长记忆理论及其在金融市场建模中的应用
|
摘要
上个世纪80年代以来,长记忆理论开始在计量经济学领域中得到迅速发展,并逐渐在金融领域中得到广泛应用。然而到目前为止,理论界关于如何准确检验一个时间序列是I(0),I(d)还是I(1)过程的问题,在检验方法和统计量上仍存在很多争论。当前的估计和检验方法也只适用于较为简单的情形,对于模型中存在短记忆成分和结构突变成分的研究方法还很不完善,迫切需要从理论上进一步深化。此外,在实证研究中,有证据表明,金融市场如股票市场中的回报率及其波动性都具有长记忆特征,这就意味着市场有效性假定不复存在。不仅如此,政府宏观政策的冲击还有可能改变市场的长期特征,忽略这一问题很可能影响到市场有效性的判定和相应政策的制定。因此,对金融市场长记忆性的精确建模对于投资组合与风险管理就有着非常重要的意义。要实现上述目的,就必须从理论上对这一领域的研究进行系统和完善,并在实践中加以正确运用。
本文立足于目前理论研究和实践应用中存在的具体问题,在总结前人研究成果基础上,对长记忆理论及其在金融市场建模中的应用进行深入了研究。其贡献主要体现在以下几点:第一,基于一个ARFIMA(p,d,q)过程,对三种分数单积参数的半参数估计方法的有限样本性质进行了比较,并用蒙特卡罗方法对备择假设下短期项对分数单积参数估计和长记忆性判别的影响进行了模拟。第二,基于一个误差项是分数白噪声过程的AO模型,对小样本下不同突变形式对分数单积参数估计和检验的影响进行了分析,并对上述情形下多个未知突变点BP检验的有限样本性质给出了模拟证据。第三,分析了结构突变对中国股市长记忆性的影响,将长记忆参数改变看作市场有效性的变化,就政府干预对股市有效性的影响进行了科学评价,并提出了政策建议。
本文对于长记忆理论及其在金融市场中应用的研究是按照一定的逻辑思路逐步展开的。全文共分为六章,第一章是引言,阐述论文的选题背景及意义、研究思路与方法、主要创新点。接下来三章是理论研究部分,其中第二章是长记忆过程及其模型研究,主要对长记忆过程的定义与特征,两种长记忆模型即ARFIMA和FIGARCH模型的设定及其估计方法进行分析。在此基础上,第三、四章分别探讨了短期项和结构突变项对于长记忆建模的影响。第五章作为实证研究部分,首先运用长记忆模型对石油期货市场中的“约瑟夫效应”进行了验证。接下来运用长记忆性与结构突变分析方法对我国股市的有效性与政府干预问题进行了研究。最后一章是结论以及对未来研究方向的展望。通过各章的分析,我们得到的主要结论有:
第一,当数据生成过程是一个一般的ARFIMA(p,d,q)而不是分数白噪声过程时,各半参数方法得到的分数单积参数估计量仍能服从正态分布,但其分布的均值却明显偏离了真实值,这就造成分数单积参数估计的偏差和长记忆性的误判。同时,自回归参数和移动平均参数往往具有不同的短期效应。此外,基于小样本下不同带宽的模拟,建议在实践中应该选择较大的带宽。
第二,通过对一个带有结构突变的长记忆过程检验其长记忆性,发现模型中的结构突变尤其是趋势突变会造成对分数单积参数的高估,导致长记忆性检验的功效和尺度扭曲都很高。这时,在各种半参数方法中,GPH方法的稳健性相对较高,造成的偏差最小。接下来,将Bai和Perron(1998,2003)多个未知突变点估计和检验方法推广到误差项是长记忆过程的情形中,发现除d接近于0.5的情形外,该方法对突变点的估计和检验具有比较好的效果。尤其是当d＜0时,突变点估计量的分布、以及相应的检验功效和实际检验水平也更优。该检验对于突变点的位置、突变幅度的变化也表现的比较稳健。
第三,对石油期货市场的实证研究发现,该市场的投资风险对新息冲击的响应是一个长记忆过程。因此在实践中,建议采用FIGARCH等模型对未来的石油价格及其波动进行预测,并采取相应策略确保宏观经济平稳运行。通过对我国股市有效性的再审视,发现我国股票价格走势的改变常常是政府政策的出台造成的。而在不同阶段,政策对于股票市场有效性的影响也不同。因此,在今后的发展中,政府要逐步抛弃依靠政策引导股市的观念,这是中国股市走向有效性的必由之路。
Since 1980s, the long memory theory has been developed very fast in the Econometrics area, and been put into application in the Finance widely. However, as far as the theoretical research is concerned, there is no consensus among the question on how to test a stochastic process is I(0), I(d), or I(1) . Moreover, the estimation and test method is only fit for the simple model, and there is no mature analysis when there is affection of short memory term and structural break. Therefore, theoretical improvement is highly needed. As to the empirical research, there is evidence that the return series and its volatility show long memory properties in the financial market, for example, the stock market, which means that the Efficient Market Hypothesis (EMH) is no longer existed. In addition, shock from government policy may change the long term characteristics of the market, lose sight of which may influence the EMH judgment and the related policy. So it is great importance of exact model specification on the investment portfolio and risk management. To make all of the above goals into realization, the research on long memory theory must be systematical improved and then put into application in a correct way.
Based on the available literatures, this thesis aims at the solution of the present theoretical and empirical problems and makes research on the long memory theory with its application in the financial market. The main contribution of the thesis are as follows: Firstly, based on an AKFIMA(p, d, q) process, it compares the finite sample properties of three semiparametic estimator on the fractional integration parameter d, and makes simulations on the influence of the short memory term on the estimation of d and the test of long memory under the alternative hypothesis by Monte Carlo approach. Secondly, the influence of the different structural breaks terms is analyzed based on an AO model in which the error term is a fractional noise. And then, the performance of BP test on unknown muli-break points is simulated via Monte Carlo method under small samples. Thirdly, taking the change of the long memory parameter as the market efficiency's altering, the latent impact of structural breaks to China's stock market is studied. At last, the effect of the government policy is evaluated and advices are given.
The study on the long memory theory with its application on financial market is deployed gradually. The whole thesis comprises six chapters. The first one is introduction, which briefly demonstrates the background and significance of the research, the goals and method used, and the innovations. The next three are theoretical study. Thereinto, the second chapter is the long memory process and model study in which we introduce the definition, characteristics, and ARFIMA and FIGARCH model with the estimation. Above on this, the third and forth chapter analyze the impact of short term and structural breads on the modeling of long memory process. The fifth chapter is empirical research which starts with an verification of the "Joseph effect" in the oil futures market, and then the EMH in China's stock and government intervention is researched via the long memory and struck breaks approach. The last chapter is conclusion and future research interest. The main conclusions of the thesis are as follows:
Firstly, When the process is a general ARFIMA(p, d, q) rather than fractional noise, the distributions of the semi-estimators are still normal, however, the mean has some deviation from the real value, which cause power and size problems in the test. And the AR and MA parameters have different short term effects. Moreover, we suggest chose longer bandwidth in the practice.
Secondly, through the long memory test on an AO model with fractional noise errors, we find that the structural breaks especially the trend breaks can cause overestimate of the long memory parameter, and lead to high power and size. In this situation, the GPH estimator is more robust with smaller bias. Next, the Bai and Perron(1998, 2003) test is extended into long memory conditions and shows a fairly well performance except that d is close to 0.5. Especially when d<0, the distribution of the estimator and the test are even better. Besides, the test is relatively robust to the location and change of the break points.
Lastly, through the empirical research on the oil futures market, it is indicated that the shock of innovation to the investment risk is a long memory process. Therefore, the FIGARCH model should be used in practice to forecast the oil futures price and its volatility and the related policy should be make to keep the stable economic growth. Through the retest of the long memory properties in our stock market, it shows that the change of the price have something to do with the government policy, which has different impact to the market efficiency in different periods. Therefore, the policy-leading way on management of the stock market should be abandoned in the future which is the exact way for the market to go to efficiency.
