.i.d., E[z t ]=0,Var[z t ]=1. { F IGARCH(1,d,0) : σt 2 = ω [1 β 1 (L)] [1 β 1 (L)] 1 (1 L) d} ɛ 2 t. F IEGARCH(1,d,0) : ln (σt 2 )=ω +[1 β 1 (L)] 1 (1 L) d g(z t 1 ), g(z t 1 )=θz t 1 + γ[ z t 1 E z t 1 ]. F IGARCH(1,d,0) F IEGARCH(1,d,0) n t GED skt n t GED skt μ (1.190) (1.191) (1.183) (1.168) (0.853) (1.365) (0.825) (1.789) λ ( 1.011) ( 0.981) ( 0.986) ( 1.013) ( 0.683) ( 1.492) ( 0.606) ( 0.611) ω (2.562) (2.498) (2.568) (2.581) (1.678) (1.58) (1.716) (1.782) d (5.425) (5.171) (5.423) (5.180) (5.910) (4.195) (5.961) (5.901) β (3.687) (3.565) (3.699) (3.710) (2.356) (2.460) (2.372) (2.292) θ (0.496) ( 0.189) (0.503) (0.480) γ (0.729) (0.136) (0.730) (0.703) ν (2.250) (19.32) (2.418) (1.998) (19.28) (2.275) ln(ξ) ( 3.205) ( 2.559) Log-lik Q(20) Q 2 (20) * 5%. t. 23

25 5: R t = μ + λσ t + ɛ t, ɛ t = σ t z t, σ t > 0, z t i.i.d., E[z t ]=0,Var[z t ]=1. { F IGARCH(1,d,0) : σt 2 = ω [1 β 1 (L)] [1 β 1 (L)] 1 (1 L) d} ɛ 2 t. F IEGARCH(1,d,0) : ln (σt 2 )=ω +[1 β 1 (L)] 1 (1 L) d g(z t 1 ), g(z t 1 )=θz t 1 + γ[ z t 1 E z t 1 ]. F IGARCH(1,d,0) F IEGARCH(1,d,0) n t GED skt n t GED skt μ (1.999) (1.986) (1.983) (1.976) (10.74) (2.268) (2.009) (3.190) λ ( 1.096) ( 1.082) ( 1.123) ( 1.038) ( 1.098) (1.109) ( 1.373) ( 1.621) ω (1.365) (1.401) (1.334) (1.119) (0.143) (3.053) (3.386) (3.154) d (5.930) (6.500) (5.466) (5.901) (3.804) (9.836) (7.479) (2.222) β (7.008) (7.805) (5.466) (7.369) (5.102) (13.51) (2.365) (16.55) θ ( 0.105) ( 1.502) ( 0.840) ( 0.512) γ (0.098) (2.627) (0.709) (5.877) ν (5.267) (16.48) (6.091) (2.647) (16.87) (4.838) ln(ξ) ( 3.774) ( 2.805) Log-lik Q(20) Q 2 (20) * 5%. t. 24

26 6: R t = μ + λσ t + ɛ t, ɛ t = σ t z t, σ t > 0, z t i.i.d., E[z t ]=0,Var[z t ]=1. { F IGARCH(1,d,0) : σt 2 = ω [1 β 1 (L)] [1 β 1 (L)] 1 (1 L) d} ɛ 2 t. F IEGARCH(1,d,0) : ln (σt 2 )=ω +[1 β 1 (L)] 1 (1 L) d g(z t 1 ), g(z t 1 )=θz t 1 + γ[ z t 1 E z t 1 ]. F IGARCH(1,d,0) F IEGARCH(1,d,0) n t GED skt n t GED skt μ (0.798) (0.399) (0.168) (0.355) (0.515) (1.971) (0.019) (2.404) λ (0.180) (0.918) (1.055) (0.598) (0.441) (1.009) (1.331) (0.568) ω (2.641) (2.570) (2.588) (2.601) (2.151) (2.437) (2.160) (2.245) d (5.983) (6.316) (6.349) (6.301) (4.174) (5.419) (5.084) (2.734) β (3.981) (4.526) (4.352) (5.856) (12.42) (5.324) (17.07) (4.029) θ (0.265) (0.921) (0.204) (0.485) γ (0.614) (0.547) (0.534) (0.610) ν (6.216) (21.14) (5.856) (5.899) (20.76) (4.540) ln(ξ) ( 3.960) ( 3.970) Log-lik Q(20) Q 2 (20) * 5%. t. 25

27 1: (2003/4/1 2010/12/30) 125 NCF Index TC Index time time 400 NTC Index Gold time time 26

28 2: (2003/4/2 2010/12/30) % 20 NCF Index % 5.0 TC Index % 5 NTC Index time % 5 Gold time time time 27

29 3: (2003/4/2 2010/12/30) 0.4 NCF N(s=1.32) 0.3 TC N(s=1.34) NTC N(s=1.55) 0.4 Gold N(s=1.32)

30 4: (2003/4/2 2010/12/30) % 75 NCF Index % TC Index % time % time 10.0 NTC Index Gold time time 29

31 5: (2003/4/2 2010/12/30) NCF Index Standardized residuals skt (0,1, 0.109,5.353) 0.4 TC Index Standardized residuals skt (0,1, 0.152,7.353) Standardized residuals skt (0,1, 0.113,48.58) NTC Index Standardized residuals skt (0,1, 0.117,5.856) Gold

32 Research Institute of Economic Science College of Economics, Nihon University Misaki-cho, Chiyoda-ku, Toyko JAPAN Phone: Fax:

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

Working Paper Series No March 2012 日本の商品先物市場におけるボラティリティの 長期記憶性に関する分析 三井秀俊 Research Institute of Economic Science College of Economics, Nihon Un

Working Paper Series No March 2012 日本の商品先物市場におけるボラティリティの長期記憶性に関する分析三井秀俊 Research Institute of Economic Science College of Economics, Nihon Un