Wavelet-based multi-resolution GARCH model for financial spillover effects

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 81 (2011) 2529–2539

Wavelet-based multi-resolution GARCH model for financial spillover effects夽 Shian-Chang Huang ∗ Department of Business Administration, College of Management, National Changhua University of Education, No. 2, Shi-Da Road, Changhua 500, Taiwan Received 24 June 2009; received in revised form 29 January 2011; accepted 25 April 2011 Available online 18 May 2011

Abstract This study proposes a wavelet-based multi-resolution BEKK-GARCH model to investigate spillover effects across financial markets. Compared with traditional multivariate GARCH analysis, the proposed model can identify or decompose cross-market spillovers on multiple resolutions. Taking two highly correlated indices, the NASDAQ (U.S.) and TWSI (Taiwan composite stock index) for analysis, the empirical results show that the NASDAQ returns strongly predict the movements of TWSI on the raw data level, but via wavelet-based multi-resolution analysis we find that the prediction power unevenly spreads over each time scale, and the spillover patterns are totally different as that revealed on the raw data level. The direction and magnitude of return and volatility spillovers significantly vary with their time scales. Considering the fact that heterogeneous groups of investors trade on different time horizons, the results of this study help investors to uncover the complex pattern of return and volatility spillovers on their own horizon, and make a good hedge on their risk. © 2011 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Wavelet analysis; Spillover effects; BEKK-GARCH model; Multi-resolution decomposition; Financial risk

1. Introduction With the continued liberalization of cross-border cash flows, international financial markets have become increasingly interdependent. Investors are highly exposed to the exchange risk and equity price fluctuations over the world. In order to manage such risks, return and volatility spillovers across financial markets are the most important mechanism to analyze. However, international investors are heterogeneous in their trading strategies. Each group of investors operate on their only time horizon. As a result, the transmission and causal relationship between stock markets are different on each time scale. Prior research adopted multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models (McAleer [24]; McAleer et al. [25]) for the analysis. Multivariate GARCH models capture market information on aggregate level. To address these issues, this study proposes a new strategy based on wavelet analysis to improve multivariate GARCH models on the investigation of complex transmission or spillover mechanism across financial markets. 夽 ∗

The author wishes to thank two anonymous referees for their valuable comments and suggestions. Tel.: +886 4 7232105 7420; fax: +886 4 7211292; mobile: +886 953092968. E-mail address: [email protected]

0378-4754/$36.00 © 2011 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2011.04.003

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Recently, it has been widely accepted that most high technology investments over the world concentrate on U.S. and Eastern Asia, especially in U.S. Silicon Valley and Taiwan. In addition, owing to the popularization of computer technology and world-wide network, the quick processing of news has shorten the reaction time for domestic stock markets to new information from international markets. As a consequence, an increasing attention has been given in recent literature to the topic of international transmission of stock market returns and volatility (Eun and Shim [9]; Maasoumi and McAleer [23]; Huang et al. [16]; Hakim and McAleer [15]). This study takes two highly correlated indices, the NASDAQ and TWSI (Taiwan composite stock index), as an illustration to investigate the financial spillover effects. First, the returns series of the NASDAQ and TWSI are decomposed into four time scales by a wavelet basis, and then the return and volatility spillovers under each time scale are analyzed by a multivariate BEKK-GARCH model (Engle and Kroner [8]; Gallagher and Twomey [10]; Dunne [7]). Compared to traditional multivariate GARCH analysis, the new method can identify or decompose the market transmissions on multi-resolutions, and hence help traders to reduce risk on their investing time horizons. Reviewing the applications of wavelet analysis in economics and finance, Ramsey and Zhang [28] investigated foreign exchange data using waveform dictionaries; Davidson et al. [6] analyzed the commodity price behavior by wavelet analysis; Ramsey and Lampart [29,30] have used wavelet analysis to decompose economic relationships of expenditure and income; Pan and Wang [26] have examined the stock market inefficiency by wavelet analysis; Genc¸ay et al. [11,13,14] have used wavelet analysis to investigate scaling properties of foreign exchange volatility and systematic risk (the beta of an asset) in a capital asset pricing model; Recently, In and Kim [17] and Kim and In [19] used wavelet analysis to study the multiscale hedge ratio and relationship between financial variables and real economic activity; Lee [21] employed wavelet analysis to study the transmission of stock market movements; Yamada [31] used a wavelet-based beta estimation to investigate Japanese industrial stock prices. As indicated by Kim and In [19], due to the lack of analytical tools to decompose data into more than two time scales, financial and economic analysts often have been restricted to at most two time scales (the short-run and the long-run). The main contribution of this study lies in combining wavelet analysis and the multivariate BEKK-GARCH model to decompose the return and volatility spillovers over each time scale. Our analysis breaks down the traditional barrier, and reveals the micro transmission mechanism over each time scale. The results of this study help investors to implement good hedge strategies, and reduce their risk in the international investments. Our wavelet-based multi-resolution BEKK-GARCH model shows that the spillovers between the two indices vary with the time scale. Conventional multivariate GARCH estimate is only an “average” of the multi-scale BEKK-GARCH estimates. In the aggregate market spillover will be the outcome of various time-horizon micro-spillovers. In almost all decomposed series, the GARCH coefficients are significant which indicate volatilities of the decomposed series are still heteroscedastic. A interesting result is the prediction power of NASDAQ returns is unevenly spread over four time scales. The return spillovers from U.S. to Taiwan are significant and strong at scales 1, 3, 4, while the spillovers from Taiwan to U.S. are only weak significant at scales 2, 3, 4 and with much smaller magnitudes. The volatility spillovers show a similar pattern that the spillovers from U.S. to Taiwan in almost all components are more significant and strong than the opposite direction from Taiwan to U.S. The volatility spillovers are also unevenly spread in four time scales. The empirical results show that this study successfully decomposes the total spillover into five sub-spillovers. The remainder of the paper is organized as follows. Section 2 describes the wavelet analysis and multiresolution decomposition. Section 3 introduces the BEKK-GARCH models and its statistical properties. Section 4 describes the data used in the study and discusses the empirical findings. Finally, conclusions are given in Section 5.

