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European Journal of Operational Research 186 (2008) 1184–1200 www.elsevier.com/locate/ejor

O.R. Applications

Prediction of index futures returns and the analysis of ﬁnancial spillovers—A comparison between GARCH and the grey theorem Ling-Ming Kung *, Shang-Wu Yu

1

Graduate School of Management, National Taiwan University of Science and Technology, 43, Sec. 4, Keelung Road, Taipei 106, Taiwan Received 6 May 2006; accepted 28 February 2007 Available online 2 April 2007

Abstract This paper adopts the GM(1, 1) model to predict the rates of return of nine major index futures in the American and Eurasian markets. In a further step, by means of local grey relational analysis and by employing the GM(1, N) model for the ﬁrst time, the variation relatedness and the main inﬂuencing factor among the above mentioned targeted markets is determined. Then, a comparison between GARCH/TGARCH and the grey theory with regard to predictive power is conducted. The ﬁndings reveal that the GARCH/TGARCH model performs better than the GM(1, 1), including the optimal a method, in terms of forecasting capabilities. Meanwhile, it is also found that GARCH and spillover eﬀects indeed exist. Moreover, GM(1, N) also reveals that the daily rate of return of the Dow Jones index futures has the most inﬂuence on the rates of return of the other index futures. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Local grey relational analysis; GM(1, N); TGARCH; Spillover eﬀects; Grey prediction

1. Introduction With the ﬂourishing development of the internationalization and liberalization of ﬁnancial markets, various kinds of derivative goods are rampant in the market. This gives rise to ﬁnancial innovation, and the related analyses of the major stock indexes and futures in various countries seem to be very * Corresponding author. Tel.: +886 2 25925252x3490#30; fax: +886 2 25959665. E-mail addresses: [email protected] (L.-M. Kung), [email protected] (S.-W. Yu). 1 Tel.: +886 2 27376123; fax: +886 2 27376777.

important for enterprises or individuals in terms of global investment, hedging, and arbitrage tactics. Grubel (1968) was the ﬁrst to apply internationally diversiﬁed portfolios, and insisted that allocating funds in cross-national securities markets can avoid the risk of investment in a single country. After that, the studies on the dependence of mutual stock markets in various countries have become an important research theme. The magnitude of literatures concerning the prediction of stock prices, the rate of returns of the stock index, and the spillover eﬀect of the rate of returns of the stock price is beyond our imagination (Abhyankar, 1995; Abraham and Seyyed, 2006;

0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.02.046

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

Audrino and Trojani, 2006; Bilson et al., 2005; Chan-Lau and Ivaschenko, 2003; Chang et al., 2000; Chen et al., 2003; Chen, 1998; Darbar and Deb, 1997; Freitas and Rodrigues, 2006; Gebka and Serwa, 2006; Ghosh et al., 1999; Jang and Sul, 2002; Hsu, 2003; Jeon and Jang, 2004; Kanas, 1998; Kanas and Yannopoulos, 2001; Lee and Rui, 2002; Liu and Pan, 1997; Luca et al., 2006; Mun, 2005; Neaime, 2006; Ng, 2000; Wen, 2001; Wu et al., 2005; Zheng, 2004). Eun and Shim (1989), whose research is based on nine stock markets, namely, those in New York, Paris, London, Frankfurt, Toronto, Zurich, Sydney, Hong Kong, and Tokyo, used the vector autoregressive regression (VAR) model to investigate the international transmission mechanism of stock market movements from January 1980 to December 1985. The results revealed that the US stock market had the most inﬂuence on each spot market. Moreover, the variation of stock prices in the US would be transmitted to the other markets in the world as soon as possible. That is, the US stock market has the largest impact on all the other stock markets. On the contrary, the inﬂuence of the Japanese stock market on the other stock markets is minor. According to Theodossiou and Lee (1993), who used the GARCH-M method, the weak statistically signiﬁcant mean spillovers radiate from the stock market of the US to that of Japan; they also radiate to the stock market of UK, Canada, and Germany after which they radiate from the stock market of Japan to that of Germany. However, the volatility spillover between the spot market of the UK and Canada could be shaken by the abroad market, especially by that of the US. Furthermore, the wave of apparent ﬂuctuation can be felt from (1) the US to the UK, Canada, Germany and Japan; (2) the UK to Canada; and (3) Germany to Japan, etc. Yu (2002) listed the ﬁndings of his research as follows: The impact of the returns of American stock markets on Taiwan is signiﬁcant, and the impact of the volatility of American stock markets on Taiwanese stock markets is also signiﬁcant. Further, the grey relational analysis could be a good supplement to traditional statistical methods. Additionally, in the recent research of Yang and Bessler (2004), who explored dynamic price relationships among nine major stock index futures markets by combining an error–correction model with directed acyclic graph (DAG) analysis, it was revealed the Japanese market is isolated from other major stock index futures markets. Meanwhile, the US and the

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UK appear to share leadership roles in the stock index futures markets. Furthermore, the markets of the UK and Germany rather than that of the US exert signiﬁcant inﬂuences on most European markets. Moreover, in Wang et al. (2005) research, in which the return and volatility spillovers from the US and Japanese stock markets to three South Asian capital markets were examined by univariate EGARCH, the return spillovers were discovered in all the three markets. There were also volatility spillovers from the US to the Indian and Sri Lankan markets, and from the Japanese to the Pakistani market. Moreover, according to Gannon (2005), by employing the SIMULT model and MA method to test the volatility transmission and spillover eﬀect between the stock and futures markets of the US and Hong Kong, the volatility rate of the Hang Seng index futures has a one-way transmission eﬀect on the ﬂuctuation of the Hang Seng spot index. In a further step, a signiﬁcant spillover eﬀect could be found from the US index futures market. However, the past researches still mostly focused on the issue regarding the price transmission eﬀect and volatility spillover of international stock markets as well as the international forward markets. In sum, in researches on time series, although genetic algorithms, neural networks, and GARCH are adopted and also yield a good result, the theory of artiﬁcial intelligence has rarely been applied with the grey model theory of artiﬁcial intelligence in a big range step to forecast and conduct a relationship analysis of the intercontinental and intracontinental rates of return of stock price index futures. The proposed aims of this research can be stated in terms of three aspects. (1) One of the aims of this research is to predict the variation of the intercontinental rates of return of the index futures in the American and Eurasian markets by utilizing residual check, rolling check, and optimal a methods of GM(1, 1). Another aim is to explore the forecasting capability of the GM(1, 1) model in comparison with the proposed GARCH/TGARCH. This research also presents a minor discussion on the prediction of the intracontinental rates of return by simulated GARCH/TGARCH. (2) This research also aims to further the relational analysis of the ﬂuctuation of the rates of return of index futures in the three above mentioned major global markets by means of local grey relational theory as well as to diﬀerentiate between the proposed GARCH/ TGARCH model and local grey relational theory. It also aims to test if the grey relational theorem

