Comparing the performances of GARCH models in capturing the stock market volatility in Malaysia

P rocedia Economics and Finance 5 ( 2013 ) 478 – 487

2212-5671 ? 2013 The Authors. Published by Elsevier B.V.Selection and/or peer-review under responsibility of the Organising Committee of ICOAE 2013d oi: 1

0.1016/S2212-5671(13)00056-7

Available online at https://www.360docs.net/doc/aa15963483.html,

ScienceDirect

Open access under CC BY -NC-ND license.

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C hing Mun Lim and Siok Kun Sek / P rocedia Economics and Finance 5( 2013 )478 – 487

stock market provide useful information in measuring risk, many models/ theory are applied in forecasting stock market movement and evaluating the performance of the stock market. Many studies show that random walk model is superior in explaining the stock market movement. However, more recent studies reveal results that stock market deviates from the random walk behaviour. We further the investigation on stock market movement by looking at different aspects (i.e. different models and different time frames).

Among the models that have been applied to capture the stock market volatility include ARCH which was proposed by Engle (1982) and generalized ARCH proposed by Bollerslev (1986) and Taylor (1986). After the introduction of ARCH and GARCH models, many researchers have proposed the extensions and alternative specifications on the models such as GARCH-M, IGARCH, EGARCH (Nelson (1991)), Threshold GARCH (Glosten et al. (1993)), Asymmetric GARCH model AGARCH (Engle (1990)) and Fractionally Integrated FIGARCH (Baillie et al. (1996)). These alternative models seek to improve the GARCH model in capturing the characteristics of return series. However, previous studies show no consensus on the best model in capturing volatility. Some studies show preferable results using simple GARCH (p,q) models but some show extensions of GARCH models perform better. The performance of these models varies across markets and time period. The performance of these forecast models is affected by error measures.

In this study, we seek to identify the superior model in capturing the characteristics of Malaysia’s stock market. The models to be compared are symmetric GARCH and asymmetric GARCH (EGARCH and TGARCH). In particular, we evaluate the performance of these models using the error measurement approaches such as MSE, RMSE and MAPE. Apart from this, we also seek to investigate if exchange rate can influence the stock market movement in Malaysia. The results are compared for three different time frames

i.e. pre-crisis, crisis period and post-crisis periods. Our results show that the performance of GARCH-type models is dependent on the time/ periods, i.e. during pre-crisis, crisis and post-crisis and also the error measures. In general, the total rank show that GARCH/ TGARCH model perform the best in the pre-crisis period while GARCH model works well during the crisis and TGARCH model work well in the post-crisis period in capturing the stock market volatility in Malaysia.

The remaining paper is organized as follows: section two provides some concepts and literature reviews. Section three is about the data and methodology. Section four is about model evaluations while section five summarizes the findings. Section six concludes.

2.Literature review

The study on stock market volatility is broad. Empirical studies apply GARCH and ARCH models in capturing the stock market volatility. These studies cover different regions/ countries. The studies that focus

on stock market in developed countries (United States, United Kingdom, Germany, and Japan) include Claessen & Mittnik (2008), Ou & Wang (2011), Choo & Lee (2011), and Mootamri (2011). These studies applied different frequency data. Among the studies that focused on the stock market in developing countries (Malaysia, Singapore, India, Saudi, China, Egypt and Vietnam) are Mishra (2010), Alshogeathri (2011), Abdalla (2012), Wong & Kok (2005) and Liu et al. (2009), Hien (2008).

In evaluating the performance of models, previous studies apply different evaluation measures. The most widely applied measures include Mean Square Error (MSE), Root Mean Square Error (RMSE), and Mean Absolute Percent Error (MAPE). In practice, when comparing the different models, it is rarely the case that one model dominate the other with respect to all evaluation measures. The common way to solve the problem

is to carry out the average figures of some statistical measures and then compare the forecast models based on the parameter obtained.

