AGARCH模型及多维常数相关GARCH模型的统计分析
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摘要
近几十年来，关于时间序列分析的研究得到了迅速发展，特别是对于线性时间序列，取得了系统而丰富的成果。但是，对非线性时间序列的研究，仅在近二十年来才逐渐被重视起来。
对非线性时间序列的研究，近几十年来，有两条研究路线非常活跃，其一是自回归条件异方差(ARCH)模型，其二是非平稳(单位根)时间序列模型。具有自回归条件异方差(ARCH)的时间序列模型，首先是由Engle(1982)提出，这类模型在金融和经济领域有着广泛的应用。此后，经济学家和数学家又根据在实际研究中的需要，提出了许多GARCH模型的变异，形成了一个ARCH模型体系。
本文进行了极限理论的研究，主要解决了以下问题：
1．对于吴硕思和方兆本(2000)提出的非对称广义自回归条件异方差新模型，证明了它的极大似然估计(MLE)的渐近正态性和相合性。
2．在多维的情形下，着重研究Bollerslev(1990)提出的常数相关的多维GARCH模型的极大似然估计(MLE)的渐近正态性和相合性。
3．本文提出了非正态的AGARCH模型，并进行了实证研究，实证结果表明这样的方法是可行的和较优的。
In recent several decades, the development of time series analysis is rapid. Especially ,the investigation of linear time series, the systematic and abundant achievements have been obtained. However, the statisticians and econometricians have gradually paid attention to investigation of nonlinear time series for the last two decades.
In the last decade, there exist two active lines on the investigation of nonlinear time series. One is the autoregressive conditional heteroscedasticity(ARCH) model, the another is the nonstationary(unit root) time series model.
The concept of ARCH, which stands for autoregressive heteroscedasticity, was first introduced by Engle(1982) to handle time series with a changing conditional variance. Thereafter, mathematician and econometricians brought forward various subsidiaries of ARCH models according to the request in the practical research and formed a ARCH model system .
In this paper, the limit theory is discussed and the main problems are solved as followed:
1 .We will obtain asymptotic normality and consistency of MLE for AGARCH model introduced by Wu shuosi and Fang zhaoben (2000).
2. We will give the asymptotic normality and consistency of MLE for multivariate model with constant correlation introduced by Bollerslev(1990).
3. Moreover we will introduce the AGARCH model with non-normal distribution and have empirical study shows that the model suggested is feasible.
对非线性时间序列的研究，近几十年来，有两条研究路线非常活跃，其一是自回归条件异方差(ARCH)模型，其二是非平稳(单位根)时间序列模型。具有自回归条件异方差(ARCH)的时间序列模型，首先是由Engle(1982)提出，这类模型在金融和经济领域有着广泛的应用。此后，经济学家和数学家又根据在实际研究中的需要，提出了许多GARCH模型的变异，形成了一个ARCH模型体系。
本文进行了极限理论的研究，主要解决了以下问题：
1．对于吴硕思和方兆本(2000)提出的非对称广义自回归条件异方差新模型，证明了它的极大似然估计(MLE)的渐近正态性和相合性。
2．在多维的情形下，着重研究Bollerslev(1990)提出的常数相关的多维GARCH模型的极大似然估计(MLE)的渐近正态性和相合性。
3．本文提出了非正态的AGARCH模型，并进行了实证研究，实证结果表明这样的方法是可行的和较优的。
In recent several decades, the development of time series analysis is rapid. Especially ,the investigation of linear time series, the systematic and abundant achievements have been obtained. However, the statisticians and econometricians have gradually paid attention to investigation of nonlinear time series for the last two decades.
In the last decade, there exist two active lines on the investigation of nonlinear time series. One is the autoregressive conditional heteroscedasticity(ARCH) model, the another is the nonstationary(unit root) time series model.
The concept of ARCH, which stands for autoregressive heteroscedasticity, was first introduced by Engle(1982) to handle time series with a changing conditional variance. Thereafter, mathematician and econometricians brought forward various subsidiaries of ARCH models according to the request in the practical research and formed a ARCH model system .
In this paper, the limit theory is discussed and the main problems are solved as followed:
1 .We will obtain asymptotic normality and consistency of MLE for AGARCH model introduced by Wu shuosi and Fang zhaoben (2000).
2. We will give the asymptotic normality and consistency of MLE for multivariate model with constant correlation introduced by Bollerslev(1990).
