ウェーブレット分数を用いた金融時系列の長期記憶性の分析

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1 TOPIX

2 wavelet TOPIX Baillie Gourieroux and Jasiak Elliott and Hoek

3 TOPIX I (0) I (1) I (0) I (1) TOPIX ADFAugmented Dickey-Fuller testppphillips-perron test I (1) I (0) Beran I (0) I (1) ni (n) n >1 I (n)

4 ADFPP1% I (0) ADFPPMaddala and KimKPSS Kwiatkowski et al.kpss I (0) ADFPP I (0) I (1) I (0) I (1) I (0) KPSS

5 I (0) I (1) I (0) I (1) I (0) I (1) I (0) (h) h = 0 ( h) =. I (1) I (1) (h) t I (0) x t = 0.8x t 1 + t I (1) x t = x t 1 + t {x t } d = 0.{x t }

6 (h) (h) (h) x x x 1, x,, x n xn x = x n n var[ x ] ( s, t) n. = s= 1 t 1 (s, t ) x s x t {x t } (s, t ) = ( t s ) = (h) h = t s x n var[ x ] = n n s= 1 t = 1 ( t s ) = x n n 1 h [(0) + (1 ) (h) ] n h = 1 x n = [ 1+ n ()]. n () n n 1 h ( ) = (1 ) (h). n h = 1 n n () var[ x ]n 1 n var[ x ] var[ x ] x c ( ) n. c () c ( ) = lim n n ( s, t). s t var[ x ]n 0 < 1 = 1 > 1 var[ x ] = x /n = 1

7 0 < < 1 var[ x ]n 1 c () (s, t ) n (h) n 1 (s, t ) n 1 h= (n 1) (h) 0 < < 1n 1 c (h) = c h, 0 < <1. {Y t } t X s t {Y t } t = s s = 1 Y t X. ab(a<b)y b Y a Y b Y a Y b Y a m m = E [Y b Y a ]=0, = E [(Y b Y a ) ]=A (b a). A (b a) 1/. Y b Y a b a

8 E [(Y b Y a ) ]=A b a H. H0 < H< 1 Y t {Y t } H= 1/{Y t } X t Y t Y t 1 {X t }Fractional Gaussian Noise FGNX t Y 0 = 0 a > 0E [Y a ]= A a H E [ Y Y ]= a b A ( a H H H + b b a ), X t A h b a X t (h) (h) = Cov( X, X var( X ) var ( X a a b ) b ) E [ Xa X = A b ] 1 = E [( Y 1)( 1)] a Ya Yb Yb A (h) hx t (h) hh H h = 1 A { E [ Y a 1 = {( h + 1) Y ] E [ b H h Ya 1 H Y ] [ Y a Y b 1] + E [ b + ( h 1) E Ya 1Y b 1 H ]} }. (h) h H, FGN1/ < H< 1 H= 0E [(Y b Y a ) ]H< 0 E [(Y b Y a ) ] H= 1(h) 1H>1h (h)= 0 < H< 1

9 {x t }(t = 1,, ) d (1 L) d x t = t. LL h x t = x t h t (1 L) d d(1 d) d(1 d)( d) 3 ( 1 L) d = 1 dl L L. 3 I (d) I (d) d d 1/ < d< 1/Granger and JoyeuxHosking Γ(1 d) Γ( h + d) (h) =. Γ( d) Γ( h + 1 d) Γ(.) h Γ( h + a) h Γ( h + b) a b. I (d) h ( 1 d) d 1 Γ (h) h. Γ( d) FGNI (d) d FGNH d =H 1/. d0 < d< 1/I (d) x t } dd

10 FGNI (d) FGNHI (d) d d=h 1/I(d) I (d) d 0<d<1/ d I (d) S(f ) S( f ) i fh d e = (h) cd f = h. f 1/ < f < 1/c d 0< d< 1/ f 0 d ln S(f )=const.+ ( d )ln f. I (f k ) i fk t 1 N I ( f k ) = ( xt x) e N t = 1. N{x t } x {x t }f k f k =k/nk 1/ < f k < 1/ I (f k ) d Beran

11 ln I (f k )=const.+ ( d )ln f k. k = 1,, k max Geweke and Porter-HudakI (d) d ln I (f k )=const. d ln( 4sin ( f k )). dgphgph estimator sin ( f k ) f k dk max k max d Lardic, Mignon, and MurtinGPH k max N 0.5 Hurvich, Deo, and Brodskey dgphk max k max N 0.8 d k max Percival and Waldenj j j j 1. LDWTDiscrete Wavelet Transform j {w j,t }DWT (w j )

12 j ( w ) = j N j N t = L j j w L j, t j + 1. NL j (L )(1 j )+1 {w j,t }DWT j L j DWTS( f ) {x t } x S( f ) x 1/ = 1/ 1/ 0 S( f ) df = S( f ) df. {x t } x DWT x = ( w j ). j = 1 DWT DWT DWT

13 f1/4(=1/ ) 1/ f 1/ 3 1/ DWT ( w j ) j 1/ j + 1 1/ S( f ) df. I (d) S( f ) DWT j (w j ) j d 1. ln (w j )=const.+ (d 1)ln j. j = 1,, j max d DWTdj max j max k max j max j max j max f Gençay Percival and Walden

14 = 13 8,19=16 9 j max 8,19=3 8 j max d = 0.I(d) x t = t, (1 L) d x t = t. t I (d) d JensenGeweke and Porter-Hudak d

15 ARFIMAAutoregressive Fractional Integration Moving Average I(d) d ARMA I(0.) DWT dd d 4, =644, ARMA ARMA d

16 d DWT 4,096=16 8 D(4) D(1) D(1) dd=0 I(0.) d=0 d I(d) d D(4) D(1) d dd=0 LD(L)

17 d d d d I(0.) dd = 0 d DWTd d=0 I(0.) d=0

18 d

19 DWT I (d) d I (0 ) KPSSI (0) I (d) d DWT

20 I (d) d d TOPIX =17 9 d d = d d d d =3 7 d d d d

22 , {x 1, x }, {x 3, x 4 },{x 5, x 6 },{x 7, x 8 },{x 9, x 10 },., {x, x 3 }, {x 4, x 5 },{x 6, x 7 },{x 8, x 9 },. x 4 x 5, {x 1, x, x 3, x 4 }, {x 5, x 6, x 7, x 8 },., {x 3, x 4, x 5, x 6 }, { x 7, x 8, x 9, x 10 },.

