Volatility Analysis For Chinese Stock Market Using Garch Model

Volatility Analysis for Chinese Stock Market Using GARCH Model

Jia Geng December, 2006

Abstract In this paper, I apply the GARCH-class models to Chinese stock market. And I analyze the characteristics of the volatility of Chinese stock market .By comparing the models, I conclude that EGARCH model and EGARCH-M model have almost the same efficiency in Shanghai Stock Exchange (SHSE) and Shenzhen Stock Exchange (SZSE). Then I use the estimated model to forecast the volatilities for these two stock exchanges.

1. Introduction

Since

Robert

Engle

wrote

his

famous

paper

Autoregressive

conditional

heteroscedasticity with estimates of the variance of United Kingdom inflation in 1982, ARCH-class models have been developed into the most highly used models in volatility analysis. GARCH model was presented by Bollserslev in 1986. It was based on ARCH model but it has some advantages compared to its predecessor. During the past twenty years, GARCH model has been developed into a class of models. Those models have been applied to stock markets, foreign exchange markets and future markets and they are proven to be relatively accurate and easy to use. The purpose of this paper is to apply GARCH into Chinese stock market. Since Chinese stock market has some unique characteristics, I need to do some modification before I use the general GARCH model.

The rest of the paper contains 4 sections. Section2 is a brief review of ARCH and GARCH models. Section 3 is the empirical analysis and prediction using the models. Section 4 is the conclusion.

2. ARCH and GARCH models

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Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, Robert Engle presented a class of processes called autoregressive conditional heteroscedasticity (ARCH). These are zero mean, serially uncorrelated processes with nonconstatnt variance conditional on the past.

A useful generalization of this model is the GARCH parameterization introduced by Bollerslev (1986). This model is also a weighted average of past squared residuals, but it has declining weights that never go completely to zero. Below is the original GARCH model:

′ y t = xt β + ε t

(1)

ε t = ht ⋅ vt

(2)

ht = α 0 + α 1ε t2−1 + " + α q ε t2− q +θ 1ht −1 + " + θ p ht − p

(3)

E (vt ) = 0, D(vt ) = 1 , E (vt v s ) = 0(t ≠ s );α 0 > 0,α i ≥ 0,θ j ≥ 0, ∑i =1α i + ∑ j =1θ j < 1 q

p

The above process is called GARCH(p,q) process. In the third equation ht = var(ε t ϕ t −1 ), ϕ t −1 ， it is the information before time t-1.

Because GARCH(p,q) is an extension of ARCH model, it has all the characteristics of the original ARCH model. And because in GARCH model the conditional variance is not only the linear function of the square of the lagged residuals, it is also a linear

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function of the lagged conditional variances, GARCH model is more accurate than the original ARCH model and it is easier to calculate.

The most widely used GARCH model is GARCH(1,1) model. The (1,1) in parentheses is a standard notation in which the first number refers to how many autoregressive lags, or ARCH terms, appear in the equation, while the second number refers to how many moving average lags are specified, which here is often called the number of GARCH terms. Sometimes models with more than one lag are needed to find good variance forecasts. GARCH(1,1) is the most widely used GARCH model because it is accuracy and simplicity.

Although GARCH model is very useful in the forecasting of volatility and asset pricing, there are still several problems GARCH model cannot explain. The biggest problem is that standard GARCH models assume that positive and negative error terms have a symmetric effect on the volatility. In other words, good and bad news have the same effects on the volatility in this model. In practice this assumption is frequently violated, in particular by stock returns, in that the volatility increases more after bad news than after good news.

According to the problems in the standard GARCH model, a number of parameterized extensions of the standard GARCH model have been suggested recently. In the following I will discuss two of the most important ones: the exponential GARCH

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(EGARCH) model and GARCH-M model.

