汇率时间序列非线性动力学特征及组合预测研究
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摘要
汇率是一国经济的重要变量,既决定着经济的对内均衡,又影响着经济的对外均衡。随着经济全球化的不断推进和国际资本流动的日益加剧,汇率对于投资者选择正确的投资策略、企业对外汇风险的规避和防范以及中央银行对外汇市场的有效干预和制定正确的货币政策,都有着非常重要的影响。因此,关于汇率的行为描述和预测问题一直是国内外理论界关注的焦点。
传统的资本市场理论构建在理性投资人、有效市场假说和随机游动三大假设条件基础之上。然而,这种基于线性研究范式的均衡分析体系无法对外汇市场的一些异象给出合理的解释,如汇率收益率分布的“尖峰厚尾”特征、波动的集群性、外汇市场中汇率波动的“无链接”问题等。资本市场在其本质上是非线性的。因此,引入非线性研究范式,对金融变量进行分析和预测研究是金融市场理论发展的必然结果。
本文基于非线性研究范式和投资者异质预期的假设,以5种主要货币的汇率时间序列为研究对象,检验其非线性动力学特征,研究对象包括英镑(GBP)、瑞士法郎(CHF)、日元(JPY)、瑞典克朗(SEK)和加拿大元(CAD)兑美元(USD)的日汇率数据,样本区间选自1975年1月1日至2007年5月31日。实证检验表明,在研究的汇率时间序列中,均找到了非线性依赖性、长记忆性、混沌动力学特征以及多重分形性的判据:
非线性依赖性的检验借助于BDS统计量,所有汇率序列的检验结果均拒绝独立同分布(iid)的原假设,数据中存在非线性依赖特征。同时,通过对子样本序列和标准序列进行BDS检验,其结果支持“数据中的非线性依赖性可能源于低维混沌”的假设;
长记忆性特征分析借助于重标极差分析方法(R/S)、对数周期图法(GPH)和高斯半参数估计法(GSP),估计长记忆参数d和Hurst指数,结果表明5种汇率序列的每日、每周和每月的数据均具有长记忆特征,在统计意义上存在标度不变的自相似结构,汇率的波动服从分形布朗运动,即有偏的随机游动,从而否定了传统线性研究范式的随机游动假说;
混沌动力学特征分析借助相空间重构技术,在判定非线性依赖性特征存在的前提下,对汇率时间序列的混沌特征量进行了估计。其中,重构参数的选择借助于Kim提出的C-C算法。实证结果显示,5种汇率时间序列的最大Lyapunov指数λ1均大于0,相关维的估计值均为分数,从而得到了确定性混沌存在的判据。这一结论为运用混沌理论对汇率变量进行解释以及短期预测提供了可能;
多重分形理论是分形理论的重要组成部分。本文在获取汇率时间序列存在长记忆性和分数维等分形特征的基础之上,进一步对其多重分形特征进行了分析:一方面借助配分函数分布图进行存在性的判定,另一方面通过多重分形谱对该特征进行了描述和分析比较。
本文通过对汇率时间序列的非线性依赖性、长记忆性、混沌动力学特征以及多重分形特征进行检验,得到汇率时间序列存在非线性动力学特征的证据。这不仅为人们更好地认识资本市场的本质特征提供了理论依据,同时也为投资者制定投资策略、企业规避外汇风险、货币当局制定货币政策和对外汇市场进行有效干预提供了技术支持。汇率时间序列存在非线性动力学特征表明,复杂系统内部的常态波动性来源于汇率系统的内随机性,是系统内部非线性机制导致的必然结果,而不必依赖于外部随机事件的冲击。因此,将汇率变量偏离均衡的原因完全归结于外部干扰,试图通过强制干预令其回归均衡的努力是无效的。
在得到汇率时间序列存在分形和混沌动力学特征的实证结果后,本文利用混沌时间序列预测方法对汇率变量进行预测研究。考虑到单个预测模型的局限性,本文利用组合预测误差最小化原理,将三种常见的非线性混沌时间序列预测模型,即基于径向基函数的神经网络模型、基于Lyapunov指数的预测模型和基于Volterra级数展开的自适应预测模型,集成构建了动态组合预测模型。D-M检验和H-M检验均表明构建的组合预测模型具有较好的市场预测能力,能取得显著优于随机游走模型的预测效果。通过误差性能指标和方向统计量指标分析可知,组合模型较之单独的预测模型均具有较明显的优势。
As an important variable in the international financial field, the exchange rate could determine the inner equilibrium and also influence the outer equilibrium of the national economy. With the development of the economic globalization and the acceleration of the international capital’s movement, the exchange rate is becoming more and more important for the investor’s making correct strategies, the enterprise’s risk aversion, and the central bank’s intervention in international exchange market. So, the behavior description and the forecasting’s validity of the exchange rate variable has been becoming one of the most popular topics and focus in the financial research field.
The traditional capital market theory has been established on the foundation of three hypotheses: Rational Investors, Efficient Market, and Random Walk. However, such equilibrium analysis system, which belongs to linearity methodology, couldn’t provide reliable explanation of many odds lying in exchange market. Such as the“fat tails”in returns distribution and autocorrelation in time series, the volatility clustering, and the irrelevance problem between the exchange rate and the interrelated economic variables. The intrinsic characteristic of the capital market is not linearity, but nonlinearity. Therefore, the phenomenon that a new research trend comes to being could be taken for granted, which bases the point of nonlinearity and system evolution instead of in a linear and equilibrium view.
This paper investigates the intrinsic nonlinear dynamics complex behaviors of five main exchange rates thoroughly, basing on the nonlinearity methodology and the hypothesis of heterogeneous expectations. The empirical sample consists of five daily exchange rates time series, including GBP/USD, CHF/USD, JPY/USD, SEK/USD and CAD/USD. The period covers more than 30 years, from Jan 1st 1975 to May 31st 2007. Finally, empirical researches have revealed that there really existing the nonlinear dependence, the long-memory property, the chaotic dynamic characters, and the multi fractal property.
The BDS testing has been applied to detect whether the nonlinearity properties existing in exchange rates time series. Luckily, all of the empirical results refuse the null hypothesis (independent and identically distributed, iid). Besides the discovering of the nonlinearities dependence lying in the time series, the testing has been also put on the sub-sample series and the normalized series, to explore an intrinsic explanation for the existence of nonlinearity. At last, the results revealed that the nonliterary lying in the data may be caused by chaos.
In order to detect the Long-Memory property, this paper applies the Rescaled Range Analysis(R/S), the Log Period-gram Estimation(GPH) and the Gaussian Semi parametric Estimation(GSP) on fifteen time series, including daily, weekly and monthly data of five exchange rate time series, by estimating the parameter d and Hurst exponents. The results tell us that there has do exist the Long-Memory property in all of the chosen exchange rates. It can be concluded that there is multi-scale self-similarity structure lying in the exchange rate time series. Therefore, the variable should follow Fractal Brown Movement, not the pure random walk (hypothesizes under the traditional capital market theory).
In order to judge the existence of chaotic dynamical features in exchange rate time series, this paper applies the technique of phase space reconstruction and the algorithm of small data sets on the exchange rates. The empirical results show that all of the largest Lyapunov exponentsλ1 are above zero, and all fractal dimensions are no-integer, which suggest the existence of chaos. It not only helps us for knowing the behavior of exchange rates better, but also providing the possibility for short-term forecasting.
Multi fractal theory is an important composition of fractal theory. After detecting the long-memory and fractal properties in the exchange rate time series, this paper puts the multi fractal analysis on exchange rates for more details. Firstly, the distribution diagrams of partition function have been applied for diagnosing the existence of multi fractal characteristic. After then, the multi fractal spectra have been used to describe the multi fractal characteristics. The discovery of nonlinear dynamic properties lying in exchange rate time series means that the volatility of the variables should origins from the nonlinear system itself, not only depending on the outer interrupting. Therefore, it’s invalid for the national bank’s forcibly intervention in the exchange market to make the price return equilibrium.
Furthermore, after detecting the nonlinear dynamic properties lying in the exchange rate time series, this paper integrates the RBF Neural Network Model, the Lyapunov Exponent Predicting Model and the Volterra Adaptive Predicting Model into an Adjustable Component Predicting Model, in order to make a good predicting of the chaotic variable of exchange rate, and the weights of which could be adjusted by the time series themselves. The results of the empirical research on five exchange rate time series show that both the forecasting errors and the direction statistics of the Component Predicting Model could obtain better results than individual models, especially for the JPY/USD and the SEK/USD time series. Moreover, a compare of the predicting performance has been taken between the Adjustable Component Predicting Model and the Random Walk Model. Luckily, both of the D-M testing and H-M testing refuse the original hypnosis and the empirical results show that the Adjustable Component Predicting Model could obtain obvious advantages over the Random Walk Model as expected.
传统的资本市场理论构建在理性投资人、有效市场假说和随机游动三大假设条件基础之上。然而,这种基于线性研究范式的均衡分析体系无法对外汇市场的一些异象给出合理的解释,如汇率收益率分布的“尖峰厚尾”特征、波动的集群性、外汇市场中汇率波动的“无链接”问题等。资本市场在其本质上是非线性的。因此,引入非线性研究范式,对金融变量进行分析和预测研究是金融市场理论发展的必然结果。
本文基于非线性研究范式和投资者异质预期的假设,以5种主要货币的汇率时间序列为研究对象,检验其非线性动力学特征,研究对象包括英镑(GBP)、瑞士法郎(CHF)、日元(JPY)、瑞典克朗(SEK)和加拿大元(CAD)兑美元(USD)的日汇率数据,样本区间选自1975年1月1日至2007年5月31日。实证检验表明,在研究的汇率时间序列中,均找到了非线性依赖性、长记忆性、混沌动力学特征以及多重分形性的判据:
非线性依赖性的检验借助于BDS统计量,所有汇率序列的检验结果均拒绝独立同分布(iid)的原假设,数据中存在非线性依赖特征。同时,通过对子样本序列和标准序列进行BDS检验,其结果支持“数据中的非线性依赖性可能源于低维混沌”的假设;
长记忆性特征分析借助于重标极差分析方法(R/S)、对数周期图法(GPH)和高斯半参数估计法(GSP),估计长记忆参数d和Hurst指数,结果表明5种汇率序列的每日、每周和每月的数据均具有长记忆特征,在统计意义上存在标度不变的自相似结构,汇率的波动服从分形布朗运动,即有偏的随机游动,从而否定了传统线性研究范式的随机游动假说;
混沌动力学特征分析借助相空间重构技术,在判定非线性依赖性特征存在的前提下,对汇率时间序列的混沌特征量进行了估计。其中,重构参数的选择借助于Kim提出的C-C算法。实证结果显示,5种汇率时间序列的最大Lyapunov指数λ1均大于0,相关维的估计值均为分数,从而得到了确定性混沌存在的判据。这一结论为运用混沌理论对汇率变量进行解释以及短期预测提供了可能;
多重分形理论是分形理论的重要组成部分。本文在获取汇率时间序列存在长记忆性和分数维等分形特征的基础之上,进一步对其多重分形特征进行了分析:一方面借助配分函数分布图进行存在性的判定,另一方面通过多重分形谱对该特征进行了描述和分析比较。
本文通过对汇率时间序列的非线性依赖性、长记忆性、混沌动力学特征以及多重分形特征进行检验,得到汇率时间序列存在非线性动力学特征的证据。这不仅为人们更好地认识资本市场的本质特征提供了理论依据,同时也为投资者制定投资策略、企业规避外汇风险、货币当局制定货币政策和对外汇市场进行有效干预提供了技术支持。汇率时间序列存在非线性动力学特征表明,复杂系统内部的常态波动性来源于汇率系统的内随机性,是系统内部非线性机制导致的必然结果,而不必依赖于外部随机事件的冲击。因此,将汇率变量偏离均衡的原因完全归结于外部干扰,试图通过强制干预令其回归均衡的努力是无效的。
在得到汇率时间序列存在分形和混沌动力学特征的实证结果后,本文利用混沌时间序列预测方法对汇率变量进行预测研究。考虑到单个预测模型的局限性,本文利用组合预测误差最小化原理,将三种常见的非线性混沌时间序列预测模型,即基于径向基函数的神经网络模型、基于Lyapunov指数的预测模型和基于Volterra级数展开的自适应预测模型,集成构建了动态组合预测模型。D-M检验和H-M检验均表明构建的组合预测模型具有较好的市场预测能力,能取得显著优于随机游走模型的预测效果。通过误差性能指标和方向统计量指标分析可知,组合模型较之单独的预测模型均具有较明显的优势。
As an important variable in the international financial field, the exchange rate could determine the inner equilibrium and also influence the outer equilibrium of the national economy. With the development of the economic globalization and the acceleration of the international capital’s movement, the exchange rate is becoming more and more important for the investor’s making correct strategies, the enterprise’s risk aversion, and the central bank’s intervention in international exchange market. So, the behavior description and the forecasting’s validity of the exchange rate variable has been becoming one of the most popular topics and focus in the financial research field.
The traditional capital market theory has been established on the foundation of three hypotheses: Rational Investors, Efficient Market, and Random Walk. However, such equilibrium analysis system, which belongs to linearity methodology, couldn’t provide reliable explanation of many odds lying in exchange market. Such as the“fat tails”in returns distribution and autocorrelation in time series, the volatility clustering, and the irrelevance problem between the exchange rate and the interrelated economic variables. The intrinsic characteristic of the capital market is not linearity, but nonlinearity. Therefore, the phenomenon that a new research trend comes to being could be taken for granted, which bases the point of nonlinearity and system evolution instead of in a linear and equilibrium view.
This paper investigates the intrinsic nonlinear dynamics complex behaviors of five main exchange rates thoroughly, basing on the nonlinearity methodology and the hypothesis of heterogeneous expectations. The empirical sample consists of five daily exchange rates time series, including GBP/USD, CHF/USD, JPY/USD, SEK/USD and CAD/USD. The period covers more than 30 years, from Jan 1st 1975 to May 31st 2007. Finally, empirical researches have revealed that there really existing the nonlinear dependence, the long-memory property, the chaotic dynamic characters, and the multi fractal property.
The BDS testing has been applied to detect whether the nonlinearity properties existing in exchange rates time series. Luckily, all of the empirical results refuse the null hypothesis (independent and identically distributed, iid). Besides the discovering of the nonlinearities dependence lying in the time series, the testing has been also put on the sub-sample series and the normalized series, to explore an intrinsic explanation for the existence of nonlinearity. At last, the results revealed that the nonliterary lying in the data may be caused by chaos.
In order to detect the Long-Memory property, this paper applies the Rescaled Range Analysis(R/S), the Log Period-gram Estimation(GPH) and the Gaussian Semi parametric Estimation(GSP) on fifteen time series, including daily, weekly and monthly data of five exchange rate time series, by estimating the parameter d and Hurst exponents. The results tell us that there has do exist the Long-Memory property in all of the chosen exchange rates. It can be concluded that there is multi-scale self-similarity structure lying in the exchange rate time series. Therefore, the variable should follow Fractal Brown Movement, not the pure random walk (hypothesizes under the traditional capital market theory).
In order to judge the existence of chaotic dynamical features in exchange rate time series, this paper applies the technique of phase space reconstruction and the algorithm of small data sets on the exchange rates. The empirical results show that all of the largest Lyapunov exponentsλ1 are above zero, and all fractal dimensions are no-integer, which suggest the existence of chaos. It not only helps us for knowing the behavior of exchange rates better, but also providing the possibility for short-term forecasting.
Multi fractal theory is an important composition of fractal theory. After detecting the long-memory and fractal properties in the exchange rate time series, this paper puts the multi fractal analysis on exchange rates for more details. Firstly, the distribution diagrams of partition function have been applied for diagnosing the existence of multi fractal characteristic. After then, the multi fractal spectra have been used to describe the multi fractal characteristics. The discovery of nonlinear dynamic properties lying in exchange rate time series means that the volatility of the variables should origins from the nonlinear system itself, not only depending on the outer interrupting. Therefore, it’s invalid for the national bank’s forcibly intervention in the exchange market to make the price return equilibrium.
Furthermore, after detecting the nonlinear dynamic properties lying in the exchange rate time series, this paper integrates the RBF Neural Network Model, the Lyapunov Exponent Predicting Model and the Volterra Adaptive Predicting Model into an Adjustable Component Predicting Model, in order to make a good predicting of the chaotic variable of exchange rate, and the weights of which could be adjusted by the time series themselves. The results of the empirical research on five exchange rate time series show that both the forecasting errors and the direction statistics of the Component Predicting Model could obtain better results than individual models, especially for the JPY/USD and the SEK/USD time series. Moreover, a compare of the predicting performance has been taken between the Adjustable Component Predicting Model and the Random Walk Model. Luckily, both of the D-M testing and H-M testing refuse the original hypnosis and the empirical results show that the Adjustable Component Predicting Model could obtain obvious advantages over the Random Walk Model as expected.
引文
[1] Frankel J., Froot K. A. Forward discount bias: is it an exchange risk premium[J]. Quarterly Journal of Economics, 1989, 104(1): 139-161.
[2] Hsieh D. A. The statistical property of daily foreign exchange rates: 1974-1983[J]. Journal of International Economics, 1988, 24(6): 132-145.
[3] Hsieh D. A. Testing for nonlinear dependence in daily foreign exchange rates[J]. Journal Business, 1989, 62(3): 339-368.
[4] Mandelbrot B. B. The fractal geometry of nature[M]. New York: Freeman and Co Press, 1982: 334-345.
[5] Winkler R. L. Combining forecasts: a philosophical basis and some current issues[J]. Forecasting, 1985, 5(4): 605-609.
[6] MacDonald R., MaRsh I. W. Combining exchange rate forecasts: what is the optimal consensus measure[J]. Forecasting, 1994, 13(3): 313-332.
[7]唐小我,滕颖,曾勇.汇率的组合预测模型及应用[J].预测, 1996, 15(3):31-36.
[8] Thomas H. L., Kyung D. N. Combining foreign exchange rate forecasts using neural networks[J]. Global Finance Journal, 1998, 9(1): 5-27.
[9] Fama E. F. The behavior of stock markets prices[J]. Journal of Business, 1965, 38(1): 34-105.
[10] Goodhart C. E. The evolution of central banks[M]. Cambridge: MIT Press, 1988: 21-39.
[11] Faust J., Rogers J., Wright J. Exchange rate forecasting: the errors we've really made[J]. Journal of International Economics, 2003, 60(1): 35-59.
[12] Hsieh D. A. Chaos and nonlinear dynamics: application to financial markets[J]. Finance, 1991, 46(18): 39-77.
[13] Scheinkman J. A., LeBaron B. Nonlinear dynamics and stock returns[J]. Business, 1989, 62(3): 11-37.
[14] Paul DeGrauwe, Hans Dewachter, Mark Embrechts. Exchange rate theory: chaotic models of foreign exchange markets[M]. Blackwells: Cambridge Press, 1993: 27-58.
[15] Ming Chya Wu. Phase correlation of foreign exchange time series[J]. Physica A, 2007, 375(2): 633-642.
[16] Strozzi F., Zaldivar J. M. Non-linear forecasting in high-frequency financial timeseries[J]. Physica A, 2005, 353(2): 463-479.
[17] Johnson N. F., Jefferies P., Ming Hui P. Financial market complexity[M]. Oxford: Oxford University Press, 2003: 78-84.
[18] Fama E. F. Efficient Capital Market: A review of theory and empirical work[J]. Journal of Finance, 1970, 25(2): 43-56.
[19] Lux T., Marchesi M. Volatility clustering in financial markets: a micro-simulation of interacting agents[J]. International Journal of Theoretical and Applied Finance, 2000, 13(4): 675-702.
[20] Paul D. G., Marianna G. Exchange rate puzzles: a tale of switching attractors[J]. European Economic Review, 2006, 50(1): 1-33.
[21] Meese R. A., Rogoff K. Empirical exchange rate models of the seventies: do they fit out of sample[J]. Journal of international economics, 1983, 14(2): 3-24.
[22] Paul D. G., Vansteenkiste I. Exchange rates and fundamentals: A non-linear relationship[A] In: CESifo Working Paper[C]. Berlin: The CESifo Group, 2001: 24-37.
[23] Obstfeld M., Rogoff K. Foundations of international macroeconomics[M]. Cambridge: MIT Press, 1996: 78-85.
[24] Paul DeGrauwe, Marianna Grimaldi. The exchange rate in a behavioral finance framework[M]. Princeton: Princeton University Press, 2006: 38-51.
[25] Campbell D. K. Nonlinear science: from paradigms to practicalities[J]. Los Alamos Science, 1987, 15(1): 218-262.
[26]吕金虎,陆君安,陈士华.混沌时间序列分析及其应用[M].武汉:武汉大学出版社, 2002: 36-51.
[27] Lorenz E. D. Deterministic non periodic flow[J]. Journal of the Atmospheric Sciences, 1963, 20(3): 130-141.
[28] Li T. Y., York J. A. Periods 3 implies chaos[J]. Amer Math Monthly, 1975, 82(1): 985-992.
[29] Stutzer M. Chaotic dynamics and bifurcations in a macro model[J]. Journal of Economic Dynamics and Control, 1980, 4(3): 353-376.
[30] Boldrin M., Montrucchio L. On the indeterminacy of capital accumulation paths[J]. Journal of Economic Theory, 1986, 40(1): 26-29.
[31] Barnett W., Chen P. Deterministic chaos and fractal attractors as tools for non- parametric dynamical econometric inference: with an application to the division monetary aggregates[J]. Mathematical and Computer Modeling, 1988, 10(4): 275- 296.
[32] Edgar E. Peters. Chaos and order in the capital markets[M]. New York: John Wiley & Sons Press, 1991: 82-121.
[33] Mirowski P. Mandebrot’s economics after a quarter century[J]. Fractals, 1995, 3(1): 581-600.
[34] Brock W. A., Sayers C. L. Is the business cycle characterized by deterministic chaos[J]. Monetary Econ, 1988, 22(7): 71-90.
[35] Frank M. Z., Gencay R., Stengos T. International chaos[J]. Eur Econ Rev, 1988, 32(15): 69-84.
[36] LeBaron B. Chaos and nonlinear forecast ability in economics and finance[J]. Phil Trans Royal Soc, London, Series A, 1994, 348(1): 397-404.
[37] Bajo Rubio O. Chaotic behaviour in exchange-rate series: First results for the Peseta-US dollar case[J]. Economics Letters, 1992, 39(2): 207-211.
[38] Aydin Cecen, Cahit Erkal. Distinguishing between stochastic and deterministic behavior in high frequency foreign exchange rate returns: can non-linear dynamics help forecasting[J]. Economics Letters, 1996, 51(3): 323-329.
[39] Aydin A. Cecen, Cahit Erkal. Distinguishing between stochastic and deterministic behavior in foreign exchange rate returns: Further evidence[J]. International Journal of Forecasting, 1996, 12(2): 465-473.
[40] Jonsson M. Studies in business cycles[J]. Institute for International Economic Studies, 1997, 5(4): 15-32.
[41] Atin Das, Pritha Das. Chaotic analysis of the foreign exchange rates[J]. Applied Mathematics and Computation, 2007, 185(1): 388-396.
[42] Marites A. Khanser. The application of chaos theory in the Philippine foreign Exchange Market[J]. Khanser, 1999, 42(1): 12-25.
[43] Mikael Bask. A positive Lyapunov exponent in Swedish exchange rates?[J]. Chaos, Solitons and Fractals, 2002, 15(5): 1295-1304.
[44] Fernanda Strozzi, José-Manuel Zaldívar, Joseph P. Zbilut. Application of non- linear time series analysis techniques to high-frequency currency exchange data[J]. Physica A, 2002, 312(3): 520-538.
[45] Brian Schwartz, Shahriar Yousefi. On complex behavior and exchange rate dynamics[J]. Chaos, Solitons and Fractals, 2003, 18(3): 503-523.
[46] Weston R., Premachandran P. A chaotic analysis of the new Zealand exchange rate 1985-2004[J]. Emerging Financial Markets and Services Conference, 2004, 22(5): 261-268.
[47] Mikael Bask. Long swings and chaos in the exchange rate in a DSGE model with aTaylor rule[A] In: Bank of Finland Research Discussion Papers[C]. Helsinki: Bank of Finland, 2007: 1-28.
[48] Scarlat E. I., Cristina Stan, Cristescu C. P. Chaotic features in Romanian transition economy as reflected onto the currency exchange rate[J]. Applied Mathematics and Computation, 2007, 185(5): 388-396.
[49] Sprott J., Rowlands G. Chaos data analyzer. The Professional Version User’s Manual[M]. New York: American Institute of Physics Press, 1995: 27-31.
[50] Mikael Bask. Chartism and exchange rate volatility[A] In: Helsinki Center of Economic Research Discussion Paper[C]. Helsinki: Helsinki Center of Economic Research, 2005: 301-316.
[51] Ken Arrow. Heterogeneous agent models in economics and finance[A] In: Tinbergen Institute Discussion Paper Cars Hommes[C]. Ann Arbor: The University of Michigan Press, 2004: 21-34.
[52] Yu Yingjin. Competitions hatch butterfly attractors in foreign exchange markets[J]. Physica A, 2005, 348(3): 380-388.
[53] Wang Yajie, Hui Xiaofeng, Abdol S. Soofi. Estimating renminbi(RMB) equilibrium exchange rate[J]. Journal of Policy Modeling, 2007, 29(1): 417-429.
[54]张永安.汇率波动的奇异吸引子测定及其应用分析[J].西安交通大学学报, 1998, 32(8): 92-95.
[55]崔孟修.汇率决定的混沌分析方法[J].管理世界, 2001, 1(6): 198-200.
[56]陆前进.汇率理论研究的新方法——混沌理论及其评述[J].国际金融研究, 2000, 2(3): 21-25.
[57]方锦清.非线性控制与混沌控制论——略谈与现代控制论的结合[J].自然杂志, 1994, 3(1): 147-152.
[58]戴国强,徐龙炳,陆蓉.国际汇率波动的非线性探索及其政策意义[J].国际金融研, 1999, 10(2): 9-15.
[59]谢赤,杨妮.汇率行为的混沌性及其分形维描述[J].湖南大学学报社会科学版, 2005, 19(5): 50-56.
[60]王海燕,盛昭瀚.多变量时间序列复杂系统的相空间重构[J].东南大学学报, 2003, 33(1): 115-118.
[61]郝柏林.分叉、混沌、奇怪吸引子、湍流及其它[J].物理学进展, 1987, 3(3): 392-443.
[62] Mandelbrot B. B. The variance of certain speculative prices[A] In: The Random Character of Stock Prices[C]. Cambridge: MIT Press, 1964: 18-27.
[63]埃德加·彼得斯.分形市场分析——将混沌理论应用到投资与经济理论上[M].北京:经济科学出版社, 2002: 27-50.
[64] Gordon R. Richards. The fractal structure of exchange rates: measurement and forecasting[J]. Journal of International Financial Markets, Institutions and Money, 2000, 25(10): 163-180.
[65] Tsonis A. A., Heller F., Takayasu H., Marumo K., Shimizu T. A characteristic time scale in dollar-yen exchange rates[J]. Physica A, 2001, 291(3): 574-582.
[66] Muniandy S. V., Lim S. C., Murugan R. Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates[J]. Physica A, 2001, 301(1): 407-428.
[67] Andres Fernandez Diaz, Pilar GrauCarles, Lorenzo Escot Mangas. Nonlinearities in the exchange rates returns and volatility[J]. Physica A, 2002, 316(1): 469-482.
[68] Mandelbrot B. B., Fisher A., Calvet L. Cowles foundation for research and economics[M]. New Haven: Yale University Press, 1997: 27-47.
[69] Kyungsik Kim, Seong-Min Yoon. Multifractal features of financial markets[J]. Physica A, 2004, 344(2): 272-278.
[70] Norouzzadeh P., Rahmani B. A multifractal detrended fluctuation description of Iranian rial-US dollar exchange rate[J]. Physica A, 2006, 367(2): 328-336.
[71] Ruipeng Liu, Di Matteo T., Thomas Lux. True and apparent scaling: The proximity of the Markov-switching multifractal model to long-range dependence[J]. Physica A, 2007, 383(6): 35-42.
[72] Abhyankar A., Copeland L. S., Wong W. Nonlinear dynamics in real time equity market indices: evidence from the United Kingdom[J]. Econ, 1995, 105(8): 64-80.
[73] Abhyankar A., Copeland L.S., Wong W. Uncovering nonlinear structure in real- time stock market indexes: the S&P 500, the DAX, the Nikkei 225, and the FTSE- 100[J]. Business Econ Statist, 1997, 15(1): 1-14.
[74] Arvind Mahajan, Andrew J. Wagner. Nonlinear dynamics in foreign exchange rates[J]. Global Finance Journal, 1999, 10(1): 1-23.
[75] Frankel J., Froot K. A. Understanding the U.S.Dollar in the eighties: the expectation of chartists and fundamentals[J]. Ecnomic Record, Supplement, 1986, 97(10): 24-38.
[76] Meese R. A. An empirical assessment of nonlinear times in models of exchange rate determination[J]. Review of Economic Studies, 1991, 58(3): 603- 619.
[77] Paul DeGrauwe, Hans Dewachter. Chaos in the Dornbusch model of the exchange rate[J]. Review of economic studies, 1992, 25(1): 26-54.
[78] Paul DeGrauwe, Hans Dewachter. Deterministic assessment of onlinearities in models of exchangerate determination[J]. Review of economic studies, 1991, 58(3):603-619.
[79] Cassel. Abornormal deviations in inernational exchanges[J]. Economic Journal, 1918, 28(4): 413-415.
[80] Keynes J. M. A tract on monetary reform[M]. London: St Martin’s Press, 1923: 27-41.
[81] Frankel J. A monetary approach to the exchange rate: doctrinal aspects and empirical evidence Scandinavian[J]. Journal of Economics, 1976, 78(2): 220-224.
[82] Dornbusch R. Expectations and exchange rate dynamics[J]. Journal of Political Economy, 1976, 84(1): 1161-1176.
[83] Kouri P. J. The exchange rate and the balance of payments in the short run and in the long run: a monetary approach[J]. Scandinavian Journal of Economics, 1976, 78(2): 280-303.
[84] Branson W. H. Asset markets and relative prices in exchange rate determination [M]. Prineston: Prineston University Press, 1977: 39-51.
[85] Allen Polly R., Peter B., Kenen, Asset market, exchange rate and economic integration[M]. Cambridge: Cambridge University Press, 1980: 25-38.
[86] Frankel J. A flexible exchange rate, price and the role of news: lessons from the 1970s[J]. Journal of Political Economics, 1981, 89(4): 51-62.
[87] Garman M. Market microstructure[J]. Journal of Financial Economics, 1976, 3(3): 257-275.
[88] Lyons R. K. Test of Microstructure Hypotheses in the Foreign Exchange Market[J]. Journal of Financial Economics, 1995, 39(2): 321-351.
[89] Engel R. Auto regressive conditional heteroskedasticity with estimates of the variance of UK inflation [J]. Econometrica, 1982, 50(1): 987-1008.
[90] Bollerslev Tim. Generalized autoregressive conditional heteroskedasticity[J]. Journal of Econometrics, 1986, 31(2): 307-327.
[91] Nelson B. Conditional heteroskedasticity in asset returns: a new approach[J]. Econometrica, 1991, 59(2): 347-370.
[92] Tong H. On a threshold model[A] In C H Chen(ed), Pattern Recognition and Signal Processing[C]. Amsterdam: Sijthoff and Noordhoff Press, 1978: 18-29.
[93] Tong H., Lim K. Threshold autoregression, limit cycles and cyclical data(with discussion)[J]. Journal of the Royal Statistical Society, Series B, 1980, 42(2): 245- 292.
[94] Hamiton, James D. Time series analysis[M]. Princeton: Princeton University Press, 1994: 37-46.
[95] Granger C. W. J. Modeling nonlinear economic relationships[M]. Oxford: Oxford University Press, 1993: 29-36.
[96]唐国兴,徐剑刚.现代汇率理论及模型研究[M].北京:中国金融出版社, 2003: 96-112.
[97] Dornbusch R. The Exchange Market: Discussion[A] In: V Grilli and R Hamaui (eds) Financial Markets’Liberalization and the Role of Banks[C]. Cambridge: Cambridge University Press, 1993: 24-36.
[98] Frankel J., Froot K. L. Using survey data to test standard propositions on exchange rate expectations[J]. American Economic Review, 1987, 77(1): 135-153.
[99] Mikael Bask, Gencay R. Testing chaotic dynamics via Lyapunov exponents[J]. Physica D, 1998, 114(3): 1-12.
[100] Meese R. A., Rose R. K. An empirical assessment of nonlinearities in models of exchange rate determination[J]. Review of Economic Studies, 1991, 58(3): 603- 619.
[101] Schwartz, Yousefi. On complex behavior and exchange rate dynamics[J]. Chaos, Solitons and Fractals, 2003, 18(3): 503-523.
[102] Shuheng C. Testing for non-linear structure in an artificial financial market[J]. Journal of Economic Behavior & Organization, 2001, 46(3): 327-342.
[103] Brock W. A., Hsieh D., LeBaron B. Nonlinear dynamics, chaos and instability: theory and evidence[M]. Cambridge: MIT Press, 1991: 47-56.
[104] Marcelo Soto. Capital flows and growth in developing countries: recent empirical evidence[J]. OECD Development Center, 2000, 9(1): 160-164.
[105] Shuheng Chen, Chia Hsuan Yeh. Evolving traders and the business school with genetic programming: a new architecture of the agent-based artificial stock market[J]. Journal of Economic Dynamics and Control, 2001, 25(1): 363-393.
[106] Brock W., Dechert W., Scheinkman J. A Test for Independence Based on Correlation Dimension[J]. Econometric Reviews, 1996, 15(3): 197-235.
[107] Grassberger P., Procaccia I. Dimensions and entropies of strange attractors form a fluctuating dynamics approach [J]. Physics, 1984, 13(1): 34-54.
[108] Scheinkman A., LeBaron B. Nonlinear dynamics and GNP data[A] In: Barnett W, Greweke J, Shell K Economic Complexity[C]. Cambridge: Cambridge University Press, 1989: 56-64.
[109] Brock W., Hsieh D., LeBaron B. Nonlinear dynamics, chaos and instability[M]. Cambridge: MIT Press, 1991: 37-48.
[110] Barnett A., Gallant R., Hinich J., Kaplan T. A single-blind controlled competitionamong tests for nonlinearity and chaos[J]. Journal of Econometrics, 1997, 82(1): 157-192.
[111] Kocenda E. Optimal range for the iid test based on integration across the correlation integral[J]. Econometric Reviews, 2005, 24(3): 265-296.
[112] Brockwell P. J., Davis R. A. Time series: theory and methods[J]. Springerverlag, 1991, 24(1): 577-591.
[113] Hurst H. E. The long-term storage capacity of reservoirs[J]. Transactions of the American Society of Civil Engineers, 1951, 116(1): 47-59.
[114] Mandelbrot B. B., Wallis J. R. Robustness of the rescaled range R/S in the measurement of non cyclic long-run statistical dependence[J]. Water Resour Res, 1969, 5(5): 967-988.
[115] Lo A.W. Long-term memory in stock market prices[J]. Econometrical, 1991, 59(5): 1279-1313.
[116] Geweke J. F., Porter-Hudak S. The estimation and application of long memory time series models[J]. Journal of Time Series Analysis, 1983, 4(1): 221-238.
[117] Robinson P. M. Gaussian semi-parametric estimation of long-range dependence [J]. The Annals of Statistics, 1995, 23(3): 1630-1661.
[118] Robinson P.M. Log period gram regression of time series with long range dependence[J]. Annals of Statistics, 1995, 23(3): 1048-1072.
[119] Sowell F. B. Maximum likelihood estimation of stationary unvaried fractionally integrated time series models[J]. Journal of Econometrics, 1992, 53(3): 165-188.
[120] Doornik J. A., Ooms M. Computational aspects of maximum likelihood estimation of autoregressive fractionally integrated moving average models[J]. Computational Statistics & Data Analysis, 2003, 42(3): 333-348.
[121] Cox D. R., Reid N. Parameter orthogonality and approximate conditional inference[J]. Journal of the Royal Statistical Society Series B, 1987, 39(1): 1-39.
[122] Granger C. W., Newbold P. Forecasting economic time series[M]. New York: Academic Press, 1986: 32-43.
[123] Teverovsky V., Taqqu M. S., Willinger W. A critical look at Lo's modified R/S statistic[J]. Journal of Statistical Planning and Inference, 1999, 80(1): 211-227.
[124] Doornik J. A. Object-Oriented Matrix Programming using Ox 5th edn[M]. London: Timberlake Consultants Press, 2006: 18-27.
[125] Davis M. Louis Bachelier’s theory of speculation[M]. Princeton: Princeton University Press, 2006: 29-36.
[126] Eckmann J. P., Oliffson Kamphorst S., Ruelle D., Scheinkman J. A. Lyapunovexponents for stock returns[A] In Anderson P W, Arrow K J, Pines D editors, The economy as an evolving complex system[C]. New York: Addison-Wesley Press, 1988: 25-37.
[127] Mizrach B. Determining delay times for phase space reconstruction with application to the FF/DM exchange rate[J]. Econ Behav Organ, 1996, 30(3): 69-81.
[128] Jiaorui Li, Wei Xu, Wenxian Xie, Zhengzheng Ren. Research on nonlinear stochastic dynamical price model[J]. Chaos, Solitons and Fractals, 2006, 42(7): 1381-1393.
[129] Orlowski A., Struzik Z.R., Syczewska E., Zaluska Kotur M.A. Fluctuation dynamics of exchange rates on Polish financial market[J]. Physica A, 2004, 344(1): 184-189.
[130] Marotto F. R. Snap-back repellers imply chaos[J]. Math Anal Appl, 1978, 63(1): 199-223.
[131] Devaney R. L. An introduction to chaotic dynamical systems[M]. Redwood City: Addison-Wesley Publishing Company, 1987: 29-40.
[132] Wolf A., Swift J., Swinney H. Determining Lyapunov exponents from a time series[J]. Physica D, 1985, 16(3): 285-317.
[133] Barana G., Tsuda L. A new method for computing Lyapunov exponents[J]. Physics Letter A, 1993, 175(6): 421-427.
[134] Rosenstein M. T., Collins J. J., De Luca C. J. A practical method for calculating largest Lyapunov exponents from small data sets[J]. Physica D, 1993, 65(1): 117-134.
[135] Packard N. H., Crutchfield J. D., Farmer J. D., Shaw R. S. Geometry from a Time Series[J]. Phys Rev Lett, 1980, 45(9): 712-716.
[136] Takens F., Mane. In dynamical systems of turbulence[A] In: Rand D A, Young L S, eds, Lecture Notes in Mathematics[C]. Berlin: Springer Press, 1986: 366-381.
[137] Fraser A. M., Swinney H. L. Independent coordinates from mutual information [J]. Phys Rev A, 1986, 33(2): 1134-1140.
[138] Tsonis A. A. Chaos: from theory to application[M]. New York: Plenum Press, 1992: 39-47.
[139] Broomhead D. S., King G. P. Extracting qualitative dynamics from experimental data[J]. Physica D, 1986, 20(2): 217-236.
[140] Kugiumtzis D. Correction of the correlation dimension for noisy time series: the role of the time window length[J]. Physica D, 1996, 95(13): 13-28.
[141] Kim H. S., Eykholt R., Salas J. D. Nonlinear Dynamics, Delay times, andEmbedding windows[J]. Physica D, 1999, 127(2): 48-60.
[142] Xie Chi, Yang Ni. Analysis and empirical research on the long-memory property of exchange rate time series[J]. Wireless Communications, Networking and Mobile Computing 2007, 2007, 8(1): 5906-5909.
[143] Galluccio S., Caldarelli G., Marsili M. Scaling in currency exchange[J]. Physica A, 1997, 245(4): 423-436.
[144]谢赤,熊一鹏.深圳股票市场多标度分形特征分析[J].财经理论与实践, 2005, 26(13): 53-55.
[145] Bacry E., Delour J., Muzy J. F. Modeling financial time series using multifractal random walks[J]. Physica A, 2001, 299(1): 84-92.
[146] Michele P., Maurizio S. Multiscaling and clustering of volatility[J]. Physica A, 1999, 269(1): 140-147.
[147] Mandelbrot B. B., Calvet L., Fisher A. Large deviation theory and the multifractal spectrum of financial prices[A] In Working Paper of Yale University[C]. Yale: Yale University, 1997: 12-22.
[148] Fisher A., Calvet L., Mandelbrot B. B. Multifractality of deutschmark/US dollar exchange rates[A] In Working Paper of Yale University[C]. Yale: Yale University, 1997: 3-12.
[149] Calvet L., Fisher A., Mandelbrot B. B. A multifractal model of asset returns[A] In Working Paper of Yale University[C]. Yale: Yale University, 1997: 5-16.
[150] Muniandy S. V., Lim S. C., Murugan R. Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates[J]. Physica A, 2001, 301(1): 407-428.
[151]韩敏.混沌时间序列预测理论与方法[M].北京:中国水利水电出版社, 2007: 57-71.
[152] Mizrach B. Multivariate nearest-neighbor forecasts of EMS Exchange Rates[J]. Journal of Applied Econometrics, 1992, 7(1): 151-163.
[153] Kenya A., Alvaro V., Elton C. Using artificial neural networks to forecast chaotic time series[J]. Physica A, 2000, 284(9): 393-404.
[154] Timo T., Dick D., Marcelo C. Linear models, smooth transition auto regressions, and neural networks for forecasting macroeconomic time series: A reexamination [J]. International Journal of Forecasting, 2005, 21(1): 755-774.
[155] Francesco L., Rosa A. S. A comparison between neural networks and chaotic models for exchange rate prediction [J]. Computational Statistics & Data Analysis, 1999, 30(1): 87-102.
[156]余乐安,汪寿阳,黎建强.外汇汇率与国际原油价格波动预测——TEI@I方法论[M].长沙:湖南大学出版社, 2006: 67-81.
[157] Diebold F. X., Mariano R. S. Comparing predictive accuracy[J]. Journal of Business and Economic Statistics, 1995, 13(3): 253-263.
[158] Henriksson R. D., Merton R. C. On market timing and investment performance: statistical procedures for evaluating forecasting skill[J]. Journal of Business, 1981, 54(4): 513-533.
[2] Hsieh D. A. The statistical property of daily foreign exchange rates: 1974-1983[J]. Journal of International Economics, 1988, 24(6): 132-145.
[3] Hsieh D. A. Testing for nonlinear dependence in daily foreign exchange rates[J]. Journal Business, 1989, 62(3): 339-368.
[4] Mandelbrot B. B. The fractal geometry of nature[M]. New York: Freeman and Co Press, 1982: 334-345.
[5] Winkler R. L. Combining forecasts: a philosophical basis and some current issues[J]. Forecasting, 1985, 5(4): 605-609.
[6] MacDonald R., MaRsh I. W. Combining exchange rate forecasts: what is the optimal consensus measure[J]. Forecasting, 1994, 13(3): 313-332.
[7]唐小我,滕颖,曾勇.汇率的组合预测模型及应用[J].预测, 1996, 15(3):31-36.
[8] Thomas H. L., Kyung D. N. Combining foreign exchange rate forecasts using neural networks[J]. Global Finance Journal, 1998, 9(1): 5-27.
[9] Fama E. F. The behavior of stock markets prices[J]. Journal of Business, 1965, 38(1): 34-105.
[10] Goodhart C. E. The evolution of central banks[M]. Cambridge: MIT Press, 1988: 21-39.
[11] Faust J., Rogers J., Wright J. Exchange rate forecasting: the errors we've really made[J]. Journal of International Economics, 2003, 60(1): 35-59.
[12] Hsieh D. A. Chaos and nonlinear dynamics: application to financial markets[J]. Finance, 1991, 46(18): 39-77.
[13] Scheinkman J. A., LeBaron B. Nonlinear dynamics and stock returns[J]. Business, 1989, 62(3): 11-37.
[14] Paul DeGrauwe, Hans Dewachter, Mark Embrechts. Exchange rate theory: chaotic models of foreign exchange markets[M]. Blackwells: Cambridge Press, 1993: 27-58.
[15] Ming Chya Wu. Phase correlation of foreign exchange time series[J]. Physica A, 2007, 375(2): 633-642.
[16] Strozzi F., Zaldivar J. M. Non-linear forecasting in high-frequency financial timeseries[J]. Physica A, 2005, 353(2): 463-479.
[17] Johnson N. F., Jefferies P., Ming Hui P. Financial market complexity[M]. Oxford: Oxford University Press, 2003: 78-84.
[18] Fama E. F. Efficient Capital Market: A review of theory and empirical work[J]. Journal of Finance, 1970, 25(2): 43-56.
[19] Lux T., Marchesi M. Volatility clustering in financial markets: a micro-simulation of interacting agents[J]. International Journal of Theoretical and Applied Finance, 2000, 13(4): 675-702.
[20] Paul D. G., Marianna G. Exchange rate puzzles: a tale of switching attractors[J]. European Economic Review, 2006, 50(1): 1-33.
[21] Meese R. A., Rogoff K. Empirical exchange rate models of the seventies: do they fit out of sample[J]. Journal of international economics, 1983, 14(2): 3-24.
[22] Paul D. G., Vansteenkiste I. Exchange rates and fundamentals: A non-linear relationship[A] In: CESifo Working Paper[C]. Berlin: The CESifo Group, 2001: 24-37.
[23] Obstfeld M., Rogoff K. Foundations of international macroeconomics[M]. Cambridge: MIT Press, 1996: 78-85.
[24] Paul DeGrauwe, Marianna Grimaldi. The exchange rate in a behavioral finance framework[M]. Princeton: Princeton University Press, 2006: 38-51.
[25] Campbell D. K. Nonlinear science: from paradigms to practicalities[J]. Los Alamos Science, 1987, 15(1): 218-262.
[26]吕金虎,陆君安,陈士华.混沌时间序列分析及其应用[M].武汉:武汉大学出版社, 2002: 36-51.
[27] Lorenz E. D. Deterministic non periodic flow[J]. Journal of the Atmospheric Sciences, 1963, 20(3): 130-141.
[28] Li T. Y., York J. A. Periods 3 implies chaos[J]. Amer Math Monthly, 1975, 82(1): 985-992.
[29] Stutzer M. Chaotic dynamics and bifurcations in a macro model[J]. Journal of Economic Dynamics and Control, 1980, 4(3): 353-376.
[30] Boldrin M., Montrucchio L. On the indeterminacy of capital accumulation paths[J]. Journal of Economic Theory, 1986, 40(1): 26-29.
[31] Barnett W., Chen P. Deterministic chaos and fractal attractors as tools for non- parametric dynamical econometric inference: with an application to the division monetary aggregates[J]. Mathematical and Computer Modeling, 1988, 10(4): 275- 296.
[32] Edgar E. Peters. Chaos and order in the capital markets[M]. New York: John Wiley & Sons Press, 1991: 82-121.
[33] Mirowski P. Mandebrot’s economics after a quarter century[J]. Fractals, 1995, 3(1): 581-600.
[34] Brock W. A., Sayers C. L. Is the business cycle characterized by deterministic chaos[J]. Monetary Econ, 1988, 22(7): 71-90.
[35] Frank M. Z., Gencay R., Stengos T. International chaos[J]. Eur Econ Rev, 1988, 32(15): 69-84.
[36] LeBaron B. Chaos and nonlinear forecast ability in economics and finance[J]. Phil Trans Royal Soc, London, Series A, 1994, 348(1): 397-404.
[37] Bajo Rubio O. Chaotic behaviour in exchange-rate series: First results for the Peseta-US dollar case[J]. Economics Letters, 1992, 39(2): 207-211.
[38] Aydin Cecen, Cahit Erkal. Distinguishing between stochastic and deterministic behavior in high frequency foreign exchange rate returns: can non-linear dynamics help forecasting[J]. Economics Letters, 1996, 51(3): 323-329.
[39] Aydin A. Cecen, Cahit Erkal. Distinguishing between stochastic and deterministic behavior in foreign exchange rate returns: Further evidence[J]. International Journal of Forecasting, 1996, 12(2): 465-473.
[40] Jonsson M. Studies in business cycles[J]. Institute for International Economic Studies, 1997, 5(4): 15-32.
[41] Atin Das, Pritha Das. Chaotic analysis of the foreign exchange rates[J]. Applied Mathematics and Computation, 2007, 185(1): 388-396.
[42] Marites A. Khanser. The application of chaos theory in the Philippine foreign Exchange Market[J]. Khanser, 1999, 42(1): 12-25.
[43] Mikael Bask. A positive Lyapunov exponent in Swedish exchange rates?[J]. Chaos, Solitons and Fractals, 2002, 15(5): 1295-1304.
[44] Fernanda Strozzi, José-Manuel Zaldívar, Joseph P. Zbilut. Application of non- linear time series analysis techniques to high-frequency currency exchange data[J]. Physica A, 2002, 312(3): 520-538.
[45] Brian Schwartz, Shahriar Yousefi. On complex behavior and exchange rate dynamics[J]. Chaos, Solitons and Fractals, 2003, 18(3): 503-523.
[46] Weston R., Premachandran P. A chaotic analysis of the new Zealand exchange rate 1985-2004[J]. Emerging Financial Markets and Services Conference, 2004, 22(5): 261-268.
[47] Mikael Bask. Long swings and chaos in the exchange rate in a DSGE model with aTaylor rule[A] In: Bank of Finland Research Discussion Papers[C]. Helsinki: Bank of Finland, 2007: 1-28.
[48] Scarlat E. I., Cristina Stan, Cristescu C. P. Chaotic features in Romanian transition economy as reflected onto the currency exchange rate[J]. Applied Mathematics and Computation, 2007, 185(5): 388-396.
[49] Sprott J., Rowlands G. Chaos data analyzer. The Professional Version User’s Manual[M]. New York: American Institute of Physics Press, 1995: 27-31.
[50] Mikael Bask. Chartism and exchange rate volatility[A] In: Helsinki Center of Economic Research Discussion Paper[C]. Helsinki: Helsinki Center of Economic Research, 2005: 301-316.
[51] Ken Arrow. Heterogeneous agent models in economics and finance[A] In: Tinbergen Institute Discussion Paper Cars Hommes[C]. Ann Arbor: The University of Michigan Press, 2004: 21-34.
[52] Yu Yingjin. Competitions hatch butterfly attractors in foreign exchange markets[J]. Physica A, 2005, 348(3): 380-388.
[53] Wang Yajie, Hui Xiaofeng, Abdol S. Soofi. Estimating renminbi(RMB) equilibrium exchange rate[J]. Journal of Policy Modeling, 2007, 29(1): 417-429.
[54]张永安.汇率波动的奇异吸引子测定及其应用分析[J].西安交通大学学报, 1998, 32(8): 92-95.
[55]崔孟修.汇率决定的混沌分析方法[J].管理世界, 2001, 1(6): 198-200.
[56]陆前进.汇率理论研究的新方法——混沌理论及其评述[J].国际金融研究, 2000, 2(3): 21-25.
[57]方锦清.非线性控制与混沌控制论——略谈与现代控制论的结合[J].自然杂志, 1994, 3(1): 147-152.
[58]戴国强,徐龙炳,陆蓉.国际汇率波动的非线性探索及其政策意义[J].国际金融研, 1999, 10(2): 9-15.
[59]谢赤,杨妮.汇率行为的混沌性及其分形维描述[J].湖南大学学报社会科学版, 2005, 19(5): 50-56.
[60]王海燕,盛昭瀚.多变量时间序列复杂系统的相空间重构[J].东南大学学报, 2003, 33(1): 115-118.
[61]郝柏林.分叉、混沌、奇怪吸引子、湍流及其它[J].物理学进展, 1987, 3(3): 392-443.
[62] Mandelbrot B. B. The variance of certain speculative prices[A] In: The Random Character of Stock Prices[C]. Cambridge: MIT Press, 1964: 18-27.
[63]埃德加·彼得斯.分形市场分析——将混沌理论应用到投资与经济理论上[M].北京:经济科学出版社, 2002: 27-50.
[64] Gordon R. Richards. The fractal structure of exchange rates: measurement and forecasting[J]. Journal of International Financial Markets, Institutions and Money, 2000, 25(10): 163-180.
[65] Tsonis A. A., Heller F., Takayasu H., Marumo K., Shimizu T. A characteristic time scale in dollar-yen exchange rates[J]. Physica A, 2001, 291(3): 574-582.
[66] Muniandy S. V., Lim S. C., Murugan R. Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates[J]. Physica A, 2001, 301(1): 407-428.
[67] Andres Fernandez Diaz, Pilar GrauCarles, Lorenzo Escot Mangas. Nonlinearities in the exchange rates returns and volatility[J]. Physica A, 2002, 316(1): 469-482.
[68] Mandelbrot B. B., Fisher A., Calvet L. Cowles foundation for research and economics[M]. New Haven: Yale University Press, 1997: 27-47.
[69] Kyungsik Kim, Seong-Min Yoon. Multifractal features of financial markets[J]. Physica A, 2004, 344(2): 272-278.
[70] Norouzzadeh P., Rahmani B. A multifractal detrended fluctuation description of Iranian rial-US dollar exchange rate[J]. Physica A, 2006, 367(2): 328-336.
[71] Ruipeng Liu, Di Matteo T., Thomas Lux. True and apparent scaling: The proximity of the Markov-switching multifractal model to long-range dependence[J]. Physica A, 2007, 383(6): 35-42.
[72] Abhyankar A., Copeland L. S., Wong W. Nonlinear dynamics in real time equity market indices: evidence from the United Kingdom[J]. Econ, 1995, 105(8): 64-80.
[73] Abhyankar A., Copeland L.S., Wong W. Uncovering nonlinear structure in real- time stock market indexes: the S&P 500, the DAX, the Nikkei 225, and the FTSE- 100[J]. Business Econ Statist, 1997, 15(1): 1-14.
[74] Arvind Mahajan, Andrew J. Wagner. Nonlinear dynamics in foreign exchange rates[J]. Global Finance Journal, 1999, 10(1): 1-23.
[75] Frankel J., Froot K. A. Understanding the U.S.Dollar in the eighties: the expectation of chartists and fundamentals[J]. Ecnomic Record, Supplement, 1986, 97(10): 24-38.
[76] Meese R. A. An empirical assessment of nonlinear times in models of exchange rate determination[J]. Review of Economic Studies, 1991, 58(3): 603- 619.
[77] Paul DeGrauwe, Hans Dewachter. Chaos in the Dornbusch model of the exchange rate[J]. Review of economic studies, 1992, 25(1): 26-54.
[78] Paul DeGrauwe, Hans Dewachter. Deterministic assessment of onlinearities in models of exchangerate determination[J]. Review of economic studies, 1991, 58(3):603-619.
[79] Cassel. Abornormal deviations in inernational exchanges[J]. Economic Journal, 1918, 28(4): 413-415.
[80] Keynes J. M. A tract on monetary reform[M]. London: St Martin’s Press, 1923: 27-41.
[81] Frankel J. A monetary approach to the exchange rate: doctrinal aspects and empirical evidence Scandinavian[J]. Journal of Economics, 1976, 78(2): 220-224.
[82] Dornbusch R. Expectations and exchange rate dynamics[J]. Journal of Political Economy, 1976, 84(1): 1161-1176.
[83] Kouri P. J. The exchange rate and the balance of payments in the short run and in the long run: a monetary approach[J]. Scandinavian Journal of Economics, 1976, 78(2): 280-303.
[84] Branson W. H. Asset markets and relative prices in exchange rate determination [M]. Prineston: Prineston University Press, 1977: 39-51.
[85] Allen Polly R., Peter B., Kenen, Asset market, exchange rate and economic integration[M]. Cambridge: Cambridge University Press, 1980: 25-38.
[86] Frankel J. A flexible exchange rate, price and the role of news: lessons from the 1970s[J]. Journal of Political Economics, 1981, 89(4): 51-62.
[87] Garman M. Market microstructure[J]. Journal of Financial Economics, 1976, 3(3): 257-275.
[88] Lyons R. K. Test of Microstructure Hypotheses in the Foreign Exchange Market[J]. Journal of Financial Economics, 1995, 39(2): 321-351.
[89] Engel R. Auto regressive conditional heteroskedasticity with estimates of the variance of UK inflation [J]. Econometrica, 1982, 50(1): 987-1008.
[90] Bollerslev Tim. Generalized autoregressive conditional heteroskedasticity[J]. Journal of Econometrics, 1986, 31(2): 307-327.
[91] Nelson B. Conditional heteroskedasticity in asset returns: a new approach[J]. Econometrica, 1991, 59(2): 347-370.
[92] Tong H. On a threshold model[A] In C H Chen(ed), Pattern Recognition and Signal Processing[C]. Amsterdam: Sijthoff and Noordhoff Press, 1978: 18-29.
[93] Tong H., Lim K. Threshold autoregression, limit cycles and cyclical data(with discussion)[J]. Journal of the Royal Statistical Society, Series B, 1980, 42(2): 245- 292.
[94] Hamiton, James D. Time series analysis[M]. Princeton: Princeton University Press, 1994: 37-46.
[95] Granger C. W. J. Modeling nonlinear economic relationships[M]. Oxford: Oxford University Press, 1993: 29-36.
[96]唐国兴,徐剑刚.现代汇率理论及模型研究[M].北京:中国金融出版社, 2003: 96-112.
[97] Dornbusch R. The Exchange Market: Discussion[A] In: V Grilli and R Hamaui (eds) Financial Markets’Liberalization and the Role of Banks[C]. Cambridge: Cambridge University Press, 1993: 24-36.
[98] Frankel J., Froot K. L. Using survey data to test standard propositions on exchange rate expectations[J]. American Economic Review, 1987, 77(1): 135-153.
[99] Mikael Bask, Gencay R. Testing chaotic dynamics via Lyapunov exponents[J]. Physica D, 1998, 114(3): 1-12.
[100] Meese R. A., Rose R. K. An empirical assessment of nonlinearities in models of exchange rate determination[J]. Review of Economic Studies, 1991, 58(3): 603- 619.
[101] Schwartz, Yousefi. On complex behavior and exchange rate dynamics[J]. Chaos, Solitons and Fractals, 2003, 18(3): 503-523.
[102] Shuheng C. Testing for non-linear structure in an artificial financial market[J]. Journal of Economic Behavior & Organization, 2001, 46(3): 327-342.
[103] Brock W. A., Hsieh D., LeBaron B. Nonlinear dynamics, chaos and instability: theory and evidence[M]. Cambridge: MIT Press, 1991: 47-56.
[104] Marcelo Soto. Capital flows and growth in developing countries: recent empirical evidence[J]. OECD Development Center, 2000, 9(1): 160-164.
[105] Shuheng Chen, Chia Hsuan Yeh. Evolving traders and the business school with genetic programming: a new architecture of the agent-based artificial stock market[J]. Journal of Economic Dynamics and Control, 2001, 25(1): 363-393.
[106] Brock W., Dechert W., Scheinkman J. A Test for Independence Based on Correlation Dimension[J]. Econometric Reviews, 1996, 15(3): 197-235.
[107] Grassberger P., Procaccia I. Dimensions and entropies of strange attractors form a fluctuating dynamics approach [J]. Physics, 1984, 13(1): 34-54.
[108] Scheinkman A., LeBaron B. Nonlinear dynamics and GNP data[A] In: Barnett W, Greweke J, Shell K Economic Complexity[C]. Cambridge: Cambridge University Press, 1989: 56-64.
[109] Brock W., Hsieh D., LeBaron B. Nonlinear dynamics, chaos and instability[M]. Cambridge: MIT Press, 1991: 37-48.
[110] Barnett A., Gallant R., Hinich J., Kaplan T. A single-blind controlled competitionamong tests for nonlinearity and chaos[J]. Journal of Econometrics, 1997, 82(1): 157-192.
[111] Kocenda E. Optimal range for the iid test based on integration across the correlation integral[J]. Econometric Reviews, 2005, 24(3): 265-296.
[112] Brockwell P. J., Davis R. A. Time series: theory and methods[J]. Springerverlag, 1991, 24(1): 577-591.
[113] Hurst H. E. The long-term storage capacity of reservoirs[J]. Transactions of the American Society of Civil Engineers, 1951, 116(1): 47-59.
[114] Mandelbrot B. B., Wallis J. R. Robustness of the rescaled range R/S in the measurement of non cyclic long-run statistical dependence[J]. Water Resour Res, 1969, 5(5): 967-988.
[115] Lo A.W. Long-term memory in stock market prices[J]. Econometrical, 1991, 59(5): 1279-1313.
[116] Geweke J. F., Porter-Hudak S. The estimation and application of long memory time series models[J]. Journal of Time Series Analysis, 1983, 4(1): 221-238.
[117] Robinson P. M. Gaussian semi-parametric estimation of long-range dependence [J]. The Annals of Statistics, 1995, 23(3): 1630-1661.
[118] Robinson P.M. Log period gram regression of time series with long range dependence[J]. Annals of Statistics, 1995, 23(3): 1048-1072.
[119] Sowell F. B. Maximum likelihood estimation of stationary unvaried fractionally integrated time series models[J]. Journal of Econometrics, 1992, 53(3): 165-188.
[120] Doornik J. A., Ooms M. Computational aspects of maximum likelihood estimation of autoregressive fractionally integrated moving average models[J]. Computational Statistics & Data Analysis, 2003, 42(3): 333-348.
[121] Cox D. R., Reid N. Parameter orthogonality and approximate conditional inference[J]. Journal of the Royal Statistical Society Series B, 1987, 39(1): 1-39.
[122] Granger C. W., Newbold P. Forecasting economic time series[M]. New York: Academic Press, 1986: 32-43.
[123] Teverovsky V., Taqqu M. S., Willinger W. A critical look at Lo's modified R/S statistic[J]. Journal of Statistical Planning and Inference, 1999, 80(1): 211-227.
[124] Doornik J. A. Object-Oriented Matrix Programming using Ox 5th edn[M]. London: Timberlake Consultants Press, 2006: 18-27.
[125] Davis M. Louis Bachelier’s theory of speculation[M]. Princeton: Princeton University Press, 2006: 29-36.
[126] Eckmann J. P., Oliffson Kamphorst S., Ruelle D., Scheinkman J. A. Lyapunovexponents for stock returns[A] In Anderson P W, Arrow K J, Pines D editors, The economy as an evolving complex system[C]. New York: Addison-Wesley Press, 1988: 25-37.
[127] Mizrach B. Determining delay times for phase space reconstruction with application to the FF/DM exchange rate[J]. Econ Behav Organ, 1996, 30(3): 69-81.
[128] Jiaorui Li, Wei Xu, Wenxian Xie, Zhengzheng Ren. Research on nonlinear stochastic dynamical price model[J]. Chaos, Solitons and Fractals, 2006, 42(7): 1381-1393.
[129] Orlowski A., Struzik Z.R., Syczewska E., Zaluska Kotur M.A. Fluctuation dynamics of exchange rates on Polish financial market[J]. Physica A, 2004, 344(1): 184-189.
[130] Marotto F. R. Snap-back repellers imply chaos[J]. Math Anal Appl, 1978, 63(1): 199-223.
[131] Devaney R. L. An introduction to chaotic dynamical systems[M]. Redwood City: Addison-Wesley Publishing Company, 1987: 29-40.
[132] Wolf A., Swift J., Swinney H. Determining Lyapunov exponents from a time series[J]. Physica D, 1985, 16(3): 285-317.
[133] Barana G., Tsuda L. A new method for computing Lyapunov exponents[J]. Physics Letter A, 1993, 175(6): 421-427.
[134] Rosenstein M. T., Collins J. J., De Luca C. J. A practical method for calculating largest Lyapunov exponents from small data sets[J]. Physica D, 1993, 65(1): 117-134.
[135] Packard N. H., Crutchfield J. D., Farmer J. D., Shaw R. S. Geometry from a Time Series[J]. Phys Rev Lett, 1980, 45(9): 712-716.
[136] Takens F., Mane. In dynamical systems of turbulence[A] In: Rand D A, Young L S, eds, Lecture Notes in Mathematics[C]. Berlin: Springer Press, 1986: 366-381.
[137] Fraser A. M., Swinney H. L. Independent coordinates from mutual information [J]. Phys Rev A, 1986, 33(2): 1134-1140.
[138] Tsonis A. A. Chaos: from theory to application[M]. New York: Plenum Press, 1992: 39-47.
[139] Broomhead D. S., King G. P. Extracting qualitative dynamics from experimental data[J]. Physica D, 1986, 20(2): 217-236.
[140] Kugiumtzis D. Correction of the correlation dimension for noisy time series: the role of the time window length[J]. Physica D, 1996, 95(13): 13-28.
[141] Kim H. S., Eykholt R., Salas J. D. Nonlinear Dynamics, Delay times, andEmbedding windows[J]. Physica D, 1999, 127(2): 48-60.
[142] Xie Chi, Yang Ni. Analysis and empirical research on the long-memory property of exchange rate time series[J]. Wireless Communications, Networking and Mobile Computing 2007, 2007, 8(1): 5906-5909.
[143] Galluccio S., Caldarelli G., Marsili M. Scaling in currency exchange[J]. Physica A, 1997, 245(4): 423-436.
[144]谢赤,熊一鹏.深圳股票市场多标度分形特征分析[J].财经理论与实践, 2005, 26(13): 53-55.
[145] Bacry E., Delour J., Muzy J. F. Modeling financial time series using multifractal random walks[J]. Physica A, 2001, 299(1): 84-92.
[146] Michele P., Maurizio S. Multiscaling and clustering of volatility[J]. Physica A, 1999, 269(1): 140-147.
[147] Mandelbrot B. B., Calvet L., Fisher A. Large deviation theory and the multifractal spectrum of financial prices[A] In Working Paper of Yale University[C]. Yale: Yale University, 1997: 12-22.
[148] Fisher A., Calvet L., Mandelbrot B. B. Multifractality of deutschmark/US dollar exchange rates[A] In Working Paper of Yale University[C]. Yale: Yale University, 1997: 3-12.
[149] Calvet L., Fisher A., Mandelbrot B. B. A multifractal model of asset returns[A] In Working Paper of Yale University[C]. Yale: Yale University, 1997: 5-16.
[150] Muniandy S. V., Lim S. C., Murugan R. Inhomogeneous scaling behaviors in Malaysian foreign currency exchange rates[J]. Physica A, 2001, 301(1): 407-428.
[151]韩敏.混沌时间序列预测理论与方法[M].北京:中国水利水电出版社, 2007: 57-71.
[152] Mizrach B. Multivariate nearest-neighbor forecasts of EMS Exchange Rates[J]. Journal of Applied Econometrics, 1992, 7(1): 151-163.
[153] Kenya A., Alvaro V., Elton C. Using artificial neural networks to forecast chaotic time series[J]. Physica A, 2000, 284(9): 393-404.
[154] Timo T., Dick D., Marcelo C. Linear models, smooth transition auto regressions, and neural networks for forecasting macroeconomic time series: A reexamination [J]. International Journal of Forecasting, 2005, 21(1): 755-774.
[155] Francesco L., Rosa A. S. A comparison between neural networks and chaotic models for exchange rate prediction [J]. Computational Statistics & Data Analysis, 1999, 30(1): 87-102.
[156]余乐安,汪寿阳,黎建强.外汇汇率与国际原油价格波动预测——TEI@I方法论[M].长沙:湖南大学出版社, 2006: 67-81.
[157] Diebold F. X., Mariano R. S. Comparing predictive accuracy[J]. Journal of Business and Economic Statistics, 1995, 13(3): 253-263.
[158] Henriksson R. D., Merton R. C. On market timing and investment performance: statistical procedures for evaluating forecasting skill[J]. Journal of Business, 1981, 54(4): 513-533.