## The Effects of Asian Stock Indexes on

## Conditional Volatility Thai Stock Index

Pathairat Pastpipatkul

Faculty of Economics

Chiang Mai University

Songsak Sriboonchitta

Faculty of Economics

Chiang Mai University

Abstract

The relationship of the Asia stock Index affects on conditional volatility of the SETI (SET Composite Index) Stock Exchange of Thailand by adding explanatory variables into the Fractionally Integrated EGARCH (FIEGARCH) model are proposed in this paper. The modified information criterion (MIC) shows the proper lag length in unit root test. The rates of return are stationary. The leverage effects from both Models are significant. The estimates of d (fraction) are all in the region of 0.1 to 0.25. They show the long run property.

The result shows impact of the Asia stock Index on conditional volatility of the SETI the Stock Exchange of Thailand with. However, The PSI (PSE Composite Index) Philippine Stock Exchange does not impact on conditional volatility of the SETI the Stock Exchange of Thailand.

JEL classification codes: Stock market volatility; Long memory; Fractionally integrated EGARCH

Acknowledgement

I am very grateful for recommendation of prof.Cathy W.S. Chen, Department of Statistics, Feng Chia University who has a lot of kindness.

### Introduction

The several countries in Asia observed steep falls in their exchange rates in late 1997 and early 1998. The collapse of the Thai baht's peg is believed that the rapid spread of the currency and stock market crisis, from one country in the region to another in July 1997. There are the evidence to support the existence of relationship between the stock markets of Thailand and Indonesia, and between Thailand and the Philippines, over both the pre- and post-1997 crisis periods (Daly, 2003). The Singapore and Taiwan are cointegrating with Japan while Hong Kong is cointegrating with the United States and the United Kingdom. There are no long run equilibrium relationship between Malaysia, Thailand and Korea and the developed markets of the United States, the United Kingdom and Japan. The relationship between the developed and emerging markets also change over time (Wong, Penm ,Terrell, Lim, 2004). Evaluating if contagion occurs is important for several reasons. Firstly, if contagion occurs after a negative shock, then market correlations would increase, which then undermines much of the rationale for international diversification. Secondly, the investors react differently after a large negative shock. Thirdly, many international institutions and policy-makers worry that a negative shock to one country can have a negative impact on financial flows to another country. (Daly, 2003)

From the uncertain situation, the many economic and financial time series lie on the borderline separating stationary from non-stationary. The ARFIMA model has become a tool in the analyses of time series in different fields such as economic time series, astronomy, computer science and many others. It can characterize “long-range dependence or positive memory” when d lies (0.0, 0.5), and “intermediate or negative memory” when d lies (−0.5, 0.0). A good review of long memory process may be found in Beran (Sowel, 1992) and (Beran, 1994). However, between I(0) and I(1) processes can be far too restrictive. In the discrete time long-memory fractionally integrated I(d) class of processes, the propagation of shocks to the mean occurs at a slow hyperbolic rate of decay when 0 < d < 1, as opposed to the extremes of I(0) exponential decay associated with the stationary and invertible ARMA class of processes, or the infinite persistence resulting from an I(1) process. An overview can be found in Adenstedt (1974), Granger (1980, 1981), Granger and Joyeux (1980), Hosking (1981), Cathy and Tiffany (2005).

The proper modeling of the long-run dependencies in the conditional mean of economic time series, similar questions therefore become relevant in the modeling of conditional variances. The Fractionally Integrated Generalized Auto Regressive Conditionally Heteroskedastic, or FIGARCH, class of processes is to develop a more flexible class of processes for the conditional variance that are more capable of explaining and representing the observed temporal dependencies in financial market volatility (Baillie, Bollerslev and Mikkelsen, 1996).

The FIEGARCH model directly extends the EGARCH model, to a fractionally integrated model. The FIEGARCH (fractionally integrated EGARCH) model allows the existence of leverage effects. Adding explanatory variables into the conditional variance formula might have impacts on conditional volatility. This research finds the effect of the rate of return of Asia stock indexes to Thailand conditional volatility.

The outline of this paper is as follows: in Section 2 we summarize some Model of EGARCH and FIEGARCH are estimated using the BFGS algorithm (Zivot, 2006). Section 3 we show model over view for estimation of coefficients. The concluding is given in Section 4.

### The modified information criteria (MIC) for choosing lag lengths in testing unit root

The good size and power properties of the all the efficient unit root test rely on the proper choice of the lag length p used for specifying the regression (Nelson and Perron, 2001).