23 x 5 x 5 {x 5, x 6 }{x 4, x 5 } x 5 =x 6 = d = 7

24 I (0) I (1)

25 d d 1 d 0 d d 1 >d 0 d 1 <d 0 d Gourieroux and Jasiak d

26 d d DWT MODWTMaximal Overlap Discrete Wavelet TransformDWT MODWT DWT MODWT MODWT MODWT MODWT

28 TOPIX Realized Volatility

29 Discrete Wavelet Transform DWTDWTMODWTMaximal Overlap Discrete Wavelet Transform DWT h L h i i = 1 = 0, A-1 L h i i = 1 = 1, A- L h i hi+ n = 0. n0 i = 1 A-3 L high-pass filter g ç

30 i i 1 gi = ( 1) hl i+ 1 hi = ( 1) gl i+ 1. A-4 L gi = i = 1, A-5 L g = i 1, i = 1 A-6 L g g i i + n 0 i = 1 =. n0 A-7 low-pass filter L g h i i + n 0 i = 1 =. n A h =,, g = 1 1,. A-9 1 h = 4 1+ g = ,,, ,,, , A-10 NxDWTDWT j LD(L)

31 w j v j v 0 xj 1v j 1 hjw j j 1v j 1 g j DWT DWT L w j, t = hk v j 1, {( t k mod ( N/ j 1 ) )} + 1 k= 1, A-11 v L j, t = gk vj 1, {( t N k mod ( / j 1 ) )} + 1. k= 1 A-1 j DWT j DWT j A mod BAB0B 1

32 DWTMODWTMODWTh g DWThg h = h, g = g. A-13 j w j v j v 0 x L w j, t = hk v j 1, {( t ( k 1) 1) mod N} + k= 1. j 1 1 A-14 L v j, t = gk k= 1 v 1, j. j {( t 1( k 1) 1) mod N} + 1 A-15 MODWTDWT DWT MODWT MODWT MODWT {a t } a a = E [( a E [ a ]) ]. A-16 t t {w j, t } { w j, t }E [w j, t ] E [ w j, t ] DWTMODWT J I(d) L>d Percival and Waldend =1/

33 x x wj, t + vj,, = J t t j = 1 w j, t + vj,. = J t t j = 1 A-17 A-18 =, t j = 1 x t w j, =, t j = 1 x t w j. A-19 A-0 MODWT MODWTE w j, t DWT DWT E w j, t DWTE w j, t E w j, t / j L MODWT L j ( j 1)(L 1)+1 ˆ ( w ) = j N t = L j N L j w j, t + 1. A-1 DWT(L ) (1 j )+1L j j N / t = L w j, t j j ˆ ( wj) = j N/ L + 1 j. A-

34 A ( w j ) MODWT (w j ) DWT DWTE [w j, t ] DWTMODWT DWTMODWT

35 Baillie, R. T., Long Memory Processes and Fractional Integration in Econometrics, Journal of Econometrics, 73 (1), 1996, pp Beran, J., Statistics for Long-Memory Processes, Chapman and Hall, Elliott, R. J., and J., V. der Hoek, A General Fractional White Noise Theory and Applications to Finance, Mathematical Finance, 13 (), 003, pp Gençay, R., F. Selçuk, and B. Whitcher, An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press, 00. Geweke, J., and S. Porter-Hudak, The Estimation and Application of Long-Memory Time Series Models, Journal of Time Series Analysis, 4 (4), 1983, pp Gourieroux, C., and J. Jasiak, Memory and Infrequent Breaks, Economics Letter, 70 (1), 001, pp Granger, C. W. J., and R. Joyeux, An Introduction to Long Memory Ttime Series Models and Fractional Differencing, Journal of Time Series Analysis, 1 (1), 1980, pp Hosking, J. R. M., Fractional differencing, Biometrika, 68 (1), 1981, pp Hurvich, C. M., R. Deo, and J. Brodsky, The Mean Squared Error of Geweke and Poter-Hudak s Estimator of the Memory Parameter of a Long-Memory Time Series, Journal of Time Series Analysis, 19 (1), 1998, pp Jensen, M. J., Using Wavelets to Obtain a Consistent Ordinary Least Squares Estimator of the Long-Memory Parameter, Journal of Forecasting, 18 (1), 1999, pp Kwiatkowski, D., P. C. B. Phillips, P. Schmidt, and Y. Shin, Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root, Journal of Econometrics, 54 (1-3), 199, pp Lardic, S., V. Mignon, and F. Murtin, Frequency-Domain Estimation of Fractionally Integrated Process: Impact of Short-Term Components on the Bandwidth Choice, Working Paper , University of Paris X.

36 Maddala, G. S., and In-Moo Kim, Unit Roots, Cointegration, and Structural Change, Cambridge University Press, Percival, D. B., and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 000. Schleicher, C., An Introduction to Wavelets for Economists, Working Paper 00-3, Bank of Canada, 00.

seminar0220a.dvi

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 }