2. Wavelet analysis Wavelet theory is a comparatively new and powerful mathematical tool for time series analysis. This section reviews two basic tools of wavelet analysis: the discrete wavelet transform (DWT) and the multiresolution decomposition (MRD). For a thorough review of wavelet analysis please refer to Chui [3], Daubechies [5], and Percival and Walden [27]. Practical applications of wavelet analysis is given in Lee [20] and Genc¸ay et al. [12]. Technical details of wavelet analysis are discussed in Bruce and Gao [2].

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First, we introduce the description of a signal in terms of wavelets. There are two types of wavelets defined on different normalization rules. Father wavelets ␾ and mother wavelets ψ. The father wavelet integrates to 1 and the mother wavelet integrates to 0:  φ(t)dt = 1, (1)  ψ(t)dt = 0.

(2)

The father wavelets are good at representing the smooth and low-frequency parts of a signal, and the mother wavelets are useful in describing the detail and high-frequency components. 2.1. Multi-resolution decomposition Any function y(t) in L2 (R) (space for square summable functions) to be represented by a wavelet analysis can be built up as a sequence of projections onto father and mother wavelets generated from ␾ and ψ through scaling and translation as follows:   t − 2j k −j/2 −j −j/2 (3) φj,k (t) = 2 φ(2 t − k) = 2 φ 2j   t − 2j k ψj,k (t) = 2−j/2 ψ(2−j t − k) = 2−j/2 ψ . (4) 2j They form a basis for functional analysis. The wavelet representation of the signal y(t) in L2 (R) can be written as     y(t) = sJ,k φJ,k (t) + dJ,k ψJ,k (t) + dJ−1,k ψJ−1,k (t) + · · · + d1,k ψ1,k (t). k

k

k

k

In the representation J is the number of multi-resolution components, and sJ,k are called the smooth coefficients, and dj,k are called the detailed coefficients. They are defined by  (5) sJ,k = y(t)φJ,k (t)dt  dj,k =

y(t)ψj,k (t)dt,

j = 1, 2 . . . , J.

(6)

The magnitude of these coefficients reflects a measure of the contribution of the corresponding wavelet function to the total signal. The scale factor 2j is also called the dilation factor, and the translation parameter 2j k refers to the location. For the larger the index j, the larger the scale factor 2j , and hence the function gets wider and more spread out. The translation parameter 2j k is matched to the scale parameter 2j in that as the functions ␾J,k (t) and ψj,k (t) get wider, their translation steps are correspondingly larger. For multi-resolution decomposition, the decomposed signals are defined as follows:  SJ (t) = sJ,k φJ,k (t) (7) k

Dj (t) =



dj,k ψj,k (t)

for j = 1, 2, . . . , J.

(8)

k

SJ (t) and Dj (t)|Jj=1 are called the smooth signal and the detail signals, respectively, which constitute a decomposition of a signal into orthogonal components at different scales. A signal y(t) can thus be expressed in terms of these signals: y(t) = SJ (t) + DJ (t) + DJ−1 (t) + · · · + D1 (t).

(9)

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We can represent the discrete wavelet transform in matrix form. Let y be the observation of length T. The string of wavelet coefficients can be ordered from fine scales to coarse scales as ⎞ d1 ⎜d ⎟ ⎜ 2⎟ ⎜ ⎟ ⎜ · ⎟ ⎟ w=⎜ ⎜ · ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ dJ ⎠ ⎛

(10)

sJ where dj and sJ are column vectors of the detailed coefficients (dj,k ) and the smooth coefficients (sJ,k ), respectively. Taking the case that the sample size T is divisible by 2J for example, we have T/2 coefficients at d1,k (i.e. the finest scale), T/4 at d2,k (the next finest scale), and so forth until we find T/2J coefficients for dJ,k and sJ,k (the coarsest scale), for a total amount of coefficients equal to T = T/2 + T/4 + · · · + T/2J + T/2J (the number of coefficients is approximate if T is not divisible by 2J ). The discrete wavelet transform (DWT) can then be represented by a matrix from as follows: w = Wy,

(11)

where W is an T × T real-valued orthonormal matrix defining the DWT which satisfies WT W = IT (T × T identity matrix). We refer the details of W on different wavelet bases and corresponding wavelet filter banks to Percival and Walden [27]. Using the DWT, we may formulate an additive decomposition of y by reconstructing the wavelet coefficients at each scale independently. Component Dj can be represented as Dj = WTj wj which define the j th level wavelet detail associated with changes in y at the scale j. The wavelet coefficients wj = Wj y represent the portion of the wavelet analysis (decomposition) attributable to scale j, while WTj wj is the portion of the wavelet synthesis (reconstruction) attributable to scale j.

2.2. The choice of φ, ψ, and J For the choice of ␾ and ψ, a traditional and popular wavelet function, the daublet with length 8 (designated as “d8”), is used for the study. Alternative choices for the basic wavelet such as “haar”, “symmlet”, and “coiflet” are tried for comparison, but the results are not much affected. For the multi-resolution level J, this study sets J = 4 in empirical analysis, because decomposing raw data into four level is matched with our convection in the investigation of spillover effect; namely, in daily, weekly, and monthly time horizons. Here the highest frequency component D1 represents short-term variations due to shocks occurring at a time scale of 21 = 2 days (daily spillovers), and the next highest component D2 accounts for variations at a time scale of 22 = 4 days, near the working days of a week (weekly spillovers). Similarly, D3 and D4 components represent the mid-term variations at time scale of 23 = 8 and 24 = 16 days (weekly to monthly spillovers), respectively. S4 is the residual of original signal after subtracting D1 , D2 , D3 , and D4 . On the other hand, if we set J too small, only daily effects can be investigated. The maximum J for analysis must satisfy the condition that 2J ≤ T. Too large J subjects to the following drawbacks: typical markets are efficient so that financial time series are difficult to forecast. The difficulties result from the spread of most information or signal energy in high frequency fluctuations (high frequency fluctuations behave like white noises, and thus are difficult to forecast). Consequently, too large J cannot help our investigation, because most energy spread in D1 , D2 , D3 or D1 + D2 , D2 + D3 , not DJ , SJ . The variances in DJ−1 , DJ , SJ are too small to make GARCH-type model significant.

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3. Modeling market dynamics and spillovers by BEKK-GARCH models This research employs a multivariate BEKK-GARCH models (Engle and Kroner [8]) to model the time-varying dynamics and spillovers between the two indices. The BEKK-GARCH model is specified as follows: the conditional mean part,







C1 A11 A12 R1,t−1 ε1,t R1,t = + + , (12) Rt = R2,t C2 A21 A22 R2,t−1 ε2,t where Rt is an 2 × 1 vector of daily returns at time t for each index and εt |Ft−1 ∼N(0, t ). The 2 × 1 vector of random errors, εt , is the innovation or shock for each market at time t. The time varying covariance matrix of BEKK-GARCH is specified as follows:









 2 2 σ12,t σ1,t  α11 α12 w1 0 w1 0 ε1,t−1  α11 α12 ε1,t−1 ε2,t−1 t = = + 2 2 w2 w3 w2 w3 α21 α22 ε2,t−1 α21 α22 σ21,t σ2,t





