Stock market Volatility Persistence Performance of 2008 Crash: Evidence from the BRIC Markets

Transcription

1 Stock market Volatility Persistence Performance of 2008 Crash: Evidence from the BRIC Markets Master Thesis Finance School of Economics and Management Han Cong ANR: August 2017 Supervisor: B.J.M. Werker 1

2 Abstract The stock market volatility is believed to be the barometer of the financial market. The major purpose of this paper is to analyze the different volatility persistence performance of stock returns around the 2008 market crisis which is proved to be a terrible financial disaster. In this paper, I compare the different persistence behaviors of China, Brazil, Russia and India s (i.e. the BRIC countries) stock markets which are dynamic developing economies in the current world s financial market. The daily data ranges from Dec.30 th 1994 to June 13 th 2017 and the whole crisis period is divided into three subperiods: pre-crisis, during-crisis and post-crisis. The research is processed mainly by GARCH (Generalized Autoregressive Conditional Heteroskedasticity) and EGARCH (Exponential Generalized Autoregressive Conditional Heteroskedasticity) models. Based on the empirical results, both GARCH and EGARCH model demonstrate a universal trend that the volatility persistence becomes stronger when the 2008 crisis occurs, and turns weaker when the crisis begins to recover. Also, the long-term unconditional volatility is higher during the crisis period. The EGARCH model fits the sample better than GARCH model, which conveys the information that the negative shocks have stronger effect on the conditional volatility than positive shocks in the 2008 crisis. The empirical results are robust. Keywords: volatility persistence, GARCH model, EGARCH model, asymmetric response, conditional volatility, long term unconditional volatility, BRIC countries. 2

3 Contents 1.Introduction Literature Overview BRIC countries stock markets overview Stock returns, volatility and the crash relationship Volatility modeling Data sources and descriptive statistics Return definition BRIC Price index Crisis Period division, the crash and crisis definitions High and low volatility separation Methodology and empirical results Test for stationary, normality and ARCH effect of data Stationary test Normality Test Autocorrelation Test LM test for ARCH effect and The Serial Correlation The ARMA model for autocorrelation of data The GARCH model for volatility persistence The EGARCH model adjusting for asymmetric response Long term volatility calculations and comparison Summary and Conclusion References

4 1.Introduction The 2008 market crash has proved to be most horrible disaster derived from the subprime mortgage bubbles in the United States. This was started academically started with the remarkable bankruptcy of the Lehman Brothers on September 15 th 2008, which swept throughout the whole world in the next five years not only in the real estate field like declining housing price, but also the housing related securities and broad investment assets. There have been a number of academic literatures studying the crisis, not only focusing on the relationship between stock returns and volatility, but also studying the fluctuation of stock market before, during, and after the market crisis. Qi Li, Jian Yang, Cheng Hsiao, Yung-Jae Chang (2012) studied the relationship between the stock returns and volatility and found that stock market returns are negatively correlated with volatility. After the ARCH model and extension versions of autoregressive conditional heteroskedastic models are introduced by Engle (1982), Tim BOLLERSLEV (1986) and Nelson (1991), the studies on the conditional and unconditional market volatilities boomed. Taufiq Choudhry (1996) investigates the 1987 crash and the market volatility performance of six emerging markets with a conclusion that ARCH parameters change and even disappear after the 1987 crash, but the results differ in six countries and there are no regular laws to capture the performance of volatility persistence. Jon Danielson, Marcela Valenzuela and Ilknur Zer (2016) studies the financial crisis and find that the low volatility incurs risk-taking actions by agencies which can result in an increasing likelihood of crisis especially in less regulated markets. However, the conclusions by now are far from satisfying to prove the great destruction effect of the crisis as a result of limitation of data after the crisis occurs. The comparison between several similar developed or developing countries to find the laws of stock volatility fluctuation movements is not enough neither. 4

5 This thesis examines the persistence performance of stock returns volatility for four BRIC countries: China, Brazil, Russia, India. The 2008 crisis period is divided into three subperiods based on a certain criterion derived from Sandeep Patel J. P. Morgan and Asani Sarkar (1998). It is aimed at proving the assumption that the volatility persistence performance would increase after the crisis occurs and drop back to a normal level after the crisis begins to recover. The GARCH model adopted firstly by Tim BOLLERSLEV (1986) and the EGARCH (1,1) model is applied to capture the volatility persistence performance and other characteristics around the 2008 market crisis. The major questions to be answered in the paper are as follows: (1) Is there a significant ARCH effect as well as GARCH effect in the three subperiods of 2008 market crisis in the BRIC countries? Is there significant evidence that the stock market volatility present different persistence pre, during and post the 2008 market crisis? The significant ARCH effect has been proved in some of the countries samples in previous study but not the sample of BRIC emerging market. Does the effect hold in all of the BRIC countries stock markets? Does the volatility persistence vary significantly before, during and after the 2008 crisis in all the four countries? Is there an obvious regular trend of the movement of volatility persistence performance based on the four countries data across the crisis? These will be discussed with vertical and horizontal comparisons in section 4. (2) Is there an asymmetric response of the conditional volatility to the impact of unexpected excess return? (i.e. the market shocks across the 2008 market crisis) Do positive and negative market shocks have different impacts on the stock market volatility before and after the market crisis? Are the impacts robust and consistent? The literatures state that the GARCH model cannot capture the conditional volatility response that is not asymmetric and the signs of market 5