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can be useful in relatedness analysis. (3) In addition, this research aims to identify the main inﬂuencing factor among the targeted intercontinental markets by applying the GM(1, N) model for the ﬁrst time to ﬁnd out whether in contrast with the past researches, the scarcely applied GM(1, N) can work. In this paper, for the purpose of our study, the American markets cover the Dow–Jones Industrial Average (DJ), Nasdaq (NDQ), and S&P500 (SP) index futures. In the European market, we include the 100 kinds of indexes of the Financial Times of London (FTSE) and France CAC40 (CAC) indexes futures. In the Asian market, the Nikkei 225 (NK) index futures, the Taiwan weighted stock index futures (TX), the ﬁnancial insurance-type stock index futures of Taiwan (TF), and the electronictype stock index futures of Taiwan (TE) are the targets for the analysis. The remainder of this article is organized as follows: Section 2 will describe the unit root test, grey prediction theorem, grey relational analysis theory, and GARCH model including multivariate TGARCH (namely, GJR-GARCH). Section 3 will present the empirical results, and the ﬁnal section provides concluding remarks on this research. 2. Methodology From January 1, 2002 to December 22, 2004, after deleting the data on diﬀerent bargain days for the three major markets, we obtain 628 daily closing prices of major stock price index futures of America, Europe, and Asia, oﬀered by Polaris Refco Futures. The preliminary tests involve the following: transformation of the daily closing price of stock price index futures into the rate of return per day (the index futures closing price of the calculated day is divided by the price on the day before calculation, after which the log values of the transformed items are obtained). Next, it involves conducting a unit root test on the major daily rates of return of index futures to determine whether the time series are stable enough, i.e., to ensure that in the analyzed ﬁnancial time series, no root results exist. After the above mentioned test, we continue to conduct analyses of GM(1, 1) grey predictions, localized grey relations, and GARCH/TGARCH and GM(1, N) for multivariate analysis. Finally, we evaluate the prediction performance between GM(1, 1) and GARCH/TGARCH and conduct a relational analysis of the targeted markets.

2.1. Unit root test Before analyzing the time series, it is necessary to examine the daily rates of return of index futures that appeared to be stationary. If the result matches the null hypothesis H0 that some root results exist, it shows that the analyzed returns are unstable and random; this should be dealt with by means of ﬁrst-order diﬀerentiation steps to make it stationary. Otherwise, no root results exist. 2.2. Grey prediction theory The grey theory deals mainly with the uncertainty of the system model as well as with incomplete information cases. Moreover, in some circumstances, it can deal with the relational analysis concerning the system and model construction. Additionally, the systematic phenomena can be explored and determined by means of prediction and decision making (Wen et al., 2003). Prediction implies making future inferences about uncertain events; this implies that one can estimate the future on the basis of past events, and the results can oﬀer information as a reference during decision making. However, according to Tsao and Hu (1993), the grey prediction is characterized by the following three points: 1. The construction of the grey model by generating methods based on initial sequences to accomplish the goal of regularity without utilizing the original sequences: The generated new sequences are termed ‘‘model’’. The purpose of this method lies in deleting the randomness of the primitive sequences. 2. The building up of the diﬀerential dynamic model: This implies setting up a valid model in an appropriate manner despite insuﬃcient information. 3. The use of the relational analysis instead of regression analysis. A rule should be adopted to deal with initial data in order to make the data regular at the time of grey system modeling. That is, a way of data processing should be adopted as an information supplement and a regular rule to search for the data via data. According to the grey theory, GM model generation can be divided into accumulated generating operation (AGO), inverse accumulated generating operation, and localized generating (Wen et al.,

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

2002). In this research, we adopt AGO, which is described as follows: Assume x(0) is the original sequence (non-negative time series of n periods) xð0Þ ¼ ðxð0Þ ð1Þ; xð0Þ ð2Þ; xð0Þ ð3Þ; . . . ; xð0Þ ðkÞ; k ¼ 1; 2; 3; . . . ; LÞ: Deﬁne the AGO of x(0) equals x(1) as follows: AGOfxð0Þ g ¼ xð1Þ ðkÞ " # 1 2 L X X X ¼ xð0Þ ðkÞ; xð0Þ ðkÞ;.. .; xð0Þ ðkÞ : k¼1

k¼1

k¼1

ð1Þ In addition, this research will utilize both GM(1, 1) to carry out the prediction of the variation of the rates of return as well as the GM(1, N) model for conducting a relationship analysis on the major rates of return of index futures. Meanwhile, GM(1, 1) is composed of residual check, rolling check, optimal a, etc., while GM(1, N) is a commonly used method for grey model construction. 2.2.1. GM(1, 1) model The GM(1, 1) model means ﬁrst diﬀerential and one variable, which is generally used in making a prediction. xð0Þ ðkÞ þ azð1Þ ðkÞ ¼ b;

k ¼ 1; 2; 3; . . . ; L;

zð1Þ ðkÞ ¼ 0:5½xð1Þ ðkÞ þ xð1Þ ðk 1Þ;

ð2Þ

k ¼ 2; 3; . . . ; L;

ð3Þ where z is background value, a is development coefﬁcient, b is grey function coeﬃcient. Generally, we use the least squares method to estimate parameters a and b; moreover, the commonly used prediction equation is b ak ð0Þ a ð0Þ ^x ðk þ 1Þ ¼ ð1 e Þ x ð1Þ e : a 2.2.2. GM(1, N) model The GM(1, N) model implies ﬁrst diﬀerential and N variables and is generally used in a multivariable analysis N X ð0Þ ð1Þ ð1Þ xi ðkÞ þ azi ðkÞ ¼ bj xj ðkÞ; i ¼ 1; 2; 3; . . . ; N ; j¼1

and i 6¼ j; k ¼ 2; 3; . . . ; L:

ð4Þ

Furthermore, the deﬁnition of a and b is the same as that in Eq. (2). The GM(1, N) model is analyzed through the following steps:

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1. Setting up original sequences. ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ T xi ¼ ½xi ð1Þ; xi ð2Þ; xi ð3Þ; . . . ; xi ðkÞ , where i ¼ 1; 2; 3; . . . ; N ; k ¼ 1; 2; 3; . . . ; L. When ð0Þ i ¼ 1; x1 is assumed as the major sequence, the other sequences ði ¼ 2; 3; . . . ; N Þ are named the ð0Þ inﬂuencing sequences. That is, if the ith xi is considered as the main factor, then the others ði ¼ 1; 2; . . . ; i 1; i þ 1; . . . ; N Þ are named the inﬂuence factors. 2. Setting up AGO as follows: ð1Þ

ð1Þ

ð1Þ

ð1Þ

xi ¼ ½xi ð1Þ; xi ð2Þ; . . . ; xi ðkÞT ; i ¼ 1; 2; 3; . . . ; N ; k ¼ 1; 2; 3; . . . ; L:

ð5Þ

3. Combining the AGO sequences with the major sequence: N X ð0Þ ð1Þ ð1Þ xi ðkÞ þ azi ðkÞ ¼ bj xj ðkÞ; i 6¼ j; j¼1

where ð1Þ

ð1Þ

ð1Þ

zi ðkÞ ¼ 0:5xi ðkÞ þ 0:5xi ðk 1Þ; i ¼ 1; 2; 3; . . . ; N ; k P 2:

ð6Þ

4. Using the inverse and matrix method to ﬁnd the values of bN by using the following method: P ¼ ðBT BÞ1 BT YN ; ð0Þ ð0Þ ð0Þ ð0Þ YN ¼ ½yð0Þ y2 yi1 yiþ1 yN T ; 1 ð0Þ

ð0Þ

ð1Þ

ð1Þ

ð0Þ ð0Þ ð0Þ T yi ¼ yi ðkÞ ¼ ½xð0Þ i ð2Þ xi ð3Þ xi ð4Þ xi ðLÞ ; ð0Þ ð0Þ ð0Þ T yi ¼ yi ðkÞ ¼ ½xð0Þ j ð2Þ xj ð3Þ xj ð4Þ xj ðLÞ ; ð1Þ ð1Þ ð1Þ ð1Þ T Zi ¼ ½zð1Þ 1 ðkÞ z2 ðkÞ zi1 ðkÞ ziþ1 ðkÞ zN ðkÞ ; 2 3 ð1Þ ð1Þ 0:5ððxi ð2Þ þ xi ð1ÞÞ 6 7 6 7 ð1Þ 6 0:5ððxð1Þ ð3Þ þ xi ð2ÞÞ 7 i ð1Þ 6 7; zi ðkÞ ¼ 6 7 6 7 4 5 ð1Þ ð1Þ 0:5ððxi ðLÞ þ xi ðL 1ÞÞ ð1Þ B ¼ ½zð1Þ i ðkÞ xj ðkÞ ; where i ¼ 1;2;3;...;N ;

i 6¼ j; k ¼ 2;3;...;L; P ¼ ½a b1 b2 bi1 biþ1 bN T :

