JOURNAL OF MANAGEMENT SCIENCES IN CHINA. R t. t = 0 + R t - 1, R t - 2, ARCH. j 0, j = 1,2,, p

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6 2 2003 4 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 6 No 2 Apr 2003 GARCH ( 300072) : ARCH GARCH GARCH GARCH : ; GARCH ; ; :F830 :A :1007-9807 (2003) 02-0068 - 06 0 2 t = 0 + 2 i t - i = 0 +( L) 2

1 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 6 No 2 Apr 2003 GARCH ( ) : ARCH GARCH GARCH GARCH : ; GARCH ; ; :F830 :A : (2003) t = i t - i = 0 +( L) 2 t i = 1 i 0 ( i = 0 1 ) ( L) 1 :L ( L) = 1 L + 2 L Engle 1982 L ; t t X t R t (ARCH) ARCH R t - 1 R t GARCH ARCH ARCH Bolleslev [2 ] Generalized2ARCH( GARCH) GARCH t t - 1 N (0 2 t) 2 t = i t - i + GARCH i =1 1 ARCH 1 1 ARCH ARCH 20 Engle [1 ] (ARCH) ARCH R t = ( X t ; b) + t t t - 1 n i d (0 2 t) p 2 j t - j = 0 + ( L) 2 t + ( L) 2 t j = i 0 i = 1 2 j 0 j = 1 2 p (1) (2) : ; : : ( ) : (1976 ) China Academic Journal Electronic Publishing House All rights reserved

2 2 : GARCH 69 ARCH-M [3 ] EGARCH [4 ] IGARCH [5 ] FIGARCH [6 ] [7 ] 2 GARCH ( B ik g B ik ) (5) k =1 : g ronecker (3) [9 ] BE ARCH 2 1 GARCH( p ) BE H t Bollerslev GARCH GARCH GARCH [8 ] Y t = M t + t t t - 1 N (0 H t ) Vech( H t ) = W + A i Vech( t - i t - i) + i = 1 p B j Vech( H t - j ) = W + (3) 2 3 BE GARCH A i = k = 1 B i = [8 ] ( A ik g A ik ) (4) Vech( H t ) = W + p i =1 k =1 (A ik g A ik ) Vech( t - i t - i) + ( B ik g B ik ) Vech( H t - j ) (6) j = 1 k = GARCH GARCH Y it Y jt Y t i j t ijt = h ijt / h iit h jjt (7) : h ijt H t ( i j) Y j t - 1 ; h iit h jjt j =1 H t i j Y it A ( L) Vech( t t) + B ( L) Vech( H t ) Y jt t - 1 : Y t : N ; M t Y t ijt ; A i B j : N ( N + 1) A ( L) B ( L) Bollerslev [10 ] 2 p ; W : N ( N + 1) ijt = ij it jt (8) 2 H t ( ) ; H t = DD t t (9) Vech ( ) : N : D t = diag( 1 t 2 t Nt ) ; = ( ij ) N N N N ( N + 1) GARCH( p ) 1 GARCH GARCH( p ) Bollerslev A i B j GARCH 1 GARCH Y i GARCH GARCH BE GARCH H t China Academic Journal Electronic Publishing House All rights reserved

3 GARCH 3 1 GARCH (3) TN GARCH L ( ) = - ln T (ln H t + t H - 1 t t ) (10) t =1 A : H t = ( ijt ) N N ; W A ( L) B ( L) A = 1 2 (12) 2 N ( N + 1) + N 2 ( N + 1) 2 ( p + ) 4 : ; k [min k m x k ] Y t [min k m x k ] k GARCH(1 1) 21 ;Y t GARCH k [ ] k = GARCH M i P( i) = { A (1 i) A (2 i) A (M i)} (13) Berndt (BHHH ) [11 ] n - 1 n ( i +1) = ( i) 9L t 9L t 9L k i t =1 9 9 (11) (i) t =1 9 = : : n ; k i i (11) A L ( A) F( A) GARCH BHHH [12 ] F(A) = L (A) - C min L (A) > C min (14) GARCH 0 C min BHHH GARCH C min GARCH 3) M = GARCH A ( m i) ARCH [13 ARCH ] GARCH p i F( A ( m i) ) m = (15) : M F( A ( m i) ) m = 1 1) 2) China Academic Journal Electronic Publishing House All rights reserved