6 shocks would affect the estimation of conditional volatility. These questions will be answered in Section 4. (3) What is the difference for long-term unconditional volatility in the BRIC countries before and after the 2008 crisis? Does the occurrence of the market crisis affect the long-term average volatility, and how? Methodology papers introduce the calculations function for the long-term average volatility which is not affected by the time, t. The parameters from the GARCH and EGARCH can be used to generate the value. But would the long-term unconditional volatility change before and after the market crisis? Is there a law for the movements of long-term volatilities to follow? These questions can be answered in Section 4. The thesis is organized as follows. Section 2 overviews previous important literatures relative to the topic of the thesis, for example the BRIC stock markets overview, the relationship between stock returns volatility and the market crash as well as the volatility modeling. Section 3 introduces the descriptive statistical results and primary analysis based on the daily stock returns data of four countries from December 12 th 1994 to June 12 th, Section 4 describes the methodology frameworks which will be used in this thesis and records the empirical results with deeper econometric analysis. Section 5 summarizes and makes conclusions of the opinions of the thesis and point out the limitations of the study which could be improved and perfected by further research. 6

7 2. Literature Overview 2.1 BRIC countries stock markets overview The emerging economies of China, Brazil, Russia and India are defined as the BRIC countries, which are characterized as low-income and rapidgrowth (O Neill. J 2001). In the year 2003, the research report of Goldman Sachs used the acronym BRICs. Leslie Elliott Armijo (2007) examined the BRIC concept and think that there are dissimilarities in the four countries politics and economic environment. They can only be considered as an analytical set if they offer usual opportunity for foreign direct investment. The four large emerging countries are divided into two group according to the demographic statistics. Brazil and Russia are countries with large land and small population while China and India are countries with large population and relative small land per capita (David Broker 2012). There is a dynamic linkage between the stock market returns and some other economic measurement values for the BRICs countries. Walid Chkili and Duc Khuong Nguyen (2014) investigated the correlation between the four countries and found that the interactions change from the high volatility regime to low volatility regime. 2.2 Stock returns, volatility and the crash relationship William Schwert (1989) studied on the stock volatility performance during 87 crash and found out several important relationships between return and volatility. He concluded that the stock volatility jumped during and after the crash, which confirm the significant impact that the crash has on the market volatility. Engle (1987) investigated the time-varying volatility with ARCH model. Indika Karunanayake (2009) express the positive relationship between stock market volatility and returns using the MGARCH model. Hyman P. Minsky (1992) expressed his theoretical conclusion that a crisis is more likely to happen after the market experienced a period of low volatility, Danielson, Jon, Marcela Valenzuela, and Ilknur Zer (2016) formulate that only the extended low volatility eras (from 5 to 10 years) can predict crisis. Sandeep A. Patel and Asani Sarkar (1999) expressed emerging markets are more volatile 7

8 and there is an increase of price level ahead of a crash and a decrease behind that. In Ho Lee (1997) is objected to the main research stream based on history price study to explain the price volatility. He focused on the accumulated hidden information and concluded it is the information cascade and avalanche that generate the crisis stage. Joseph Zeira (1998) used the informational dynamics theory to explain the market boom and crash. TAUFIQ CHOUDHRY (1996) found that the empirical results varied among countries and proved the relationship between volatility and returns is a negative one. Laura L. Veldkamp (2005) found that insufficient endogenous information and misty expectation restrict the financial institutes market conduction and result in the gradual boom and sudden crash, due to unconditional asymmetry. Jozef Barunik and Jiri Kukacka (2014) adopt the catastrophe theory and find that the endogenous factors result in the fluctuation in the first half period, and the exogenous factors for the last half period. 2.3 Volatility modeling As S.T. Rachev, Y.S. Kim, M.L. Bianchi, S. Stoyanov, S. Altobelli, and F.J. Fabozzi expressed in their thesis, there are three traditional statistical ways to measure the volatile of the market, namely the Value at risk, the Markowitz mean-variance framework and the Black-Scholes option pricing model. But all of the traditional ways of volatility modeling don t take the timevarying effect into account and set it as a constant value which is far from the fact of volatile financial market. Eagle (1982) firstly model the time-varying volatility of the stock market which has not been considered by traditional econometric models for a long time. He introduced the ARCH (autoregressive conditional heteroskedastic) model to capture the volatility clustering performance in time-series of data. After that, Bollerslev (1986) introduce another more general model for volatility clustering analysis which is called the GARCH (generalized autoregressive conditional heteroskedastic) model. The GARCH model is different from the previous model for its past condition volatility term into the 8