本文立足于目前理论研究和实践应用中存在的具体问题,在总结前人研究成果基础上,对长记忆理论及其在金融市场建模中的应用进行深入了研究。其贡献主要体现在以下几点:第一,基于一个ARFIMA(p,d,q)过程,对三种分数单积参数的半参数估计方法的有限样本性质进行了比较,并用蒙特卡罗方法对备择假设下短期项对分数单积参数估计和长记忆性判别的影响进行了模拟。第二,基于一个误差项是分数白噪声过程的AO模型,对小样本下不同突变形式对分数单积参数估计和检验的影响进行了分析,并对上述情形下多个未知突变点BP检验的有限样本性质给出了模拟证据。第三,分析了结构突变对中国股市长记忆性的影响,将长记忆参数改变看作市场有效性的变化,就政府干预对股市有效性的影响进行了科学评价,并提出了政策建议。
本文对于长记忆理论及其在金融市场中应用的研究是按照一定的逻辑思路逐步展开的。全文共分为六章,第一章是引言,阐述论文的选题背景及意义、研究思路与方法、主要创新点。接下来三章是理论研究部分,其中第二章是长记忆过程及其模型研究,主要对长记忆过程的定义与特征,两种长记忆模型即ARFIMA和FIGARCH模型的设定及其估计方法进行分析。在此基础上,第三、四章分别探讨了短期项和结构突变项对于长记忆建模的影响。第五章作为实证研究部分,首先运用长记忆模型对石油期货市场中的“约瑟夫效应”进行了验证。接下来运用长记忆性与结构突变分析方法对我国股市的有效性与政府干预问题进行了研究。最后一章是结论以及对未来研究方向的展望。通过各章的分析,我们得到的主要结论有:
第一,当数据生成过程是一个一般的ARFIMA(p,d,q)而不是分数白噪声过程时,各半参数方法得到的分数单积参数估计量仍能服从正态分布,但其分布的均值却明显偏离了真实值,这就造成分数单积参数估计的偏差和长记忆性的误判。同时,自回归参数和移动平均参数往往具有不同的短期效应。此外,基于小样本下不同带宽的模拟,建议在实践中应该选择较大的带宽。
第二,通过对一个带有结构突变的长记忆过程检验其长记忆性,发现模型中的结构突变尤其是趋势突变会造成对分数单积参数的高估,导致长记忆性检验的功效和尺度扭曲都很高。这时,在各种半参数方法中,GPH方法的稳健性相对较高,造成的偏差最小。接下来,将Bai和Perron(1998,2003)多个未知突变点估计和检验方法推广到误差项是长记忆过程的情形中,发现除d接近于0.5的情形外,该方法对突变点的估计和检验具有比较好的效果。尤其是当d＜0时,突变点估计量的分布、以及相应的检验功效和实际检验水平也更优。该检验对于突变点的位置、突变幅度的变化也表现的比较稳健。
第三,对石油期货市场的实证研究发现,该市场的投资风险对新息冲击的响应是一个长记忆过程。因此在实践中,建议采用FIGARCH等模型对未来的石油价格及其波动进行预测,并采取相应策略确保宏观经济平稳运行。通过对我国股市有效性的再审视,发现我国股票价格走势的改变常常是政府政策的出台造成的。而在不同阶段,政策对于股票市场有效性的影响也不同。因此,在今后的发展中,政府要逐步抛弃依靠政策引导股市的观念,这是中国股市走向有效性的必由之路。
Since 1980s, the long memory theory has been developed very fast in the Econometrics area, and been put into application in the Finance widely. However, as far as the theoretical research is concerned, there is no consensus among the question on how to test a stochastic process is I(0), I(d), or I(1) . Moreover, the estimation and test method is only fit for the simple model, and there is no mature analysis when there is affection of short memory term and structural break. Therefore, theoretical improvement is highly needed. As to the empirical research, there is evidence that the return series and its volatility show long memory properties in the financial market, for example, the stock market, which means that the Efficient Market Hypothesis (EMH) is no longer existed. In addition, shock from government policy may change the long term characteristics of the market, lose sight of which may influence the EMH judgment and the related policy. So it is great importance of exact model specification on the investment portfolio and risk management. To make all of the above goals into realization, the research on long memory theory must be systematical improved and then put into application in a correct way.
Based on the available literatures, this thesis aims at the solution of the present theoretical and empirical problems and makes research on the long memory theory with its application in the financial market. The main contribution of the thesis are as follows: Firstly, based on an AKFIMA(p, d, q) process, it compares the finite sample properties of three semiparametic estimator on the fractional integration parameter d, and makes simulations on the influence of the short memory term on the estimation of d and the test of long memory under the alternative hypothesis by Monte Carlo approach. Secondly, the influence of the different structural breaks terms is analyzed based on an AO model in which the error term is a fractional noise. And then, the performance of BP test on unknown muli-break points is simulated via Monte Carlo method under small samples. Thirdly, taking the change of the long memory parameter as the market efficiency's altering, the latent impact of structural breaks to China's stock market is studied. At last, the effect of the government policy is evaluated and advices are given.
The study on the long memory theory with its application on financial market is deployed gradually. The whole thesis comprises six chapters. The first one is introduction, which briefly demonstrates the background and significance of the research, the goals and method used, and the innovations. The next three are theoretical study. Thereinto, the second chapter is the long memory process and model study in which we introduce the definition, characteristics, and ARFIMA and FIGARCH model with the estimation. Above on this, the third and forth chapter analyze the impact of short term and structural breads on the modeling of long memory process. The fifth chapter is empirical research which starts with an verification of the "Joseph effect" in the oil futures market, and then the EMH in China's stock and government intervention is researched via the long memory and struck breaks approach. The last chapter is conclusion and future research interest. The main conclusions of the thesis are as follows:
Firstly, When the process is a general ARFIMA(p, d, q) rather than fractional noise, the distributions of the semi-estimators are still normal, however, the mean has some deviation from the real value, which cause power and size problems in the test. And the AR and MA parameters have different short term effects. Moreover, we suggest chose longer bandwidth in the practice.
Secondly, through the long memory test on an AO model with fractional noise errors, we find that the structural breaks especially the trend breaks can cause overestimate of the long memory parameter, and lead to high power and size. In this situation, the GPH estimator is more robust with smaller bias. Next, the Bai and Perron(1998, 2003) test is extended into long memory conditions and shows a fairly well performance except that d is close to 0.5. Especially when d<0, the distribution of the estimator and the test are even better. Besides, the test is relatively robust to the location and change of the break points.
Lastly, through the empirical research on the oil futures market, it is indicated that the shock of innovation to the investment risk is a long memory process. Therefore, the FIGARCH model should be used in practice to forecast the oil futures price and its volatility and the related policy should be make to keep the stable economic growth. Through the retest of the long memory properties in our stock market, it shows that the change of the price have something to do with the government policy, which has different impact to the market efficiency in different periods. Therefore, the policy-leading way on management of the stock market should be abandoned in the future which is the exact way for the market to go to efficiency.
引文
1 参见Hannan,E.J.,and M.Deistler,The Statistical Theory of Linear Systems.[J]1988,New York:John Wiley.
1 参见Elliott,G.,T.J.Rothenberg,and J.H.Stock,Efficient Tests for an Autoregressive Unit Root.[J]Econometrica,1996,64,813-836.
2 Dufour,J.M.,and M.King,Optimal Invariant Tests for the Autocorrelation Coefficient in Linear Regressions with Stationary and Non-stationary Errors.[J]Journal of Econometrics,1991,47,115-143.
3 Perron,P.,and S.,Ng,Useful Modifications to Unit Root Tests with Dependent Errors and their Local Asymptotic Properties.[J]Review of Economic Studies,1996,63,435-465.
1 见赵留彦,王一鸣,《A、B股之间的信息流动与波动溢出》,金融研究,2003(10):37-52
[1] Agiakloglou C and P Newbold. Empirical Evidence on Dickey-Fuller Type Tests. Journal of Time series Analysis, 1992,13,471-483.
[2] Agiakloglou C and P Newbold. Lagrange multiplier tests for fractional difference. Journal of Time Series Analysis, 1994, 15,253 262.
[3] Agiakloglou C, P Newbold, and M Wohar. Bias in an Estimator of the Fractional Difference Parameter. Journal of Time Series Analysis, 1993, 14: 235-246.
[4] Akaike H. Information Theory and an Extension of the Maximum Likelihood Principle, in 2nd International Symposium on Information Theory, 1973, eds. B. N. Petrov and F. Csaki, Budapest: Akademiai Kiado, pp. 267-281.
[5] Anderews D W K and Fair R C. Inference in Nonlinear Econometric Models with Structural Change, Review of Economic Studies, 1988, 55, 615-640.
[6] Andersen T G, Bollerslev T, Diebold F X and Labys P. Modelling and forecasting realized volatility. Econometrica, 2003, 71, 579-625.
[7] Andrews D W K and Guggenberger P. A bias-reduced log-periodogram regression estimator for the long-memory parameter, Econometrica, 2003, 71, 675-712.
[8] Andrews D W K and Ploberger W. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica, 1994, 62, 1383-1414.
[9] Andrews D W K. Tests for parameter instability and structural change with unknown change point. Econometrica, 1993, 61, 821-856.