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shocks) and GARCH term 1E (long-run persistency of shocks) for GARCH (1, 1) are highly significant in all

periods, indicating the impacts of shocks on stock market volatility (the results is not provide here due to space constraint). Comparing the results of asymmetric and asymmetric models, we observe that the coefficient of J for TGARCH and EGARCH are insignificant at pre-crisis period. The result implies no significant asymmetric effect of shocks on stock market volatility. Therefore, linear model is sufficient to model the volatility of stock market in the pre-crisis period. On the other hand, there are significant asymmetric effects of shocks during the crisis and post-crisis periods in Malaysia. Therefore, asymmetric models are the better models used to represent the stock market volatility during the economic fluctuation periods (crisis period) and economic recovery period (post crisis). The effect of asymmetric shocks is larger during the crisis period, indicating larger leverage effect during crisis period relative to post-crisis period. Next, comparing the effects of short run and long run persistency of shocks (1D and 1E ), our results reveal that long-run shocks are more persistent compared to short-run shocks, indicating larger impacts of long-run shocks on stock market volatility. The result holds for all three periods and three models.

Investigating the impacts of exchange rate and crude oil price on stock market volatility, we observe that both variables only have small impacts or non-significance impacts on stock market volatility in Malaysia. The only exception is the result reported by EGARCH for the pre and post-crisis periods. Our results imply that both variables fail to explain the stock market volatility during the crisis period when the stock index is highly fluctuated. During the stable periods (pre and post-crisis), exchange rate and crude oil price have some powers on predicting stock market volatility.

The three GARCH-type models are evaluated using three measurements, i.e. mean square error (MSE), root mean squared error (RMSE) and mean absolute percent error (MAPE). 5.1. In sample analysis

For in sample analysis, the models are estimated using the full sample with one month shorter, i.e. up to end of November 2010. The performance of the models is ranked from the lowest error. The total rank is the summation values of rank from the three measurements.

Table 1. Overall Model Evaluation (In Sample) Model

Errors Measures MSE

Ranking RMSE Ranking MAPE(%) Ranking Total Rank Pre Crisis Period GARCH 0.000157775 1 0.012560851 1 109.4740 2 4 TGARCH 0.000157776 2 0.012560891 3 107.3696 1 6 EGARCH 0.000157778

0.012560871

106.5497

Crisis Period GARCH 0.001184216 1 0.034412439 1 147.6658 1 3 TGARCH 0.001208099 3 0.034757718 3 222.9189 3 9 EGARCH 0.001189480 2 0.034488838 2 178.0566 2 6 Post Crisis Period

GARCH 0.000172664 2 0.013140167 2 107.7394 2 6 TGARCH 0.000172662 1 0.013140091 1 107.8859 3 5 EGARCH

0.000172887

0.013148650

101.2583

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C hing Mun Lim and Siok Kun Sek / P rocedia Economics and Finance 5( 2013 )478 – 487

Comparing the performance of symmetric and asymmetric GARCH models, our results of total ranks show that symmetric GARCH model is the best model to capture the stock market volatility in Malaysia during the pre- and crisis periods. On the other hand, TGARCH model is superior in the post-crisis period. However, one has to be caution because the ranking for these three measurements are different under three time frames. The values of errors do not vary much for MSE and RMSE but when the error is measured in percentage, the values are much more different.

5.2.Out of sample analysis

In this part, the out-of-sample forecast is performed for one month ahead from the full sample generating 20 observations.

Table 2 describes the overall evaluation for three models by using three error measures. The MSE statistic indicates that the TGARCH model provides the most accurate forecast for the pre-crisis period. The next ranked models are EGARCH and GARCH in that order. Besides, the RMSE implies that the TGARCH model provides the most accurate forecast for the pre-crisis period. The next ranked models are EGARCH and TGARCH in that order. The MAPE statistic indicates that EGARCH provides the most accurate forecast for the pre-crisis period. The next ranked models are TGARCH and the worst performing model is GARCH. In order to choose the best performing models, we used the total rank which generated from the ranking of three statistics. From the ranking of the three statistics, the total rank of TGARCH is the lowest in the pre- and post-crisis periods. Therefore, we indicate that TGARCH model outperform the other models in the pre- and post-crisis periods. On the other hand, GARCH model performs the best in the crisis-period.