5. Comparing the value of bN in order to determine the relationship between the major sequence and the inﬂuencing sequence. 2.2.3. Error analysis of the GM(1, 1) prediction model (1) Traditional formula The GM(1, 1) model is deﬁned as

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b ð0Þ ðkÞÞ=xð0Þ ðkÞj 100%; eðkÞ ¼ jðxð0Þ ðkÞ X

ð7Þ

b ð0Þ ðkÞ: predicted value. xð0Þ ðkÞ: true value; X (2) Optimal a (error analysis toolbox) To reduce the calculation time for the purpose of reducing error analysis, this research uses the optimal a toolbox ushered by Wen (2004), in Grey Systems: Modeling and Prediction. 2.2.4. GM(1, 1) rolling model In the grey theory, AGO and the ﬁrst diﬀerential model have been used to process the data; then, the least squares method was used to solve the ﬁrstorder derivative coeﬃcient, and ﬁnally, to assess and predict the system. In Liu (1998) and Tsai et al.’s (1997) research, the grey rolling model is proposed as follows: Each time the prediction model is built up, it is important to replace and rearrange backward information with forward data in sequence. That is, it is important to input a new datum and in the meantime, update the oldest datum for retaining the numbers of the data into the original model sequence and simultaneously formulate a new numerical sequence. In other words, the GM(1, 1) rolling model can predict the next datum x(0)(5) on the basis of the GM(1, 1) model, which consists of forward data of the sequence ½xð0Þ ð1Þ; xð0Þ ð2Þ; xð0Þ ð3Þ; xð0Þ ð4Þ. Every time the result is found, the above mentioned steps should be repeated; however, the ﬁrst datum of forward sequence x(0)(1) should be replaced by x(0)(2), resulting in the backward data of consequence becoming ½xð0Þ ð2Þ; xð0Þ ð3Þ; xð0Þ ð4Þ; xð0Þ ð5Þ. Then, the next datum x(0)(6) should be predicted. The procedures mentioned above should be repeated continuously until the last datum. The above method is called rolling check.

under incomplete information by means of data processing. The purpose of grey relational analysis is to modify the traditional grey relational grade and to transform the latter from qualitative analysis into the quantitative analytic type. Hence, it quantiﬁes the grey relational grade in terms of the inﬂuencing factors aﬀecting the theme. The grey system and grey prediction theory lay the basis for the grey relational analysis, which deals with the ranks, and not the real value of the grey relational grade. This study regards the grey relational coeﬃcient of the localization grey relational grade as the reference sequence, and regards the remaining coeﬃcients as inspected/ compared sequences. The steps of the grey relational analysis are as follows: 1. Deﬁning the inspected sequences and the reference sequence. Let x0 ¼ ðx0 ðkÞ; k ¼ 1; 2; . . . ; LÞ reference sequence; and xi ¼ ðxi ðkÞ; k ¼ 1; 2; . . . ; LÞ inspected sequences. 2. Conducting data preprocessing: the process methods contain the following contents: ð0Þ ð0Þ (1) Initial method: xi ðkÞ ¼ xi ðkÞ=xi ð1Þ. ð0Þ ð0Þ (2) Maximum method: xi ðkÞ ¼ xi ðkÞ= max xi for all i. ð0Þ ð0Þ (3) Minimum method: xi ðkÞ ¼ xi ðkÞ= min xi for all i, where xi ðkÞ is the generated value. In our research, the original data (the rate of return of index futures) already satisfy comparability; therefore, we do not need any data preprocessing. 3. Finding the diﬀerence sequences D0i ðkÞ among the inspected sequences and the reference sequence. D0i ðkÞ ¼ jx0 ðkÞ xi ðkÞj;

2.3. Grey relational analysis There are some commonly used quantitative methods in system analysis, such as regression analysis, relevance analysis, principal component analysis, and factor analysis. The common characteristics in the aforecited methods are that they require a large sample size and a typical probability distribution. However, in practice, there do exist some diﬃculties. Moreover, grey relational analysis is not restricted by the conditions just mentioned; it can get the relatedness from random factor sequences

i ¼ 1; 2; . . . ; N ;

k ¼ 1; 2; . . . ; L: 4. Finding the max and min D0i ðkÞ. max D0i ðkÞ ¼ Maxi Maxk D0i ðkÞ; min D0i ðkÞ ¼ Mini Mink D0i ðkÞ: 5. Taking distinguishing coeﬃcient 1 (the traditional value equals 0.5, which can also be adopted by the fuzzy method). The main purpose of the distinguishing coeﬃcient is to contrast the background value with the predicted objects. Furthermore, the change

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of the distinguishing coeﬃcient aﬀects only the size of the relative value, no more than to say the inﬂuence on the rank of the grey relational grade (Wen et al., 2003). This research aims to adopt both the traditional and fuzzy approaches. After comparing the results, it can be argued that the distinguishing coeﬃcient 1 determined by the fuzzy method and the traditional approaches lead to the same ranks of grey relational grade. 6. Calculating the grey relational coeﬃcient rðx0 ðkÞ; xi ðkÞÞ by using Deng’s formula. rðx0 ðkÞ; xi ðkÞÞ ¼ ½Dmin þ 1Dmax =½D0i ðkÞ þ 1Dmax : ð8Þ 7. Calculating the grey relational grade (based on equal weights) L 1X Cðx0 ; xi Þ ¼ rðx0 ðkÞ; xiðkÞÞ: ð9Þ L k¼1 8. Ranking according to the magnitude of the grey relational grade. 2.4. ARCH and the multivariate GARCH model The returns of ﬁnancial assets are usually inﬂuenced by the past information; this leads to the variation of the variance of the rate of return and the phenomena of volatility clustering. Engle (1982) once proposed the autoregressive conditional heteroscedasticity (ARCH) model to explain this phenomenon. In his model, the conditional variance is simultaneously aﬀected by both realized error terms of past q periods, and changes with time. Therefore, Engle proves the existence of volatility clustering and the fat tail of the rate of return in contrast with that of the normal distribution. The general form of ARCH(q) is as follows: Rt ¼ aX t þ et ; et ¼ Rt aX t ; et jXt1 N ð0; ht Þ; q X ht ¼ a0 þ bi e2ti ;

ð10Þ

i¼1

where Rt is the dependent variable; Xt is the variable vector, including the lag periods items of independent variables and other simultaneous exogenous variables; Xt1 is available information set until time t 1; ht is conditional variance of heteroscedasticity, which is the linear combination of the past e2ti , where the et jXt1 N ð0; ht Þ denotes that the

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residual items et obey the normal distribution with the mean 0 and the variance ht until time t 1. In terms of expectation value, the ARCH model implies that the contemporary variation is strongly aﬀected by the past variation. In other words, the previous large variation will be followed by a large variation in the same direction in the next period and vice versa. The above characteristic obeys the rule proposed by Fama and French (1992); according to the rule, the volatility clustering often exists in ﬁnancial markets. Engle and Bollerslev (1986) added the past squares of error terms and the past conditional variance into the variance equation. Thus, this enabled the ARCH model to be more ﬂexible and general and it was named the GARCH model. GARCH uses a brief model and few parameters to estimate volatility, enabling the accomplishment of the goal of the parameters condensed in ARCH, proposed by Engle (1982). In addition, the GARCH model can simulate and forecast the future results in a more concise way. Owing to the strength of being able to catch the systematic volatility clustering, until now, the GARCH model has been the main stream tool for variance estimation. The GARCH(p, q) model assumes that the variance of the rate of assets returns obeys a given path, and that the conditional variance of the current rate of return is determined by the sum of the past squares of residual terms of the rate of return and the previous conditional variance of the rate of return. This could be stated as follows: ht ¼ a0 þ

q X j¼1

bj htj þ

p X

a1 e2ti ;

i¼1

p P 0; q P 0; a1 P 0; bj P 0:

ð11Þ

where ht is the conditional variance of the current period; htj is the conditional variance of the rate of return of the past time t j; j ¼ 1 q; e2ti is the squares of the residual terms of the rate of return of the past time t i; i ¼ 1 p. 2.5. Multivariate TGARCH/GJR GARCH model Engle and Ng (1993) utilized Japanese stock prices data to compare the strengths and weaknesses of EGARCH, AGARCH, NGARCH, VGARCH, and GJR-GARCH (namely, Threshold GARCH and TGARCH) models in catching the asymmetry of conditional volatility. They concluded that GJR-GARCH is the best model for asymmetric

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parameter volatility. This research applies the structure of multivariate GJR-GARCH and GARCH to simulate the relative best ﬁt model and then analyzes the prediction eﬀects by combining both EViews 5.0 and the daily rates of return of the stock prices of the major index futures in the American and Eurasian markets,. The following explanation is aimed at the GJR-GARCH time series model. Glosten et al. (1993) made a simple revision to the conventional GARCH model. Thus, the GJRGARCH model is established as follows and has the following implication:

2.6.1. ARCH-LM test Assume that the empirical model is et ¼ y t xt a, and et ¼ vt r2t ; vt N ð0; 1Þ

Rt ¼ bX t þ et ; et jXt1 N ð0; r2 Þ; q p r X X X 2 ht ¼ a0 þ bj htj þ ai e2ti þ ck I tk etk ; j¼1

i¼1

a0 > 0; ai P 0; bi P 0:

items. Second, if the p-value of the JB value is not signiﬁcant; that is, if we cannot reject H0, the residuals obey the normal distribution. Third, if in the meantime, according to the ARCH-LM test, the residuals have no ARCH situation, that is, if we cannot reject H0, no ARCH eﬀect exists in the squared residuals. When the conditions ﬁt, it implies that the estimated model can be accepted; otherwise, the estimative results of the ﬁtted model cannot be accepted. The ARCH-LM test (Engle, 1982) and Ljung–Box Q2 statistics test are stated as follows:

(1) If there is an ARCH(q) eﬀect, the variance equation of the above model will be

k¼1

ð12Þ

In the conditional variance equation, ht, if et1 < 0, it implies bad news and dummy variable I tk ¼ 1. On the other hand, if et1 > 0, it implies good news and the dummy variable I tk ¼ 0. The good and bad news have diﬀerent eﬀects on conditional variance; the good news has ai impact, while the bad news has ai þ ck impact. If ck > 0, it implies that bad news increases volatility. In addition, when ck 6¼ 0, the impact of the news is asymmetric. Moreover, GARCH is a special case of the TGARCH model, where the threshold term is set to zero. 2.6. The goodness of ﬁt of the GARCH/TGARCH model When the conditional variance of the residuals of the regression model appears to be heterogeneous, the estimated coeﬃcients are not eﬀective, and it is necessary to perform a test of autocorrelational conditional variance of heteroscedasticity. In general, there are two methods—the ARCH-LM test (Lagrange Multiplier Test) and the Ljung–Box Q2 test (abbreviated as Q2-stat)—concerning the ARCH/GARCH test. In our research, to conﬁrm the goodness of ﬁt of the models, a Q-stat of standardized residuals, Q2-stat of squares of residuals and the JB value of the Jarque–Bera normality test (JB statistics) are used. First, if the Q-stat of the correlogram of residuals and the Q2-stat of the correlogram of residuals squared are not autocorrelated, that is, if we cannot reject the H0, there is no autocorrelation eﬀect in the residuals/residuals squared

r2t ¼ a0 þ a1 e2t1 þ a2 e2t2 þ þ aq e2tq :

ð13Þ

(2) If there is no ARCH(q) eﬀect, the variance will be constant, that is, r2t ¼ a0 . Therefore, the null hypothesis about whether the ARCH eﬀect exists in conditional variance turns out to be H 0 : a1 ¼ a 2 ¼ a3 ¼ ¼ aq (there is no ARCH eﬀect). (3) The steps of the ARCH-LM test are as follows (Yang, 2005): a. Using of the ordinary least squares (OLS) method to estimate the better/best ﬁt mean equation (the sample size equals to N) y t ¼ xt ^a, where ^a indicates the coeﬃcient from OLS, and the residuals ^et ¼ y t xt ^a; in a further step, the time series ^e2t will be the squared value of ^et . b. Making a supplementary regression of ^e2t , which consists of intercept a0 and q lag periods items, in addition to calculating the determinant coeﬃcient R2, that is, estimating ^e2t ¼ a0 þ^e2t1 þ^e2t2 þ^e2t3 þ þ^e2tq :

ð14Þ

c. Calculating ARCH-LM statistics, namely, N R2 v2 ðqÞ:

2.6.2. Ljung–Box Q2 test The null hypothesis of the Q2 test, which is the same as that of the ARCH-LM test, is written as follows:

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

H 0 : a1 ¼ a2 ¼ a3 ¼ ¼ aq (in other words, there is no ARCH eﬀect) According to Yang (2005), the calculation and test procedures of Q2 test statistics are listed as follows: (1) Utilizing the OLS method to estimate the beﬁtting mean equation (the sample size is assumed to be N), we will obtain residuals ^et and the time series ^e2t . The sample variance of ^e2t can be written as follows: N X ^e2t =N : ^2 ¼ ð^e21 þ^e22 þ^e23 þ þ^e2N Þ=N ¼ r t¼1

ð15Þ (2) Calculating the autocorrelation coeﬃcient q(i) of ^e2t , i is the lag order. , N N X X 2 2 2 2 ^ Þð^eti r ^Þ ^e2 Þ: ð^et r ð^e2t r P ðiÞ ¼ t¼iþ1

t¼1

ð16Þ 2

(3) Calculating the Q (q) statistics, q is the order of the ARCH items in the null hypothesis. Q2 ðqÞ ¼ N ðN þ 2Þ ½qð1Þ=ðN 1Þ þ qð2Þ=ðN 2Þ þ þ qðqÞ=ðN qÞ ¼ N ðN þ 2Þ q X ½qðiÞ=ðN iÞ v2 ðqÞ: ð17Þ i¼1

2.7. Prediction capability analysis of the GARCH model The familiar indicators of the prediction capability of diﬀerent models are root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). The commonly used RMSE and the MAE indexes adopted in our study are deﬁned as follows: " #1=2 X 1 MþN 2 RMSE ¼ ðy ^y t Þ ; ð18Þ N t¼Mþ1 t " # þN 1 MX MAE ¼ jy ^y t j : ð19Þ N t¼Mþ1 t 3. Empirical results The total sample size is 628. The rates of return from 2002/01/01 to 2004/11/30 are referred to as in-the-sample data, which includes 612 data for