1. EGARCH model:

Exponential GARCH (EGARCH) model was first presented by Nelson in 1991. The main purpose of EGARCH model is to describe the asymmetrical response of the market under the positive and negative shocks. In EGARCH model, I have

p

q

j =1

i =1

ln(ht ) = α 0 + ∑θ j ln(ht − j ) + ∑ α i g (vt −i ) g (v t ) = ϕ i v t +

εt ht

−E

εt ht

(4)

(5)

We can see in equation (7), the Nelson rewrote the conditional variance with the natural log of the conditional variance. When ϕ ≠ 0 , the effects of information are

asymmetry. When ϕ < 0 , there is a significant leverage effect. If we compared the above equations with the definition of the original GARCH mode, we can see that there are no constraints for the parameters. This is one of the biggest advantages of EGARCH model compared to the standard GARCH model.

2. GARCH-M(GARCH-in-mean) model: In GARCH-M(GARCH-in-mean) ht is added in the right hand side of equation (1):

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′ y t = xt β + γht + ε t

(6)

ε t = ht ⋅ vt

where ht follows the ARCH or GARCH process. This new term is introduced to measure the response of the dependent variable to the "volatility" of the time series.

3.

TARCH model:

Threshold ARCH (TARCH) model was first presented by Zakoian in 1990. It has the following conditional variance:

ht = α 0 + ∑i =1α i ε t2−i + ϕε t2−1 d t −1 + ∑ j =1θ j ht − j q

p

(7)

Where d t is latent variable ⎧1 ε t < 0 dt = ⎨ ⎩0 ε t ≥ 0

(8)

Because d t is included, the increase ( ε t < 0 ) and decrease ( ε t > 0 ) of stock prices will have different effects on conditional variance. When the stock prices increase,

ϕε t2−1 d t −1 = 0 the effects can be described by the parameter decrease, the effects can be described by the parameter

∑

∑

q i =1

q i =1

α i ; when the prices

α i + ϕ . If ϕ ≠ 0 we

conclude that the information has asymmetrical effects. If ϕ > 0 we say that there is leverage effect.

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3. Empirical analysis and prediction

1. Data:

There are two stock exchanges in China: Shanghai Stock Exchange (SHSE) and Shenzhen Stock Exchange (SZSE). I use the stock indices from both markets in this paper. The data was downloaded from Yahoo! Finance. Each dataset contains 997 1

daily close prices for each market from Jan 04, 2000 to Feb 17, 2004.

Before I estimate the ARCH-GARCH model, I calculated the natural log of the return y t : y t = ln(indext ) − ln(indext −1 ) , where indext is the close price in tth day. I used Eviews4.0 to do the estimation.

2. ARCH test:

Before we use ARCH-GARCH to estimate the model, we need to test whether the data has ARCH effect. The most widely used method is Lagrange Multiplier test (LM). When I do the LM test to the residuals of the returns for both indices, I found that when q=12, the p-value of χ 2 is still less than α = 0.05 . That means the

1

Stock markets are closed during holidays in China. Those observations are skipped during regression. 7

residuals have high order ARCH effect.

3. Estimation and prediction:

From the time series graph of the returns for both markets, we can see that high volatilities are followed by high volatilities and low volatilities are followed by low volatilities. That means both time series have significant time varying variances. Furthermore, it is appropriate to put conditional variance into the function to describe the effects of risk on the returns. Therefore, GARCH class model is the model we should use.

.12

.08

.04

.00

-.04

-.08 100

200

300

400

500

600

700

800

900

1000

Figure1 Time-series graph for returns in Shanghai Stock Exchange (SHSE)

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.12

.08

.04

.00

-.04

-.08 100

200

300

400

500

600

700

800

900

1000

Figure2 Time-series graph for returns in Shenzhen Stock Exchange (SZSE)

I used GARCH、GARCH-M、TARCH、TARCH-M、EGARC and EGARCH-M to estimate the data. Below is the table with the results estimated from different models2. From this table, we can choose the best model for the further prediction.