 2 2 σ12,t−1 σ1,t−1 β11 β12 β11 β12 + , (13) 2 2 β21 β22 β21 β22 σ21,t−1 σ2,t−1 where  is the operator for matrix transpose. The elements of the covariance matrix are expressed below 2 = w2 + α 2 ε2 2 2 2 2 2 2 σ1,t 1 11 1,t−1 + 2α11 α12 ε1,t−1 ε2,t−1 + α12 ε2,t−1 + β11 σ1,t−1 + 2β11 β12 σ21,t−1 + β12 σ2,t−1 2 2 σ12,t = σ21,t = w1 w2 + α11 α21 ε21,t−1 + α11 α22 ε1,t−1 ε2,t−1 + α21 α12 ε1,t−1 ε2,t−1 + α22 α12 ε22,t−1 2 2 + β21 β12 σ21,t−1 + β11 β22 σ12,t−1 + β22 β12 σ2,t−1 +β11 β21 σ1,t−1 2 2 2 2 2 2 2 = (w2 + w2 ) + α2 ε2 σ2,t 2 3 21 1,t−1 + 2α22 α21 ε1,t−1 ε2,t−1 + α22 ε2,t−1 + β21 σ1,t−1 + 2β22 β21 σ12,t−1 + β22 σ2,t−1

With the assumption that the random errors, εt , are normally distributed, the log-likelihood function for the BEKKGARCH model is: T Tn 1 L(θ) = − ln(2π) − (ln |t | + εt |t |−1 εt ), 2 2

(14)

t=1

where T is the number of observations, n = 2 is the number of indices, ␪ is the vector of parameters to be estimated, and all other variables are as previously defined. 3.1. Statistical properties of the model Statistical properties of multivariate GARCH models are only partially known. Available results are richer in the constant conditional correlation vector GARCH (CCC-GARCH) model. Ling and McAleer [22] have proven asymptotic normality of its quasi-maximum-likelihood estimator (QMLE). They also established the conditions for the strict stationarity, the ergodicity, and the higher order moments of the model. McAleer et al. [25] investigated the structure and asymptotic theory for a constant conditional correlation vector ARMA-asymmetric GARCH (VARMAAGARCH) model. They established its underlying structure, including the unique, strictly stationary, and ergodic solution, its causal expansion, and convenient sufficient conditions for the existence of moments. They also established alternative empirically verifiable sufficient conditions for the consistency and asymptotic normality of the QMLE under non-normality of the standardized shocks. The results of BEKK-GARCH model are available only based on some strong assumptions (as stated below). Comte and Lieberman [4] have studied the statistical properties of BEKK-GARCH model. Relying on a result in Boussama [1], they give sufficient, but not necessary conditions for strict stationarity and ergodicity of the model. Based on the existence of the sixth moment and applying the results of Jeantheau [18], they provide conditions for the strong consistency of the QMLE. Furthermore, based on the existence of the eighth moment, they proved the asymptotic normality of the QMLE when the initial state is either stationary or fixed. In our case, each decomposed component Dj = WTj wj is just a linear transformation of the original time series. If the original time series meet the

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Table 1 Descriptive statistics of daily returns for NASDAQ and TWSI indices. All index data encompass the period from January 1998 to December 2004. Each series contains 1663 observations. Series

Mean

Std. deviation

Minimum

Maximum

Skewness

Kurtosis

JBstat

NASDAQ TWSI

0.0187 −0.0161

2.1401 1.8178

−10.1684 −12.7777

13.2546 8.5198

0.1030 −0.1717

2.6203 3.1131

478.6914 679.7176

strong assumptions mentioned above, the statistical properties of Comte and Lieberman [4] are reserved under similar conditions including the strict stationarity and ergodicity of the model, and the strong consistency and asymptotic normality of the QMLE.

4. Empirical application 4.1. Data description The empirical data used in the study are daily NASDAQ (U.S.) and TWSI (Taiwan composite stock index) indices. All of the data are collected from Datastream, which encompass the period from January 1998 to December 2004. Because the two markets operate with different holidays, some daily observations are deleted for matching the two time series. Totally we have 1664 observations in the sampling period. These stock market indices are then transformed to daily returns (by 100 times the log difference). Selected descriptive statistics of daily returns for the two indices are presented in Table 1. Sample means, standard deviations, maximums, minimums, skewness, kurtosis and the Jarque–Bera statistic are reported. In this period NASDAQ index is more volatile than TWSI index, but the Jarque–Bera statistics indicate that the distribution of TWSI is more leptokurtic and skew than that of NASDAQ. From Jarque–Bera statistics these two indices all strongly reject the null hypothesis that their distributions are normal. Therefore, this study adopts a bivariate BEKK-GARCH model for their dynamics and spillovers effects.

