ウェーブレットによる経済分析

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2 wavelet J.J. Morlet D. D. Gabor uncertanty prncple Conway and Frame Schlecher

4 wave let

7 DWT: dscrete wavelet transformcwt: contnuous wavelet transform AUSSMATLABS-PlusMathematca CD-ROMMathematca

9 MODWT h L h k =. k = L h k =. k = L h kh k+n =. n k = L Percval and Walden Gençay, Selçuk and Whtcher k+ n >L,,, h,, h L,,,

10 orthonormalty gquadrature mrror relatonshp g =( ) h L + h =( ) g L +. L g k =. k = L g k =. k = L g kg k +n =. n k = L g kh k +n =. n k = h =,, g =,. g k =

11 L L DLDL L / L (k ) h k =. =,, L / k = vanshng momentsl / D4 = D4 h = ,,, 4 4 4, g + = ,,, D4 D

12 D D LA MB

13 D / 4 LAL least asymmetrc wavelet MBL mnmum bandwdth wavelet

14 D D LA MB Gençay, Selçuk and Whtcher

15 S. S. Mallat x = (x,,x N )N h w L, t = k x{( t k) mod N } + k= w h. Nt N/A mod N AN N xg v v L g x, = t k k= {( t k) mod N } +. N/ N

16 w x x, x, x 3, x 4 g h v x + x x, 4 + x 3 w x x, x 4 x 3 g h v x 4 + x 3 + x + x w x 4 + x 3 x x g = [, ], h = [, ]

17 v v w v v v x v h w w,t = L h k v, {( N/ } + k= t k ) mod. v g v v, t L = g kv, {( t k) mod N/ } + k=. w v N/ w v v w 3 v 3 + w + v + v w + v + w + = (, w +,,, w +,,)v + = (, v +,,, v +,,)

18 + w + h + v + g v v = L L, t hk w + g +,{( t + k )mod N / } + kv k= k= +,{( t+ k )mod N / } +.

19 (, 3, 3, 3,,, 3, 4) g,h ( 4,,, 7) (, 6,, ) g,h ( 4, 9) (4, 5) g,h (5) (3) g = [, ], h = [, ]

20 v w v N/ N/ W V h h4 g g 4 h h g g 3 3 h g h g g h 4 4 h g 3 3 h h g g 4 4 h h g g 3 3 h h g 4 4 g h3 h g3 g,. w = W v, v = V v. W V U N/ N/ w = U v v, W U. V w v

21 UU T = U T U = I U T UI +U T + w U = T + v v T T + + w+ + V+ v+ + W = v. U T + h g U T + ( N/ N/ )w + v + w + v + N/ U + T h = h4 h3 h h h h h 4 3 h h h h 4 3 h h h h 4 3 g g g g 4 3 g g g g 4 3 g g g g 4 3, g 4 g 3 g g w v + + w + w = +, v+ v +,, N/, N/. U T + T w U + + = v v +. U T + N / N / W T + V T + W T + w + +V T +v + = v.

22 v w w v w d d v v w s v s w d s d s s d x s x s = d + s. x x, x, x 3, x 4 s x + x x, + x x, 4 + x 3 x, 4 + x 3 x d + x x, x x x, 4 + x 3, 4 x 3 x 4 + x 3 + x + x s, 4 x 4 + x x x, 4 + x x x, 4 + x 3 + x + x 4 4 x 4 + x 3 + d x 4 x 3 + x + x, x 4 x 3 + x + x, 4 4 x 4 + x 3 x x, 4 x 4 + x 3 x x 4 g = [, ], h = [, ]

23 r = k= d kwavelet rough x = r + s. s d s d s s 3 s d 3

24 (, 3, 3, 3,,, 3, 4) g,h ( 4, 4,,,,, 7, 7) g,h (,, 6, 6,,,, ) 4 ( 4, 4, 4, 4, 9, 9, 9, 9) 4 ( 4, 4, 4, 4, 5, 5, 5, 5) g,h 8 (5, 5, 5, 5, 5, 5, 5, 5) 8 ( 3, 3, 3, 3, 3, 3, 3, 3) g = [, ], h = [, ]

25 HT: hard thresholdng HT( w, j w, j for w, j >, ) =. soft thresholdng unversal thresholdng Gençay, Selçuk and Whtcher

26 w, j j =.5 HT w w HT w HT d HT s HT (= s ) x HT HT k = for for k, k +. = (w ). Walker

27 g,h ( 4,,, 7) (, 6,, ) g,h ( 4, 9) (4, 5) g,h (5) (3) g = [, ], h = [, ]

28 zz z z T z + UU T = U T U = I v = v T = T v w +, v ( T T +) U +U + w v + + = wt T w v +v + = w + v + +. v x = x = w + v. w = = w = s = V T V T v. VV T = I s =( V T T T T V v ) V V T v T T T T V v = v v v = v V V V =. d = V T V T W T w. WW T = I VV T = I

29 T T T T T T d = ( V V W w ) V V T T T T T = w WV V V V W w = w w = w W T w. x x x

30 w w w 3 w 4 j MODWTmaxmal overlap dscrete wavelet transform MODWT h = h/, g = g /. MODWT v x