1191

the purpose of model estimation. The rates of return from 2004/12/01 to 2004/12/22 are out-of-sample data, consisting of 16 data, for the comparison of prediction abilities. 3.1. Grey relational analysis In this research, the localization grey relational analysis has been employed. That is, only one sequence x0(k) is deﬁned as the reference sequence, whereas the others are referred to as the inspected sequences. Besides, in our study, the distinguishing coeﬃcient 1 is determined by the fuzzy f vv g max max

and traditional methods, wherein 1 = 0.5. The summarized results are as follows: (1) If the daily rate of return of the DJ is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9320), the ﬁndings reveal that the SP, FTSE, NDQ, CAC, and TF have signiﬁcant return spillover eﬀects on the DJ; on the other hand, when 1 equals 0.5, the result will be the same as that obtained by the fuzzy method. (2) If the daily rate of return of the CAC is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1is determined by the fuzzy method (1 = 0.9987), the ﬁndings reveal that the FTSE, MAX (the maximum daily rate of return among the major index futures), DJ, TF, and SP have signiﬁcant return spillover eﬀects on the CAC; on the other hand, when 1 = 0.5, the result is identical. (3) If the daily rate of return of the FTSE is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9915), the ﬁndings reveal that the DJ, SP, NDQ, CAC, and TF have signiﬁcant return spillover eﬀects on the FTSE; on the other hand, when the 1 = 0.5 method is used, the result is the same. (4) If the daily rate of return of the NK is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9651), the ﬁndings reveal that the MAX, CAC, FTSE, TE, and TF have signiﬁcant return spillover eﬀects on the NK; on the other hand, when the traditional way 1 = 0.5 method is adopted, the result is not the same as that obtained by the fuzzy

1192

(5)

(6)

(7)

(8)

(9)

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

method. That is, the MAX, CAC, TE, TX, and FTSE have signiﬁcant return spillover eﬀects on the NK. If the daily rate of return of the Taiwanese weighted stock index futures (TX) is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9950), the ﬁndings reveal that the TF, TE, FTSE, CAC, and MAX have signiﬁcant return spillover eﬀects on the TX; on the other hand, when the conventional way 1 = 0.5 method is adopted, the result is the same. If the daily rate of return of the TE is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9288), the ﬁndings reveal that the TX, MAX, TF, FTSE, and DJ have significant return spillover eﬀects on the TX; on the other hand, when the 1 = 0.5 method is adopted, the outcome is the same as that obtained by the fuzzy method. If the daily rate of return of the NDQ is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9031), the ﬁndings reveal that the DJ, SP, FTSE, CAC, and TF have signiﬁcant return spillover eﬀects on the NDQ; on the other hand, when the 1 = 0.5 method is adopted, the result is not the same as that obtained by the fuzzy method. That is, the DJ, SP, FTSE, CAC, and MAX have signiﬁcant return spillover eﬀects on the NDQ. If the daily rate of return of the SP is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9828), the ﬁndings reveal that the DJ, NDQ, FTSE, CAC, and TF have signiﬁcant return spillover eﬀects on the SP; on the other hand, when the 1 = 0.5 method is adopted, the result is identical. If the daily rate of return of the TF is deﬁned as the reference sequence, and the distinguishing coeﬃcient 1 is determined by the fuzzy method (1 = 0.9864), the outcomes reveal that the TX, TE, FTSE, CAC, and DJ have signiﬁcant return spillover eﬀects on the TF; on the other hand, when the 1 equals to 0.5, the result is as the same as that obtained by the fuzzy method.

The above ﬁndings are summarized in Table 1, which ranks the ﬁrst ﬁve inﬂuencing factors represented by v.

As can be observed from Table 1, the return of the NK index futures has the least impact on the other targeted markets; on the contrary, the return of the DJ index futures has a stronger inﬂuence on the other analyzed markets. On the other hand, the return of the FTSE index futures seems to have the most signiﬁcant spillover eﬀect on the other targeted index futures markets. These results are the same as those by Yang and Bessler (2004). That is, the US and the UK appear to share leadership roles in the stock index futures markets. What is mentioned above implies that the local grey relation theory seems to work when conducting a relationship analysis for index futures returns. 3.2. Grey model construction

1. The performance analysis of the GM(1, 1) prediction model, shown in Table 2. From Table 2, it is clear that the average prediction errors are so signiﬁcant that the GM(1, 1) model has a poor performance with regard to forecasting the rates of return of the indexes futures. 2. The GM(1, N) model for main inﬂuence factor analysis. (1) By considering the daily rate of return of the DJ as the major sequence, the results reveal that the SP (ranked as the top inﬂuencing factor), FTSE (ranked as the second inﬂuencing factor) and the TE, NDQ, and TX (ranked as the third, fourth, and ﬁfth inﬂuencing factors, respectively) have a signiﬁcant impact on the DJ. (2) When the daily rate of return of the CAC is considered as the major sequence, the results show that the DJ (ranked as the top inﬂuencing factor), FTSE (ranked second), MAX, SP, and TE (ranked third, fourth, and ﬁfth, respectively) have a signiﬁcant impact on the CAC. (3) By considering the daily rate of return of the FTSE as the major sequence, the results reveal that the DJ (ranked as the top inﬂuencing factor), SP (ranked second), and the TF, TX, and CAC (ranked third, fourth, and ﬁfth, respectively) have a signiﬁcant impact on the FTSE. (4) When the daily rate of return of the NK is considered as the major sequence, the ﬁndings are that the DJ (ranked as the top

Reference sequence

Return of DJa Return of CACa Return of FTSEa Return of NKb Return of TXa Return of TEa Return of NDQc Return of SPa Return of TFa a

Inﬂuencing factors Return of the DJ

Return of the CAC

Return of the FTSE

Return of the NK

Return of the TX

Return of the TE

Return of the NDQ

Return of the SP

Return of the TF

Return of the MAXd

the

–

v

v

–

–

–

v

v

v

–

the

v

–

v

–

–

–

–

v

v

v

the

v

v

–

–

–

–

v

v

v

–

the

–

v

v

–

–

v

–

–

v

v

the

–

v

v

–

–

v

–

–

v

v

the

v

–

v

–

v

–

–

–

v

v

the

v

v

v

–

–

–

–

v

v

–

the

v

v

v

–

–

–

v

–

v

–

the

v

v

v

–

v

v

–

–

–

–

The result (when 1 equals 0.5) is the same as that obtained by the fuzzy model. When the traditional way 1 = 0.5 method is adopted, the result is not the same as that obtained by the fuzzy method; that is, the MAX, CAC, TE, TX, and FTSE have a signiﬁcant return spillover eﬀects on the NK. c When the traditional way 1 = 0.5 method is adopted, the result is not the same as that obtained by the fuzzy method; that is, the DJ, SP, FTSE, CAC, and MAX have a signiﬁcant return spillover eﬀects on the NDQ. d The return of the MAX is the maximum daily rate of return of index futures among the American and Eurasian markets. b

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

Table 1 Analysis of localized grey relational return spillover eﬀect (distinguishing coeﬃcient 1 is determined by fuzzy method)

1193

1194

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

Table 2 Prediction performance analysis of the GM(1, 1) model Predicted target

Return Return Return Return Return Return Return Return Return

(5)

(6)

(7)

(8)

(9)

of of of of of of of of of

the the the the the the the the the

DJ CAC FTSE NK TX TE NDQ SP TF

Prediction method Average optimal alpha error (%)

Average residual check error (%)

Average rolling check error (%) (5 points)

Average rolling check error (%) (4 points)