[10] Avram F and Taqqu M S. Noncentral limit theorems and Appell polynomials. Annals of Probability, 1987, 15, 767-775.
[11] Bachelier L. Theorie de la Speculation (Random Character of Stock Market Prices). Annales Scientifiques de l'Ecole normale superieure, 3e Series, 1900, 17: 21-86.
[12] Bai J and Perron P. Estiamting and testing linear models with multiple structural changes. Econometrica, 1998, 66, 47-78.
[13] Bai J and Perron P. Computation and analysis of multiples structural change models. Journal of applied econometrics, 2003, 18, 1-22.
[14] Bai J. Estimating multiple breaks one at a time. Econometric Theory, 1997, 13, 315-352.
[15] Baillie R T, Bollerslev T and Mikkelsen H O. Fractionally integrated generalized autoregressive conditional heteroskedasiticity, Journal of Econometircs, 1996, 74, 3-30.
[16] Baillie R T. Long memory processes and fractional integration in econometrics. Journal of Econometrics, 1996, 73, 5-59.
[17] Banerjee A and Lumsdaine R L and Stock J H. Recursive and sequential tests of the unit root and trend break hypothesis: Theory and international evidence. Journal of Business and Economic Statistics, 1992, 10, 271-287.
[18] Batten J and Ellis C. Statistical long-term dependence in currency futures markets. Advances in Pacific Basin Financial Markets ZV, 1998, 159-170.
[19] Benoit B. Mandelbrot. Forecasts of future prices, unbiased markets and martingale models. Journal of Business, 1966, 39: 242-255.
[20] Beran J and Terrin N. Testing for a change of the long-memory parameter. Biometrika, 1996, 83, 627-638.
[21] Bollerslev, T. and J.M. Wooldridge, 1992, Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, Econometric Reviews i 1, 143-172.
[22] Bollerslev T. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 1986,31,307-327.
[23] Bos C S, Franses Ph H B F and Ooms M. Long memory and level shifts: Re-analyzing inflation rates. Tinbergen Institute Discussion Paper, 1998, TI98-039/4.
[24] Breidt F J, Crato N. and de Lima P. The detection and estimation of long-memory in stochastic volatility. Journal of Econometrics, 1998 83, 325-348.
[25] Brockwell, P J and R Davis. Time series: Theory and methods. Springer-Verlag, 1987, New York,
[26] Cajueiro D O and Tabak B M. Testing for long-range dependence in world stock markets. Chaos, Solitions and Fractals, 2006, 9, 1-10.
[27] Caporate G M and Gil-Alana L A. Long memory and structural breaks in hyperinflation countries. Journal of Economics and Finance, 2003,27, 136-152.
[28] Carles G P. Empirical evidence of long-range correlations in stock returns. Physica A, 2000, 287, 396-404.
[29] Cbeung Y.-W. Long memory in foreign-xchange rates. Journal of Business and Economic Statistics, 1993a, II, 93-101.
[30] Chang C.-F. and R T Baillie. Small sample bias in conditional sum of squares estimators of fractionally integrated ARMA models. Empirical Economics, 1993, 18, 791-806.
[31] Chaudhuri K and Wu Y. Random walk versus breaking trend in stock prices: evidence from emerging markets. Journal of Banking and Finance, 2003, 27, 575-592.
[32] Cheong C W. Time-varing volatility in Malaysian stock exchange: An empirical study using multiple-volatility-shift fractionally integrated model. Physica A, 2007, 1-10.
[33] Cheung Y.-W. Tests for fractional integration: A Monte Carlo investigation. Journal of Time Series Analysis, 1993b, 14,331-345.
[34] Christos Floros. Modelling Volatility using GARCH Models: Evidence from Egypt and Israel. Middle Eastern Finance and Economics, 2008, Issue 2, 31-41.
[35] Chu C S J. and White H. A direct test for changing trend. Journal of Business and Economic Statistics, 1992, 10,289-300.
[36] Chu C S J. New Tests for Parameter Constancy in Stationary and Nonstationary Regression Models. 1989, Mimeo, University of California at San Diego.
[37] Connolly R. A, Guner Z N and Hightower K N. Evidence on the extent and potential sources of long memory in US treasury security returns and yields. Journal of Money, Credit and Banking, 2007, 39, 689-702.
[38] Cunado J, Gil-Alana L A and Gracia F P. Is the US fiscal deficit sustainable? a fractionally integrated approach. Journal of Economics and Business, 2004, 56, 501-526.
[39] Dahlhaus R. Efficient parameter estimation for self-similar processes. 1989, The Annals of Statistics,1989,17,1749-1766
[40] Davies R and Harte D. Tests for the Hurst effect. Biometrica, 1987, 74,95-101.
[41] Davydov Y A. The invariance principle for stationary processes. Theory of Probability and its Applications, 1970, 15,487-489.
[42] Diebold F X and G D Rudebasch. On the power of Dickey Fuller tests against fractional alternatives. Economics Letters, 1991b, 35, 155-160,
[43] Diebold F X and Inoue A. Long memory and regime switching, Journal of Econometrics, 2001, 105, 131-159.
[44] Ding Z and Granger C W J. Modelling volatility persistence of speculative returns: A new approach. Journal of Econometrics, 1996, 73, 185-215.
[45] Dittmann I, Granger C W J. Properties of nonlinear transformations of fractionally integrated processes. Journal of Econometrics, 2002, 110, 113-133.
[46] Djankov S, Glaeser E, La Porta R, Lopez-de-Silanes F and Shleifer A. The new comparative economics. Journal of Comparative Economics, 2003, Vol.31:595-619.
[47] Dolado J J, Gonzalo J and Mayoral L. A fractional Dickey-Fuller test for unit roots. Econometrica, 2002, 70, 1963-2006.
[48] Eillis C. Estimation of the ARFIMA(p, d, q) fractional differencing parameter (d) using the classical rescaled adjusted range technique. International Review of Financial Analysis, 1999, 8, 53-65.
[49] Ellis C. Estimation of the ARFIMA(p, d, q) fractional differencing parameter d using the classical rescaled adjusted range technique. International Review of Financial Analysis, 1999,8:1,53-65.
[50] Ellis C. Modelling the expected value of the classical resealed adjusted range for long-term dependent series. Working Paper Series. No 79, School of Finance and Economics, University of Technology, Sydney, 1998.
[51] Engle R F and Smith A D. Stochastic permanent breaks. Review of Economics and Statistics, 1999,81,553-574.
[52] Engle R F. Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica, 1982, 50, 987-1008.
[53] Fabrizio Lillo and J Doyne Fanner. The long memory of the effcient market. Studies in Nonlinear Dynamics & Econometrics, 2004, volume, Issue3.
[54] Fama E F. Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance, 1970, 5, 383-423.
[55] Fama E F. The Behavior of Stock Market Prices. Journal of Business, 1965, 38, 34-105.
[56] Fong W M, See K H. A markov switching model of the conditional volatility of crude oil futures prices. Energy Economics, 2002, 24 :71-95
[57] Fox R and M S Taqqu. Large sample properties of parameter estimates for strongly dependen', stationary Gaussian time series. Annals of Statistics, 1986,14, 517-532.
[58] Gawon and Yoon. Long-memory property of nonlinear transformations of break process. Economics Letters, 2005, 87, 373-377.
[59] Geweke J and S Porter-Hudak. The estimation and application of long memory time series models. Journal of Time Series Analysis, 1983,4: 221-238.
[60] Gil-Alana L A. Fractional integration and structural breaks at unknown periods of time. Journal of Time Series Analysis, 2008,29, 163-185.
[61] Gil-Alana L A. Fractional integration in daily stock market indexes. Review of Financial Economics, 2005, 15, 28-48.
[62] Gil-Alana L A. Modelling the US interest rate in terms of I(d) statistical models. The Quarterly Review of Economics and Finance, 2004,44,475-486.
[63] Gilles Teyssiere. Modelling exchange rates volatility with multivariate long-memory ARCH process. GREQAMDT, 97B03, 1997.
[64] Giraitisl and Surgailisd. A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle's estimate. Probab. Theory Related Fields, 1990, 86, 87-104.
[65] Gourieroux C, Jasiak J. Memoryand infrequent breaks. Economics Letters, 2001, 70, 29-41.
[66] Granger C W J and Hyung N. Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance, 2004, 11, 339-421.
[67] Granger C W J and Hyung N. Occasional structural breaks and long memory. Discussion paper 99-14, University of California San Diego, 1999
[68] Granger C W J and R Joyeux. An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis I, 1980, 15 39.
[69] Granger C W J and Terasvirta T. A simple nonlinear time series model with misleading linear properties. Economics Letters, 1999, 62,161-165.
[70] Granger C W J. Long memory relationships and the aggregation of dynamic models. Journal of Econometrics, 1980, 14,227-238.
[71] Granger C W J. Real and spurious long-memory properties of stock market data: comment. Journal of Business and Economic Statistics, 1998, 16,268-269.
[72] Granger C W J. Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 1981, 16, 121-130.
[73] Hall A. Testing for a Unit Root in Time Series With Pretest Data Based Model Selection. Journal of Business & Economic Statistics, 1994, 12,461-470.
[74] Hansen B E. Approximate asymptotic P values for structural-change tests. Journal of Business & Economic Statistics,1997,15,60-67.
[75]Harris R I D.Testing for Unit Roots Using the Augmented Dickey-Fuller Test.Economic Letters,1992,38,381-386.
[76]Harry V Roberts.Stock-Market "Patterns" and Financial Analysis:Methodological Suggestions.The Journal of Finance,1959,Vol.14,1-10.
[77]Harvey A C.Long-memory in stochastic volatility.LSE,Discussion paper 10,SPS,1993.
[78]Hassler U and J Wolters.On the power of unit roots against fractionally integrated alternatives.Economies Letters,1994,45,1-5.
[79]Hauser M A and Reschenhofer E.Estimation of the fractionally differencing parameter with the R/S method,Computational Statistics & Data Analysis,1995,20,569-579.
[80]Hauser M A.Semiparametric and nonparametric testing for long memory.Working paper,1994,(University of Vienna,Vienna).
[81]Helson J and Sarason Y.Past and future.Mathematica Scandanavia,1967,21,5-16.
[82]Henry P B.Stock market liberalization,economic reform and emerging market equity price.Journal of finance,2000,55,529-564.
[83]Hidalgo J and Robinson P M.Testing for structural change in a long-memory environment.Journal of Econometrics,1996,70,159-174.
[84]Holbrook Working.A Random Difference Series for Use in the Analysis of Time Series.Journal of the American Statistical Association,1934,29,11-24.
[85]Horvath L and Kokoszka P.The effect of long-range dependence on change-point estimators.Journal of Statistical Planning and Inference,1997,64,57-81.
[86]Hosking J R M.Fractional differencing,Biometrika,1981,68,165-176.
[87]Hosking J R M.Modeling persistence in hydrological time series using fractional differencing.Water Resources Research,1984,20,1898-1908.
[88]Hurst H E.A suggested statistical model of some time series that occur in nature.Nature,1957,180,494.
[89]Hurst H E.Long-term storage capacity of reservoirs.Transactions of the American Society of Civil Engineers,1951,116,770-799.
[90]Hurvich C M,R S Deo and J Brodsky.The Mean Squared Error of Geweke and Porter-Hudak's Estimator of the Memory Parameter of a Long-Memory Time Series.Journal of Time Series Analysis,1998,19,19-46.
[91]Jacobsen B.Long term dependence in stock returns.Journal of Empirical Finance,1996,3,393-417.
[92]Kaen F R.and Rosenman R E.Predictable behavior in financial markets:Some evidence in support of Heinber's hypothesis.American Economic Review,1986,76,212-220.
[93]Kang S H and Yoon S M.Long memory properties in retum and volatility:evidence from the Korean stock Market.Physica A,2007,591-600.
[94]Kendall M.The Analysis of Economic Time-Series-Part Ⅰ:Prices.Journal of the Royal Statistical Society A(General),1953,116(1):11-34.
[95] Kim H J and Siegmund D. The likelihood ratio test for a change-point in simple linear regression. Biometrika, 1989, 76,409-423.
[96] Kramer W and Sibbertsen P. Testing for structural change in the presence of long memory. International Journal of Business and Economics, 2003, 1, 235-242.
[97] Kramer W, Ploberger W. and Alt R. Testing for structural change in dynamic models. Econometric,1988, 56, 1355-1369.
[98] Kuan, Chung-Ming and Hsu, Chih-Chiang. Change-point estimation of fractionally integrated process. Journal of Time Series Analysis, 1998, 19, 693-708.
[99] Lavielle M and Moulines E. Least-squares estimation of an unknown number of shifts in a time series. Journal of Time Series Analysis, 2000,21, 33- 59.
[100] Lazarova S. Testing for structural change in regression with long memoryprocesses. Journal of Econometrics, 2004. 129,329-372.
[101] Lee D and Schmidt P. On the power of the KPSS test of stationarity against fractionally-integrated alternatives. Journal of Econometrics, 1996, vol. 73(1), 285-302.
[102] Lee S W and B E Hansen. Asymptotic theory for the GARCH(I,I) quasi-maximum likelihood estimator, Econometric Theory, 1994, 10,29-52.
[103] Lewis R and G C Reinsel. Prediction of Multivariate Time Series by Autoregressive Model Fitting. Journal of Multivariate Analysis, 1985, 16, 393-411.
[104] Li Xiaoming. China's economic growth: what do we learn from multiple-break unit root tests? Scottish Journal of Political Economy, 2005, 52,261-81.
[105] Li Xiaoming. The great leap forward, Economic reforms and the unit root hypothesis: testing for breaking trend functions in China's GDP data. Journal of Comparative Economics, 2000, 28, 814-17.
[106] Liu J and Wu S and Zidek J. V. On Segmented Multivariate Regressions. Statistics Since, 1997,7,497-525.
[107] Liu M. Modelling long memoryin stock market volatility. Journal of Econometrics, 2000, 99, 139-171
[108] Lo A W. Long-term memory in stock market prices. Econometrica, 1991, 59, 1279-1313.
[109] Lobato I N and Robinson P M. A nonparametric test for I(0). Review of Economic Studies, 1998,65,475-495.
[110] Lobato I N and Savin. N E. Real and spurious long-memory properties of stock-market data, Jouranl of Business & Economic Statistics, 1998, 16, 261-268.
[111] Lobato I N and Velasco C. Efficient wald tests for fractional unit roots. Econometrica, 2007, 75, 575-589.
[112] Lobatoa I and Robinsonb P M. The averaged periodogram estimate of long memory. Journal of Econometrics, 1996,vol.73, Issue 1, 303-324
[113] Louis Bachelier. Theorie de la Speculation(Random Character of Stock Market Prices), Annales de l'Ecole normale superiure, 1900.
[114] Mandelbrot B B and Taqqu M. Robust R/S analysis of long run serial correlation. Bulletin of International Statistical Institute, 1979,48, book 2, 59-104.
[115] Mandelbrot B B and Wallis J. Noah, Joseph and operational hydrology. Water Resources Research, 1968,4, 909-918.
[116] Mandelbrot B B and Wallis J. Robustness of the rescaled range R/S in the measurement of noncyclic long-run statistical dependence. Water Resources Research, 1969b, 5, 967-988.
[117] Mandelbrot B B. A fast fractional Gaussian noise generator. Water Resources Research, 1975, 7, 543-553.
[118] Mandelbrot B B. Forecasts of future prices, unbiased markets and martingale models. Journal of Business, 1966, 39: 242-255.
[119] Mandelbrot B B. Statistical methodology for non periodic cycles: From the covariance to R/S analysis. Annals of Economic and Social Measurement, 1972,1, 259-290.
[120] Mandelbrot B B. Statistical methodology for non periodic cycles: From the covariance to R/S analysis. Annals of Economic and Social Measurement, 1972,1,259-290.
[121] Mandelbrot, B.B. and J.W. Van Ness, 1968, Fractional Brownian motions, fractional Brownian noises and applications, SIAM Review IO, 422-437.
[122] Martin V L and Wilkins N P. Indirect estimation of ARFIMA and VARFIMA models. Journal of Econometrics, 1999, 93, 149-175.
[123] Maurice Kendall. The Analysis of Economic Time-Series-Part I: Prices. Journal of the Royal Statistical Society A (General), 1953, 116 (1), 11-34.
[124] McLeod A I and Hipel K W. Preservation of the rescaled adjusted range, 1: A reassessment of the Hurst phenomenon. Water Resources Research, 1978, 14,491-508.
[125] Ng S and P Perron. Lag Length Selection and the Construction of Unit Roots Tests With Good Size and Power. Econometrica, 2001, 69, 1519-1554.
[126] Ng S and P Perron. Practitioners' Corner: A Note on the Selection of Time Series Models. Oxford Bulletin of Economics and Statistics, 2005, 67, 115-134.
[127] Ng S and P Perron. Unit Root tests in ARMA Models With Data-Dependent Methods for the Selection of the Truncation Lag. Journal of the American Statistical Association, 1995, 90,268-281.