Table 2. Overall Model Evaluation (Out of Sample)

Model Errors Measures

MSE Ranking RMSE Ranking MAPE(%) Ranking Total Rank

Pre Crisis Period

GARCH 0.000157564 3 0.012552449 3 109.4903 3 9

TGARCH 0.000157560 1 0.012552303 1 107.3061 2 4

EGARCH 0.000157561 2 0.012552322 2 106.4940 1 5

Crisis Period

GARCH 0.0011400015 1 0.033763909 1 144.0898 1 3

TGARCH 0.0011681337 3 .0341779715 3 217.4841 3 9

EGARCH 0.0011470467 2 0.033868078 2 173.6390 2 6

Post Crisis Period

GARCH 0.0001716995 2 0.013103418 2 107.8626 2 6

TGARCH 0.0001716994 1 0.013103413 1 107.8752 3 5

EGARCH 0.0001719257 3 0.013112042 3 101.2477 1 7

From the result of both in sample forecast and out of sample forecast, we found that the performance of GARCH-type models is affected by the period of time, i.e. pre-crisis, crisis and post-crisis periods. Using

486C hing Mun Lim and Siok Kun Sek / P rocedia Economics and Finance 5( 2013 )478 – 487 different error measures also give different evaluations on the ranking of performance. MSE and RMSE show very small differences in the performance of these three models. However, when the measurement is made in percentage using MAPE, the differences on the performance of these models are larger and more apparent. In general, the total rank show that GARCH/ TGARCH model perform the best in the pre-crisis period while GARCH model works well during the crisis and TGARCH model work well in the post-crisis period in capturing the stock market volatility in Malaysia. The evaluation results are consistent to the estimation results that show no significant asymmetric effect during the pre-crisis period, i.e. asymmetric GARCH model is sufficient to capture volatility in the pre-crisis period; there are significant asymmetric effect in the crisis and post-crisis period, implying possibility that asymmetric GARCH models work better than symmetric GARCH model.

6.Conclusion

We conduct comparative analyses in evaluating the forecasting performance of symmetric and asymmetric GARCH-type models in capturing stock market volatility in Malaysia. These models include GARCH, TGARCH and EGARCH models. These models are evaluated using three error measures which are MSE, RMSE and MAPE. We divide the data into three sub-periods, i.e. pre-crisis, crisis and post-crisis of 1Asia financial periods. Our results show that the performances of these models vary across periods and error measurement methods. . In general, the total rank show that GARCH/ TGARCH model perform the best in the pre-crisis period while GARCH model works well during the crisis and TGARCH model work well in the post-crisis period in capturing the stock market volatility in Malaysia. The evaluation results are coincide with the results of estimation which imply symmetric GARCH works well in the pre-crisis period and asymmetric GARCH model(s) can be the better model in capturing volatility of stock market Malaysia in the crisis and post-crisis periods.

References

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Alberg, D., Shalit, H., Yosef, R. 2008. Estimating Stock Market Volatility Using Asymmetric GARCH Models. Applied Financial Economics. 1 (18), p.1201.

Alshogeathri, M.A.M. 2011. Macroeconomic Determinants of the Stock Market Movements: Empirical Evidence from the Saudi Stock Market. Ph.D thesis, Kansas State University.

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Journal of Econometrics, 74, p. 3.

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用GARCH模型预测股票指数波动率

用GARCI模型预测股票指数波动率目录 Abstract ......................................................................... 1.引言........................................................................... 2.数据........................................................................... 3.方法........................................................................... 3.1.模型的条件平均............................................................ 32模型的条件方差............................................................... 3.3预测方法.................................................................... 3.4业绩预测评价............................................................... 4.实证结果和讨论................................................................. 5.结论........................................................................... References ....................................................................... Abstract This paper is designed to makea comparison between the daily conditional varianee through seven GRAChhodels. Through this comparison, to test whether advaneed GARCH models are outperform ing the sta ndard GARCH models in predict ing the varia nee of stock in dex. The database of this paper is the statistics of 21 stock in dices around the world from 1 January to 30 November 2013. By forecast ing one —step-ahead con diti onal varia nee within differe nt models, the n compare the results within multiple statistical tests. Throughout the tests, it is found that the sta ndard GARCH model outperforms the more adva need GARCH models, and recomme nds the best

用GARCH模型预测股票指数波动率

用GARCH模型预测股票指数波动率目录Abstract (2) 1.引言 (3) 2.数据 (6) 3.方法 (7) 3.1．模型的条件平均 (7) 3.2.模型的条件方差 (8) 3.3预测方法 (9) 3.4业绩预测评价 (9) 4.实证结果和讨论 (12) 5.结论 (16) References (18)

Abstract This paper is designed to make a comparison between the daily conditional variance through seven GRACH models.Through this comparison,to test whether advanced GARCH models are outperforming the standard GARCH models in predicting the variance of stock index.The database of this paper is the statistics of21stock indices around the world from1January to30 November2013.By forecasting one–step-ahead conditional variance within different models, then compare the results within multiple statistical tests.Throughout the tests,it is found that the standard GARCH model outperforms the more advanced GARCH models,and recommends the best one-step-ahead method to forecast of the daily conditional variance.The results are to strengthen the performance evaluation criteria choices;differentiate the market condition and the data-snooping bias. This study impact the data-snooping problem by using an extensive cross-sectional data establish and the advanced predictive ability test.Furthermore,it includes a13years’period sample set, which is relatively long for the unpredictability forecasting studies.It is part of the earliest attempts to inspect the impact of the market condition on the forecasting performance of GARCH models.This study allows for a great choice of parameterization in the GARCH models,and it uses a broad range of performance evaluation criteria,including statistical loss function and the Mince-Zarnowitz regressions.Thus,the results are more robust and diffusely applicable as compared to the earliest studies. KEY WORDS:GARCH models;volatility,conditional variance,forecast,stock indices.

基于GARCH模型的沪深300指数收益率波动性分析

摘要股票市场的价格波动研究，不仅具有重要的学术意义，而且具有重要的实际意义。股价的波动给投资者带来了获利的机会。因此，金融市场的波动性一直以来都是投资者和经济研究人员关注的焦点。本文以对沪深300指数2005年1月4日到2014年6月11日每个交易日收盘价为原始数据，对其收益率进行了研究分析。研究结果表明：日收益率序列的波动表现出时变性、突发性和集簇性等特征。序列分布呈现出尖峰厚尾的特点，并且存在明显的GARCH效应，表明过去的波动对于未来的影响是持久的，同时也是逐渐衰减的。而且，沪深300指数波动幅度大。沪深300指数的频繁交易使得股指期货市场具有高流动性，这种高流动性也是造成指数波动的一大成因。关键字：收益率；ARCH模型；GARCH模型

一、前言 1.1研究意义股票市场的价格波动研究，不仅具有重要的学术意义，而且具有重要的实际意义。股价的波动给投资者带来了获利的机会。投资者可以通过对度量波动率来猜测股市的风险有多大，同时，了解波动性有助于投资者更好的理解和把我股票市场的运行规律，将股价界定在一个可能的范围内，当投资者认识到股价波动的规律，就可以帮助其做出明智的投资，以获取更多的利益。因此，金融市场的波动性一直以来都是投资者和经济研究人员关注的焦点。 1.2 研究对象和方法沪深300指数是由上海和深圳证券市场中选取300只A股作为样本编制而成的成份股指数。沪深300指数选取规模大、流动性好的股票作为样本, 覆盖了沪深市场六成左右的市值,具有良好的市场代表性，所以有必要对其进行深入分析。本文以中国金融期货交易所的沪深300指数2005年1月4日到2014年6月11日每个交易日收盘价为原始数据，共2288个数据样本。就沪深300指数收益率的波动性研究方法而言，国内外的研究结果表明，许多金融时间序列都将GARCH模型作为解释金融数据的经验方法。因此，本文采用GARCH模型检验沪深300指数日收益率的波动性变化，希望可以发现沪深300指数的波动性特征。二、GARCH模型介绍 2.1 ARCH模型 ARCH模型由Engle（1982）提出，并由Bollerslev（1986）发展成为GARCH-广义自回归条件异方差。这些模型广泛的应用与经济性的各个领域，尤其是金融时间序列中。 ARCH的核心是(1)式中t时刻的随机误差项ε的方差（σ2）依赖于t-1时刻的平方误差的大小，即依赖于ε2t?1。 Y t=β0+β1X1t+?+βk X kt+εt (1)