24.14 21.57 22.85 35.83 94.45 33.47 503.97 97.12 73.66

139.80 53.15 143.67 15,120.81 1693.26 155.88 996.00 1038.79 322.66

91.86 62.52 1320.71 11,057.65 419.83 185.20 2775.45 244.75 68.46

38.69 31.19 96.96 5383.34 709.11 55.00 277.79 1521.32 73.73

inﬂuencing factor), MAX (ranked second), and the FTSE, TE, and SP (ranked third, fourth, and ﬁfth, respectively) have a signiﬁcant impact on the NK. By considering the daily rate of return of the TX as the major sequence, the ﬁndings are that the DJ (ranked as the top inﬂuencing factor), SP (ranked second), and the FTSE, TF, and MAX (ranked third, fourth, and ﬁfth, respectively) have a signiﬁcant impact on the TX. When the rate of daily return of the TE is considered as the major sequence, the outcomes reveal that the DJ (ranked as the top inﬂuencing factor), the SP (ranked second), and the FTSE, TX, and TF (ranked third, fourth, and ﬁfth) have a signiﬁcant impact on the TE. By considering the daily rate of return of the NDQ as the major sequence, the results show that the DJ (ranked as the top inﬂuencing factor), SP (ranked second), and the FTSE, TX, and TF (ranked third, fourth, and ﬁfth, respectively) have a great impact on the NDQ. When the daily rate of return of the S&P is considered as the major sequence, the ﬁndings reveal that the DJ (ranked as the top inﬂuencing factor), TX (ranked second), and the FTSE, TF, and MAX (ranked third, fourth, and ﬁfth, respectively) have a great impact on the SP. By considering the daily rate of return of the TF as the major sequence, the results are that the DJ (ranked as the top inﬂuencing factor), SP (ranked second), and the TX, FTSE, and TE (ranked third, fourth, and ﬁfth, respectively) have a great impact on the TF.

The detailed ﬁndings are shown in Table 3. According to our research results shown in Table 3, we can conclude that in contrast to the return of the NK, the return of the DJ index futures has the largest impact on the other indexes futures. 3.3. GJR-GARCH/TGARCH This study begins by performing a unit root test on the sample data of the rates of return of major index futures in three markets. When the analyzed data appear to be stationary; that is, when the null hypothesis that a unit root eﬀect exists is rejected, the time series of the daily rates of return of America, Europe, and Asia will be simulated and tested. After the simulation analysis, the regression models of the daily rates of return of the index futures are summarized as follows: 1. The spillover eﬀect among cross-national major index futures returns. The research results are listed in Table 4 (intercontinental analysis). The above mentioned research results reveal that the daily rates of return of the three major index futures of America, the two major index futures of Europe, and the TX index futures of Asia have a two-way return spillover eﬀect on the TE and TF, with the exception of the NK. In addition, the CAC and NK, the CAC and SP, and the NDQ and TX index futures returns also have a signiﬁcant two-way spillover eﬀect (shown in Table 4). 2. The spillover eﬀect of the daily rates of return of major index futures return in each (intracontinental analysis). (1) The spillover eﬀect of the daily rates of return of the important index futures of America.

Main sequence

Return of DJ Return of CAC Return of FTSE Return of NK Return of TX Return of TE Return of NDQ Return of SP Return of

Main inﬂuence factor Return of the DJ

Return of the CAC

Return of the FTSE

Return of the NK

Return of the TX

Return of the TE

Return of the NDQ

Return of the SP

Return of the TF

Return of the MAX

the

–

6

2

8

5

3

4

1

7

9

the

1

–

2

9

6

5

8

4

7

3

the

1

5

–

6

4

8

9

2

3

7

the

1

6

3

–

7

4

9

5

8

2

the

1

7

3

8

–

6

9

2

4

5

the

1

7

3

9

4

–

6

2

5

8

the

1

8

3

9

4

6

–

2

5

7

the

1

9

3

6

2

8

7

–

4

5

TF

1

8

4

9

3

5

7

2

–

6

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

Table 3 Main inﬂuence factor analysis of the GM(1, N) model (the priority is ranked ﬁrst, second, third, etc.) according to the impact scale on the main sequence/dependent variable

1195

1196

Table 4 Spillover eﬀect analysis of the daily rates of return of major index futures (intercontinental analysis) Indep. var.

Dep. var. Return of the DJ

of of of of of of of of of of of of of of of

the the the the the the the the the the the the the the the

DJ CAC FTSE NK TX TE NDQ SP TF FTSE(-1) TE(-1) TE(-2) CAC(-3) TF(-1) TF(-3)

Variance equation C RESID(-1)^2 RESID(-2)^2 RESID(-3)^2 RESID(-4)^2 RESID(-5)^2 GARCH(-1) GARCH(-2) GARCH(-3) GARCH(-4) GARCH(-5) * ** ***

Return of the FTSE 0.631472*

0.017868**

Return of the NK

Return of the TX

Return of the NDQ

Return of the SP

Return of the TF

0.559506*

0.811524* 0.033526* 0.028089** – – – 0.138529* – –

1.544804* 0.564246* 0.124904* 0.134237** –

0.164825 0.798875* 0.080108 0.798284*

0.048187** 0.045202** 0.631948* 0.904533* 0.625259* 0.917470*

0.158030*

1.009821* 0.071199* 0.388057*

0.058223* 1.001049*

Return of the TE

1.345095* 0.580920* 0.124167* 0.142078* 0.286206*

0.224573*

0.061416*

0.048684* 1.878088* 0.236012

*

0.034500 0.038471* 0.047863* 0.057476*

7.217662E08* 0.087167* 0.130917* 0.058669**

0.0639567* 0.121013* 0.855209*

Signiﬁcant at the 0.01 signiﬁcance level. Signiﬁcant at the 0.05 signiﬁcance level. Signiﬁcant at the 0.1 signiﬁcance level.

0.713134* 0.881134* 0.535124* 0.693448* 0.167125*

0.211017* 0.845998* 0.276114* 0.914406*

8.65E08 0.016038 0.031424 0.046660 0.203877* 0.127926 0.624874 0.118889 0.179880 0.384238 0.062974

8.59E08 0.064472** 0.069858** 0.130937*

0.998603* 0.946517* 0.812405*

1.16E07 0.006136* 0.004563*

0.300959* 0.709808*

3.68E07*** 0.354423* 0.228340*

6.86E07* 0.179210*

1.04E07 0.094645*

1.29E08*** 0.105040*

0.802624*

0.784520*

0.907626*

0.887687*

3.11E07** 0.036357**

2.040823* 1.398315** 0.308793

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

Return Return Return Return Return Return Return Return Return Return Return Return Return Return Return AR(1) AR(2) MA(1) MA(2) MA(3)

Return of the CAC

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

a. If the DJ return is deﬁned as the dependent variable (dep. var.), it will be found that the daily returns of the NDQ and SP have a spillover eﬀect on the DJ. A 1% increase of the NDQ will result in a 0.057598% decrease of the DJ; a 1% increase of the SP will also result in a 1.014496% increase of the DJ. b. If the NDQ return is deﬁned as the dep. var., it will be found that the daily returns of the DJ and SP have a spillover eﬀect on the NDQ. A 1% increase of the DJ will result in a 0.541486% decrease of the NDQ; a 1% increase of the SP will also result in a 1.877678% increase of the NDQ. c. If the SP return is deﬁned as the dep. var., it will be found that the daily returns of the DJ and NDQ have a spillover eﬀect on the SP. A 1% increase of the DJ will result in a 0.829715% increase of the SP; a 1% increase of the NDQ will also result in a 0.135028% increase of the SP. The above results are displayed in Table 5. The summarized results show that the three major index futures returns of America have a mutual

1197

spillover eﬀect on each other (the ﬁndings are the same as those obtained in the cross-national analyses). (2) The spillover eﬀect of the daily rates of return of the important index futures of Europe. a. If the FTSE return is deﬁned as the dep. var., it will be found that the daily return of the CAC has a spillover eﬀect on the FTSE. A 1% increase of the CAC will result in a 0.619931% increase of the FTSE. b. If the CAC return is deﬁned as the dep. var., it will be found that the daily return of the FTSE has a spillover eﬀect on the CAC. A 1% increase of the FTSE will also result in a 1.134304% increase of the CAC. The research results are shown in Table 6. The above results displayed in Table 6 are that the two major index futures returns of Europe have a two-way spillover eﬀect (same as the results of the intercontinental analysis). (3) The spillover eﬀect of the daily rates of return of the important index futures of Asia. a. Based on the NK return as the dep. var., it will be found that the daily return of the NK is aﬀected only by the TX, and not by

Table 5 Spillover eﬀect analysis of the DJ, NDQ, and SP index futures returns—GARCH(1, 1) Indep. var.