The traditional model selection criteria such as AIC and BIC are not well suited for determining p with integrated data. The modified information criteria (MIC) that selects p as pmic = arg minp<Pmax MIC(P).

with and as . The maximum lag, pmax, may be set . The modified AIC (MAIC) results when CT = 2, and the modified BIC (MBIC) results when CT = ln(T-pmax). The selecting the lag length p is minimizing the MAIC. This testing for unit root is base on efficient unit root tests of Nelson and Perron (2001).

### FIEGARCH

The ARCH model is a systematic framework for volatility of Engle (Engle, 1982). The shock of ARCH models is serially uncorrelated, but dependent, and the dependence can be described by a simple quadratic function of its lagged values. Speciﬁcally, an ARCH (m) model assumes that

(1)

where {εt} is a sequence of independent and identically distributed (iid) random variables with mean zero and variance 1, α0 > 0, and αi ≥ 0 for i >0. The coefficients αi must satisfy some regularity conditions to ensure that the unconditional variance of at is ﬁnite. The structure of the model, it is seen that large past squared shocks imply a large conditional variance for the innovation at. Under the ARCH framework, large shocks tend to be followed by another large shock. A large variance does not necessarily produce a large realization.

The ARCH model is often requires many parameters to adequately describe the volatility. Bollerslev proposes a useful extension known as the generalized ARCH (GARCH) model. Let at be the innovation at time t (Bollerslev, 1986). Then at follows a GARCH (m, s) model

(2)

where {εt} is a sequence of independent and identically distributed (iid) random variables with mean zero and variance 1, α0 0, αi ≥ 0, βj ≥ 0, and . It is αi = 0 for i > m and βj = 0 for j > s. The constraint on implies that the unconditional variance of at is finite, whereas its conditional variance evolves over time. The is assumed to be a standard normal or standardized Student-t distribution or generalized error distribution. Equation (2) reduces to a pure ARCH (m) model if s = 0. The αi and βj are referred to as ARCH and GARCH parameters.

Nelson (1991) proposes the exponential GARCH (EGARCH) model to allow for asymmetric effects between positive and negative asset returns, by weighted innovation (Nelson and CAO, 1992).

(3)

where θ and γ are real constants. Both and are zero-mean iid sequences with continuous distributions. The asymmetry of can be seen as

(4)

For the standard Gaussian random variable , For the standardized Student-t distribution, there are

(5)

An EGARCH (m,s) model is

(6)

where is a constant, is the back-shift (or lag) operator such that ,and and are polynomials with zeros outside the unit circle and have no common factors. By outside the unit circle, the absolute values of the zeros are greater than 1. The simple model with order (1,1):

(7)

In case, and the model for becomes

(8)

where . This is a nonlinear function similar to that of the threshold autoregressive (TAR) model of Tong (Tong, Non-Linear Time Series: A Dynamical System Approach, 1990) (Tong, On a threshold model, 1978). The simple EGARCH model the conditional variance evolves in a nonlinear manner depending on the sign of .

(9)

The coefficients (γ +θ) and (γ −θ) show the asymmetry in response to positive and negative. The model is, therefore, nonlinear if θ = 0. Since negative shocks tend to have larger impacts, the θ to be negative.

An alternative form for the EGARCH (m,s) model is

(10)

Here a positive . Contributes to the log volatility, whereas a negative gives, where .The parameter thus signiﬁes the leverage effect of . The are negative in the application.

To allow for high persistence and long memory in the conditional variances the FARIMA (m,d,q) process as

(11)

Where all the root of and lie outside unit circle. When d = 0 It reduces to the GARCH model; When d =1 It becomes The IGARCH model; When 0<d<1, the fractionally differenced squared residuals,, follow a stationary ARMA(m,q) process. The conditional variance is

(12)

Baillie, Bollerslev and Mikkelsen (1996) referred the model as FIGARCH (m,d,q). When 0<d<1, the coefficients in and capture the short run dynamics of volatility, the fractional difference parameter d models the long run characteristics of volatility.

To guarantee a general FIGARCH model is stationary and the conditional variance is always positive, the restrictions have to be imposed on the model coefficients. The EGARCH model can be represented as an ARMA process in terms of the logarithm of conditional variance and always guarantees that the conditional variance is positive, the fractionally integrated EGARCH (EFIGARCH) model can be presented

(13)

Where is defined as the FIGARCH model, allows the existences of the leverage effects, and is the standardized residual:

The FIEGARCH model is stationary is 0<d<1. The FIEGARCH model is estimated using the BFGS algorithm.