Model SHSE

GARCH-M

RSS 0.182229

A- R 2 0.990901

AIC

SC

-5.944449 -5.919431

2

Because of the space of this paper, I only listed the best four models for each market based on the performance of the estimation. 9

TARCH

0.183343

0.990845

-5.957579 -5.932561

EGARCH

0.183336

0.990845

-5.972198 -5.947180

EGARCH-

0.181927

0.990906

-5.963084

M SZSE

-5.933063

GARCH-M

0.207526

0.993848

-5.814886 -5.789868

TARCH

0.208405

0.993822

-5.823784 -5.798766

EGARCH

0.208346

0.993824

-5.836708 -5.811690

EGARCH-

0.207514

0.993842

-5.826283 -5.796262

M

Table 1

Estimates using different models

From table 1, we can see that for both markets EGARCH(1,1)-M have the lowest RSS and the relative high adjusted R 2 . That means, EGARCH(1,1)-M is better than other models in the estimation. From the standard of AIC and SC, we can see that EGARCH(1,1) has the lowest value. That means EGARCH(1,1) is also a relative good model for the estimation.

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Furthermore, when I use GARCH(1,1) to estimate the data, I found that the α 1 and

θ 1 for both markets are 0.9733 and 0.9716. They are very close to 1. This indicates that there are high durabilities of the volatilities in both markets. That means if there is an expected shock in these markets, the fluctuations will not die out in the short run. That is a sign for high risk. At the same time, I found that the summation of the parameters is less than 1, which indicates that the GARCH process for the stock return is wide-sense stationary.

When I use TARCH(1,1) to estimate the model, I found that the estimate of ϕ s are greater than 0 for both markets. When I use EGARCH(1,1), I found the estimates of

ϕ s are less than 0 for both markets. Then we can conclude that there are leverage effects in both markets. That is to say the volatilities caused by negative shocks is greater than that caused by positive shocks. This is consistent with most of the research. 7 .8 7 .6 .1 2

7 .4

.0 8

7 .2

.0 4 7 .0 .0 0 -.0 4 -.0 8 2 5 0

5 0 0

R e s id u a l

Figure 3

A c tu a l

7 5 0 F it t e d

Prediction of returns in SHSE

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8 .6 8 .4 8 .2 .1 2

8 .0

.0 8 7 .8

.0 4 .0 0 -.0 4 -.0 8 2 5 0

5 0 0

R e s id u a l

Figure 4

7 5 0

A c tu a l

F it t e d

Prediction of returns in SZSE

I also use the estimated EGARCH(1,1) to predict the volatilities for SHSE and SZSE. In figure 3 and 4, we can see that the model did a good job. Also we can see that these two markets are highly correlated and there is a significant synchronization in their movements. This is not surprising because these two stock exchanges are the only two stock exchanges in mainland China and they are highly intervened by the government.

4. Conclusion

From this research we can see that both markets have significant ARCH effects and it is appropriate to use ARCH/GARCH models to estimate the process. I found that both EGARCH(1,1) and EGARCH(1,1)-M did good jobs in fitting the process for both markets. Because α 1 and θ 1 are 0.9733 and 0.9716, which are close to 1, we can

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conclude that both markets will fluctuate dramatically with new shocks and this is a sign of high risk in the markets. I also proved that there are leverage effects in the markets. That means the investors in those markets are not matured and they will be influenced by information (good or bad) very easily.

REFERNCES

[1].

Bollerslev,

Tim.

1986.

“Generalized

Autoregressive

Conditional

Heteroskedasticity.” Journal of Econometrics. April, 31:3, pp. 307–27.

[2]. Engle, Robert F. 1982. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica. 50:4, pp. 987–1007.

[3]. Nelson, Daniel B. 1991. “Conditional Heteroscedasticity in Asset Returns: A New Approach.” Econometrica. 59:2, pp. 347–70.

[4]. Engle, Robert F. 2001. “GARCH 101: The Use of ARCH/GARCH Models” Applied Econometrics Journal of Economic Perspectives 15(4):157-168, 2001.

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