Dep. var. Return of the DJ

Return of the DJ Return of the NDQ Return of the SP Variance equation C RESID(-1)^2 GARCH(-1) * **

Return of the NDQ *

0.541486 – 1.877678*

– 0.057598* 1.014496* 7.38E08* 0.189077* 0.763739*

1.27E07 0.095855* 0.905329*

Return of the SP 0.829715* 0.135028* – 2.08E08** 0.130870* 0.860399*

Signiﬁcant at the 0.01 signiﬁcance level. Signiﬁcant at the 0.05 signiﬁcance level.

Table 6 Spillover eﬀect analysis of the FTSE and CAC index futures returns—GARCH(1, 1) Indep. var.

Dep. var. Return of the FTSE

Return of the CAC

Return of the FTSE Return of the CAC

– 0.619931*

1.134304* –

Variance equation C RESID(-1)^2 GARCH(-1)

6.05E08 0.091914* 0.904233*

7.33E08 0.081005* 0.916936*

*

Signiﬁcant at the 0.01 signiﬁcance level.

1198

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

the TE and TF. That is, the daily return of the TX has a spillover eﬀect only on the NK. A 1% increase of the TX will result in a 0.402643% increase of the NK. b. Based on the TX return as the dep. var., it will be found that the NK, TE, and TF have a spillover eﬀect on the TX. A 1% increase of the NK results in a 0.043320% increase of the TX; a 1% increase of the TE also results in a 0.576780% increase of the TX; a 1% increase of the TF results in a 0.285716% increase of the TX. c. Based on the TE return as the dep. var., it will be found that the NK, TX, and TF have a spillover eﬀect on the TE. A 1% increase of the NK results in a 0.084487% increase of the TE; a 1% increase of the TX also results in a 1.291801% increase of the TE; a 1% increase of the TF results in a 0.223965% decrease of the TE. d. Based on the TF return as the dep. var., it will be found that the TX and TE have a spillover eﬀect on the TF. A 1% increase of the TX results in a 1.567460% increase of the TF. A 1% increase of the TE also results in a 0.572525% decrease of the TF. The results are presented in Table 7.

The results of (3)a–d summarized in Table 7 reveal that the major index futures returns of Taiwan and Japan—TX and NK—in Asian market have a mutual spillover eﬀect. Furthermore, the TX, TE, and TF have a two-way spillover eﬀect on each other (a result that is partially diﬀerent from the cross-national analysis). The above simulated models related to the return spillover eﬀect among major index futures are tested via the Q-Stat test, Q2-Stat test, and ARCH test; besides, the predicted error values of the simulated model are acceptable in spite of the error items not appearing to be normal. Therefore, the estimation eﬀectiveness of the models adopted above is acceptable. 3. The GARCH eﬀect analyses Concerning the issue of volatility of the daily rates of return of the index futures of America, Europe, and Asia, we can conclude that a signiﬁcant GARCH eﬀect in the intercontinental analysis exists in all the index futures, with the exception of the CAC index futures (shown in Table 4). Furthermore, there is a signiﬁcant GARCH(1, 1) eﬀect in intracontinental markets (revealed in Tables 5–7). 4. The forecasting power of the GARCH/ TGARCH model: the predictive errors, which are shown in Table 8.

Table 7 Spillover eﬀect analysis of the NK, TX, TE, and TF index futures returns—GARCH(1,1) Indep. var.

Return Return Return Return

of of of of

the the the the

Dep. var.

NK TX TE TF

Variance equation C RESID(-1)^2 GARCH(-1) *

Return of the NK

Return of the TX

– 0.402643* 0.052815 0.048694

0.043320* – 0.576780* 0.285716*

1.68E07* 0.001307* 1.002695*

3.32E06* 0.155230* 0.047390

Return of the TE 0.084487* 1.291801*

Return of the TF 0.042604 1.567460* 0.572525* –

– 0.223965* 5.03E07* 0.217842* 0.783516*

1.18E07 –0.011151* 1.003001*

Signiﬁcant at the 0.01 signiﬁcance level.

Table 8 Prediction errors (%) of the GARCH/TGARCH model Return of xxx

RMSE MAE

Predicted targets DJ

CAC

FTSE

NK

TX

TE

NDQ

SP

TF

0.0595 0.0364

0.2220 0.1495

0.1505 0.1157

0.3700 0.3086

0.0976 0.0738

0.1301 0.1120

0.1416 0.1260

0.0440 0.0342

0.1939 0.1284

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4. Summary and conclusion The purpose of this study was to employ the GM(1, 1) model for the prediction of the variation of the returns of major stock index futures, namely, the American, European, and Asian index futures (the DJ, CAC, FTSE, NK, TX, TE, NDQ, SP, and TF index futures), and to then compare the performance between GM(1, 1) and GARCH/ TGARCH. By taking a step further, the GM(1, N) multivariate relatedness model, local grey relational analysis, and the GARCH/TGARCH methods are adopted to determine the spillover eﬀect of the daily mean and volatility of the returns of the major index futures. The ﬁndings are as follows: 1. The prediction ability of the GARCH/TGARCH model concerning the cross-national daily rate of return signiﬁcantly outperforms the GM(1, 1 mid a) model (as shown in Tables 2 and 8). 2. It is revealed thru the simulated GARCH/ TGARCH model, by using EViews 5.0, that the daily rates of return of the major index futures of America, Europe, and Asia (the TX, TE, and the TF) have a two-way return spillover eﬀect. In addition, in the cross-continental analysis, the CAC and NK, the CAC and SP, the SP and TF, and the NDQ and TX index futures returns also have a signiﬁcant spillover eﬀect. Meanwhile, the same results are found in the intracontinental analysis of the American, European, and Asian (the TX, TE, and TF) index futures. However, there are a few diﬀerences between the TX and NK. That is, the TX also has a two-way spillover eﬀect on the NK in the intracontinental analysis; on the contrary, the TX index futures has a oneway spillover eﬀect on the NK only in the intercontinental analysis. 3. With regard to the theme of volatility of the daily rates of return of the index futures of America, Europe, and Asia, the results show that there is a signiﬁcant GARCH eﬀect in the cross-continental markets; this eﬀect, however, does not exist with regard to the CAC index futures. Furthermore, there is a signiﬁcant GARCH(1, 1) eﬀect in each continental market. 4. The use of the grey theory for the analysis of the daily return spillover eﬀect of major index futures in the continents yields the following ﬁndings. First, by the local grey relational analysis, we can ﬁnd the partial factors that have the same