9 estimation of current condition volatility, which can promote the estimation for condition volatility as well as the uncertainty of finance markets. Tim Bollerslev, Ray Y. Chou, Kenneth F. Kroner (1992) overview the developments of the ARCH model and relative extensions, like ARCH-M, IGARCH, multivariate ARCH, etc. 3.Data sources and descriptive statistics 3.1 Return definition The sample of the thesis consists a time-series data including the date and the stock market price index in China, Brazil, Russia and India over the research period. We should calculate the stock returns from the original data of price index to process deeper analysis. The stock return according to the market index data is defined as follows: R i,t = MSCI i,t MSCI i,t 1 MSCI i,t 1 ln ( MSCI i,t MSCI i,t 1 )= ln (MSCI i,t ) ln (MSCI i,t 1 ) (1) R i,t stands for the daily stock return for country, i at time, t. The term MSCI t represents the MSCI price index for country, i at time, t. MSCI t 1 represents the MSCI price index for country, i at time, t-1. From the function above I can calculate the values of stock returns from the MSCI price index I get from a database. 3.2 BRIC Price index To capture the integral performance of the emerging countries in the 2008 market crisis, I analyze the MSCI stock returns data of the emerging countries over 30 years covering the 2008 market crash briefly. As Sandeep A. Patel and Asani Sarkar (1998) defined in their papers, a market crash starts after a peak value of the stock return index according to its historical trend, when the index suddenly declines. The EM index trend for 20 years is as the Figure 1 shows. 9

10 Figure1.1: The time-series line for MSCI stock price index of China, Brazil, Russia and India from December 30 th, 1994 to June 13 th, 2017 Figure 1.2: The Stock Prices index performance of the emerging market for 30 years from 1987 to 2017 Data source: Thomson Reuters DataStream The set of Figure 1.1 shows the price index trend for the BRIC countries. Firstly, the trends for the price index in four countries are similar which proves that the correlation among the world economy has been tighter. All of the four 10

11 countries have a peak value of price index around the year 2003 and then drop to a valley bottom around the year Before this time point, the price index rises mildly as a result of economic development. After the valley bottom value, the price index begins to recover and rise slowly. The price index of four country are all faced with a more volatile trend from 2005 to 2010 though the integral trend of price index is upward. Secondly, when we analysis the price index separately, the specific time point for the peak value and valley value differ in the four countries which shows a time lag for the spread of market shocks. China has an abnormal boom of price index around the year 2000, which can be explained by China joining the WTO on November 10 th, The big event promotes the confidence of investors at home and abroad as well as rise the investment risks. We can observe from Figure 1.2 that the whole emerging market experienced there is a severe market crash after the peak value in the year 2007 and the price index suddenly decrease sharply into a valley value in The stock market returns of emerging countries markets suddenly decreased to a certain level around the year Next, according to the stock daily returns from 1987 to 2017 of the four countries China, Brazil, Russia and India, I selected 10 max values and 10 min values for each country and calculate the differences between them to show a brief capture of the volatility for these countries stock returns over 30 years. I used the MSCI stock index of each country as the data source. Date China MAX Date China MIN Difference 1997/9/ /9/ /2/ /9/ /9/ /1/ /11/ /8/ /4/ /1/ /1/ /9/ /9/ /1/ /10/ /10/ /10/ /10/ /10/ /11/ Table 1.1 Summaries for China stock price index 11

12 Date Brazil MAX Date Brazil MIN Difference 1995/2/ /10/ /3/ /10/ /3/ /11/ /9/ /8/ /9/ /9/ /9/ /10/ /1/ /9/ /10/ /10/ /10/ /10/ /11/ /5/ Table 1.2 Summaries for Brazil stock price index Date Russia MAX Date Russia MIN Difference 1997/10/ /10/ /6/ /1/ /7/ /8/ /8/ /8/ /10/ /10/ /10/ /5/ /11/ /9/ /12/ /10/ /9/ /10/ /10/ /11/ Table 1.3 Summaries for Russia stock price index Date India MAX Date India MIN Difference 1998/6/ /3/ /3/ /10/ /1/ /4/ /4/ /5/ /5/ /5/ /6/ /5/ /1/ /1/ /10/ /10/ /10/ /10/ /5/ /1/ Table 1.4 Summaries for India stock price index Table 1: China, Brazil, India and Russia Daily stock returns Max and Min and the differences 12

13 From table 1, we can notice that the maximum and the minimum stock daily returns for the four countries almost occur around 2008 over the 30 years research period. For China, the max value occurs on 2003/9/19 (13.71%) and the min value occurs on 2008/10/27(-12%). Brazil has the highest stock daily return on 1999/1/15 (27.96%) and lowest on 2008/10/15 (- 13%). Russian highest stock return is on 2008/9/19 (27%) and lowest one is on 2008/10/6 (-22.34%). When it comes to India, the maximum value of returns is on 2009/5/18(17.85%) and minimum value is on 2008/10/24(-11%). Most positive and negative values located on the years around 2008 which implies that the standard deviation should be large when there comes the 2008 market crash which is a signal of the crisis. Next, we look into the time series line of the four countries daily stock returns as Figure 2 shows. We set the time variable as date and the delta is 1 day, draw the trend lines of China daily stock returns, Brazil daily returns, Russia daily stock returns and Indian daily stock returns using STATA and get the macro performance of the daily stock returns over the research period. Figure 2.1 China Daily Stock Returns Figure 2.2 Brazil Daily Stock Returns 13

14 Figure 2.3 Russia Daily Stock Returns Figure 2.4 India Daily Stock Returns Figure 2: The time series line for the daily stock returns of China, Brazil, Russia and India from 1987 to 2017 From Figure 2, we are obviously conscious of the two most volatile periods of daily stock returns from 1998 to 1999 and on There is an almost 10 years of market stability before the year 2008, which is exactly the conclusion we want to get through our empirical analysis-- that there is a period of low volatility of stock returns before the market crash and then the volatility would peak to its highest value after the crisis occurs and sustain the high level to its end. 14

15 In order to check the linkages of the four BRIC countries, I summarized the correlations coefficients estimations of the four countries daily stock returns from December 30 th, 1994 to June 13 th, 2017 as follows. To check the relationship between the four countries market stock returns and the world stock markets, I add the S&P 500 index and calculate the daily stock returns of S&P500 from Dec.30 th 1994 to June.13 th 2017 in the same period. The coefficients estimations are as follows. China Brazil Russia India S&P500 China Brazil * Russia * * India * * * S&P * * * * *stands for Statistically significant at the 1% significant level Table 3: The correlations between China, Brazil, Russia and India markets as well as world stock market from Dec.30 th 1994 to June.13 th 2017 As stated in the correlation chart, we can see that all of the four countries daily stock returns have significant relationships between each other. Among them, India returns is most correlated with China market (0.3537) and least correlated with Brazil market (0.1965). The results imply that India s stock returns are most correlated to China while Brazil is least correlated with Chinese. This implies that the geographic factors are still somewhat important in the current economic relationships. The correlation coefficients increase as the geographic location getting closer. It is interesting to find that, China s stock market return has lowest correlation with S&P500 (0.1421) significant at 1% level. Brazil has the highest correlation (0.5234) with the world stock market. 3.3 Crisis Period division, the crash and crisis definitions In this section, I will set the pre-crisis, during-crisis and post-crisis sample periods of each country according to the four countries stock index 15

16 Victoria Ivashina and David Scharfstein (2010) define the peak of the credit boom as second quarter of 2007 and the peak period of the 2008 financial crisis as the fourth quarter of 2008 after analyzing the bank loans fluctuation. Michael D. Bordo (2008) states the begin of the 2008 crisis as early 2007 as a result of the collapse of subprime mortgage market. The three subperiods of the 2008 market crash, pre-crash, during-crash and post-crash are defined according to Sandeep Patel J. P. Morgan and Asani Sarkar (1998) who studied nine market crises in developed and developing countries. There is a difference between the crash and crisis definitions. A crash occurs when the daily stock returns drop to the lowest level and it is a time point. A crisis occurs when the economic condition begins to worsen and the index decreases to a certain level and maintains the level until it jumps back to a normal level before the crisis period. Firstly, the crash date of the market is set as the date when the most negative stock returns occur. According to the maximum and the minimum summarize of the four countries data, China is 2008/10/27( ), Brazil is 2008/10/15( ), Russia is 2008/10/6 ( ) and India is 2008/10/24( ). The crash period for the four countries are as follows: China Brazil Russia India crash date 2008/10/ /10/ /10/6 2008/10/24 pre-period 1994/12/ /10/ /12/ /10/ /12/ /10/6 1994/12/ /10/24 Post-period 2008/10/ /10/ /10/ /10/27- Table 4: The separation of three sub-periods, pre-crisis, during-crisis and post-crisis in the 2008 market crisis for China, Brazil, Russia and India. Secondly, the stock market crisis occurs when the MSCI index decrease to a certain level and a recovery takes place when the index is bound back to its normal level before the crash. The beginning of a crisis is when the index 16