Table 2 Estimated coefficients of the BEKK-MGARCH model under raw data. Coefficient C1 C2 A11 A21 A12 A22 w1 w2 w3 α11 α21 α12 α22 β11 β21 β12 β22 * ** ***

***

0.1023 0.0075 −0.0181 0.2013 *** 0.0190 0.0369 0.1295 ** 0.2153 ** 0.3586 *** 0.2640 *** 0.0979 *** −0.0390 * 0.2774 *** 0.9622 *** −0.0303 *** 0.0178 * 0.9263 *** Significant at 10% level. Significant at 5% level. Significant at 1% level.

Std. error

t-Stat

p-Value

0.0397 0.0401 0.0267 0.0202 0.0242 0.0265 0.0604 0.1237 0.0648 0.0169 0.0180 0.0247 0.0225 0.0047 0.0061 0.0114 0.0116

2.5802 0.1861 −0.6769 9.9608 0.7832 1.3912 2.1441 1.7405 5.5374 15.6630 5.4438 −1.5814 12.3466 204.9912 −4.9258 1.5593 79.7463

4.979e−003 4.262e−001 2.493e−001 0.000e+000 2.168e−001 8.217e−002 1.608e−002 4.098e−002 1.782e−008 0.000e+000 2.999e−008 5.699e−002 0.000e+000 0.000e+000 4.620e−007 5.956e−002 0.000e+000

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Table 3 Estimated coefficients of the BEKK-MGARCH model under D1 data.

C1 C2 A11 A21 A12 A22 w1 w2 w3 α11 α21 α12 α22 β11 β21 β12 β22 ***

Coefficient

Std. error

t-Stat

p-Value

0.0000 0.0000 −0.5904*** 0.1427 *** −0.0034 −0.5256 *** 0.0003 *** 0.0006 *** 0.0018 *** 0.6026 *** 0.2400 *** 0.0018 0.8038 *** 0.8043 *** −0.2025 *** −0.01918 *** 0.3475 ***

2.039e−5 6.090e−5 1.676e−2 2.241e−2 4.943e−3 1.338e−2 1.407e−5 9.895e−5 4.503e−5 2.278e−2 3.347e−2 7.468e−3 2.907e−2 7.678e−3 2.786e−2 7.330e−3 3.211e−2

0.3004 −0.0600 −35.2374 6.3701 −0.6942 −39.2723 21.5063 5.6053 40.4660 26.4461 7.1715 0.2372 27.6519 104.7561 −7.2686 −2.6160 10.8235

7.639e−001 9.521e−001 0.000e+000 2.441e−010 4.876e−001 0.000e+000 0.000e+000 2.431e−008 0.000e+000 0.000e+000 1.114e−012 8.125e−001 0.000e+000 0.000e+000 5.573e−013 8.978e−003 0.000e+000

Significant at 1% level.