4 2 : GARCH SH SZ { P t } { R t } R A k k t = ln P t - ln P t - 1 (17) 2 : k k k [ ] (16) ( 0 3) J - B ( 300 J - B 2 (2) 2 (2 0 01) = 9 21) J - B SH e e SZ e e Q Box2Pierce Q B P Ljung2Box Q LB ( VR) ( VD) [14 ] 4 3 Q Q B P SH SZ VR VD Q LB SH SZ VR VD SH e e e e SZ e e e e GARCH GARCH China Academic Journal Electronic Publishing House All rights reserved

5 ( ) (1 8 3 ) (1 5 3 ) GARCH(1 1) GARCH(1 1) (0 19) (7 7 3 Vech( H t - 1 ) ) (0 47) SH R t = t ( ) ( ) (2 6 3 ) V r( t t - 1 ) = h t (18) (23) h t = t h t - 1 : ( ) t ; SZ R t = t i) (24) GARCH(1 1) : R SH t R SZ t = SH t (20) SZ t t = ( SH t SZ t ) (21) V r( t t - 1 ) = H t (22) Vech( H t ) = (9 0 3 ) (1 8 3 ) (2 5 3 ) (117 3 ) ( ) ( ) (22 3 ) (- 13) (18 3 ) (701 3 ) (19 3 ) (164 3 ) Vech( t - 1 t - 1) + t GARCH (19) V r( t t - 1 ) = h t h t = t h t - 1 GARCH GARCH [15 16 ] GARCH( IGARCH : integrated GARCH) (volatility persistence) 4 4 GARCH GARCH : 1 = i ; GARCH 2 = i ; 3 = GARCH v 1 = ( i v 2 = ( i i) (25) v 3 = ( ) (26) = (1 1 ) Vec 2 ( ) v 3 = 0 (27) = 0 (28) [ ] GARCH 5 GARCH GARCH China Academic Journal Electronic Publishing House All rights reserved

6 2 : GARCH 73 GARCH GARCH : [ 1 ] Engle R F Autoregressive conditional heteroskedasticity with estimates of the variances of U inflation[j ] Econometrica : [2 ]Bolleslev T Generalized autoregressive conditional heteroskedasticity[j ] Journal of Econometrics : [3 ]Engle R et al Estimating time varying risk premia in the term structure : The ARCH-M model[j ] Econometrica : [4 ] Nelson D B Conditional heteroskedasticity in asset returns : A new approach[j ] Econometrica : [5 ]Engle R F Bolleslev T Modeling the persistence of conditional variances[j ] Econometric Reviews :1 50 [ 6 ]Baillie R F Bollerslev T Mikkelsen H O Fractionally integrated generalized autoregressive conditional heteroskedasticity[j ] Jour2 nal of Econometrics :3 30 [7 ] ARCH [J ] (2) :12 18 [8 ]Bollerslev T et al A capital-asset pricing model with time-varying coefficients[j ] Journal of Political Economy : [9 ]Engle R F roner F Multivariate simultaneous generalized ARCH[J ] Econometric Theory : [ 10 ]Bollerslev T Modeling the coherence in short- run nominal exchange rates : A multivariate generalized ARCH model[j ] The Re2 view of Economics and Statictics : [ 11 ]Berndt E et al Estimation inference in nonlinear structural models[j ] Annals of Economic and Social Measurement : [ 12 ]Dunne P G Size and book-to- market factors in a multivariate GARCH-in- mean asset pricing application[j ] International Review of Financial Analysis :35 52 [13 ] - [J ] (4) : [14 ] [M] : 1998 [15 ]Bollerslev T Engel R F Common persistence in conditional variances[j ] Econometrica (1) : [16 ]Li Han- dong Zhang Shi- ying Common persistence and error- correction model in conditional variance[j ] Journal of System Sci2 ence and Systems Engineering (3) : Multivariate GARCH modeling and its application in volatility analysis of Chinese stock markets FAN Zhi ZHANG Shi- ying School of Management Tianjin University Tianjin China Abstract : This paper first briefly reviews the evolvement of univariate ARCH class models and introduces several multivariate GARCH class models Considering the shortage of traditional estimation methods for multivariate GARCH based on gradient information we give out the likelihood estimating method based on genetic algorithm Fi2 nally the paper presents the demonstration of Chinese stock markets : Both Shanghai and Shenzhen stock markets show volatility persistence in variance When combined together the two markets show obvious bivariate GARCH effect and there is no common persistence in Shanghai and Shenzhen stock markets ey words : financial volatility ; multivariate GARCH ; genetic algorithm ; Chinese stock markets China Academic Journal Electronic Publishing House All rights reserved

13. Estimation and Extensions in the ARCH model. MA6622, Ernesto Mordecki, CityU, HK, References for this Lecture:

13. Estimation and Extensions in the ARCH model MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Robert F. Engle. GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics,