内生结构突变理论与应用研究
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摘要
内生结构突变理论与应用是最近十几年时间序列分析领域的热点研究问题。在时间序列分析中,基本假定是序列的平稳性和遍历性。经典的单位根理论只考虑了序列的平稳性,当序列中不存在结构突变时可以满足遍历性假定,但是当序列中存在结构突变时就不满足遍历性假定,因此需要对经典的单位根理论进行扩展以包含序列中可能存在的结构变化。
本文从内生结构突变理论中的估计方法与理论假定开始,重点研究了一个或多个内生结构突变点的检验方法,含结构突变的趋势平稳过程的单位根检验和含结构突变的AOADF统计量中其它参数的极限分布,并结合具体的宏观经济和金融数据对如何进行内生结构突变检验和含结构突变的单位根检验进行说明。
在理论研究方面,本文的主要贡献在于：
(1)推导了在无漂移项的截距突变,有漂移项的截距突变,无漂移项的斜率突变和有漂移项的斜率突变四种情况下,含结构突变的AOADF单位根检验中第一步回归时常数项,时间趋势项和突变幅度项d1或d2的极限分布并给出了对应的t统计量的临界值表。无论在哪种数据生成过程下,参数对应的t统计量都要除以样本容量T之后才有分布。经过样本容量标准化之后：(a)在无漂移项的截距突变时,常数项对应的t统计量的临界值随着突变点位置参数向样本末端移动而不断减小,突变幅度项d1对应的t统计量的临界值在突变点处于样本中间时最大,然后随着突变点向两端移动而减小；(b)在有漂移项的截距突变时,常数项对应的t统计量的临界值随着突变点位置参数变化没有明显的特征,突变幅度项d1对应的t统计量的临界值在突变点位于[0.3,0.7]时随着突变点位置向中间移动而增大,而时间趋势项对应的t统计量的临界值在突变点位于样本中间时达到最小值,然后随着突变点向两端的移动而增大；(c)在无漂移项的斜率突变时,常数项和时间趋势项对应的t统计量的临界值具有相似的特征,随着突变点位置参数的增大而不断减小,突变幅度项d2对应的t统计量的临界值在突变点处于样本中间时最大,然后随着突变点向两端移动而减小：(d)在有漂移项的斜率突变时,常数项对应的t统计量的临界值随着突变点位置参数向样本末端的移动而减小,突变幅度项d1对应的t统计量的临界值在突变点位于样本中间时取最小值,然后随着突变点位置参数向两端移动而增加,突变幅度项d2对应的t统计量的临界值在突变点位于样本中间时取最大值,然后随着突变点位置参数向两端移动而减小,时间趋势项对应的t统计量的临界值随着突变点位置参数向样本末端移动而增加；
(2)研究了在无趋势的截距突变,有趋势的截距突变和斜率突变三种情况下,结构突变对趋势平稳过程的ADF单位根检验的影响。发现当数据生成过程是截距突变的平稳或趋势平稳过程时,自回归系数估计量α收敛于1的速度为T1/2,自回归系数估计量α的分布和,α的分布受到突变点位置参数,突变幅度和误差项的标准差的影响,随着突变幅度的增大逐渐向右移动,而误差项的标准差的增大会抵消突变幅度的影响：当数据生成过程是斜率突变的趋势平稳过程时,自回归系数估计量α收敛于1的速度为T3/2,自回归系数估计量a的分布受到突变点位置参数,突变幅度和误差项的标准差的影响,tα的分布受到突变点位置参数和突变幅度的影响；
(3)把内生结构突变点的最优检验统计量Exp-LM和含结构突变的AOADF统计量结合起来,使得在突变点位置参数未知的情况下可以先通过Exp-LM统计量确定突变是否存在,如果存在突变,可以得到突变点位置参数的一致估计和突变类型,然后根据估计得到的突变点位置参数来进行AOADF检验。
在应用研究方面,本文的贡献在于：
(1)应用参数变化的GARCH (1,1)模型分析我国的股票市场,发现在上证综指收益率序列的GARCH (1,1)模型中不存在结构突变,而在深证成指收益率序列的GARCH (1,1)模型中存在结构突变,而且是ARCH项的突变,这一突变可以解释深证收益率序列中存在的波动持续和高峰厚尾现象；
(2)应用内生结构突变检验对我国的GDP,通货膨胀率和人民币对美元的汇率序列进行了检验,发现GDP序列中存在斜率突变,其潜在数据生成过程是含斜率突变的单位根过程；通货膨胀率序列中存在截距突变,其潜在数据生成过程是含截距突变的平稳过程：人民币对美元的汇率序列中不存在结构突变,其潜在数据生成过程是单位根过程。
The study of the theory and application of endogenous structure break is a hotspot in the field of time series analysis among the last decades. In the analysis of time series, the basic assumption is the stationarity and ergodicity of the seires. The classic unit root theory only consider the stationarity of the siries, when there is no structural breaks in the series the ergodicity assumption can be satisfied, but when there is structural breaks the ergodicity assumption cann't be satisfied, so the classic unit root theory needs to be extended to cover structural breaks that might exist.
This article begins with the estimation method and assumption of the theory of endogenous structure break, focuses on the testing method of one or more structure break points, the unit root test of trend stationary process with structural break and the limiting distribution of the other parameters of the AOADF test of unit root process with structural break, combined with specific macroeconomic and financial data on how to execute the testing of endogenous structure breaks and unit root test with structural break.
In theoretical research, the main contribution of this article is as follows:
(1) Deriving the limit distribution of the intercept, the trend and the break magnitude term under four situations such as intercept break without drift, intercept with drift, trend break without drift and trend break with drift. Under any data generating process, the parameter's t statistic need to be divided by T to get limiting distribution. Scaled by the sample size:(a) Under intercept break without drift, the critical values of the t ststistic of the intercept decreases as the break position shift to the end of the sample, the critical values of the t ststistio of the break magnitude d1reaches maximum at the intermediate then decrease to the ends.(b) Under intercept break with drift, the critical values of the t ststistic of the intercept doesn't show any regular character as the break position shifts, the critical values of the t ststistic of the break magnitude d1increases as the break position lying [0.3,0.7] move to the middle of the sample, the critical values of the t ststistic of the trend reaches maximum at the intermediate then decrease to the ends.(c) Under trend break without drift, the critical values of the t ststistic of the intercept and the trend term share the similar character, which decreases as the break position shifts to its end, the critical values of the t ststistic of the break magnitude d2reaches maximum at the intermediate then decrease to the ends.(d) Under trend break with drift, the critical values of the t ststistic of the intercept decreases as the break position shift to the end; the critical values of the t ststistic of the break magnitude d1reaches minimum at the intermediate then increases to the ends; the critical values of the t ststistic of the break magnitude d2reaches maximum at the intermediate then decrease to the ends; the critical values of the t ststistic of the trend term increases as the break position shift to the ends.
(2) Study the effect of structural break on the ADF unit root test under three processes such as intercept break without trend, intercept break with trend and trend break. When the data generating process is intercept break stationary or trend stationary process, the autoregressive coefficient estimates α converges to1with the speed of f T1/2, the limit distribution of α and ta contains the position parameter of the structural break, the magnitude of the break and the standard deviation of the error term, shift to right when the magnitude increases and left when the standard deviation increases. When the data generating process is trend stationary process with trend break, α converges to1with the speed of T3/2, the limit distribution of α contains the position parameter of the structural break, the magnitude of the break and the standard deviation of the error tenn while t(?) don't contain the standard deviation of the error tenn.
(3) Combined the optimal test statistic Exp-LM of endogenous structural break with the AOADF test statistic of structural break, such that in the case of unknown break position parameter, we can make advantage of the Exp-LM statistic to determine whether there is a break point, if there does exist one, then can get the consist estimate of the break location parameter and the type of structural break. Based on the estimated break position parameter, the AOADF test can be used.
In application study, the main contribution is
(1) Investigate the stock market with the structural break GARCH (1,1) model, the result suggests that there is no structural break in the GARCH (1,1) model of Shanghai Composite Index return series, but there is an ARCH term break in the GARCH(1,1) model of the Shenzhen Component Index return series, which could explain the persistence in the volatility and the high excess kurtosis;
(2) Take advantage of the test of endogenous structural break to analysis the series of GDP, inflation and the exchange rate. It is suggested that there is a trend break in the GDP series, the potential data generating process is unit root with trend break; there is intercept breaks in the inflation series, the potential data generating process is stationary process with intercept break; there is no structural break in the exchange rate, the potential data generating process is unit root.
本文从内生结构突变理论中的估计方法与理论假定开始,重点研究了一个或多个内生结构突变点的检验方法,含结构突变的趋势平稳过程的单位根检验和含结构突变的AOADF统计量中其它参数的极限分布,并结合具体的宏观经济和金融数据对如何进行内生结构突变检验和含结构突变的单位根检验进行说明。
在理论研究方面,本文的主要贡献在于：
(1)推导了在无漂移项的截距突变,有漂移项的截距突变,无漂移项的斜率突变和有漂移项的斜率突变四种情况下,含结构突变的AOADF单位根检验中第一步回归时常数项,时间趋势项和突变幅度项d1或d2的极限分布并给出了对应的t统计量的临界值表。无论在哪种数据生成过程下,参数对应的t统计量都要除以样本容量T之后才有分布。经过样本容量标准化之后：(a)在无漂移项的截距突变时,常数项对应的t统计量的临界值随着突变点位置参数向样本末端移动而不断减小,突变幅度项d1对应的t统计量的临界值在突变点处于样本中间时最大,然后随着突变点向两端移动而减小；(b)在有漂移项的截距突变时,常数项对应的t统计量的临界值随着突变点位置参数变化没有明显的特征,突变幅度项d1对应的t统计量的临界值在突变点位于[0.3,0.7]时随着突变点位置向中间移动而增大,而时间趋势项对应的t统计量的临界值在突变点位于样本中间时达到最小值,然后随着突变点向两端的移动而增大；(c)在无漂移项的斜率突变时,常数项和时间趋势项对应的t统计量的临界值具有相似的特征,随着突变点位置参数的增大而不断减小,突变幅度项d2对应的t统计量的临界值在突变点处于样本中间时最大,然后随着突变点向两端移动而减小：(d)在有漂移项的斜率突变时,常数项对应的t统计量的临界值随着突变点位置参数向样本末端的移动而减小,突变幅度项d1对应的t统计量的临界值在突变点位于样本中间时取最小值,然后随着突变点位置参数向两端移动而增加,突变幅度项d2对应的t统计量的临界值在突变点位于样本中间时取最大值,然后随着突变点位置参数向两端移动而减小,时间趋势项对应的t统计量的临界值随着突变点位置参数向样本末端移动而增加；
(2)研究了在无趋势的截距突变,有趋势的截距突变和斜率突变三种情况下,结构突变对趋势平稳过程的ADF单位根检验的影响。发现当数据生成过程是截距突变的平稳或趋势平稳过程时,自回归系数估计量α收敛于1的速度为T1/2,自回归系数估计量α的分布和,α的分布受到突变点位置参数,突变幅度和误差项的标准差的影响,随着突变幅度的增大逐渐向右移动,而误差项的标准差的增大会抵消突变幅度的影响：当数据生成过程是斜率突变的趋势平稳过程时,自回归系数估计量α收敛于1的速度为T3/2,自回归系数估计量a的分布受到突变点位置参数,突变幅度和误差项的标准差的影响,tα的分布受到突变点位置参数和突变幅度的影响；
(3)把内生结构突变点的最优检验统计量Exp-LM和含结构突变的AOADF统计量结合起来,使得在突变点位置参数未知的情况下可以先通过Exp-LM统计量确定突变是否存在,如果存在突变,可以得到突变点位置参数的一致估计和突变类型,然后根据估计得到的突变点位置参数来进行AOADF检验。
在应用研究方面,本文的贡献在于：
(1)应用参数变化的GARCH (1,1)模型分析我国的股票市场,发现在上证综指收益率序列的GARCH (1,1)模型中不存在结构突变,而在深证成指收益率序列的GARCH (1,1)模型中存在结构突变,而且是ARCH项的突变,这一突变可以解释深证收益率序列中存在的波动持续和高峰厚尾现象；
(2)应用内生结构突变检验对我国的GDP,通货膨胀率和人民币对美元的汇率序列进行了检验,发现GDP序列中存在斜率突变,其潜在数据生成过程是含斜率突变的单位根过程；通货膨胀率序列中存在截距突变,其潜在数据生成过程是含截距突变的平稳过程：人民币对美元的汇率序列中不存在结构突变,其潜在数据生成过程是单位根过程。
The study of the theory and application of endogenous structure break is a hotspot in the field of time series analysis among the last decades. In the analysis of time series, the basic assumption is the stationarity and ergodicity of the seires. The classic unit root theory only consider the stationarity of the siries, when there is no structural breaks in the series the ergodicity assumption can be satisfied, but when there is structural breaks the ergodicity assumption cann't be satisfied, so the classic unit root theory needs to be extended to cover structural breaks that might exist.
This article begins with the estimation method and assumption of the theory of endogenous structure break, focuses on the testing method of one or more structure break points, the unit root test of trend stationary process with structural break and the limiting distribution of the other parameters of the AOADF test of unit root process with structural break, combined with specific macroeconomic and financial data on how to execute the testing of endogenous structure breaks and unit root test with structural break.