[128] Paul A Samuelson. Proof that Properly Anticipated Prices Fluctuate Randomly, Industrial Management Review, 1965, 6: 41-49.
[129] Perron P and Bai J. Estimating and testing linear models with multiples structural changes. Econometrica, 1998, 66,47-48
[130] Perron P and Qu Z. An analytical evaluation of the log-periodogram estimate in the presence of level shifts and its implications for stock return volatility. Manuscript, 2004.
[131] Perron P. A Test for Changes in a Polynomial Trend Function for a Dynamic Time Series, 1991, Mimeo. University de Montreal.
[132] Peters E E. Fractal market analysis. New York: Wiley, 1994.
[133] Phillips P C B and P Perron. Testing for a Unit Root in Time Series Regression. Biometrika, 1988, 75, 335-346.
[134] Phillips P C B. Time series regression with a unit root. Econometrica, 1987, 55, 277-301.
[135] Rapach D E and Strauss J K. Structural breaks and garch models of exchange rate volatility, Journal of Applied Econometrics, 2008,23,65-90.
[136] Resnick. Extreme values, regular variation and point processes. Springer-Vcrlag, 1987, New York.
[137] Roberts H V. Stock-Market 'Patterns' and Financial Analysis: Methodological Suggestions. The Journal of Finance, 1959, 14: 1-10.
[138] Roberts H. Statistical versus clinical prediction of the stock market. Unpublished manuscript, 1967.
[139] Robinson P M. [Real and spurious long-memory properties of stock-market data]:Comment. Journal of Business & Economic Statistics, 1998, 16,276-279.
[140] Robinson P M. Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association, 1994b, 89, 1420-1437.
[141] Robinson P M. Gaussian Semiparametric Estimation of Long Range Dependence. The Annals of Statistics, 1995a, 23, 1630-1661.
[142] Robinson P M. Log-periodogram regression for time series with long range dependence. 1992, Unpublished manuscript.
[143] Robinson P M. Log-periodogram regression of time series with long range dependence, The Annals of Statistics, 1995b, 23, 1048-1072.
[144] Robinson P M. Nonparametric methods in specification. The Economic Journal, 1986, 96, 134-141.
[145] Robinson P M. Semiparametric analysis of long-memory time series. The Annals of Statistics, 1994a, 22, 515-539.
[146] Robinson P M. Semiparametric econometrics: a survey. Journal of Applied Econometrics, 1988,3,35-51.
[147] Robinson P M. Testing for strong serial correlation and dynamic conditional heteroskedasticity. Journal of Econometrics, 1991,47, 67-84.
[148] Robinson P M. The memory of stochastic volatility models. Journal of Econometrics, 2001,101, 195-218.
[149] Said S E and D A Dickey. Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order. Biometrika, 1984, 71, 599-607.
[150] Samuelson P A. Proof that Properly Anticipated Prices Fluctuate Randomly. Industrial Management Review, 1965, 6: 41-49.
[151] Sang Hoon Kang and Yoon S M. Long memory properties in return and volatility: Evidence form the Korean stock market. Physica A, 2007, 385, 591-600.
[152] Schwarz GW. Estimation the Dimension of a Model. The Annals of Statistics, 1978, 6, 461-464.
[153] Schwert G W. Tests for Unit Roots: A Monte Carlo Investigation. Journal of Business & Economic Statistics, 1989, 7, 147-160.
[154]Soweil F B.The fractional unit root distribution,Econometrica,1990,58,495-505.
[155]Sowell F B.Fractionally integrated vector time series.Ph.D.dissertation,1986,Duke University,Durham,NC.
[156]Sowell F B.Maximum likelihood estimation of stationary univariate fractionally integrated time series models.Journal of Econometrics,1992,53,165-188.
[157]Sowell F.Modeling Long-run behavior with the fractional ARIMA model.Journal of Monetary Economics,1992,29,277-302.
[158]Vogelsang T J.Wald type tests for detecting breaks in the trend function of a dynamic time series.Econometric Theory,1997,13,818-849.
[159]Wallis J R and Matalas N C.Small sample properties of H and K-estimators of the Hurst coefficienth.Water Resources Research,1970,6(6),1583-1594.
[160]Weber C E.Data-Dependent Criteria for Lag Length Selection in Augmented Dickey-Fuller Regressions:a Monte Carlo Analysis.Unpublished Working Paper,Seattle University,1998.
[161]Weber C E.F-tests for Lag Length Selection in Augmented Dickey-Fuller Regressions:Some Monte Carlo Evidence.Applied Economics Letters,2001,8,455-458.
[162]Weiss A A.Asymptotic theory for ARCH models:Estimation and testing,Econometric Theory,1986,2,107-131.
[163]Whittle P.Hypothesis testing in time series analysis(Almquist and Wiksells,Uppsala),1951.
[164]Working H.A Random Difference Series for Use in the Analysis of Time Series.Journal of the American Statistical Association,1934,29:11-24.
[165]Wright.Testing for a structural break at unknown date with long-memory disturbances.Journal of Times Series Analysis,1996,19,369-376.
[166]Wu P.Testing fractionally integrated time series.Working paper,1992,(Victoria University,Wellington).
[167]Xiu J and Jin Y.Empirical study of ARFIMA model based on fractional differencing.Physica A,2007,377,138-154.
[168]Yoon G.Long-memory property of nonlinear transformations of break processes.Economics letters,2005,87,373-377.
[169]邓露,张晓峒.ADF检验中滞后长度的选择——基于ARIMA(0,1,q)过程的模拟证据.数量经济技术经济研究,2008(9):126-138
[170]胡怀国.石油价格波动及其宏观经济影响——兼论供求冲击的不同后果.中国社会科学院
[171]黄后川等.中国股票市场波动率的高频估计与特性分析.经济研究,2003(2):75-82
[172]黄晓薇.关于一类β-ARCH模型参数估计的研究.吉林大学博士论文,2004
[173]姜仁娜等.沪深两市股票指数的长记忆性.清华大学学报(自然科学版),2004(44):30-31
[174]李海奇等.中国股票市场波动长记忆建模研究.统计于与决策,2006(4下):17-20
[175]李伟.国际股市波动的长期记忆性实证研究.中国经济教育科研网,2006年工作论文
[176]罗登跃等.上海股市收益率和波动性长记忆特征实证研究.金融研究,2005(11):109-116
[177]马丹.中国证券市场高频数据的动态特征.统计与决策,2005(4下):70-73
[178]施红俊等.中国股市长记忆性实证研究.同济大学报,2004(3):416-420
[179]史代敏等.股票市场收益率波动长记忆性的分解及实证研究.数量经济技术经济研究,2006(8):136-151
[180]汤果等.FIGARCH模型对股市收益长记忆性的实证分析.统计研究,1999(7):39-42
[181]王春峰等.中国股市波动性过程中的长期记忆性实证研究.系统工程,2003(1):22-28
[182]王明照、郭冰.金融资产波动性特征研究回顾.经济学动态,2005(1)
[183]徐梅等.非平稳和长记忆时间序列主频率估计方法研究.天津大学学报,2003(36):507-511
[184]赵桂芹等.证券市场长期记忆特征的实证分析.管理科学2003(2):40-44
[185]赵巍等.基于股市高频数据的半参数估计方法.数理统计与管理,2007(26):112-118
[186]祝金荣.石油期货价格预测研究.冶金工业出版社,2008年
[187]邹艳芬,陆宇海.基于GARCH模型的石油价格变动模拟.数理统计与管理,2006(6):640-644
1 参见Elliott,G.,T.J.Rothenberg,and J.H.Stock,Efficient Tests for an Autoregressive Unit Root.[J]Econometrica,1996,64,813-836.
2 Dufour,J.M.,and M.King,Optimal Invariant Tests for the Autocorrelation Coefficient in Linear Regressions with Stationary and Non-stationary Errors.[J]Journal of Econometrics,1991,47,115-143.
3 Perron,P.,and S.,Ng,Useful Modifications to Unit Root Tests with Dependent Errors and their Local Asymptotic Properties.[J]Review of Economic Studies,1996,63,435-465.
1 见赵留彦,王一鸣,《A、B股之间的信息流动与波动溢出》,金融研究,2003(10):37-52
[1] Agiakloglou C and P Newbold. Empirical Evidence on Dickey-Fuller Type Tests. Journal of Time series Analysis, 1992,13,471-483.
[2] Agiakloglou C and P Newbold. Lagrange multiplier tests for fractional difference. Journal of Time Series Analysis, 1994, 15,253 262.
[3] Agiakloglou C, P Newbold, and M Wohar. Bias in an Estimator of the Fractional Difference Parameter. Journal of Time Series Analysis, 1993, 14: 235-246.