波动率于garch模型剖析

1.1.波动率波动率是用来描述证券价格、市场指数、利率等在它们均值附近上下波动幅度的术语，是标的资产投资回报率的变化程度的度量。股票的波动率σ是用于度量股票所提供收益的不确定性。股票通常具有15%-50%之间的波动率。股票价格的波动率可以被定义为按连续复利时股票在1年内所提供收益率的标准差。当?t 很小时，2t σ?近似的等于在?t 时间内股票价格变化百分比的方差。这说明σ√?t 近似的等于在?t 时间内股票价格变化百分比的标准差。由标准差来表述股票价格变化不定性的增长速度大约为时间展望期长度的平方根（至少在近似意义下）。 1.2.由历史数据来估计波动率为了以实证的方式估计价格的波动率，对股票价格的观察通常是在固定的时间区间内（如每天、每星期或每个月）。定义 n+1——观测次数； S i ——第i 个时间区间结束时变量的价格，i =0，1，…n ； τ——时间区间的长度，以年为单位。令 1ln ,0,1, ,;i i i S u i n S -?? == ??? 1.2.1 u i 的标准差s 通常估计为 s = 1.2.2 或 s = 1.2.3 其中u ?为i u 的均值。由于i u 的标准差为。因此，变量s 是的估计值。所以σ本身可以被估计σ∧ ，其中 σ∧ = 可以证明以上估计式的标准差大约为σ∧ 。在计算中选择一个合适的n 值并不很容易。一般来讲，数据越多，估计的精确度也会越高，但σ确实随时间变化，因此过老的历史数据对于预测将来波动率可能不太相干。一个折中的方法是采用最近90～180天内每天的收盘价数据。另外一种约定俗成成俗的方法是将n 设定为波动率所用于的天数。因此，如果波动率是用于计算量年期的期权，在计算中我们可以采用最近两年的日收益数据。关于估计波动率表较复杂的方法涉及GARCH 模型与EWMA 模型，在下文中将进行详细介绍。 1.3.隐含波动率首先对于一个无股息股票上看涨期权与看跌期权，它们在时间0时价格的布莱克-斯科尔斯公式为 012()()rT c S N d Ke N d -=- 1.3.1 201()()rT p Ke N d S N d -=--- 1.3.2 式中 21d =

基于GARCH模型族的中国股市波动性预测

基于GARCH 模型族的中国股市波动性预测 2005级数量经济学专业倪小平摘要：本文采用上证综合指数和深证成份指数2000年1月4日—2006年12月27日的每日收盘价对数百分收益率为样本采用GARCH 模型对我国股市波动性进行实证分析。关键词：GARCH 模型波动性预测一、引言波动性是金融市场最为重要特性之一。金融市场在一些时间段内显得非常平静，而在另外一些时间段内剧烈波动。描述波动性的时变特性是非常重要，因为第一，资产风险是资产价格的重要决定因素，投资者要求更高的预期收益作为持有更高风险资产的补偿，因此回报方差的变化对于理解金融市场是非常重要的，事实上，波动性是证券组合理论、资本资产定价模型(CAPM)、套利定价模型(APT)及期权定价公式的核心变量。第二，它与市场的不确定性和风险直接相关，是体现金融市场质量和效率的最简洁和最有效的指标之一。另一方面波动性对企业的投资与财务杠杆决策、消费者的消费行为和模式、经济周期及相关宏观经济变量等都具有重要影响。因此，波动性的估计、预测和影响因素分析一直是金融经济学研究的持续热点。 Engle 于1982年开创性的提出ARCH 模型，Bollerslev 于1986年对其进行扩展，给出了GARCH 模型。如今GARCH 模型族已经成为度量金融市场波动性的强有力工具。本文的结构如下：首先对所选用的四种GARCH 模型给予了简单的描述；第二部分实证分析，包括：数据的选取与基本统计分析、模型参数的估计以及对波动性的预测和模型的比较；最后是本文的总结。二、模型概述 1、一般GARCH 模型 ARCH 模型的主要贡献在于发现了经济时间序列中比较明显的变化是可以预测的,并且说明了这种变化是来自某一特定类型的非线性依赖性,而不是方差的外生结构变化。GARCH 模型是ARCH 模型族中的一种带异方差的时间序列建模的方法。一般的GARCH 模型可以表示为 : 2011',t t t t t q p t i t i j t j i j y x v h h βεεααε θ--===+==++∑∑ 其中1var(|)t t t h ε?-=，1t ?-是时刻t-1及t-1之前的全部信息,其中, t v 独立同分布,且参数满足条件：这里t h 可以理解为过去所有残差的正加权平均,这与波动率的聚集效应相符合,即:大的变化后倾向于有更大的变化,小的变化后倾向于有小的变化。由于GARCH (p,q)模型是ARCH 模型的扩展,因此GARCH(p,q)同样具有ARCH(q)模型的特点。但GARCH 模型的条件方差不仅是滞后残差平方的线性函数,而且是滞后条件方差的线性函数。