In theoretical research, the main contribution of this article is as follows:
(1) Deriving the limit distribution of the intercept, the trend and the break magnitude term under four situations such as intercept break without drift, intercept with drift, trend break without drift and trend break with drift. Under any data generating process, the parameter's t statistic need to be divided by T to get limiting distribution. Scaled by the sample size:(a) Under intercept break without drift, the critical values of the t ststistic of the intercept decreases as the break position shift to the end of the sample, the critical values of the t ststistio of the break magnitude d1reaches maximum at the intermediate then decrease to the ends.(b) Under intercept break with drift, the critical values of the t ststistic of the intercept doesn't show any regular character as the break position shifts, the critical values of the t ststistic of the break magnitude d1increases as the break position lying [0.3,0.7] move to the middle of the sample, the critical values of the t ststistic of the trend reaches maximum at the intermediate then decrease to the ends.(c) Under trend break without drift, the critical values of the t ststistic of the intercept and the trend term share the similar character, which decreases as the break position shifts to its end, the critical values of the t ststistic of the break magnitude d2reaches maximum at the intermediate then decrease to the ends.(d) Under trend break with drift, the critical values of the t ststistic of the intercept decreases as the break position shift to the end; the critical values of the t ststistic of the break magnitude d1reaches minimum at the intermediate then increases to the ends; the critical values of the t ststistic of the break magnitude d2reaches maximum at the intermediate then decrease to the ends; the critical values of the t ststistic of the trend term increases as the break position shift to the ends.
(2) Study the effect of structural break on the ADF unit root test under three processes such as intercept break without trend, intercept break with trend and trend break. When the data generating process is intercept break stationary or trend stationary process, the autoregressive coefficient estimates α converges to1with the speed of f T1/2, the limit distribution of α and ta contains the position parameter of the structural break, the magnitude of the break and the standard deviation of the error term, shift to right when the magnitude increases and left when the standard deviation increases. When the data generating process is trend stationary process with trend break, α converges to1with the speed of T3/2, the limit distribution of α contains the position parameter of the structural break, the magnitude of the break and the standard deviation of the error tenn while t(?) don't contain the standard deviation of the error tenn.
(3) Combined the optimal test statistic Exp-LM of endogenous structural break with the AOADF test statistic of structural break, such that in the case of unknown break position parameter, we can make advantage of the Exp-LM statistic to determine whether there is a break point, if there does exist one, then can get the consist estimate of the break location parameter and the type of structural break. Based on the estimated break position parameter, the AOADF test can be used.
In application study, the main contribution is
(1) Investigate the stock market with the structural break GARCH (1,1) model, the result suggests that there is no structural break in the GARCH (1,1) model of Shanghai Composite Index return series, but there is an ARCH term break in the GARCH(1,1) model of the Shenzhen Component Index return series, which could explain the persistence in the volatility and the high excess kurtosis;
(2) Take advantage of the test of endogenous structural break to analysis the series of GDP, inflation and the exchange rate. It is suggested that there is a trend break in the GDP series, the potential data generating process is unit root with trend break; there is intercept breaks in the inflation series, the potential data generating process is stationary process with intercept break; there is no structural break in the exchange rate, the potential data generating process is unit root.
引文
1 参见Perron (1988) "Trends and Random Walks in Macroeconomic Time Series:Further Evidence from a New Approach," Journal of Economic Dynamics and Control,12,297-332.
1 参见Perron (1989) 1390页(A.1)。
2 Perron (1989)在证明定理一时选择了最一般化的形式,即本文第二部分中给出的模型三：y1=μ1+β1t+(μ2-μ1)DUt+(β2-β1)DTt+et(其中当TB 1 因为一般平稳过程的自回归系数收敛速度为T12,相比之下,单位根过程自回归系数收敛级数更高,故也称ADF回归式中的T(α-1)为“超一致收敛”。
2 限于篇幅,在此没有给出修正结果,详情参见文献Montanes和Reyes (1998)中357-359页。
3 参见Perron (1989) 1371页。
1 见James D. Hamilton, Time Series Analysis, Princeton University Press,1994,486页。
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1 参见Perron (1989) 1390页(A.1)。
2 Perron (1989)在证明定理一时选择了最一般化的形式,即本文第二部分中给出的模型三：y1=μ1+β1t+(μ2-μ1)DUt+(β2-β1)DTt+et(其中当TB
2 限于篇幅,在此没有给出修正结果,详情参见文献Montanes和Reyes (1998)中357-359页。
3 参见Perron (1989) 1371页。
1 见James D. Hamilton, Time Series Analysis, Princeton University Press,1994,486页。
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