2. Literature Review
There are many reports or research literatures studying the relationship between two or more time series numbers. Several multivariate GARCH models have been proposed to show the covariance of time series data in the literature, including VECH model, see Bollerslev, Engle and Wooldridge (1995), the diagonal VECH, BEKK model, see Engle and Kroner (1995), and the dynamic conditional correlation (DCC) model, see Engle (2002). The method used in this article is the last one, DCC model.
Grubel (1968) proposed the concept of internationally diversified portfolios. After that, the relationship of assets was taken into account when managing the investment portfolios. In the beginning, the coefficient of correlation of two financial instruments was used to describe their co-movement. A mathematical function is used to describe the relationship between, given one variable, and then the other is determined. It gives a curve that minimizes the overall difference. However, the method is too simplified that many determinant factors are ignored. Then, the concept of time-varying covariance was employed, see Bollerslev, Engle and Wooldridge (1995) for details.
Chou, Lin and Wu (1999) employed the multivariate GARCH model to examine the Taiwan stock market price and the volatility linkages with those of the United States.
They found a substantial spillover effect from the United States market to Taiwan market using the Taiwan close-to-open3 returns, because of the incapability of reflecting the market information into the market price when market is closed. Also, they concluded that the “US induced” volatility amounts to 12 percent to total daily stock volatilities of the Taiwan market.
Darbar and Deb (1997) employed the multivariate GARCH model to examine the linkages among four equity markets – the US market, the UK market, Canada market and the Japanese market. They concluded that the Japanese and U.S. stock markets have significant transitory covariance, but zero permanent covariance. The other pairs of markets examined display significant permanent and transitory covariance. They also concluded that, while conditional correlations between returns are generally small, they change considerably over time, and basing diversification strategies on these conditional correlations is potentially beneficial.
There are also some researches to study the linkage between Taiwan stock market and other counties’. Most of them employed the multivariate GARCH model to analyze the markets. This study differs from them in two respects. First, it uses the dynamic conditional correlation model (DCC model), see Engle (2002), which employs the
3 Close-to-open return means the overnight return, the rate of return of open price and the previous day’s close price.
time-varying dynamic structure into the model, to analyze the markets. It allows the correlation matrix to be time-varying. Second, the scope of this study is specified in two levels, the electronic industry sector index futures and the TAIEX (Taiwan Exchange Stock Index), the correlations between the indices and NASDAQ will be examined respectively. For the empirical results using DCC model, Engle and Sheppard (2001) proposed the theoretical and empirical properties of DCC multivariate GARCH model, however, they took the stock markets within the United States for reference instead of international markets.
3. Methodology
The purpose of this research is to survey the linkage of the stock markets in Taiwan and the United States. First of all, the spillover effects will be estimated. In this article, the two-step estimation method will be employed to model the price changes and volatility spillover effect between the markets. And then DCC model will be used to estimate the correlation. As implied by the name, DCC model is a correlation formula with a dynamic structure, and which takes the time-varying covariance into consideration. The details will be elaborated in the following section.
The analysis software tools used for this article are Microsoft Excel 2007 and Eviews version 5.
3.1 Definition of Rate of Return
The type of rate of return common in use is continuous compounding rate of return, and which is also used in this article. The definition of continuous compounding is shown as the below equation:
ROI ln PP (1)
In the equation above, ROI is the return of investment, defined as the natural log of the final price of investment instrument divided by the initial investment value. However,
we multiply the ROI value by 100 times to get the numerical part of the percentage into the following calculation.
In order to count the effect of cut point of stock price in the rate of return, I divided the rate of return into three types, close-to-close (C2C), close-to-open (C2O, overnight return), and open-to-close (O2C, intra-day return) returns, where the C2O return is also called overnight return. The close-to-open means the rate of return of investment is the natural log of the open price of the investment instrument (for example, Wednesday’s open price) divided by the previous trading day’s close price (for example, Tuesday’s close price). By the same token, the elements of the other types of rate of return are shown as the following table respectively.