4.2. Empirical results First, we apply a bivariate BEKK-GARCH model to the raw data. The estimated coefficients are listed in Table 2. In return part, the transmission is uni-directional from U.S. to Taiwan. It is clearly revealed in the strong significance of A21 (A21 = 0.2013∗∗∗ ) and the insignificance of A12 (A12 = 0.0190). The significant A21 indicates that the lagged NASDAQ returns (R1,t−1 ) strongly predict the future movement of TWSI returns (R2,t ). However, the lagged TWSI returns (R2,t−1 ) cannot significantly influence NASDAQ returns (R1,t ). The results are consistent with earlier research (Eun and Shim [9]) that innovations in matured stock markets are unidirectional transmitted to emerging markets. In particular, much of earlier studies found that the U.S. stock market is, by far, the most influential one in the world, and no single foreign market can effectively predict the movements of U.S. market. In volatility part, the transmission is bi-directional, and the direction from U.S. to Taiwan is stronger than that of Taiwan to U.S., because the magnitudes of α21 and β21 are larger than those of α12 and β12 ; namely, | α21 | = 0.0979∗∗∗ >|α12 | = 0.0390∗ and | β21 | = 0.0303∗∗∗ >|β12 | = 0.0178∗ . Based on raw data, Table 2 is an average result indicating the overall spillover effect. However, in order to help heterogeneous groups of investors on different trading horizons, we need further investigation on the long-run, short-run, and various time scale spillovers between the two indices. The major innovation of this paper lies in applying wavelet analysis to examine the complex multi-scale transmissions between interdependent indices. In order to investigate the interaction between the two markets, we apply the Daubechies least asymmetric filters with length 8 to do the multi-resolution decomposition. Market returns are decomposed into five mutually orthogonal different periodicity (frequency) components, ranging from the shortestperiodicity series to the longest-periodicity series; namely, from the high frequency components to low frequency ones. The volatility of the decomposed series is still heteroscedastic. A multivariate BEKK-GARCH model is also applied to evaluate the decomposed signals. The estimated coefficients of each time scale BEKK-GARCH model are shown in Tables 3–7. In these tables, we abbreviate the decomposed components D1 (t), D2 (t), D3 (t), D4 (t), S4 (t) by the following symbols D1, D2, D3, D4, S4. Here the finest scale component D1 represents short-term (or high frequency) variations due to shocks occurring at a time scale of 21 = 2 days, and the next finest component D2 accounts for variations at a time scale of 22 = 4 days, near the working days of a week. Similarly, D3 and D4 components represent the mid-term (weekly to monthly) variations at time scale of 23 = 8 and 24 = 16 days, respectively. S4 is the residual of original signal after subtracting D1, D2, D3, and D4. Variations in stock returns are mainly caused by short-term fluctuations; namely, most signal

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Table 4 Estimated coefficients of the BEKK-MGARCH model under D2 data.

C1 C2 A11 A21 A12 A22 w1 w2 w3 α11 α21 α12 α22 β11 β21 β12 β22 ** ***

Coefficient

Std. error

t-Stat

p-Value

0.0000 0.0000 0.5351 *** 0.0173 −0.0054 ** 0.5812 *** 0.0003 *** 0.0004 *** 0.0011 *** 1.1170 *** −0.1887 *** −0.0052 1.1050 *** 0.5889 *** 0.0975 *** −0.0011 0.5170 ***

1.645e−5 4.818e−5 1.615e−2 1.866e−2 2.609e−3 1.483e−2 2.152e−5 9.287e−5 4.681e−5 2.782e−2 4.822e−2 7.379e−3 1.784e−2 1.334e−2 2.113e−2 3.359e−3 1.336e−2

−0.4524 −0.6857 33.1305 0.9249 −2.0682 39.1863 12.3759 3.8932 23.3651 40.1496 −3.9141 −0.7072 61.9723 44.1472 4.6135 −0.3270 38.6937

6.511e−001 4.930e−001 0.000e+000 3.551e−001 3.878e−002 0.000e+000 0.000e+000 1.029e−004 0.000e+000 0.000e+000 9.443e−005 4.795e−001 0.000e+000 0.000e+000 4.263e−006 7.437e−001 0.000e+000

Significant at 5% level. Significant at 1% level.

power concentrates on D1 and D2. Such phenomenon is consistent with typical expectations that stock returns cannot be predicted in advance (because short-term or high frequency fluctuations are very difficult to forecast). Table 2 reports the conventional BEKK-GARCH estimates on the raw data, while Tables 3–7 report the waveletbased BEKK-GARCH estimates for the two indices. From the shortest-periodicity (or highest frequency) component to the longest-periodicity (or the lowest frequency) component, the return and volatility spillovers are unevenly spread in five components. In almost all decomposed components (except S4) the coefficients of GARCH part (α11 , α22 , β11 , and β22 ) are significant which strongly support that the volatility of each component is still heteroscedastic. The insignificance of S4 is mainly due to it low variances which make GARCH-type model insignificant. Table 5 Estimated coefficients of the BEKK-MGARCH model under D3 data.