3. Moreover we will introduce the AGARCH model with non-normal distribution and have empirical study shows that the model suggested is feasible.
引文
[1].Baillie,R.T.,Bollerslev, T.,Mikkelsen,H., Fractional integrated generalized autoregressive conditional heteroskedasticity [J].Journal of Econometrics.(74) 1996,3-30
[2].Bollerslev T. Generalized autoregressive conditional heteroskedasticity[J].Joumal of Economics,(31) 1986,307-327.
[3].Bollerslev, T.,1990.Modelling the coherence in short-run nominal exchange rates:a multivariate generalized ARCH model. Review of Economics and Statistics 72,498-505.
[4].Bollerslev, T.,Mikkelsen,H.O.,Modelling and pricing long memory in stock market volatility[J], Journal of Econometrics,(73) 1996,151-184.
[5].Changli He,Timo Tersvirta, Properties of moments of a family of GARCH processes. Journal of Econometrics (92) 1999,173-192.
[6].Daniel Straumann and Thomas Mikosch. Quasi-maximum-likelihood estimation in heteroscedastic time series:a stochastic recurrence equations approach.to be appear 2003.
[7].Duan Jin-Chun. Augmented GARCH(p,q) process and its diffusion limit[J]. Journal of Econometrics.79(1997),97-127.
[8].Ding Zhuanxin,Granger C W J,Engle R F. A long memory property of stock market returns and a new model[J].Journal of Empirical Finance, 1(1993), 83-196.
[9].Ding Zhuanxin,Granger, C.W.J.,Modelling volatility persistence of speculative
returns:A new approach[J], Journal of Econometrics,(73) 1996,185-215.
[10].Engle R F, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K..Inflation[J].Econometrica, 50 (1982) 987-1008.
[11].Engle R F, Bollerslev T. Modelling the persistence of conditional variances [J].Econometric Reviews. (5) 1986,81-87.
[12].Feike C.Drost,Chris A.J.Klaassen, Efficient estimation in semiparametric GARCH models.Journal of Econometrics (81)1997,193-221.
[13].F.Comte and O.Lieberman, Asymptotic theory for multivariate GARCH processes,J.Multivariate analysis 84 (2003),61-84.
[14].Gloria Gonz a' lez-Rivera,Feike C.Drost,Efficiency comparisons of maximum-likelihood based estimators in GARCH models.Journal of Econometrics (93)1999,93-111.
[15].Higgens M L,Bera A K. A class of nonlinear ARCH models[J].International Economic Review, 1992,33:137-158.
[16].Hong Yongmiao,One-sided testing for conditional heteroskedasticity in time series models, J.Time Ser.Anal., 18(1997),253-277.
[17].Lee,J.H.H.and King,M.L.,A locally most mean powerful based score test for ARCH and GARCH regression disturbances,J.Bus.Econ. Stat., 11(1993), 17-27.
[18].Ling.S. On the probabilistic properties of a double-threshold ARCH conditional heteroskedasticity model. Technical report. University of Hong Kong Dept.of Statistics, 1995.
[19].Nelson DB. ARCH models as diffusion approximations[J].Journal of Econometrics, 45(1990),7-38.
[20].Partha Deb,Finite sample properties of the ARCH class of models with stochastic volatility, Ecomomics Letters (55) 1997,27-34.
[21]Peter Bühlmann,Alexander J.McNeil, An algorithm for nonparametric GARCH modeling .Computational Statistics &Data Analysis 40(2002) 665-683.
[22].Robert F.Engle , Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH, Ph.D.Dissertation,University of California at San Diego.
[23].Sang-Won Lee and Bruce E.Hansen. Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory,10,1994,29-52.
[24].Shiqing Ling and Michael McAleer, Asymptotic theory for a vector ARMA-GARCH model ,To appear in Econometric Theory.
[25]S.Ling and M.McAleer, Stationarity and the existence of moments of a family of GARCH processes, Journal of Econometrics,106(2002b) 109-117.
[26]S.W. Lee,B.E. Hansen, Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator, Econometric Theory, 10(1994), 29-52.
[27].SUDUK KIM and HIROKI TSURUMI, Korean Currency Crisis and Regime Change:A Multivariate GARCH Model with Bayesian Approach. Asia-Pacific Financial Markets (7)2000,31-44.