1. C2C Rate of Return ln P P
′ 100% (2) 2. C2O Rate of Return ln P P
′ 100% (3) 3. O2C Rate of Return ln PP 100% (4)
Types of rate of return close-to-close
Initial Price (Pi) Previous trading day’s close price
Previous trading
day’s close price Open price
It deserves to be mentioned that the rate of return of NASDAQ used in the following methods is the close-to-close one. Because we aim to find out the spillover effect from the United States stock market to Taiwan market, the reference rate of return is fixed, the close-to-close rate of return of NASDAQ. For the comparison object, the various rates of return of Taiwan stock market will be considered in the following analysis respectively.
3.2 Twostep Estimation Method
In order to identify the price changes and volatility spillover effects from the United States market to Taiwan market, I use the two-step estimation method, MA(1)-GARCH(1,1) model, for the two markets, see Chou, Lin and Wu (1999) for details. For the two-step estimation method, we estimate the return and residual first and then estimate the variance. We can estimate the return and residual of the influencing economy (the United States) by uni-variate GARCH model in the first step, and the squared term of resulting residual together with the return are used as the explanatory variables in the mean and conditional variance of the influenced economy (Taiwan) in the second step. The index t is the day time, denoting either the opening or closing time. The WD dummy variable is the weekend effect dummy variable, defined to be 1 as the return covers a weekend (close-to-close and close-to-open return) or a
Monday (open-to-close return) and 0 otherwise. It helps us to identify whether the weekend effect significant or not. Weekend effect4 is a term used to describe the phenomenon in financial markets in which stock returns on Mondays are often significantly lower than those of the immediately preceding Friday. The mathematical formulas of two-step estimation method are shown as below.
R b b WD b e e (5) h E e I a a e a h a WD (6)
The first equation, we call it mean equation, is a MA(1) model. Rit is the market return for market i at day t, and eit, i=1 and 2, are white noise. The second equation, we call it variance equation, is a GARCH(1,1) model, eit-12 is the squared term of residual, and hit-1 is the conditional variance of Rit based on an information set containing all the price information at day t-1.
4 The weekend effect has been a regular feature of stock trading patterns for many years. A number of empirical results offer explanations for this market behavior. Some theories suggest that the tendency for companies to release bad news on Friday after the markets close accounts for depressed stock prices on Monday. Others state that the weekend effect might be linked to short selling, which would affect stocks with high short interest positions. Or, the effect could simply be a result of traders' fading optimism
In order to test whether Taiwan market is affected by the United States stock market or not, the most recent United States market return is introduced into the mean equation of Taiwan market return and squared term of the United States market residuals is introduced into the variance equation of Taiwan market return, too. The two markets are indexed by i: 1 for Taiwan market (domestic market), and 2 for the United States market (foreign market), and the model used in this research is defined by following:
R b b WD b e b R e (7) h E e |I a a e a h a WD a e (8)
In the two equations above, the foreign market return and foreign squared term of residual are designed to capture the return spillover and volatility spillover, respectively.
If the statistical result of b3 is significant, it implies that a change of price return of the United States stock market will cause a change of price return of Taiwan market. If the statistical result of a4 is significant, it implies that the volatility spillover effect exists from the United States market to Taiwan market.
The estimation results will be tested by maximum-likelihood estimation method.
Besides the t-statistics for each of the coefficients, the likelihood ratio statistics (LR)
will be computed for joint significance of the coefficients. The Leung-Box Q statistics for the squared normalized residuals will also be computed, denoted as LB-QS. All the estimation results will be arranged in the following chapter.
3.3 DCC Model
Engle (2002) proposed a dynamic structure to analyze the correlation of two time series data, and that is the Dynamic Conditional Correlation (DCC) model, and which is modified from the Constant Conditional Correlation (CCC) model, see Bollerslev (1990), to let the correlation matrix time-varying.. The DCC model was proposed to solve the conditional covariance problems, and that can be simplified estimated by uni-variate GARCH models for each series’ variance process. The DCC model estimates a conditional correlation by employing the transformed standardized residuals from the GARCH estimation step and then estimating the time-varying conditional estimator in the next step. An advantage of DCC model is easy-to-estimate; although it is not linear, it can be estimated simply by two stages based on the maximum likelihood method. This article will show the empirical results of using DCC model to estimate the correlation of NASDAQ and Taiwan market indexes in the chapter 4.