C1 C2 A11 A21 A12 A22 w1 w2 w3 α11 α21 α12 α22 β11 β21 β12 β22 ** ***

Coefficient

Std. error

t-Stat

p-Value

0.0000 *** −0.0001 ** 0.8136 *** 0.0637 *** −0.0034 ** 0.7749 *** 0.0002 *** 0.0002 *** 0.0007 *** 1.0116 *** −0.0790 *** 0.0091 1.1008 *** 0.5568 *** 0.0341 *** −0.0019 0.4689 ***

1.135e−005 3.496e−005 6.316e−003 8.086e−003 1.337e−003 6.605e−003 9.887e−006 5.219e−005 2.660e−005 3.636e−002 3.066e−002 5.968e−003 3.532e−002 1.018e−002 1.288e−002 2.597e−003 1.203e−002

−3.8444 −2.5642 128.8106 7.8800 −2.5117 117.3109 18.8921 4.7000 25.7166 27.8190 −2.5769 1.5321 31.1678 54.7174 2.6441 −0.7434 38.9664

1.254e−004 1.043e−002 0.000e+000 5.773e−015 1.211e−002 0.000e+000 0.000e+000 2.817e−006 0.000e+000 0.000e+000 1.000e−002 1.257e−001 0.000e+000 0.000e+000 8.268e−003 4.573e−001 0.000e+000

Significant at 5% level. Significant at 1% level.

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Table 6 Estimated coefficients of the BEKK-MGARCH model under D4 data. Coefficient C1 C2 A11 A21 A12 A22 w1 w2 w3 α11 α21 α12 α22 β11 β21 β12 β22 * ***

***

0.0000 0.0001 *** 0.8897 *** 0.0254 *** −0.0021 *** 0.8940 *** 0.0001 *** 0.0003 *** 0.0003 *** 1.1090 *** −0.0155 0.0045 * 1.0956 *** 0.3855 *** −0.0311 *** −0.0165 *** 0.3802 ***

Std. error

t-Stat

p-Value

7.767e−006 2.365e−005 2.221e−003 4.255e−003 6.415e−004 2.446e−003 3.684e−006 1.303e−005 8.975e−006 4.910e−002 1.916e−002 2.503e−003 4.925e−002 1.569e−002 1.033e−002 9.660e−004 1.550e−002

4.5044 3.1932 400.6288 5.9669 −3.3163 365.5307 29.6902 25.0126 35.4569 22.5838 −0.8095 1.8138 22.2468 24.5747 −3.0102 −17.0511 24.5338

7.121e−006 1.433e−003 0.000e+000 2.950e−009 9.319e−004 0.000e+000 0.000e+000 0.000e+000 0.000e+000 0.000e+000 4.183e−001 6.990e−002 0.000e+000 0.000e+000 2.650e−003 0.000e+000 0.000e+000

Significant at 10% level. Significant at 1% level.

The spillovers or transmissions in return part from U.S. to Taiwan (A21 ) are significant and strong in D1, D3, D4, while the transmissions from Taiwan to U.S. are only weak significant in D2, D3, D4 with much smaller magnitudes. For S4 both directions of return spillover are insignificant. This result is consistent with our intuition, since components D1, D2, D3, D4, S4 constitute the raw data R(t), and thus on the aggregate level (R(t)) the spillover from U.S. to Taiwan (A21 ) should be much stronger than the opposite direction. The five micro-spillovers constitute the macro-spillover, or the total spillover effect is multi-scale decomposed into five sub-spillovers. The spillovers or transmissions in volatility part show a similar pattern that the spillovers from U.S. to Taiwan (α21 , β21 ) in almost all components are more significant and with larger magnitudes than the opposite direction from Taiwan to U.S. (α12 , β12 ). Similarly, both directions of volatility spillover are all insignificant in S4. By similar reasoning, it is the low variances in S4 that leads to volatility clustering insignificant, and thus the volatility spillover insignificant. The volatility spillovers are also unevenly spread in five components. α21 is significantly positive for D1, and significantly Table 7 Estimated coefficients of the BEKK-MGARCH model under S4 data.

C1 C2 A11 A21 A12 A22 w1 w2 w3 α11 α21 α12 α22 β11 β21 β12 β22 ***

Significant at 1% level.

Coefficient

Std. error

t-Stat

p-Value

0.0163 −0.0012 1.0210 *** 0.0262 −0.0082 0.9808 *** 0.1651 0.0656 0.0722 0.1144 −0.0220 0.0157 0.1567 0.2197 −0.2280 0.1148 0.9818

0.0144 0.0247 0.0256 0.0614 0.0304 0.0600 0.1847 1.0334 0.1990 2.6799 1.6482 2.8424 0.8688 1.8622 7.7528 2.7000 1.3454

1.1344 −0.0490 39.8913 0.4264 −0.2707 16.3405 0.8939 0.0635 0.3631 0.0427 −0.0134 0.0055 0.1803 0.1180 −0.0294 0.0425 0.7297