[4] Akaike H. Information Theory and an Extension of the Maximum Likelihood Principle, in 2nd International Symposium on Information Theory, 1973, eds. B. N. Petrov and F. Csaki, Budapest: Akademiai Kiado, pp. 267-281.
[5] Anderews D W K and Fair R C. Inference in Nonlinear Econometric Models with Structural Change, Review of Economic Studies, 1988, 55, 615-640.
[6] Andersen T G, Bollerslev T, Diebold F X and Labys P. Modelling and forecasting realized volatility. Econometrica, 2003, 71, 579-625.
[7] Andrews D W K and Guggenberger P. A bias-reduced log-periodogram regression estimator for the long-memory parameter, Econometrica, 2003, 71, 675-712.
[8] Andrews D W K and Ploberger W. Optimal tests when a nuisance parameter is present only under the alternative. Econometrica, 1994, 62, 1383-1414.
[9] Andrews D W K. Tests for parameter instability and structural change with unknown change point. Econometrica, 1993, 61, 821-856.
[10] Avram F and Taqqu M S. Noncentral limit theorems and Appell polynomials. Annals of Probability, 1987, 15, 767-775.
[11] Bachelier L. Theorie de la Speculation (Random Character of Stock Market Prices). Annales Scientifiques de l'Ecole normale superieure, 3e Series, 1900, 17: 21-86.
[12] Bai J and Perron P. Estiamting and testing linear models with multiple structural changes. Econometrica, 1998, 66, 47-78.
[13] Bai J and Perron P. Computation and analysis of multiples structural change models. Journal of applied econometrics, 2003, 18, 1-22.
[14] Bai J. Estimating multiple breaks one at a time. Econometric Theory, 1997, 13, 315-352.
[15] Baillie R T, Bollerslev T and Mikkelsen H O. Fractionally integrated generalized autoregressive conditional heteroskedasiticity, Journal of Econometircs, 1996, 74, 3-30.
[16] Baillie R T. Long memory processes and fractional integration in econometrics. Journal of Econometrics, 1996, 73, 5-59.
[17] Banerjee A and Lumsdaine R L and Stock J H. Recursive and sequential tests of the unit root and trend break hypothesis: Theory and international evidence. Journal of Business and Economic Statistics, 1992, 10, 271-287.
[18] Batten J and Ellis C. Statistical long-term dependence in currency futures markets. Advances in Pacific Basin Financial Markets ZV, 1998, 159-170.
[19] Benoit B. Mandelbrot. Forecasts of future prices, unbiased markets and martingale models. Journal of Business, 1966, 39: 242-255.
[20] Beran J and Terrin N. Testing for a change of the long-memory parameter. Biometrika, 1996, 83, 627-638.
[21] Bollerslev, T. and J.M. Wooldridge, 1992, Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, Econometric Reviews i 1, 143-172.
[22] Bollerslev T. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 1986,31,307-327.
[23] Bos C S, Franses Ph H B F and Ooms M. Long memory and level shifts: Re-analyzing inflation rates. Tinbergen Institute Discussion Paper, 1998, TI98-039/4.
[24] Breidt F J, Crato N. and de Lima P. The detection and estimation of long-memory in stochastic volatility. Journal of Econometrics, 1998 83, 325-348.
[25] Brockwell, P J and R Davis. Time series: Theory and methods. Springer-Verlag, 1987, New York,
[26] Cajueiro D O and Tabak B M. Testing for long-range dependence in world stock markets. Chaos, Solitions and Fractals, 2006, 9, 1-10.
[27] Caporate G M and Gil-Alana L A. Long memory and structural breaks in hyperinflation countries. Journal of Economics and Finance, 2003,27, 136-152.
[28] Carles G P. Empirical evidence of long-range correlations in stock returns. Physica A, 2000, 287, 396-404.
[29] Cbeung Y.-W. Long memory in foreign-xchange rates. Journal of Business and Economic Statistics, 1993a, II, 93-101.
[30] Chang C.-F. and R T Baillie. Small sample bias in conditional sum of squares estimators of fractionally integrated ARMA models. Empirical Economics, 1993, 18, 791-806.
[31] Chaudhuri K and Wu Y. Random walk versus breaking trend in stock prices: evidence from emerging markets. Journal of Banking and Finance, 2003, 27, 575-592.
[32] Cheong C W. Time-varing volatility in Malaysian stock exchange: An empirical study using multiple-volatility-shift fractionally integrated model. Physica A, 2007, 1-10.
[33] Cheung Y.-W. Tests for fractional integration: A Monte Carlo investigation. Journal of Time Series Analysis, 1993b, 14,331-345.
[34] Christos Floros. Modelling Volatility using GARCH Models: Evidence from Egypt and Israel. Middle Eastern Finance and Economics, 2008, Issue 2, 31-41.
[35] Chu C S J. and White H. A direct test for changing trend. Journal of Business and Economic Statistics, 1992, 10,289-300.
[36] Chu C S J. New Tests for Parameter Constancy in Stationary and Nonstationary Regression Models. 1989, Mimeo, University of California at San Diego.
[37] Connolly R. A, Guner Z N and Hightower K N. Evidence on the extent and potential sources of long memory in US treasury security returns and yields. Journal of Money, Credit and Banking, 2007, 39, 689-702.
[38] Cunado J, Gil-Alana L A and Gracia F P. Is the US fiscal deficit sustainable? a fractionally integrated approach. Journal of Economics and Business, 2004, 56, 501-526.
[39] Dahlhaus R. Efficient parameter estimation for self-similar processes. 1989, The Annals of Statistics,1989,17,1749-1766
[40] Davies R and Harte D. Tests for the Hurst effect. Biometrica, 1987, 74,95-101.
[41] Davydov Y A. The invariance principle for stationary processes. Theory of Probability and its Applications, 1970, 15,487-489.
[42] Diebold F X and G D Rudebasch. On the power of Dickey Fuller tests against fractional alternatives. Economics Letters, 1991b, 35, 155-160,
[43] Diebold F X and Inoue A. Long memory and regime switching, Journal of Econometrics, 2001, 105, 131-159.
[44] Ding Z and Granger C W J. Modelling volatility persistence of speculative returns: A new approach. Journal of Econometrics, 1996, 73, 185-215.
[45] Dittmann I, Granger C W J. Properties of nonlinear transformations of fractionally integrated processes. Journal of Econometrics, 2002, 110, 113-133.
[46] Djankov S, Glaeser E, La Porta R, Lopez-de-Silanes F and Shleifer A. The new comparative economics. Journal of Comparative Economics, 2003, Vol.31:595-619.
[47] Dolado J J, Gonzalo J and Mayoral L. A fractional Dickey-Fuller test for unit roots. Econometrica, 2002, 70, 1963-2006.
[48] Eillis C. Estimation of the ARFIMA(p, d, q) fractional differencing parameter (d) using the classical rescaled adjusted range technique. International Review of Financial Analysis, 1999, 8, 53-65.
[49] Ellis C. Estimation of the ARFIMA(p, d, q) fractional differencing parameter d using the classical rescaled adjusted range technique. International Review of Financial Analysis, 1999,8:1,53-65.
[50] Ellis C. Modelling the expected value of the classical resealed adjusted range for long-term dependent series. Working Paper Series. No 79, School of Finance and Economics, University of Technology, Sydney, 1998.
[51] Engle R F and Smith A D. Stochastic permanent breaks. Review of Economics and Statistics, 1999,81,553-574.
[52] Engle R F. Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica, 1982, 50, 987-1008.
[53] Fabrizio Lillo and J Doyne Fanner. The long memory of the effcient market. Studies in Nonlinear Dynamics & Econometrics, 2004, volume, Issue3.
[54] Fama E F. Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance, 1970, 5, 383-423.
[55] Fama E F. The Behavior of Stock Market Prices. Journal of Business, 1965, 38, 34-105.
[56] Fong W M, See K H. A markov switching model of the conditional volatility of crude oil futures prices. Energy Economics, 2002, 24 :71-95
[57] Fox R and M S Taqqu. Large sample properties of parameter estimates for strongly dependen', stationary Gaussian time series. Annals of Statistics, 1986,14, 517-532.
[58] Gawon and Yoon. Long-memory property of nonlinear transformations of break process. Economics Letters, 2005, 87, 373-377.
[59] Geweke J and S Porter-Hudak. The estimation and application of long memory time series models. Journal of Time Series Analysis, 1983,4: 221-238.
[60] Gil-Alana L A. Fractional integration and structural breaks at unknown periods of time. Journal of Time Series Analysis, 2008,29, 163-185.
[61] Gil-Alana L A. Fractional integration in daily stock market indexes. Review of Financial Economics, 2005, 15, 28-48.