The conditional covariance and correlation are used to describe the relationship
between two series data (r1 and r2 with zero means) traditionally, and those are defined by:
COV E r ,, r , , (9)
ρ , E , , ,
E , E , . (10) The covariance and correlation are computed by previous data. However, there are two problems, one is too many premature data are used, and the other is that the previous lags assigned equal-weighted will cause the problem of uncoupling correlation estimation.
Here, we consider the DCC model with two assets,
H D R D , where D diag h, , (11) R diag Q / Qdiag Q / , (12) Q S ℓℓ′ A B A Z Z′ B Q , (13)
or in a bi-variate model like the empirical case in this article,
q , q ,
q , q , 1 a b 1 q
q 1 a z, z, z ,
z , z , z , b q , q ,
q , q , . (14) In the equations above, Ht is the covariance matrix, Dt is the 2 2 diagonal matrix of time varying standard deviations from uni-variate MA(1)-GARCH(1,1) models with h on the ith diagonal, and Rt is the time varying correlation matrix. A and B are parameters and is the Hadamard matrix product operator. The ℓ is a vector, S
represents the unconditional covariance of standardized residuals and Q is the conditional one. In equation (13), Z , Z D r , is the standardized residual vector, and the conditional variances of all the components of Z equal 1, where r is the return of asset that is mean zero of the residuals.
And the log-likelihood of this estimator can be written as following, and which can simply be maximized to get the parameters of the model:
L ∑T 2 log 2π log |H | r′H r (15.a) ∑T 2 log 2π log |D R D | r′D R D (15.b) ∑T 2 log 2π 2log |D | log |R | ε′R ε (15.c) In Engle (2002), Engle gave us sufficient conditions for the consistency and asymptotic normality of the estimators, see Newey and McFadden (1994) for details.
Let the parameters in the covariance matrix Dt be denoted as θ and the additional parameters in the correlation matrix Rt be denoted . The log-likelihood can be rewritten to be a summation of a volatility part and a correlation part. The new log-likelihood is:
L θ, L θ L θ, . (16) In the equation above, the volatility part is
L θ ∑ 2 log 2π log|D | r′D r , (17)
According to Engle (2002), the volatility term can be expressed as the sum of individual GARCH likelihoods:
L θ ∑ ∑ log 2π log h , ,
, . (18) And the correlation part is
L θ, ∑ log|R | ε′R ε ε′ε . (19) Owing to the estimation processes of volatility and correlation part are independent from each other; we could estimate them by maximizing the likelihood in the two-step approach,
θ arg max L θ (20) and then we can take the resulting value of θ into the second step:
arg max L θ, . (21) We can get the estimated values of θ and by the two steps above.
By the two step estimation designed in the DCC model, we can estimate for each residual series in the first step, and in the second step, residuals, transformed by their standard deviation estimated during the first step, are used to estimate the parameters of the dynamic correlation. As a result of the estimations, we can get the correlation between two markets series data; the United States and Taiwan market data were employed in this article.
3.4 DCCX Model Extreme Return Effect
Besides the correlation of the market return between the United States market and Taiwan market, we want to know furthermore if the extreme return effect exists. The extreme return effect means that when an extreme return (positive or negative) occurs in the foreign market (influencing market), an extreme return will be taken place in the domestic market (influenced market). On the other hand, the extreme return effect test will help us to check the impact of an extreme return of the influencing market.
In order to test if the extreme return effect significant, a dummy variable was designed into the DCC model, called modified DCC model (DCCX model), see Liao (2007). The dummy variable was set to 1 when the return of influencing market deviates over the criteria from the mean value of return, 0 otherwise; here in this study, the criteria are over one, two, and three times standard deviation of return, respectively.
After adding the exogenous variable of extreme return effect into the DCC model, the DCCX model will be as follows:
H D R D , where D diag h, , (22) R diag Q / Qdiag Q / , (23) Q S ℓℓ′ A B CX A Z Z′ B Q C X , (24) where the definitions and operators are as those in the standard DCC model in chapter
3.1. The estimation method is identical to the standard DCC model. The main purpose of this study of extreme return effect is to see if the coefficient C is significant or not, where the value of coefficient C indicates the offset to the original correlation derived from the standard DCC model.
