, ) ARCH (MCMC) ; ARCH : F : (2003) ARCH (APT) GARCH. (clustering effect),engle Bollerslev ARCH ].

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1 JOURNAL OF MANAGEMEN SCIENCES IN CHINA Vol 6 No 4 Aug 2003 ( ) : (MCMC) ARCH : ; ; (MCMC) ; ARCH : F :A : (2003) ARCH ; ARCH [8 10 ] ARCH (CAPM) (AP) SC GARCH [1 ] (clustering effect) Engle Bollerslev [2 3 ARCH ] ARCH ARCH (1) ARCH :1 [4 7 ] : ; : : ( ) ; ( ) ; ; : (1966 )

2 t (QML) ( GMM) (SML) t (SMM) 2 ( [4 ] ( ARMA GARCH ) [16 Wiggins ] Karpoff (indirect infer2 [17 ] ) ence) ( EMM) 3 Danielsson [6 ] (MCMC) r t = t t t = 1 t N (0 1) MCMC (1) = ( v) R = { r t } t =1 t = { t } t =1 WINBUGS ( Spiegelhalter 1996) SV SV Gibbs SV Engle Victor MCMC SV [22 ] (asymmetrics news impact curve) 1 GARCH Nelson 1991 ARCH EGARCH Nelson GARCH 1 ln ( 2 t) = + ln ( 2 t - 1) + v t t N (0 1) EGARCH EGARCH t P t r t = 100ln ( P t / P t - 1 ) r t = t t t = 1 t N (0 1) r t N (0 2 t) ln ( 2 t) = + 1 t - 1 / t t - 1 / t t ln ( 2 t - 1) + v t t N (0 1) (2) [11 15 ] :ARCH t (leverage effect) 2 t 0 ; 2 < 0 (1) SV ARCH (2) SV L

3 4 : 65 1 f ( ) = f ( f ( 0 ) 2 : MCMC f ( 2 1 t - 1 f ( r t t ) (6) t =1 0 ) t =1 [ f ( t t - 1 ) = 2 t v exp { v 2 (ln 2 t - - ln 2 t - 1) 2 } ] (4) R f ( R ) = t =1 f ( r t t ) = 1 t =1 t 2 exp - r 2 2 (5) t R ) = K f ( ) f ( 2 t 0 ) t = 1 ) t = 1 f ( K R SV f ( R ) = f ( R ) f ( ) (7) = ( v) = { t } t = 0 f ( R (4 + ) ) = ) d (8) :f ( R Kim(1998) ) f ( ) f ( ) f ( R ) = Bet (20 ; f ( 1 5) 0 86 ; ) N (0 100) ln ( v) (1) AR(1) ; ln ( v) N ( ) f ( ) ln ( t) f ( ) f ( ) = f ( ) f ( ) (3) SV L 1 N (0 100) 2 N (0 100) ; 1 2 f ( 1 2 ) = f ( 1 2 ) f ( 1 2 ) = f ( ) f ( 1 ) f ( 2 ) f ( 0 ) t = 1 f ( t t ) = f ( ) f ( 1 ) f ( 2 ) f ( 0 ) (9) 2 t =1 t v exp - 1 2v 2 (ln 2 t t - 1 / t t - 1 / t ln 2 t - 1) 2 t (3) (9) r 0 0

4 : t - 1 = r t - 1 / t - 1 ( (1) ( B) ) f ( 1 2 R ) f ( ( b) (1) g ( R ) ( b- 1) ( b+1) ( B) R ) g ( R ) Gibbs : E{ g ( R ) } = 1) (0) = ( (0) (1) (0) (B) ) i = 0 g ( R ) f ( R ) dd (10) (1) f ( (1) ( (2) i) ( ( B) i) ; R ) g ( R ) ( ( i b) +1) f ( ( b) ( (1) i +1) ( (2) i +1) ( ( i b- +1) 1) SV ( (B) i) ; R ) SV (B) f ( (B) ( i +1) ( i) ( i +1) 3) i = i + 1 2) (0) (Markov (1) ( i) chain Monte Carlo MCMC) SV ( i) ( i +1) 2 2 MCMC MCMC (Metropolis ( (2) i +1) ( ( i b- +1) 1) ( ( i) b+1) ( ( B) i) ) 1953) Hastings(1970) (12) Gelfand MCMC Smith(1990) MCMC WINBUGS(Spiegelhalter 1996) anner Wong(1987) WINBUGS Gibbs Hastings-metropolis t ln ( L) MCMC Gibbs SV Gibbs f ( 3 1 R ) { ( k) } N k = { ( k) } N k = m E( g ( R ) ) 1 N - m + 1 N k = m E( g () ) g ( ( k) R ) (11) Gibbs B := 2) ( i +1) ( i) (1) ( i +1) (2) ( (B i +1) - 1) ; R ) K G ( ( i) ( i +1) ) = B f ( ( ( i b) +1) ( (1) i +1) b = : r t = 100ln ( p t / p t - 1 ) 1 : E( g () ) g R 1 2 1

5 4 : 67 ;ARCH ; ARCH 3 Box- Pierce Q ; 2 1 (Skew) ( Kurtosis) (J - B) ARCH : () 2 Q (5) Q (10) Q (20) Q (5) Q (10) Q (20) (1 000) (1 000) (1 000) (1 000) : () 3 2 SV SV ARCH 3a 4a 3 SV MCMC MC t = r t / t MCMC 4 SV SV MC MC 1 MC MCMC ;ARCH SV ARCH 0 92