31 v h w w, t L = h k v, { t ( k ) mod N} + k=. v g v v, t = L g kv, { t ( k ) mod N} + k=. +w + h + v + g v v, = L t h kw { t + + L +, ( k ) modn } + g kv k +, { t + ( k ) mod N } + = k=. MODWT MODWT MODWT MODWT

32 4 = 5 7 D4 D

33 w w w 3 w 4 w 5 v 5

34 MODWT MODWT DWT

35 d d d 3 d 4 d 5 s 5

37 s 5 d 5 d 4 d 3 d

38 w w w 3 w 4 w 5 v 5

39 d d d 3 d 4 d 5 s 5

41 s 5 d 5 d 4 d 3 d

42 Ramsey and Lampart

44 D4 D4 = h + h + h 3 + h 4 =..h +.h +.h h 4 =. 3, 5, 7, 9 9 h h + 7 h + 5 h = ( + 9) h + h + ( + 9) h + ( + 9) h 3+ ( 3 9) 4 = ( h + h + h h4) + 9 ( h + h + h3 + h4) =. D =,,5 = D4 D4 D D D4D D4 384 = 7 3

45 D4D D4D D4 D D4 D

46 D D

47 D D

48 d d d 3 d 4 d 5 d d d 3 d 4 d 5 D D d d d 3 d 4 d 5 d d d 3 d 4 d 5 D D d d d 3 d 4 d 5 d d d 3 d 4 d 5

49 c,t =, +,. y (p),t +, t. c, t y (p),t p p y,t p D4D D4D D4D,

50 t t d d d 3 d 4 d 5 s 5 D d d d 3 d 4 d 5 s 5 D d d d 3 d 4 d 5 s 5

51 , D4D MODWTw,t ( w,t ) w,t E ( w,t ) ( w,t )=E( w,t ). N t=w,t /N x

52 MODWT L ( )(L )+w,t t = L,, N w, t t = L ( w,t) =. N L + N DWT N/ t= w,t /(N/ )MODWT L (L )( )+ w, t t = L,, N/ / N w, t t = L ( w ) =,t N / L +. DWT DWTMODWT DWT ( w,t )/ MODWT DL = ( 9 ) ( ) = 7 DMODWT

54 DWTDWTMODWT DWTMODWT ICSS: terated cumulatve sums of squaresinclán and Tao k p = L q = N/ k w t= p, t =. ( k = p,, q ) C, k q w t= p, t (w, t )=w, t C, k p q p k C,k (k p +)/(q p +) C,k (k p +)/(q p +) Percval and Walden D

55 + D = max( D, D ), + k p + = max C k q p D, k, = max k C D, k k q p p. Inclán and TaoD P( D j+ > z) ( ) exp[ j ( q p + j= ) z ]. D z p D k = k DWT MODWT p =L q = NC,k D k w t= p, t C, k = q w, t t= p. ( k p,, q ) = + D = max( D, D ), + k p + = D max C k, k q p = max k C D, k k q p p., Dk = k MODWTDWT DWT w,t MODWT w,t MODWT

56 a = p a = p c = q c = q w,t (t = a,, c) D kcw, t (t = a,, c )D kc c c b b a = c a = cc = q c = q m b b j b j b j w,t (t = b j,, b j+ )D b p b m+ q j + b j b j ICSS D DWT MODWT p DWTDWT MODWT

57 p

59 BBx t = x t ( B) x t = t. /<</ < < /x = x /<< x x Hoskng x

60 (,t w ) C ( ). CJ = + ln( ( w )),t ln( ) +. ( =,, J ) =/( +) Var ()=/4Var ( )

62 C. K. Chu, C. K., Introducton to Wavelets, New York: Academc Press, 99 Conway, P., and D. Frame, Spectral Analyss of New Zealand Output Gaps Usng Fourer and Wavelet Technques, Reserve Bank of New Zealand Dscusson Paper, DP/6,. Gençay, R., F. Selçuk, and B. Whtcher, An Introducton to Wavelets and Other Flterng Methods n Fnance and Economcs, San Dego: Academc Press,. Hoskng, J. R. M., Fractonal Dfferencng, Bometrka, 68 (), 98, pp Inclán, D., and G. C. Tao, Use of Cumulatve Sums of Squares for Retrospectve Detecton of Changes of Varance, Journal of the Amercan Statstcal Assocaton, Theory and Methods, 89, 994, pp Percval, D. B., and A. T. Walden, Wavelet Methods for Tme Seres Analyss, Cambrdge: Cambrdge Unversty Press,. Ramsey, J. B., and C. Lampart, The Decomposton of Economc Relatonshps by Tme Scale Usng Wavelets: Expendture and Income, Studes n Nonlnear Dynamcs and Econometrcs, 3 (), 998, pp Schlecher, C., An Introducton to Wavelets for Economsts, Bank of Canada Workng Paper, -3,. Walker, J. S., A Prmer on Wavelets and ther Scentfc Applcaton, Boca Raton: Chapman & Hall/CRC, 999.

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

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 TOPIX E-mail: masakazu.inada@boj.or.jp wavelet TOPIX Baillie Gourieroux and Jasiak Elliott and Hoek TOPIX I (0) I (1) I (0) I (1) TOPIX ADFAugmented Dickey-Fuller testppphillips-perron test I (1) I (0)