0.1284 0.4805 0.0000 0.3349 0.3933 0.0000 0.1858 0.4747 0.3583 0.4830 0.4947 0.4978 0.4285 0.4530 0.4883 0.4830 0.2328

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negative for D2 and D3 (but with smaller magnitudes). α12 is only weakly significant in D4, and with a small magnitude. Similarly, unevenly spread patterns are also found for β21 , β12 . For statistic characteristics of individual components, at scale 1 (the highest frequency component D1) the two series are mean-reverting with A11 = ∗∗∗ − 0.5904, A22 = ∗∗∗ − 0.5256, and return and volatility spillovers are more significant and stronger in the direction of U.S. to Taiwan. At scales 2 and 3 (the medium frequency components D2 and D3), the two series become weak persistent with A11 = ∗∗∗ 0.5351, A22 = ∗∗∗ 0.5812 for D2 and A11 = ∗∗∗ 0.8136, A22 = ∗∗∗ 0.7749 for D3. Similarly, return and volatility spillovers are more significant and stronger in the direction of U.S. to Taiwan. At scale 4 (the low frequency component D4), the return persistence are even stronger, and the spillovers on return and volatility display similar patterns that the transmissions from U.S. to Taiwan are stronger and more significant. The above phenomena result from the tradings of heterogeneous groups of investors. In the finest time scale, main traders are hedging strategists, speculators, and market makers. They trade the two markets (U.S. and Taiwan) simultaneously, and view NASDAQ as a signal containing leading information. As indicated in Kim and In [19], speculators and market makers intensively trade to realize a quick profit (or minimize a loss) over short time scales ranging from 1 or 2 days. As shown in Section 2.2, D1 represents short-term variations due to shocks occurring at a time scale of 21 = 2 days. Thus the D1 components are strong mean-reverting to reveal the speculators’ short-periodicity behavior, and the NASDAQ index can effectively predict the future movements of TWSI index. In the intermediate time scales D2, D3, the main traders are international portfolio managers who mainly follow index tracking trading strategies. As indicated in Kim and In [19], the trades typically occurs on a weekly to monthly basis, with little attention paid to daily prices. Thus the D2 and D3 components (variations of time scales ranging from 22 = 4 to 23 = 8 days) are no longer mean-reverting, but become persistent. The prediction power of NASDAQ on TWSI also substantially reduced. In the long-term trend components D4 (time scale of 24 = 16 days) and S4, the main traders are central banks which operate on longer time horizons and often consider long-term economic fundamentals for their strategy. Consequently, D4, S4 are more persistent than D2, D3. 5. Conclusion In this article we combine wavelet analysis and BEKK-GARCH models to analyze return and volatility spillovers across the NASDAQ and TWSI indices. We identify the behaviors of different groups of investors, and the results of this study can be used to implement good hedge strategies for simultaneous trading in the two markets. Unlike traditional multivariate GARCH models, the wavelet-based BEKK-GARCH approach allows us to decompose the spillover effect into many sub-spillovers on various time scales according to heterogeneous traders. The empirical results also help us to understand the detailed dynamics of return and volatility transmission on each time scale. Our empirical results show that the prediction power of lagged NASDAQ returns is unevenly spread over five frequency components. The return spillovers from U.S. to Taiwan are significant and strong in D1, D3, D4, while the spillovers from Taiwan to U.S. are only weak significant in D2, D3, D4 and with much smaller magnitudes. This is consistent with the spillovers on aggregate level. This study successfully decomposes the total spillover into five sub-spillovers. The volatility spillovers show a similar pattern that the spillovers from U.S. to Taiwan in almost all components (except S4) are more significant and strong than the opposite direction from Taiwan to U.S. The volatility spillovers are also unevenly spread in five components. The revealed transmission mechanism helps us to identify the behaviors of different groups of investors. The result therefore should be of interest to speculators, hedge managers, international investors, as well as monetary and regulatory authorities, all of whom operate on very different time scales. To summarize, the powerful framework established in this study can also be applied to other information transmission problems among international markets. For example, the problem of international hedges. A challenging future task is to model these dependence structures by wavelet-based multivariate stochastic models, a more flexible model, and test their performance on hedging. References [1] F. Boussama, Ergodicity, Mixing and Estimation in GARCH Models, Ph.D. Dissertation, University Paris 7, 1998. [2] A. Bruce, H.Y. Gao, Applied Wavelet Analysis with SPLUS, Springer-Verlag, New York, 1996.

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