6 Q-Q SV (t ) Q-Q SV ( ) ; 3 SV MCMC v ln( L) ( ) ( ) ( ) [2 846E - 4 ] [3 048E - 4 ] [ ] [ ] ( ) ( ) ( ) [4 692E - 4 ] [4 87E - 4 ] [ ] [ ] : () [ ] MC Gibbs 4 SV GARCH(1 1) SV GARCH SV GARCH (J-B) ARCH ( ) ( ) ( ) ( ) Q (5) (1 000) (1 000) Q e c r e i P - x o B Residuals Squared residuals Q (10) Q (20) Q (5) Q (10) (1 000) (0 003) ( ) ( ) (1 000) (1 000) (1 000) (0 091) ( ) ( ) (1 000) (1 000) Q (20) ( ) (1 000) ( ) (1 000) : () 3 3 SV GARCH AIC GARCH(1 1) GJ R- GARCH ARCH EGARCH ARCH GARCH( p q) GARCH(1 1) GJ R- GARCHEGARCH 5 SC

7 4 : 69 GARCH(1 1) (a : b : c : d Q-Q ( ) ) 3 SV ( ) (a : b : c : d Q-Q ( ) ) 4 GARCH ( )

8 (a : b : c : d Q-Q ( ) ) 5 SV ( ) (a : b : c : d Q-Q ( ) 6 GARCH ( ) GARCH SV GARCH Q-Q

9 4 : 71 2 t r 2 t 2 t r t 4 SV E{ ( r 2 t - < t ) u t - 1 u t - 2 } GARCH r t 4 < t = 2 t (^r 2 t - ^ 2 t) 2 MS E = MS E - 1 (^r 2 t - ^ 2 t) 2 2 : MA E = t = 1 5 SV GARCH GARCH(1 1) GARCH(1 1) SC MS E MA E L E : SC Schwarz SC = ( (ln ) ) / - 1 ^r 2 t - ^ 2 t L E = - 1 t = 1 t = 1 ln (^r2 t) - ln ( ^ 2 t) 5 SV GARCH SV GARCH MCMC SC : ARCH ARCH 4 : [1 ] [M] : 2001 [ 2 ] Engle R F Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation[j ] Econometri2 ca : [3 ]Bollerslev Generalized autogressive conditional heteroscedasticity[j ] Journal of Econometrics : [4 ]Clark P A subordinated stochastic process model with finite variance for speculative process[j ] Econometrica : [ 5 ] Epps W Epps M L he stochastic dependence of security price changes and transaction volumes : Implications for the mixture-ofdistribution hypothesis[j ] Econometrica : [ 6 ] Danielsson J Estimation of dynamic stochastic volatility model for asset price determination by simulated maximum likelihood[d] Durham : Duke University 1991 [7 ]aylor S J Modeling stochastic volatility[j ] Mathematical Finance (2) : [ 8 ] Danielsson J Stochastic volatility in asset prices estimation with simulated maximum likelihood[j ] Journal of Econometrics : [ 9 ] Kim S Shephard N Stochastic Volatility : Likelihood Inference and Comparison with ARCH Models[ R] Nuffield College London : Oxford University 1994

10 [ 10 ] Shephard N Statistical Aspects of ARCH and Stochastic Volatility[ R] Nuffield College London : Oxford University 1995 [11 ] Bollerslev A conditional heteroskedastic time series model for speculative prices and rates of return[j ] he Review of Eco2 nomics and Statistics : [ 12 ] Baillie R Bollerslev he message of daily exchange rates : A conditional variance tale[j ] Journal of Business and Economic Statistics : [ 13 ] Diebold F X Empirical Modelling of Exchange Rate[M] Berlin : Springer-Verlag 1987 [14 ] Nelson D B Conditional heteroskedasticity in asset returns : A new approach[j ] Econometrica : [15 ] Hsieh D A Statistical properties of daily exchange rates : [J ] Journal of International Economics : [ 16 ] Wiggins J B Option values under stochastic volatility : heory and empirical estimates[j ] Journal of Financial Economics : [ 17 ] Karpoff J M he relation between price changes and trading volume : A survey[j ] Journal of Financial and Quantitative Analysis : [ 18 ] Hendry D F Richard J Likelihood Evaluation for Dynamic Latent Variables Models[ R] ISDS discussion paper Durham : Duke University 1990 [ 19 ] Polson J E Rossi G E Bayesian analysis of stochastic volatility models[j ] Journal of Business & Economic Statistics : [ 20 ] Polson J E Rossi G E Bayesian Analysis of Stochastic Volatility Model with Fat-tails and Correlated errors[ R] www2 bc edu/ jacquier 2001 [21 ] Gilks W R Richardson S Spiegelhalter D J Markov Chain Monte Carlo in Practice [M] London : Chapman & Hall Press 1998 [22 ]Engle Robert F Victor K Ng Measuring and testing the impact of news on volatility[j ] Journal of Finance : Estimating volatility of Chinese stock market by stochastic volatility model WANG Chun-feng JIANG Xiang-lin LI Gang Center of Finance Engineering Management School ianjin University ianjin China Abstract :he volatility of Chinese stock market is investigated using the dynamic version of stochastic volatility model and Bayesian analysis based on MCMC is introduced to improve the parameters estimation in stochastic volatility model Empirical results on Chinese stock market indicate that stochastic volatility model outperforms the ARCH model in capturing the heteroskedasticity and serial correlation of volatility of the stock market returns Key words : stochastic volatility model ; Bayesian analysis ; MCMC ; ARCH model

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