[28].Teruo Nakatsuma,Bayesian analysis of ARMA-GARCH models:A Markov chain sampling approach, Journal of econometrics (95)2000,57-69.
[29].T.Jeantheau, Strong consistency of estimators for multivariate ARCH models, Econometrics Theory 14(1998) 70-86.
[30]William G.Jacoby. Loess:a nonparametric,graphical tool for depicting relationships between variables.Electoral Studies 19(2000) 577-613.
[31].Zakoian J M. Threshold heteroskedastic models[J].Journal of Economic Dynamics and Control, 18(1994) 931-955.
[32]刘继春著，多维广义自回归条件异方差时间序列模型，吉林大学博士论文，2001．
[33]．吴硕思，方兆本，非对称广义自回归条件异方差的新模型，应用概率统计，(16)2000．
[34]吴硕思，方兆本，带约束条件的AGARCH模型，应用概率统计，17(2001)．
[35]王德辉，宋立新，史宁中，序约束下ARCH(0，2)模型参数估计与检验，应用概率统计，18(2002)．
[36]．张世英，柯珂，ARCH模型体系，系统工程学报，(17)2002．
[37]．胡海鹏，方兆本，用AR-EGARCH-M模型对中国股市波动性的拟合分析，系统工程(20)2002．
[38]．钱敏平，龚光鲁著，应用随机过程，北京大学出版社，1998．
[39]．安鸿志，陈敏著，非线性时间序列分析，上海科学技术出版社，1998．
[40]．张尧庭，方开泰著，多元统计分析引论，科学出版社，1982．
[41]．林正炎，陆传荣，苏中根著，概率极限理论基础，高等教育出版社，1999．
[42]．田铮译，时间序列的理论与方法，高等教育出版社，2001．
[43]钱敏平，龚光鲁著，应用随机过程，北京大学出版社，1998．
[44]王永县著，运筹学，清华大学出版社，1993．
[45]．成平，陈希孺等著，参数估计，上海科学技术出版社，1985．
[46]．伍尤贵等，应用时间序列分析，广西师范大学出版社，1999．
[47]．陈懋祖著，高等时间序列经济计量学，上海人民出版社，1998．
[48]．时间序列分析，(美)詹姆斯D．汉密尔顿著，中国社会科学出版社，1999，12．
[49]．倪勤编著，SAS最优化软件速成，科学出版社，1998．
[2].Bollerslev T. Generalized autoregressive conditional heteroskedasticity[J].Joumal of Economics,(31) 1986,307-327.
[3].Bollerslev, T.,1990.Modelling the coherence in short-run nominal exchange rates:a multivariate generalized ARCH model. Review of Economics and Statistics 72,498-505.
[4].Bollerslev, T.,Mikkelsen,H.O.,Modelling and pricing long memory in stock market volatility[J], Journal of Econometrics,(73) 1996,151-184.
[5].Changli He,Timo Tersvirta, Properties of moments of a family of GARCH processes. Journal of Econometrics (92) 1999,173-192.
[6].Daniel Straumann and Thomas Mikosch. Quasi-maximum-likelihood estimation in heteroscedastic time series:a stochastic recurrence equations approach.to be appear 2003.
[7].Duan Jin-Chun. Augmented GARCH(p,q) process and its diffusion limit[J]. Journal of Econometrics.79(1997),97-127.
[8].Ding Zhuanxin,Granger C W J,Engle R F. A long memory property of stock market returns and a new model[J].Journal of Empirical Finance, 1(1993), 83-196.
[9].Ding Zhuanxin,Granger, C.W.J.,Modelling volatility persistence of speculative
returns:A new approach[J], Journal of Econometrics,(73) 1996,185-215.
[10].Engle R F, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K..Inflation[J].Econometrica, 50 (1982) 987-1008.
[11].Engle R F, Bollerslev T. Modelling the persistence of conditional variances [J].Econometric Reviews. (5) 1986,81-87.
[12].Feike C.Drost,Chris A.J.Klaassen, Efficient estimation in semiparametric GARCH models.Journal of Econometrics (81)1997,193-221.
[13].F.Comte and O.Lieberman, Asymptotic theory for multivariate GARCH processes,J.Multivariate analysis 84 (2003),61-84.
[14].Gloria Gonz a' lez-Rivera,Feike C.Drost,Efficiency comparisons of maximum-likelihood based estimators in GARCH models.Journal of Econometrics (93)1999,93-111.
