2005 7 7 100026788 (2005) 0720041208 1,2 1, (11, 300072 21, 250100) EGARCH(1,1)2M ARMA(4,3)2ARCH(1), ( R - V), ( h R - V) ( h R - h V ),, Lyapunov EGARCH2M F830 A Empirical Analysis of the Relation Between Daily Return Rate, Volatility and Trading Volume of Shanghai Stock Market Comprising Index Based on Nonlinar Dynamics LUO Deng2yue 1,2,WANG Chun2feng 1 (11School of Management,Tianjin University,Tianjin 300072,China 21School of Management,Shandong University,Jinan 250100,China) Abstract This paper investigates the relationships between return rate, return volatility and trading volume of Shanghai Stock Market Comprising Index We measure return volatility by EGARCH(1,1) 2M(exponential generalized autoregressive conditional heteroscedasticity) model and volatility of trading volume by ARMA ( 4, 3) 2ARCH ( 1) model we build two2dimensional dynamic system of return rate and trading volume ( R - V), return volatility and trading volume( h R - V),return volatility and volume volatility( h R - h V ) by using stepwise regression method,and investigate the dynamic behaviour of these systems The results indicate that there does not exist chaos in these two2 dimensional dynamic systems composed by return and trading volume,return volatility and trading volume,although many empirical analysis indicate that the time series of stock market return rate is chaotic Key words return volatility trading volume Lyapunov exponent stepwise regression EGARCH2M model chaos 1,,,, [1 9 ] Ali F Darrat,Shafiqur Rahman,Maosen Zhong(2003) 5, EGARCH2M, (MDH),, 2004207209 (70225002) (1972 - ),,,,, 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved
42 2005 7, ( SIAH) [1 ] Martin T Bohl, Harald Henke (2003) (MDH) 20,,,,, [2 ] (1998) Granger Back2Brock,, [4 ] (2000), [5 ] (2003), Granger ( ), [6,, ] (2002),,, [7, ] (2003) GARCH2M,,, [8 ] (2003) [9,, ],, ( ),, ( ), [4 ],, Var[ P t V t ] = cv t, 2 [5 ], ( ),, [6 ] [7 ] [7,, ] [10 ] (2002) [11 Lyapunov, 0109325 010002 ] (2002), Lyapunov 0176 Lyapunov [12 ] (2004), 2002,, Lyapunov,, ( )?,,, ( ),,, 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved
7 43, 2 (http ΠΠwww stockstar com) ( P t ) (L t ), 2002 3 13 2004 4 7,, 500 P t t R ( t) = R t = ln ( P t ΠP t - 1 ) 100 100 V ( t) = V t = ln ( L t ) t ( L t ) V ( t) = V t = ln ( V t ΠV t - 1 ) 10 h R ( t) h V ( t) ( R t ) ( V t ) 1 1 Eviews 410 R t V t n n 1 2 3 4 R t 0 011 0 001 0 042 0 026 R 2 t 0 053 0 007 0 019 0 043 n n 1 2 3 4 5 6 7 AC 0 837 0 774 0 736 0 692 0 644 0 608 0 594 PAC 0 837 0 245 0 152 0 044-0 005 0 016 0 085 ( 4 ) R t ADF = - 1011307 = 1 % CV = - 314458 V t ADF = - 316592 = 1 % CV = - 3 4458,,, 3, EGARCH(1,1)2M,, R t = c + t t, t t - 1 N (0, 2 t ), t = t v t, v t t - 1 t - 1 log( 2 t ) = + log( 2 t - 1) + t - 1 t - 1 2, t h R ( t) + t - 1 t - 1 ARMA(4,3)2ARCH(1) V t = c + 4 i = 1 i V t - i 3 + j = 1 j t - j + t, t t - 1 N (0, 2 t ), 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved, v t N (0,1)
44 2005 7 2 t = + 2, t h V ( t) Eviews410, 2 2 t - 1 2 EGARCH(1,1)2M,ARMA(4,3)2ARCH(1) Z2 p D2W AIC SC EGARCH2M ARMA(4,3)2ARCH(1) c - 0 334574 2 616111 0 0068 0 674017-2 705615 0 0089-0 195809-5 178734 0 000 0 9603 59 69217 0 000 0 190415 4 662837 0 000-0 076118-2 977262 0 0029 c 6 966083 62 69148 0 0000 1 1 981245 7 619012 0 0000 2-1 715735-3 725086 0 0002 3 1 004814 3 529784 0 0004 4-0 278586-3 052601 0 0023 1-1 449282-5 62257 0 0000 2 1 095864 3 321297 0 0009 3-0 493922-3 721101 0 0002 0 012527 14 00881 0 0000 0 300646 5 327706 0 0000 1 971456 1 468206 1 518781 1 920944-1 243438-1 158628 2, < 0,, ARMA(4,3) ARCH LM ARCH, ARMA (4,3)2ARCH(1) 4 [13, ] x n+1 = a 1 + a 2 x n + a 3 x 2 n + a 4 x n y n + a 5 y n + a 6 y 2 n ƒ A y n+1 = a 7 + a 8 x n + a 9 x 2 n + a 10 x n y n + a 11 y n + a 12 y 2 n ƒ B Z n + 1 = f ( Z n ), Z n = ( x n, y n ) x - y R - V, R - V R - V, R - V R - h V, R - h V h R - V h R - V, h R - V h R - h V 3 3 R - V R - V R - h V h R - V h R - V h R - h V 0 257 0 322 0 055 3 3 0 272-0 076 3 3 0 180 R - V R - V R - h V - h R - V - 0 358 0 418 0 048 3 3-0 115 3-3 3 3 0105 0101 3, R ( V, V),, h R V h V,, (1), x n+1 = A + y n+1 A B (1) (2) = B + x n+1 y n+1 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved (1)
7 45 SPSS 1115, (stepwise), 4 R - V R - V 4 SPSS 1115 R t + 1 = - 61068-01069V 2 t + 1135V t + 1 V t + 1 = 01044 R t + 11194V t - 01028V 2 t + 01096 R t + 1 R t + 1 = - 1161V t + 01041V 2 t + 11375V t + 1 V t + 1 = 1116V t - 01024V 2 t + 01198 R t + 1 R - V R - V R - h V R t + 1 = 01034 3 V t + 01126 V t + 1 V t + 1 = 0152 R t - 01363 V t + 01874 R t + 1 R t + 1 = 01344 R t - 01126 R 2 t + 01209 V t + 1 V t + 1 = 01495 R t - 01154 R t V t + 01396 V t + 11256 R t + 1 R - h V R ( t + 1) = 0122 R ( t) - 111395 R ( t) h V ( t) + 141176 h V ( t) - 601289 h 2 V ( t) + 71203 h V ( t + 1) h V ( t + 1) = 01013 R 2 ( t) + 01684 h V ( t) - 21518 h 2 V ( t) + 01005 R ( t + 1) h R - V h R - V h R - V h R ( t + 1) = 01558 h R ( t) - 01252 h 2 R ( t) + 01074 h R ( t) V ( t) V ( t + 1) = 11155V ( t) - 01022V 2 ( t) h R ( t + 1) = 11052 h R ( t) - 012 h 2 R ( t) + 01005 V ( t) + 01002 V 2 ( t) V ( t + 1) = - 01308 V ( t) h R ( t + 1) = 11091 h R ( t) - 01338 h 2 R ( t) + 01048 h R ( t) V ( t) - 01014 V ( t) + 01003 V 2 ( t) V ( t + 1) = 21668 h R ( t) - 11244 h R ( t) V ( t) + 0164 V ( t) h R - h V h R ( t + 1) = 01979 h R ( t) - 01109 3 3 h 2 R ( t) + 01887 h V ( t + 1) h V ( t + 1) = - 01045 h R ( t) - 01044 3 3 3 h 2 R ( t) + 01583 h V ( t) - 11937 h 2 V ( t) + 0109 h R ( t + 1) 3 3 3 3 3 3 t2 p 0 034 01022 01017, 0101, F2 p 0101,, gr 2 (2) x t +1 y t +1 = ( B + A)Π(1 - ) = ( A + B)Π(1 - ) Z n + 1 = f ( Z n ), Z n = ( x n, y n ), (3) 4 (1), 5 5, ( ),,, [13 (3) (1) f, Lyapunov ] f ( Z) f Jacobi, f ( Z) = J i = f ( Z 0 ) f ( Z 1 ) f ( Z i - 1 ) = [ f i ( Z) ] Z = Z0 ( i) 1 ( i) 2, f Lyapunov a 2 + 2 a 3 x + a 4 y a 4 x + a 5 + 2 a 6 y a 8 + 2 a 9 x + a 10 y a 10 x + a 11 + 2 a 12 y (4) 1 k = lim i i, J i 2, ln ( i) k k = 1,2 (5) Lyapunov, T ( x i, y i ),1 i T T 500 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved (3)
46 2005 7 496 ( R - h V, h R - h V ), f T Jacobi f ( Z i ),1 i T J T = f ( Z 1 ) f ( Z 2 ) f ( Z T ), J T 2 1, 2 1 2, f Lyapunov L E 1 ln 1 T MATLAB6 5 Lyapunov, 6 R - V R - V 5 R t + 1 = - 619715 + 010682 R t + 118519V t - 011227V 2 t V t + 1 = - 016693 + 010506 R t + 113718V t - 010398V 2 t R t + 1 = - 010206V t + 01011V 2 t V t + 1 = 111559V t - 010218V 2 t R - V R t + 1 V t + 1 = 010736-010132 015843-013333 R t V t = A R t V t R - V R t + 1 = 016067 R t - 011708 R 2 t - 010436 R t V t + 011122 V t V t + 1 = 11257 R t - 012146 R 2 t - 012089 R t V t + 015369 V t R - h V R ( t + 1) = 012282 R ( t) + 010971 R 2 ( t) - 11182 R ( t) h V ( t) + 1918166 h V ( t) - 811356 h 2 V ( t) h V ( t + 1) = 010011 R ( t) + 010135 R 2 ( t) - 010591 R ( t) h V ( t) + 017831 h V ( t) - 219248 h 2 V ( t) h R - V h R - V h R - V h R ( t + 1) = 01558 h R ( t) - 01252 h 2 R ( t) + 01074 h R ( t) V ( t) V ( t + 1) = 11155V ( t) - 01022V 2 ( t) h R ( t + 1) = 11052 h R ( t) - 012 h 2 R ( t) + 01005 V ( t) + 01002 V 2 ( t) V ( t + 1) = - 01308 V ( t) h R ( t + 1) = 11091 h R ( t) - 01338 h 2 R ( t) + 01048 h R ( t) V ( t) - 01014 V ( t) + 01003 V 2 ( t) V ( t + 1) = 21668 h R ( t) - 11244 h R ( t) V ( t) + 0164 V ( t) h R - h V h R ( t + 1) = 110206 h R h R ( t) - 011609 h 2 R ( t) + 015620 h V ( t) - 118672 h 2 V ( t) h V ( t + 1) = 010469 h R ( t) - 010585 h 2 R ( t) + 016336 h V ( t) - 211050 h 2 V ( t) 6 Lyapunov A 2 R - V R - V R - h V h R - V h R - h V Lyapunov - 0 1985 A 2 = 016764-0 1934-0 0603-0 0531 R - V R - V h R - V h R - V - Lyapunov - 0 1635-0 3347-0 0631-0 0399-6, R - V, A 2 [7 ] A 2 < 1, A 2 > 1,, Lyapunov, (2002) [7 ] s t +1 v t +1 = - 011320 010041 + 011533 011394-013185 019941 s t = A s t, (6) v t v t s, v ( s v ) A 2 = 110475 > [7 1,, ] x n + 1, = C + A x n y n + 1 y n, Z n + 1 = g ( Z n ), Z n = ( x n, y n ) g Jacobi A, (4) J i = A i J i 2, ( i) 1 ( i) 2 g 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved
7 47 1 Lyapunov k = lim i A i = i ln ( i) k k = 1,2 (6),A 011533 011394-013185 019941 = - 019265-011750 - 013763-019846 - 019265-011750 - 013763-019846 012099 0 0 019375 012099 i 0-019265 - 011750 0 019375 i - 013763-019846 = 019375 i, (6) Lyapunov - 1-019265 - 011750-013763 - 019846, 1 = 012099 i, 2-1 L E = 1 i ln (019375 i ) = ln (019375) = - 010645, Lyapunov - 115611 2 Lyapunov, [10 ], 5 EGARCH(1,1)2M ARMA(4,3)2ARCH(1), ( R - V), ( h R - V) ( h R - h V ),, 1996 12 16 2004 9 9, R - V h R - V h R - h V, Lyapunov - 011338-010459 - 010245,, [ 1 ] Ali F Darrat,Shafiqur Rahman,Maosen Zhong Intraday trading volume and return volatility of the DJ IA stocks A note[j ]Journal of Banking & Finance,2003,27 2035-2043 [ 2 ] Martin T Bohl,Harald Henke Trading volume and stock market volatility The Polish case[j ] International Reviews of Financial Analysis,2003,12 513-525 [ 3 ] [M],2001 Wang Chunfeng Managemtn of Financial Market Risk[M] Tianjin Tianjin University Press,2001 [ 4 ], [J ],1998,18(6) 111-114 Zhang Wei, Yan Jinan Empirical detection to the volume2price causality in Shanghai securities exchange[j ] Systems Engineering2 Theory & Practice,1998,18(6) 111-114 [ 5 ], [J ],2000,3(2) 62-68 Chen Yiling,Song Fengming An empirical study on the relationship between price changes and trading volume in China stock market [J ]Journal of Management Sciences in China,2000,3(2) 62-68 [ 6 ], [J ],2003,12 36-40 Hua Renhai,Ding Xiuling An empirical analysis of the dynamic relation between return,trading volume and volatility in China stock market[j ] Finance & Trade Economics,2003,12 36-40 [ 7 ], [J ],2002,20(3) 24-28 Zhang Yongdong,He Rongtian Empirical analysis of the relation between volatility and trading volume in Shenzhen stock market [J ] Systems Engineering,2002,20(3) 24-28 [ 8 ], [J ],2003,25(2) 82-85 Li Shuangcheng,Wang Chunfeng An empirical study of price2volume relation of China s stock market[j ]Journal of Shanxi Finance and Economics University,2003,25(2) 82-85 1995-2005 Tsinghua Tongfang Optical Disc Co, Ltd All rights reserved
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