[62] Gil-Alana L A. Modelling the US interest rate in terms of I(d) statistical models. The Quarterly Review of Economics and Finance, 2004,44,475-486.
[63] Gilles Teyssiere. Modelling exchange rates volatility with multivariate long-memory ARCH process. GREQAMDT, 97B03, 1997.
[64] Giraitisl and Surgailisd. A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle's estimate. Probab. Theory Related Fields, 1990, 86, 87-104.
[65] Gourieroux C, Jasiak J. Memoryand infrequent breaks. Economics Letters, 2001, 70, 29-41.
[66] Granger C W J and Hyung N. Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns. Journal of Empirical Finance, 2004, 11, 339-421.
[67] Granger C W J and Hyung N. Occasional structural breaks and long memory. Discussion paper 99-14, University of California San Diego, 1999
[68] Granger C W J and R Joyeux. An introduction to long memory time series models and fractional differencing. Journal of Time Series Analysis I, 1980, 15 39.
[69] Granger C W J and Terasvirta T. A simple nonlinear time series model with misleading linear properties. Economics Letters, 1999, 62,161-165.
[70] Granger C W J. Long memory relationships and the aggregation of dynamic models. Journal of Econometrics, 1980, 14,227-238.
[71] Granger C W J. Real and spurious long-memory properties of stock market data: comment. Journal of Business and Economic Statistics, 1998, 16,268-269.
[72] Granger C W J. Some properties of time series data and their use in econometric model specification. Journal of Econometrics, 1981, 16, 121-130.
[73] Hall A. Testing for a Unit Root in Time Series With Pretest Data Based Model Selection. Journal of Business & Economic Statistics, 1994, 12,461-470.
[74] Hansen B E. Approximate asymptotic P values for structural-change tests. Journal of Business & Economic Statistics,1997,15,60-67.
[75]Harris R I D.Testing for Unit Roots Using the Augmented Dickey-Fuller Test.Economic Letters,1992,38,381-386.
[76]Harry V Roberts.Stock-Market "Patterns" and Financial Analysis:Methodological Suggestions.The Journal of Finance,1959,Vol.14,1-10.
[77]Harvey A C.Long-memory in stochastic volatility.LSE,Discussion paper 10,SPS,1993.
[78]Hassler U and J Wolters.On the power of unit roots against fractionally integrated alternatives.Economies Letters,1994,45,1-5.
[79]Hauser M A and Reschenhofer E.Estimation of the fractionally differencing parameter with the R/S method,Computational Statistics & Data Analysis,1995,20,569-579.
[80]Hauser M A.Semiparametric and nonparametric testing for long memory.Working paper,1994,(University of Vienna,Vienna).
[81]Helson J and Sarason Y.Past and future.Mathematica Scandanavia,1967,21,5-16.
[82]Henry P B.Stock market liberalization,economic reform and emerging market equity price.Journal of finance,2000,55,529-564.
[83]Hidalgo J and Robinson P M.Testing for structural change in a long-memory environment.Journal of Econometrics,1996,70,159-174.
[84]Holbrook Working.A Random Difference Series for Use in the Analysis of Time Series.Journal of the American Statistical Association,1934,29,11-24.
[85]Horvath L and Kokoszka P.The effect of long-range dependence on change-point estimators.Journal of Statistical Planning and Inference,1997,64,57-81.
[86]Hosking J R M.Fractional differencing,Biometrika,1981,68,165-176.
[87]Hosking J R M.Modeling persistence in hydrological time series using fractional differencing.Water Resources Research,1984,20,1898-1908.
[88]Hurst H E.A suggested statistical model of some time series that occur in nature.Nature,1957,180,494.
[89]Hurst H E.Long-term storage capacity of reservoirs.Transactions of the American Society of Civil Engineers,1951,116,770-799.
[90]Hurvich C M,R S Deo and J Brodsky.The Mean Squared Error of Geweke and Porter-Hudak's Estimator of the Memory Parameter of a Long-Memory Time Series.Journal of Time Series Analysis,1998,19,19-46.
[91]Jacobsen B.Long term dependence in stock returns.Journal of Empirical Finance,1996,3,393-417.
[92]Kaen F R.and Rosenman R E.Predictable behavior in financial markets:Some evidence in support of Heinber's hypothesis.American Economic Review,1986,76,212-220.
[93]Kang S H and Yoon S M.Long memory properties in retum and volatility:evidence from the Korean stock Market.Physica A,2007,591-600.
[94]Kendall M.The Analysis of Economic Time-Series-Part Ⅰ:Prices.Journal of the Royal Statistical Society A(General),1953,116(1):11-34.
[95] Kim H J and Siegmund D. The likelihood ratio test for a change-point in simple linear regression. Biometrika, 1989, 76,409-423.
[96] Kramer W and Sibbertsen P. Testing for structural change in the presence of long memory. International Journal of Business and Economics, 2003, 1, 235-242.
[97] Kramer W, Ploberger W. and Alt R. Testing for structural change in dynamic models. Econometric,1988, 56, 1355-1369.
[98] Kuan, Chung-Ming and Hsu, Chih-Chiang. Change-point estimation of fractionally integrated process. Journal of Time Series Analysis, 1998, 19, 693-708.
[99] Lavielle M and Moulines E. Least-squares estimation of an unknown number of shifts in a time series. Journal of Time Series Analysis, 2000,21, 33- 59.
[100] Lazarova S. Testing for structural change in regression with long memoryprocesses. Journal of Econometrics, 2004. 129,329-372.
[101] Lee D and Schmidt P. On the power of the KPSS test of stationarity against fractionally-integrated alternatives. Journal of Econometrics, 1996, vol. 73(1), 285-302.
[102] Lee S W and B E Hansen. Asymptotic theory for the GARCH(I,I) quasi-maximum likelihood estimator, Econometric Theory, 1994, 10,29-52.
[103] Lewis R and G C Reinsel. Prediction of Multivariate Time Series by Autoregressive Model Fitting. Journal of Multivariate Analysis, 1985, 16, 393-411.
[104] Li Xiaoming. China's economic growth: what do we learn from multiple-break unit root tests? Scottish Journal of Political Economy, 2005, 52,261-81.
[105] Li Xiaoming. The great leap forward, Economic reforms and the unit root hypothesis: testing for breaking trend functions in China's GDP data. Journal of Comparative Economics, 2000, 28, 814-17.
[106] Liu J and Wu S and Zidek J. V. On Segmented Multivariate Regressions. Statistics Since, 1997,7,497-525.
[107] Liu M. Modelling long memoryin stock market volatility. Journal of Econometrics, 2000, 99, 139-171
[108] Lo A W. Long-term memory in stock market prices. Econometrica, 1991, 59, 1279-1313.
[109] Lobato I N and Robinson P M. A nonparametric test for I(0). Review of Economic Studies, 1998,65,475-495.
[110] Lobato I N and Savin. N E. Real and spurious long-memory properties of stock-market data, Jouranl of Business & Economic Statistics, 1998, 16, 261-268.
[111] Lobato I N and Velasco C. Efficient wald tests for fractional unit roots. Econometrica, 2007, 75, 575-589.
[112] Lobatoa I and Robinsonb P M. The averaged periodogram estimate of long memory. Journal of Econometrics, 1996,vol.73, Issue 1, 303-324
[113] Louis Bachelier. Theorie de la Speculation(Random Character of Stock Market Prices), Annales de l'Ecole normale superiure, 1900.
[114] Mandelbrot B B and Taqqu M. Robust R/S analysis of long run serial correlation. Bulletin of International Statistical Institute, 1979,48, book 2, 59-104.
[115] Mandelbrot B B and Wallis J. Noah, Joseph and operational hydrology. Water Resources Research, 1968,4, 909-918.
[116] Mandelbrot B B and Wallis J. Robustness of the rescaled range R/S in the measurement of noncyclic long-run statistical dependence. Water Resources Research, 1969b, 5, 967-988.
[117] Mandelbrot B B. A fast fractional Gaussian noise generator. Water Resources Research, 1975, 7, 543-553.
[118] Mandelbrot B B. Forecasts of future prices, unbiased markets and martingale models. Journal of Business, 1966, 39: 242-255.
[119] Mandelbrot B B. Statistical methodology for non periodic cycles: From the covariance to R/S analysis. Annals of Economic and Social Measurement, 1972,1, 259-290.
[120] Mandelbrot B B. Statistical methodology for non periodic cycles: From the covariance to R/S analysis. Annals of Economic and Social Measurement, 1972,1,259-290.
[121] Mandelbrot, B.B. and J.W. Van Ness, 1968, Fractional Brownian motions, fractional Brownian noises and applications, SIAM Review IO, 422-437.