### DATA DESCRIPTION

The raw daily data, The SSEC (Shanghai Composite Index) Shanghai Stock Exchange, The BSESN (Bombay SE Sensitive Index) Bombay Stock Exchange, The JKSE (Jakarta Composite) Indonesia Jakarta Composite, The N225 (Nikkei Stock Average 225) Tokyo Stock Exchange, The KLSE (KLSE Composite Index) Malaysian stock Exchange, The PSI (PSE Composite Index) Philippine Stock Exchange, The KS11 (KOSPI Index) Korean Stock Exchange, The SETI (SET Composite Index) the Stock Exchange of Thailand, The TWSE (Taiwan's composite index) Taiwan Stock Exchange are corrected from November 10, 1998 to November 10, 2008.They are corrected by Bloomberg data base.

### 4. Empirical Results

The descriptive statistics for Return Asia Indexes are presented in Table 1. It shows that the mean of Return Asia Indexes are between -0.017 and 0.052. The maximum mean is the BSESN Bombay Stock Exchange. The minimum mean is the N225 Tokyo Stock Exchange. The KS11 Korean Stock Exchange has the highest standard error at 1.96. The KLSE Malaysian stock market has the lowest standard error, 1.09. The measuring of asymmetry of the distribution of the series around its mean is computed as skewness. The Positive skewness means that the distribution has a long right tail which the PSI Philippine Stock Exchange equals 0.47 and negative skewness implies that the distribution has a long left tail which the others Return Asia Indexes in the region of -0.68 to -0.033. The reported Probability is the probability that a Jarque-Bera statistic do not exceeds (in absolute value) the observed value under the null hypothesis—a small probability value leads to the rejection of the null hypothesis of a normal distribution. For the all series displayed above, we reject the hypothesis of normal distribution at the 5% level and at the 1% significance level.

[Table 1]

### Testing for Unit root

The good size and power properties of the all the efficient unit root test rely on the proper choice of the lag length p used for specifying the regression Ng and Perron (2001). The traditional model selection criteria such as AIC and BIC are not well suited for determining p with integrated data. The modified information criteria (MIC) that selects p as pmic = arg minp<Pmax MIC(P). The modified AIC (MAIC) results when CT = 2, and the modified BIC (MBIC) results when CT = ln(T-pmax). The selecting the lag length p is minimizing the MAIC. This testing for unit root is base on efficient unit root tests of Ng and Perron (2001). It is clear that the rates of return are stationary. The results of rates of return are stationary are compared with the 1 % critical values to indicate rejection of the unit root null hypothesis.

[Table 2]

### The FIEGARCH model

The exponential GARCH (EGARCH) model allow for leverage effects. When εt-i is positive or there is “good news”, the total effect of εt-i is (1+γi)| εt-i|; in contrast, when εt-i is negative or there is “bad news”, the total effect of εt-i is (1+γi)| εt-i|. Bad news can have a larger impact on volatility, and the value of γi would be expected to be negative. The FIEGARCH model directly extends the EGARCH model, to a fractionally integrated model. The FIEGARCH (fractionally integrated EGARCH) model allows the existence of leverage effects. Adding explanatory variables into the conditional variance formula might have impacts on conditional volatility. This research finds the effect of the rate of return of Asia stock indexes to Thailand conditional volatility.

From the table[3,4], The sum of ARCH(1) and GARCH(1) from EGARCH Model is greater than one while the sum of ARCH(1) and GARCH(1) from FIEGARCH Model which capture the short run dynamics of volatility is less than one with significant at 1% .The leverage effects from both Models are significant at 1%. The leverage effects of EGARCH model are less than the leverage effects of FIEGARCH Model with negative value. The fractional difference parameter d (fraction), 0<d<1, integrated FIEGARCH model capture the long run characteristics of volatility. The estimates of d (fraction) are all in the region of 0.1 to 0.25 with significant at 1%. They show the long run property.

The estimate EGARCH model implies a 1% change in rate of return of SSEC Shanghai Stock Exchange cause about a 0.012% change in conditional variance at the same direction which is less than the estimate FIEGARCH model. The estimate FIEGARCH model implies a 1% change in rate of return of SSEC Shanghai Stock Exchange cause about a 0.027% change in conditional variance at the same direction. The BSESN Bombay Stock Exchange, The JKSE Indonesia Jakarta Composite, The N225 Tokyo Stock Exchange, The KLSE Malaysian stock Exchange, The KS11 Korean Stock Exchange, The TWSE Taiwan Stock Exchange show change in rate of return themselves cause about -0.059% to -0.012% change in conditional volatility of the SETI the Stock Exchange of Thailand at the opposite direction. The both models show impact on conditional volatility of the SETI the Stock Exchange of Thailand with significant at 1%. However, The PSI Philippine Stock Exchange does not impact on conditional volatility of the SETI the Stock Exchange of Thailand.