用GARCH模型预测股票指数波动率

用GARCH模型预测股票指数波动率目录 Abstract (2) 1.引言 (3) 2.数据 (6) 3.方法 (7) 3.1．模型的条件平均 (7) 3.2. 模型的条件方差 (8) 3.3 预测方法 (9) 3.4 业绩预测评价 (9) 4.实证结果和讨论 (12) 5.结论 (16) References (18)

Abstract This paper is designed to make a comparison between the daily conditional variance through seven GRACH models. Through this comparison, to test whether advanced GARCH models are outperforming the standard GARCH models in predicting the variance of stock index. The database of this paper is the statistics of 21 stock indices around the world from 1 January to 30 November 2013. By forecasting one –step-ahead conditional variance within different models, then compare the results within multiple statistical tests. Throughout the tests, it is found that the standard GARCH model outperforms the more advanced GARCH models, and recommends the best one-step-ahead method to forecast of the daily conditional variance. The results are to strengthen the performance evaluation criteria choices; differentiate the market condition and the data-snooping bias. This study impact the data-snooping problem by using an extensive cross-sectional data establish and the advanced predictive ability test. Furthermore, it includes a 13 years’ period sample set, which is relatively long for the unpredictability forecasting studies. It is part of the earliest attempts to inspect the impact of the market condition on the forecasting performance of GARCH models. This study allows for a great choice of parameterization in the GARCH models, and it uses a broad range of performance evaluation criteria, including statistical loss function and the Mince-Zarnowitz regressions. Thus, the results are more robust and diffusely applicable as compared to the earliest studies. KEY WORDS: GARCH models; volatility, conditional variance, forecast, stock indices.

基于非参数GARCH模型的一种波动率估计方法

案例13 基于非参数GARCH 模型的一种波动率估计方法一、文献及研究综述波动率（volatility ）是资产收益不确定性的衡量，它经常用来衡量资产的风险。一般来说，波动率越大，意味着风险越高。由于波动率在投资分析，期权定价等方面的重要性，近20年来一直是金融领域的一个研究热点，出现许多描述金融市场波动率的模型，最为典型的是Bollerslev （1986）提出的广义自回归条件异方差模型（GARCH 模型），而在实证中得到广泛应用的是其中的GARCH(1,1)模型，即条件方差不但依赖与滞后一期的扰动项的平方，而且也依赖于自身的滞后一期值，三者之间存在一种线形关系。针对三者之间的线形关系是否合适即能否用一种更有效的函数关系来描述的问题，人们进行了一些有意义的探索。Engel 和Gonzalez-Rivera(1991)采用半参数方法对条件方差进行建模，对扰动项的滞后值采取非参数形式，对条件方差自身的滞后值采用线形形式，两位的研究思路为人们以后的研究工作拓宽了思路。Peter Buhlmann 和Alexander J.MeNeil （2002）对三者之间的函数关系用一种非参数形式来描述，给出了一种全新的估计波动率的循环算法，并对这一全新的算法的可行性和有效性给出了证明，得出非参数形式的GARCH(1,1)对波动率的估计效果要强与参数形式的GARCH(1,1)。Antonio Cosma 和Fausto Galli （2005）利用Peter Buhlmann 和Alexander J.MeNeil 所提出的估计波动率的算法，对非参数形式的ACD 模型（Autoregressive Conditional Duration Model ）的久期(duration)进行估计，也得出用该估计算法的非参数形式比参数形式的ACD 模型的估计效果优越。本文采用非参数方法中的非参数可加模型，对条件方差采用非参数可加模型GARCH(1,1)形式进行建模，即对条件方差的滞后值和扰动项的滞后值分别采用不同的函数形式进行建模。估计方法是基于Peter Buhlmann 和Alexander J.MeNeil(2002)对非参数GARCH 估计时的算法思想，采取模拟数据和真实收益率数据分别同参数形式的GARCH(1,1)采用极大似然估计结果进行比较。文章下面的结构是：第二部分是有关方法的描述。第三部分是模拟实验。第四部分是实证部分。第五部分是本文结束语。二、方法描述㈠ Bollerslev （1986）提出的标准的GARCH(1,1)形式： t t z ε=