[15].Higgens M L,Bera A K. A class of nonlinear ARCH models[J].International Economic Review, 1992,33:137-158.
[16].Hong Yongmiao,One-sided testing for conditional heteroskedasticity in time series models, J.Time Ser.Anal., 18(1997),253-277.
[17].Lee,J.H.H.and King,M.L.,A locally most mean powerful based score test for ARCH and GARCH regression disturbances,J.Bus.Econ. Stat., 11(1993), 17-27.
[18].Ling.S. On the probabilistic properties of a double-threshold ARCH conditional heteroskedasticity model. Technical report. University of Hong Kong Dept.of Statistics, 1995.
[19].Nelson DB. ARCH models as diffusion approximations[J].Journal of Econometrics, 45(1990),7-38.
[20].Partha Deb,Finite sample properties of the ARCH class of models with stochastic volatility, Ecomomics Letters (55) 1997,27-34.
[21]Peter Bühlmann,Alexander J.McNeil, An algorithm for nonparametric GARCH modeling .Computational Statistics &Data Analysis 40(2002) 665-683.
[22].Robert F.Engle , Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH, Ph.D.Dissertation,University of California at San Diego.
[23].Sang-Won Lee and Bruce E.Hansen. Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory,10,1994,29-52.
[24].Shiqing Ling and Michael McAleer, Asymptotic theory for a vector ARMA-GARCH model ,To appear in Econometric Theory.
[25]S.Ling and M.McAleer, Stationarity and the existence of moments of a family of GARCH processes, Journal of Econometrics,106(2002b) 109-117.
[26]S.W. Lee,B.E. Hansen, Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator, Econometric Theory, 10(1994), 29-52.
[27].SUDUK KIM and HIROKI TSURUMI, Korean Currency Crisis and Regime Change:A Multivariate GARCH Model with Bayesian Approach. Asia-Pacific Financial Markets (7)2000,31-44.
[28].Teruo Nakatsuma,Bayesian analysis of ARMA-GARCH models:A Markov chain sampling approach, Journal of econometrics (95)2000,57-69.
[29].T.Jeantheau, Strong consistency of estimators for multivariate ARCH models, Econometrics Theory 14(1998) 70-86.
[30]William G.Jacoby. Loess:a nonparametric,graphical tool for depicting relationships between variables.Electoral Studies 19(2000) 577-613.
[31].Zakoian J M. Threshold heteroskedastic models[J].Journal of Economic Dynamics and Control, 18(1994) 931-955.
[32]刘继春著，多维广义自回归条件异方差时间序列模型，吉林大学博士论文，2001．
[33]．吴硕思，方兆本，非对称广义自回归条件异方差的新模型，应用概率统计，(16)2000．
[34]吴硕思，方兆本，带约束条件的AGARCH模型，应用概率统计，17(2001)．
[35]王德辉，宋立新，史宁中，序约束下ARCH(0，2)模型参数估计与检验，应用概率统计，18(2002)．
[36]．张世英，柯珂，ARCH模型体系，系统工程学报，(17)2002．
[37]．胡海鹏，方兆本，用AR-EGARCH-M模型对中国股市波动性的拟合分析，系统工程(20)2002．
[38]．钱敏平，龚光鲁著，应用随机过程，北京大学出版社，1998．
[39]．安鸿志，陈敏著，非线性时间序列分析，上海科学技术出版社，1998．
[40]．张尧庭，方开泰著，多元统计分析引论，科学出版社，1982．
[41]．林正炎，陆传荣，苏中根著，概率极限理论基础，高等教育出版社，1999．
[42]．田铮译，时间序列的理论与方法，高等教育出版社，2001．
[43]钱敏平，龚光鲁著，应用随机过程，北京大学出版社，1998．
[44]王永县著，运筹学，清华大学出版社，1993．
[45]．成平，陈希孺等著，参数估计，上海科学技术出版社，1985．
[46]．伍尤贵等，应用时间序列分析，广西师范大学出版社，1999．
[47]．陈懋祖著，高等时间序列经济计量学，上海人民出版社，1998．
[48]．时间序列分析，(美)詹姆斯D．汉密尔顿著，中国社会科学出版社，1999，12．
[49]．倪勤编著，SAS最优化软件速成，科学出版社，1998．