17 decreases up to 42 percent of the maximum stock index value in history for China, Brazil, Russia and India, which is a higher threshold value for developing country than developed countries. That means the threshold for beginning is 58 percent of maximum value in history. Another judge rule for the beginning date is that the index keeps declining afterwards. And the recovery period begins when the index makes it back to the average level before the market crisis begins. China Brazil Russia India Max (1997/8/25) (2008/5/20) (2008/5/21) (2008/1/7) Threshold ,389,542, Beginning (2007/8/10) (2008/10/6) (2008/9/9) (2008/9/29) Mean level Recover (2015/1/2) (2009/3/30) (2009/3/23) (2009/3/12) Table5: The pre-crisis, during-crisis and post-crisis periods of China, Brazil, Russia and India according to the judgement standard There are two special values in the above table which are already marked by bold lines that should be explained. The price indexes of Brazil and India as shown in the figure have a relative low level of price index before the year 2000 and the price index boom a lot after the 2008 market crash which can be explained by the economic growth. As a result, the average value of the pre-crisis price index is low and cannot be an appropriate standard to justify the recovery date for the 2008 crisis. So, I picked the value of the price 17

18 index just before the whole index trend begins to rise steadily and robustly according to the index data of Russia and India. From the table above, we can observe that there are some particular facts about the 2008 crisis in China, Brazil, Russia and India. First of all, the beginning date diversify among the four countries, especially in Brazil, Russia, India. These three countries have peak value of price index in the first half year of 2008, from January to May, which is considered almost the worst period of 2008 crash in the United States. From this we can conclude that the 2008 crisis doesn t boom in all the countries in the same time and there is a time-lag for it to spread from the U.S to the South America and the Asia. 3.4 High and low volatility separation The second way is to adopt the HP filter process to decompose the volatility level. The thesis aimed at the correlation between the volatility level and the occur of the 2008 market crisis. The distinction of the high volatility and low volatility level (σ t )is accomplished by HP filter which is introduced by Hodrick and Prescott (1997). The standard decomposition of time series variable is y t = τ t + c t (t=1,2,3 ) where y t is in logarithmic form, τ t is a trend component and c t is a cyclical component. σ t = τ t (λ) + c t (λ) (2) The term λ stands for the smoothing parameter to capture the trend of volatility. We define and use the one-side filter to classify volatility level here with an assumption that the observations are only affected by past data.the trend term c t (λ) is separated into two classes: the term c H t (λ) is for high volatility and the term c L t (λ) is for low volatility. c H t (λ)= σ t τ t (λ) if σ t > τ t and c L t (λ)= σ t τ t (λ) if σ t < τ t. Take China daily stock returns for example. 18

19 Figure 3: The HP-Filter cyclical component and the trend component for China Daily stock returns from December.30 th 1994 to June.13 th 2017 Figure4: High volatility and low volatility 4.Methodology and empirical results 4.1 Test for stationary, normality and ARCH effect of data Stationary test Firstly, I use the Dickey-Fuller test to check the stationary of the data series for daily stock returns in China, Brazil, Russia and India. The fuller test estimation results are as follows. 19

20 China Brazil Russia India Z(t) *MacKinnon approximate p-value for Z(t) = **1% critical value % critical value % critical value Table 6: The ADF Test Result for the unit root test of China, Brazil, Russia and India According to the estimation for the value of Z(t) for all the four countries daily returns data, the test statistics are significantly different from zero at 1% level. We can reject the unit root null hypothesis, which proves the stationary of our data Normality Test Next, we use the Quantile-Quantile test to test if the performance of the four countries daily stock returns obeys the normal distribution. Figure 4: The QQ-plot of Brazil, China, India and Russia daily stock returns in the 2008 crisis The results of Jarque-Bera type tests of H0: Normal 20

21 China Brazil India Russia skewness Kurtosis p-value Table 7: The Jarque-Bera test results for Normal Distribution of China, Brazil, India and Russia in the 2008 crisis From table 7 we can tell that the skewness and the kurtosis of the four countries are all different significantly from the normal distribution and we can reject the null hypothesis that H0: The distribution is a normal distribution. And the four countries daily stock returns are all positive skewed. As a conclusion, the distribution of four countries returns are not normal and all of them show fat tails Autocorrelation Test I recorded the results of Portmanteau test for white noise (Q test) to the four countries daily stock returns data. China Brazil Russia India Portmanteau (Q) statistic Prob > chi2(40) Table 8: The Q-test statistic estimation for the residual terms of Daily stock returns dataset for China, Brazil, Russia and India From the above table for Q statistic value, we can observe that the values of P are all nearly zero and the value of Q are more than 100 which is relative high. This can prove the autocorrelation for the four countries daily stock returns data series LM test for ARCH effect and The Serial Correlation Next, we use the Breusch-Godfrey test (LM test) to test the validity of the GARCH model and regress model used in this thesis to capture the stock market volatility performance during the 2008 market crisis, to test whether there is serial correlation in the model otherwise the results would be incorrect. The lagged values of the former dependent variables in the model might be used as the independent variables in the latter regression which can 21

22 result in an autocorrelation of the errors in the regression. 1 It is assumed that there is no serial correlation in the GARCH model and the multi-regression model. The test can be easily processed using STATA. Firstly, we draw back the model formed forehead. q h t = k + G i h t i + i=1 j=1 A j ε 2 t j +u t (3) Where the errors u t can follow an AR(q) autoregressive pattern like u t = Ω 1 u t 1 + Ω 2 u t 2 + Ω 3 u t ε (4) We can get the residuals u t to fit the model with least ordinary squares(los). Then we can check if the following model regression is fitted. u t = λ 0 + λ 1 X t,1 + λ 2 X t,2 + Ω 1 u t 1 + Ω 2 u t 2 + Ω 3 u t ε (5) Next, we can calculate the statistic R 2 for the regression and use the method Asymptotic approximation to distribute the value: p nr 2 ~X p 2 (6) If we can prove the hypothesis that Ω=0 for all the situations, there would be no serial correlation effect in the model. The LM test (Breusch-Godfrey test) is processed to evaluate the validity of the assumptions as well as the hypothesis and test the regression model to check if there is autocorrelation in the data series. The LM test for Brazil, India, Russia are the same as that of China. LM test for China autoregressive conditional heteroskedasticity (ARCH) lags(p) chi2 df Prob > chi

23 H0: no ARCH effects vs. H1: ARCH(p) disturbance Table 9: The LM test for ARCH effect of China daily stock returns 4.2 The ARMA model for autocorrelation of data The next step is to test the ARMA (p, d, q), estimate the value and sign of statistics and find the best fit model for the sample data. After the ADF unit root test, we can identify the d term to be 0 because there is no unit root in the ADF test. Then in order to determine the p and q term we need to use ACF and PACF. As Box and Jenkins (1976) defined, the autocorrelation and partial autocorrelation test is used to check the ARMA model s time series movements. Autocorrelation Partial Auto LAG AC PAC Q P>Q

24 Table 10: The AutoCorrelation and Partial Autocorrelation table for China daily returns As we can see from the ARMA table of China Daily returns, when the lag is added to 2, the value of Q increases a little magnitude from to , and the autocorrelation and partial autocorrelation are small and significant. Thus, we can set the p and q value to lag 1. The AIC and BIC estimation of China, Brazil, Russia and India daily stock returns data are as follows. Akaike's information criterion and Bayesian information criterion China Brazil Russia India Obs ll II(model) df AIC BIC AR (1) MA (1) MA (2) MA (3) MA (4) sigma Table 11: The AIC and BIC estimation of China, Brazil, Russia and India daily stock returns in the 2008 crisis From the table above, large value of LL and small value of AIC and BIC for four countries can prove that the ARMA (1,1) is the best fit model for our data. 24

25 4.3 The GARCH model for volatility persistence The methodology to investigate the persistence performance of stock market pre-crisis, during-crisis and post-crash in this paper is the GARCH (the General Autoregressive Conditional Heteroscedasticity) model. Tim BOLLERSLEV (1986) expounded a systematic explanation towards the GARCH model. The GARCH has an advantage that it can display the volatility clustering with an added past conditional volatility term which can The original ARCH model introduced by Engle (1982): y t = γy t 1 + ε t (7) ε t = z t h t 2 (8) z t ~i, i, d, E (z t )=0, VAR (z t ) = 1 (9) 2 h t = α 0 + α 1 ε t 1 (10) The first function is the first-order difference of y t. ε t is the white noise term and Var (ε)=σ 2. The conditional meaning of y t is the term γy t 1 which is a notable progress to analysis the time-series models mean value rather than a constant value. The ARCH model makes it possible to predict the underlying forecast conditional variance changing over time by past forecast errors. Thus, it is a better approximation of the financial real markets rather than making standard assumptions of the fluctuations. T. Bollerslev (1986) introduced a more general class of process, the GARCH model to allow for more flexible lag structure. The standard GARCH model function is as follows: R r t = c 1 + i=1 Φ i r t i + j=1 Φ j ε t j + ε t...(11) M ε t = z t h t...(12) h t = w + βh t i + αε 2 t j...(13) 25

26 (α, β 0) The simplest process is the GARCH (1,1) function. h t = w + βh t i + αε 2 t j (14) h t means the conditional variance at time t. The sign and magnitude of α indicate the effect imposed by the lagged error term ε 2 t i on the conditional variance h t. In other words, the size and significance of α implies the existence of the ARCH process in the error term which captures the volatility clustering. β measure the impact that past lagged conditional variance has on current condition variance. The equation (1) is called the conditional mean function. r t is the stocks return series. The empirical distribution of yield usually has leptokurtosis and heavy tails and unconditional asymmetry (Laura L. Veldkamp 2005). The error term ε t denotes a real-valued discrete-time stochastic process. In the equation (2), h t is the conditional variance of the auto regressive process which has can be related to the previous infinite order of period t. z t is the independent and identically distributed random variable which obeys the standard normal distribution N (0,1). The equation (3) is called the conditional variance function. k is the constant value and the term α + β is the GARCH fixed effect term which is used to depict the variable characteristics of conditional variance. The size and significance of term β denotes the magnitude of effect imposed by the lagged error term on the conditional variance. Next, I ran the GARCH model for each country in each period of market crisis and make a portrait comparison. Analysis the term α+β, the long-term volatility in each period for each country and the asymmetric effect using EGARCH model. 26

27 According to the three subperiods which are defined in section 3, we can run the GARCH regression before the crisis, during the crisis and post-crisis for the four countries. ε t = z t h t z t ~i, i, d E(z t ) = 0, var(z t ) = 1 (15) h t = w + βh t 1 + αε 2 t 1 (16) The residual term ε t is the error of the GARCH model at time t. α is the ARCH effect parameter which measures the impact that unexpected events has on the conditional volatility. β is the GARCH effect parameter which captures the persistence performance of volatility in time t and lagged terms of t. The addition of parameters α+β is the measurement of the integral impact of clustering characteristic of conditional volatility in a range of time China (1) (2) (3) Before crisis During crisis After crisis ARCH effect parameters α 0.102*** *** *** (15.89) (9.73) (5.41) β 0.889*** 0.920*** 0.911*** (18.2) (17.82) (9.86) α+β constant *** *** ** (7.08) (3.7) (2.83) N z-statistics in parentheses: * p < 0.05, ** p < 0.01, *** p < Table 12: the GARCH model parameters estimation for China daily stock returns in 08 crisis 27

28 GARCH parameters estimation for China in 08'crisis α β α+β pre_chi dur_chi post_chi Figure 5: GARCH parameter estimation for China in 08 crisis The table shows the value of estimated coefficients α and β as well as the sum of α + β. The estimations for statistics are statistically significantly different from zero at level. The null hypothesis α + β = 0 can be rejected. Firstly, the α, β and constant terms for China daily stock returns are all positive and significant statistically at level except the constant term post-crisis which is significant at 0.01 level. The squared conditional volatility 2 2 at time t σ t would increase if the squared volatility term σ t 1 or the squared lagged residual term ε 2 t 1 increases. What s more, the magnitude of β is larger than α throughout the crisis periods, which implies that the past condition volatility has larger impact on current volatility than the past market shocks. Secondly, the trends for the three functions α, β and α + β during the three subperiods are obvious and consistent as the Figure shows. The alpha decreases from to when the period is changed from pre-crisis to dur-crisis, which implies that the impact of the squared lagged residual term ε 2 t 1 on the squared volatility at time t σ t 2 decreases after the crisis occurs. 28

29 The alpha increases from to when dur-crisis period turns into post-crisis, which implies that the impact of the squared lagged residual term ε 2 t 1 on the squared volatility at time t σ t 2 increases after the crisis become to recover. On the contrary, from pre-crisis to dur-crisis, the value of beta rises from to although slightly. And then beta decrease to The different movements of α and β proves that, the impact of past market shocks would be weaker during a crisis and reversely the impact of past condition volatility would be stronger during a market crisis. The most important GARCH effect parameter function α + β is faced with a similar fluctuation trend during the 2008 crisis, which jumps from to and then , though the trend is slight but significant. This implies that the volatility persistence performance would increase during the market crisis and then drop back when the crisis begins to recover. The relationship between past market shocks and past condition volatility would be more tight when a crisis occurs Brazil (1) (2) (3) Before crisis During crisis After crisis ARCH effect parameters α 0.136*** * *** (18.44) (2.1) (8.72) β 0.827*** 0.917*** 0.909*** (70.5) (25.3) (85.16) α+β constant *** *** (7.39) (0.97) (4.92) N z statistics in parentheses * p < 0.05, ** p < 0.01, *** p < Table 13: the GARCH model parameters estimation for China daily stock returns in 08 crisis 29

30 GARCH parameters estimation for Brazil in 08'crisis α β α+β pre_bra dur_bra post_bra Figure 6: GARCH parameter estimation for Brazil in 08 crisis Similar to the GARCH model estimation results for China, the estimated coefficients α and β for Brazil are all significant different from zero, which can reject the null hypothesis α + β = 0. The α, β and constant terms for Brazil daily stock returns are all positive and significant statistically at level except the constant term post-crisis which is significant at 0.01 level. It implies that the squared conditional 2 2 volatility at time t σ t would increase if the squared volatility term σ t 1 or the squared lagged residual term ε 2 t 1 increases. The significance implies that the GARCH and ARCH effect coefficients are proved to be different from zero. Secondly, the trends are still significant for Brazil during the three periods. The alpha decreases from to when the crisis occurs, which implies that the impact of the squared lagged residual term ε 2 t 1on the squared volatility at time t σ t 2 decreases after the crisis occurs. The alpha increases to when dur-crisis period turns into post-crisis, which implies that the impact of the squared lagged residual term ε 2 t 1on the squared volatility at time t σ t 2 increases after the crisis become to recover. The beta 30

31 also changes among periods. From pre-crisis to dur-crisis, the value of beta rises from to and then beta decrease to when the crisis begins to recover. The GARCH effect parameter function α + β changes from 0.963(precrisis) to (dur-crisis) and then to (post-crisis), all of which are significantly different from zero. This also implies that the volatility persistence performance would increase during the market crisis and then drop back when the crisis begins to recover. The relationship between past market shocks and past conditional volatility would be more tight when a crisis occurs like the conclusion from China Russia (1) (2) (3) Before crisis During crisis After crisis ARCH effect parameters α 0.141*** * *** (20.69) (2.01) (9.51) β 0.843*** 0.902*** 0.916*** (18.76) (16.1) (13.86) α+β constant *** *** (13.79) (1.04) (9.15) N z statistics in parentheses * p < 0.05, ** p < 0.01, *** p < Table 14: the GARCH model parameters estimation for China daily stock returns in 08 crisis 31

32 GARCH parameters estimation for Russia in 08'crisis α β α+β pre_rus dur_rus post_rus Figure 7: GARCH parameter estimation for Russia in 08 crisis The estimated coefficients α and β are all significant different from zero, which can reject the null hypothesis α + β = 0. The α, β and constant terms for Russia daily stock returns are all positive and significant statistically at level. The alpha decreases from to from pre-crisis to end of crisis, which implies that the impact of the squared lagged residual term ε 2 t 1 or the market shocks on the current volatility keeps decreasing during the crisis period. The value of beta rises from to for the whole sample period, which implies that the impact of the past condition volatility on the current volatility keeps rising. The GARCH effect parameter function α + β changes from 0.984(precrisis) to 0.998(dur-crisis) and then to (post-crisis), all of which are significantly different from zero. This is consistent with the conclusion from China and Brazil India (1) (2) (3) Before crisis During crisis After crisis ARCH effect parameters 32

33 α 0.130*** * *** (15.35) (1.86) (8.32) β 0.841*** 0.954*** 0.920*** (8.76) (3.05) (9.79) α+β constant *** *** (7.75) (1.01) (4.67) N Z statistics in parentheses * p < 0.05, ** p < 0.01, *** p < Table 15: the GARCH model parameters estimation for China daily stock returns in 2008 crisis GARCH parameters estimation for India in 08'crisis α β α+β pre_ind dur_ind post_ind Figure 8: GARCH parameter estimation for Russia in 2008 crisis The estimated coefficients α and β are significantly different from zero, which can reject the null hypothesis α + β = 0. The α, β and constant terms are all positive and significant statistically. The alpha decreases from 0.13 to when the crisis occurs, which implies that the impact of the market shocks on the current condition volatility decreases after the crisis occurs. The alpha increases to when durcrisis period turns into post-crisis, which implies that the impact of the market shocks impact increases after the crisis become to recover. From pre-crisis to 33

34 dur-crisis, the value of beta rises from to and then beta decreases to 0.92 when the crisis begins to recover. The GARCH effect parameter function α + β changes from 0.971(precrisis) to (dur-crisis) and then back to (post-crisis), all of which are significantly different from zero. The value that more than one during the crisis can prove the larger persistence performance of volatility in India during the 2008 market crisis As a comparison: S&P500 index for the United States stock market To illustrate the stock persistence performance more clearly in the BRIC s markets, I add S&P 500 index as a comparison to check the difference between developing countries and developed countries. Firstly, the classification method of crisis periods like pre-crisis, duringcrisis and post-crisis for S&P 500 is like that of BRIC s markets. The difference is that the judgement standard for the beginning date of crisis is the date when the S&P index drops up to 25 percent of the maximum value in history and the index keeps declining afterwards. The fallen level is smaller than that of developing countries (42 percent), which means the developed countries like the United States has more stable and mature stock market than developing countries and the volatility is relatively lighter in the crisis. Maximum Threshold Beginning Average Recover S&P (2007/10/9) (2008/9/29) (2009/11/16) Table 16: The pre-crisis, during-crisis and post-crisis subperiods of S&P 500 After that, I run the GARCH model for S&P stock returns in each period and get the parameters estimation as follows. 34

35 (1) (2) (3) before crisis during crisis after crisis ARCH effect parameters α *** ** *** (12.69) (3.25) (10.86) β *** *** *** (162.23) (46.13) (53.07) α+β constant *** *** (6.61) (1.23) (8.07) N z statistics in parentheses * p < 0.05, ** p < 0.01, *** p < Table 17: the GARCH model parameters estimation for the United States S&P 500 index daily stock returns in 08 crisis GARCH parameter estimation for S&P 500 in 08' crisis α β α+β before crisis during crisis after crisis Figure 9: GARCH parameter estimation for S&P 500 in 2008 crisis From the result above, we can see that the ARCH effect terms α & β are all significantly different from zero. Before the crisis, α equals and then declines to when the 2008 crisis occurs, which means the influence of past residual errors or the unexpected shocks on the conditional volatility decreases during the crisis. When the economic begins to recover, the value of α increases to 0.14 in a relative large scale, which means the unexpected 35