[122] Martin V L and Wilkins N P. Indirect estimation of ARFIMA and VARFIMA models. Journal of Econometrics, 1999, 93, 149-175.
[123] Maurice Kendall. The Analysis of Economic Time-Series-Part I: Prices. Journal of the Royal Statistical Society A (General), 1953, 116 (1), 11-34.
[124] McLeod A I and Hipel K W. Preservation of the rescaled adjusted range, 1: A reassessment of the Hurst phenomenon. Water Resources Research, 1978, 14,491-508.
[125] Ng S and P Perron. Lag Length Selection and the Construction of Unit Roots Tests With Good Size and Power. Econometrica, 2001, 69, 1519-1554.
[126] Ng S and P Perron. Practitioners' Corner: A Note on the Selection of Time Series Models. Oxford Bulletin of Economics and Statistics, 2005, 67, 115-134.
[127] Ng S and P Perron. Unit Root tests in ARMA Models With Data-Dependent Methods for the Selection of the Truncation Lag. Journal of the American Statistical Association, 1995, 90,268-281.
[128] Paul A Samuelson. Proof that Properly Anticipated Prices Fluctuate Randomly, Industrial Management Review, 1965, 6: 41-49.
[129] Perron P and Bai J. Estimating and testing linear models with multiples structural changes. Econometrica, 1998, 66,47-48
[130] Perron P and Qu Z. An analytical evaluation of the log-periodogram estimate in the presence of level shifts and its implications for stock return volatility. Manuscript, 2004.
[131] Perron P. A Test for Changes in a Polynomial Trend Function for a Dynamic Time Series, 1991, Mimeo. University de Montreal.
[132] Peters E E. Fractal market analysis. New York: Wiley, 1994.
[133] Phillips P C B and P Perron. Testing for a Unit Root in Time Series Regression. Biometrika, 1988, 75, 335-346.
[134] Phillips P C B. Time series regression with a unit root. Econometrica, 1987, 55, 277-301.
[135] Rapach D E and Strauss J K. Structural breaks and garch models of exchange rate volatility, Journal of Applied Econometrics, 2008,23,65-90.
[136] Resnick. Extreme values, regular variation and point processes. Springer-Vcrlag, 1987, New York.
[137] Roberts H V. Stock-Market 'Patterns' and Financial Analysis: Methodological Suggestions. The Journal of Finance, 1959, 14: 1-10.
[138] Roberts H. Statistical versus clinical prediction of the stock market. Unpublished manuscript, 1967.
[139] Robinson P M. [Real and spurious long-memory properties of stock-market data]:Comment. Journal of Business & Economic Statistics, 1998, 16,276-279.
[140] Robinson P M. Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association, 1994b, 89, 1420-1437.
[141] Robinson P M. Gaussian Semiparametric Estimation of Long Range Dependence. The Annals of Statistics, 1995a, 23, 1630-1661.
[142] Robinson P M. Log-periodogram regression for time series with long range dependence. 1992, Unpublished manuscript.
[143] Robinson P M. Log-periodogram regression of time series with long range dependence, The Annals of Statistics, 1995b, 23, 1048-1072.
[144] Robinson P M. Nonparametric methods in specification. The Economic Journal, 1986, 96, 134-141.
[145] Robinson P M. Semiparametric analysis of long-memory time series. The Annals of Statistics, 1994a, 22, 515-539.
[146] Robinson P M. Semiparametric econometrics: a survey. Journal of Applied Econometrics, 1988,3,35-51.
[147] Robinson P M. Testing for strong serial correlation and dynamic conditional heteroskedasticity. Journal of Econometrics, 1991,47, 67-84.
[148] Robinson P M. The memory of stochastic volatility models. Journal of Econometrics, 2001,101, 195-218.
[149] Said S E and D A Dickey. Testing for Unit Roots in Autoregressive-Moving Average Models of Unknown Order. Biometrika, 1984, 71, 599-607.
[150] Samuelson P A. Proof that Properly Anticipated Prices Fluctuate Randomly. Industrial Management Review, 1965, 6: 41-49.
[151] Sang Hoon Kang and Yoon S M. Long memory properties in return and volatility: Evidence form the Korean stock market. Physica A, 2007, 385, 591-600.
[152] Schwarz GW. Estimation the Dimension of a Model. The Annals of Statistics, 1978, 6, 461-464.
[153] Schwert G W. Tests for Unit Roots: A Monte Carlo Investigation. Journal of Business & Economic Statistics, 1989, 7, 147-160.
[154]Soweil F B.The fractional unit root distribution,Econometrica,1990,58,495-505.
[155]Sowell F B.Fractionally integrated vector time series.Ph.D.dissertation,1986,Duke University,Durham,NC.
[156]Sowell F B.Maximum likelihood estimation of stationary univariate fractionally integrated time series models.Journal of Econometrics,1992,53,165-188.
[157]Sowell F.Modeling Long-run behavior with the fractional ARIMA model.Journal of Monetary Economics,1992,29,277-302.
[158]Vogelsang T J.Wald type tests for detecting breaks in the trend function of a dynamic time series.Econometric Theory,1997,13,818-849.
[159]Wallis J R and Matalas N C.Small sample properties of H and K-estimators of the Hurst coefficienth.Water Resources Research,1970,6(6),1583-1594.
[160]Weber C E.Data-Dependent Criteria for Lag Length Selection in Augmented Dickey-Fuller Regressions:a Monte Carlo Analysis.Unpublished Working Paper,Seattle University,1998.
[161]Weber C E.F-tests for Lag Length Selection in Augmented Dickey-Fuller Regressions:Some Monte Carlo Evidence.Applied Economics Letters,2001,8,455-458.
[162]Weiss A A.Asymptotic theory for ARCH models:Estimation and testing,Econometric Theory,1986,2,107-131.
[163]Whittle P.Hypothesis testing in time series analysis(Almquist and Wiksells,Uppsala),1951.
[164]Working H.A Random Difference Series for Use in the Analysis of Time Series.Journal of the American Statistical Association,1934,29:11-24.
[165]Wright.Testing for a structural break at unknown date with long-memory disturbances.Journal of Times Series Analysis,1996,19,369-376.
[166]Wu P.Testing fractionally integrated time series.Working paper,1992,(Victoria University,Wellington).
[167]Xiu J and Jin Y.Empirical study of ARFIMA model based on fractional differencing.Physica A,2007,377,138-154.
[168]Yoon G.Long-memory property of nonlinear transformations of break processes.Economics letters,2005,87,373-377.
[169]邓露,张晓峒.ADF检验中滞后长度的选择——基于ARIMA(0,1,q)过程的模拟证据.数量经济技术经济研究,2008(9):126-138
[170]胡怀国.石油价格波动及其宏观经济影响——兼论供求冲击的不同后果.中国社会科学院
[171]黄后川等.中国股票市场波动率的高频估计与特性分析.经济研究,2003(2):75-82
[172]黄晓薇.关于一类β-ARCH模型参数估计的研究.吉林大学博士论文,2004
[173]姜仁娜等.沪深两市股票指数的长记忆性.清华大学学报(自然科学版),2004(44):30-31
[174]李海奇等.中国股票市场波动长记忆建模研究.统计于与决策,2006(4下):17-20
[175]李伟.国际股市波动的长期记忆性实证研究.中国经济教育科研网,2006年工作论文
[176]罗登跃等.上海股市收益率和波动性长记忆特征实证研究.金融研究,2005(11):109-116
[177]马丹.中国证券市场高频数据的动态特征.统计与决策,2005(4下):70-73
[178]施红俊等.中国股市长记忆性实证研究.同济大学报,2004(3):416-420
[179]史代敏等.股票市场收益率波动长记忆性的分解及实证研究.数量经济技术经济研究,2006(8):136-151
[180]汤果等.FIGARCH模型对股市收益长记忆性的实证分析.统计研究,1999(7):39-42
[181]王春峰等.中国股市波动性过程中的长期记忆性实证研究.系统工程,2003(1):22-28
[182]王明照、郭冰.金融资产波动性特征研究回顾.经济学动态,2005(1)
[183]徐梅等.非平稳和长记忆时间序列主频率估计方法研究.天津大学学报,2003(36):507-511
[184]赵桂芹等.证券市场长期记忆特征的实证分析.管理科学2003(2):40-44
[185]赵巍等.基于股市高频数据的半参数估计方法.数理统计与管理,2007(26):112-118
[186]祝金荣.石油期货价格预测研究.冶金工业出版社,2008年
[187]邹艳芬,陆宇海.基于GARCH模型的石油价格变动模拟.数理统计与管理,2006(6):640-644