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impact scale that the GARCH/TGARCH model has on each targeted market. That is, the localized grey relational analysis can partly be proved useful in the relatedness analysis of the rates of return. Second, when the GM(1, N) model is adopted, the results are that the return of the DJ index futures is ranked as the most signiﬁcant inﬂuential factor, which is the similar results obtained by Gannon (2005) as well as Yang and Bessler (2004). On the other hand, the return of the NK index futures has the least signiﬁcant inﬂuence on the other targeted indexes futures, which is the same as the ﬁnding of Eun and Shim (1989) as well as Yang and Bessler (2004). The above results show that the GM(1, N) model can work well in a spillover eﬀect analysis of each targeted futures market. Additionally, since the DJ index plays a leading signal in the international futures markets, in order to formulate hedge, speculation, and arbitrage strategies, it is necessary to suggest that the international investors pay more attention to the risk/return characteristics of America in terms of diversiﬁed investments portfolios. References Abhyankar, A.H., 1995. Return and volatility dynamics in the FTSE-100 stock index and stock index futures markets. Journal of Futures Markets 15 (4), 457–488. Abraham, A., Seyyed, F.J., 2006. Information transmission between the Gulf equity markets of Saudi Arabia and Bahrain. Research in International Business and Finance 20 (3), 276–285. Audrino, F., Trojani, F., 2006. Estimating and predicting multivariate volatility thresholds in global stock markets. Journal of Applied Econometrics 21 (3), 345–369. Bilson, C., Brailsford, T., Evans, T., 2005. The international transmission of arbitrage information across futures markets. Journal of Business Finance and Accounting 32 (5–6), 973– 1000. Chang, W.C., Wen, K.L., Chen, H.S., (November 2000). The grey model toolbox: GM(0, N) and GM(1, N). In: 5th national conference on grey theory and application, pp. 115–120. Chan-Lau, J.A., Ivaschenko, I., 2003. Asian ﬂu or Wall Street virus? Tech and non-tech spillovers in the United States and Asia. Journal of Multinational Financial Management 13 (4– 5), 303–322. Chen, H.B., 1998. The integration of grey theory and artiﬁcial neural network in the development of the forecasting model with the application to SIMEX Taiwan stock index future, Master’s Dissertation, Yi-Shou University, Taiwan. Chen, C.W.S., Chiang, T.C., So, M.K.P., 2003. Asymmetrical reaction to US stock-return news: Evidence from major stock markets based on a double-threshold model. Journal of Economics and Business 55 (5–6), 487–502.

1200

L.-M. Kung, S.-W. Yu / European Journal of Operational Research 186 (2008) 1184–1200

Darbar, M., Deb, Partha, 1997. Comovement in international equity market. Journal of Financial Research 3, 305–322. Engle, R.F., 1982. Autoregressive condition heteroskedasticity with estimates of the variance of United Kingdom inﬂation. Econometrica 50, 987–1007. Engle, R.F., Bollerslev, T., 1986. Modelling the persistence of conditional variances. Econometric Reviews 5, 1–50. Engle, R.F., Ng, V.K., 1993. Measuring and testing the impact of news on volatility. Journal of Finance 48 (5), 1749–1778. Eun, C.S., Shim, S., 1989. International transmission of stock market movements. Journal of Financial and Quantitative Analysis 24, 241–256. Fama, E.F., French, K.R., 1992. The cross section of expected stock returns. Journal of Finance 47, 427–466. Freitas, P.S.A., Rodrigues, A.J.L., 2006. Model combination in neural-based forecasting. European Journal of Operational Research 173 (3), 801–814. Gannon, Gerard, 2005. Simultaneous volatility transmissions and spillover eﬀects: US and Hong Kong stock and futures markets. International Review of Financial-Analysis 14 (3), 326–336. Gebka, B., Serwa, D., 2006. Are ﬁnancial spillovers stable across regimes? Evidence from the 1997 Asian crisis. Journal of International Financial Markets, Institutions and Money 16 (4), 301–317. Ghosh, A., Saidi, R., Johnson, K.H., 1999. Who moves the Asia– Paciﬁc stock markets – US or Japan? Empirical evidence based on the theory of co-integration. The Financial Review 34, 159–170. Glosten, L.R., Jagannathan, R., Runkle, D.E., 1993. On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779– 1802. Grubel, H.G., 1968. Internationally diversiﬁed portfolios: Welfare gains and capital ﬂows. American Economic Review 58, 1299–1314. Hsu, Yu-Hung. 2003. The empirical research of macroeconomic indicators to forecast return of stock index-application of genetic algorithm and artiﬁcial neural network, Master’s Dissertation, National Taipei University, Taiwan. Jang, H., Sul, W., 2002. The Asian ﬁnancial crisis and the comovement of Asian stock markets. Journal of Asian Economics 13, 94–104. Jeon, B.N., Jang, B.-S., 2004. The linkage between the US and Korean stock markets: The case of NASDAQ, KOSDAQ, and the semiconductor stocks. Research in International Business and Finance 18 (3), 319–340. Kanas, A., 1998. Volatility spillovers across equity market: European evidence. Applied Financial Economics 8, 245–256. Kanas, A., Yannopoulos, A., 2001. Comparing linear and nonlinear forecasts for stock returns. International Review of Economics and Finance 10 (4), 383–398. Lee, B.S., Rui, O.M., 2002. The dynamic relationship between stock returns and trading volume: Domestic and cross-country evidence. Journal of Banking and Finance 26 (1), 51–78.

Liu, T.K., 1998. An evaluation study of hedging strategies of stock index futures in grey dynamic model-applied on volume weighted stock index futures in Taiwan, Master’s Dissertation, National Taiwan Normal University. Liu, Y.A., Pan, M.S., 1997. Mean and volatility spillover eﬀects in the US and Paciﬁc-Basin stock markets. Multinational Finance Journal 1, 47–62. Luca, G.D., Genton, M.G., Loperﬁdo, N., 2006. A multivariate skew-garch model. Advances in Econometrics 20 (A), 33– 57. Mun, C.K.-C., 2005. Contagion and impulse response of international stock markets around the 9-11 terrorist attacks. Global Finance Journal 16 (1), 48–68. Neaime, S., 2006. Volatilities in emerging MENA stock markets. Thunderbird International Business Review 48 (4), 455– 484. Ng, Angela, 2000. Volatility spillover eﬀects from Japan and the US to the Paciﬁc-Basin. Journal of International Money and Finance 19, 207–233. Theodossiou, P., Lee, U., 1993. Mean and volatility spillovers across major national stock market: Further empirical evidence. The Journal of Financial Research 16, 337–350. Tsai, Q.-X. et al., 1997. The coeﬃcient a adjustment of grey prediction and the confer of the rolling model. Grey System Theory and Application Seminar, 105–108. Tsao, J., Hu, W.-Yi., 1993. Grey System Theory and Method. Northeast Forest University Press, China. Wang, Y., Gunasekarage, A., Power, D.M., 2005. Return and volatility spillovers from developed to emerging capital markets: The case of South Asia. Contemporary Studies in Economic and Financial Analysis 86, 139–166. Wen, K.L., 2001. Study of GM(1, N) with data square matrix. Journal of Grey System 13 (1), 41–48. Wen, K.L. et al., 2002. Grey Prediction Theory and Application. Opentech Company, Taipei. Wen, Kun-Li et al., 2003. Grey relation: Modeling Method and Application. Kao-Li Book Company, Taipei. Wen, K.L., 2004. Grey systems: Modeling and Prediction. Yang’s Scientiﬁc Press, Tucson. Wu, C., Li, J., Zhang, W., 2005. Intradaily periodicity and volatility spillovers between international stock index futures markets. Journal of Futures Markets 25 (6), 553–585. Yang, Yi-Nung, 2005. Time Series Analysis in Economics and Finance. Yeh Yeh Book Gallery, Taipei. April. Yang, J., Bessler, D.A., 2004. The international price transmission in stock index futures markets. Economic Inquiry 42 (3), 370–386. Yu, Tzu-Yao. 2002. An application of VAR, GARCH and grey relational analysis on the relationship between American and Taiwan stock markets. Master’s Dissertation, National Taiwan University of Science and Technology, Taiwan. Zheng, Zou, 2004. An comparative study on forecasting volatility of Shanghai stock index using GARCH model with diﬀerent distributions. Value Engineering 23 (5), 70–72.