4. Empirical Results
The NASDAQ (acronym of National Association of Securities Dealers Automated Quotation) is an American stock exchange established in 1971 in New York City. It is the largest electronic screen-based equity securities trading market in the United States.
With approximately 3,200 companies, it lists more companies and on average trades more shares per day than any other U.S. market. That is the reason why NASDAQ was chosen here as a representative of high-tech stock market in the United States. For the Taiwan market, the Electronic Sector Index Futures (EXF) was chosen as the counterparty to NASDAQ.
4.1 Data Sources
The historical price data of NASDAQ employed in this article were from the Yahoo!
Finance website5, and that of EXF were downloaded from the official website of Taiwan Futures Exchange6.
4.2 Sample Selection
The NASDAQ began trading on February 8, 1971, and the EXF was first traded on
5 The address of Yahoo! Finance is http://finance.yahoo.com/
6
July 21, 1999 in Taiwan Futures Exchange. The daily return time series data of each market are used in this article, and the sample period of data employed in this research is from August 2, 1999 to December 31, 2007 of East U.S.A. time zone (GMT-5). The sample size is 1,999 for both the two markets data after deleting no-trading day’s data in any market, including holidays, events and etc.
4.3 Descriptive Statistics
First of all, the historical close price data of NASDAQ, TAIEX and EXF from August 02, 1999 to December 31, 2007 are shown in Figure 1. A quick glance at the charts, we could observe the overall depression after the 911, 2001 attack, and the gradual recovery since 2003. However, the absolute price data contain different scenarios inside;
we should take the rate of return into consideration about the further analysis.
< Figure 1 is inserted about here >
The close-to-close, close-to-open and open-to-close returns are computed by the equation (3), (4) and (5) respectively, shown in the Figure 2 below, and whose descriptive statistics are shown in Table 1. The overnight returns (close-to-open return) of TAIEX and EXF accompany the lowest mean value and volatility. The overnight return volatiles with the preceding close price and the is influenced by foreign markets over the no-trading night, and the other returns (close-to-close, open-to-close) both
contain the daily trading hours that might be influenced by all the available information no matter domestic or foreign news. For this reason, the overnight return volatiles with lower variation than the other ones.
< Figure 2 is inserted about here >
<Table 1 is inserted about here >
We can also do preliminary analysis for the series from the descriptive statistics; all the series are left-skewed except for close-to-close return of NASDAQ and open-to-close return of TAIEX, because of the negative value of skewness, and all the stock return time series appear to be leptokurtic because the values of kurtosis are larger than 3.
<Table 2 is inserted about here >
Table 2 shows us the simple unconditional correlation between each two series. We can conclude some findings from the simple correlation. First, the close-to-close return of NASDAQ did influence the overnight return (close-to-open return) of TAIEX and EXF, and the correlation coefficients are 0.573 and 0.554 respectively, however, relatively lower correlation exist between NASDAQ and other returns of TAIEX and EXF. It reveals that the foreign market returns did influence the domestic overnight returns more than other types of returns.
4.4 Results of Twostep Estimation
In the two-step estimation, we use the MA(1)-GARCH(1) model to fit the influencing market (NASDAQ) in the first step and the squared term of the resulting residual together with the return are respectively used as explanatory variables in the second step. Our purpose is to model the price changes and volatility spillover effect between the two countries. Table 3 shows us the two-step estimation results of foreign and domestic markets.
<Table 3 is inserted about here >
The estimation model is shown in the description panel of Table 3. As is expected, the time-varying conditional heteroskedasticity is shown by the significant t-statistics of the coefficients of the lagged squared residuals (a1) except for the C2O return of EXF and the lagged conditional variance terms (a2). The b3 and a4 are coefficients of price changes and volatility spillover effects respectively. By the estimation results in Table 3, we can conclude some inferences. First of all, the price change spillover effect (coefficient b3) from NASDAQ return to the C2C returns of both TAIEX and EXF is the largest. It means that among the three types of returns, the C2C returns of TAIEX and EXF were affected by NASDAQ most. The market value of Taiwan Stock Exchange market is about NTD (New Taiwan Dollar) 22 trillion (or U.S. Dollar 0.73