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Wen-Hsiu Kuo al./AsiaManagement Pacific Management Review (2005) 10(1), 131-143 AsiaetPacific Review (2005) 10(1), 131-143 Price Volatility, Trading Activity and Market Depth: Evidence from Taiwan and Singapore Taiwan Stock Index Futures Markets Wen-Hsiu Kuoa,*, Hsinan Hsub, and Chwan-Yi Chiangc a Doctoral Student in Department of Business Administration, National Cheng Kung University, Tainan, Taiwan ; and a lecturer in Department of Finance, Ling Tung College, Taichung, Taiwan b Department of Finance, Southern Taiwan University of Technology, Tainan County, Taiwan. c Department of Business Administration, National Cheng Kung University, Tainan, Taiwan. Accepted March 2004 Available online Abstract This study empirically investigates the relations among price volatility, trading activity and market depth for some selected futures contracts traded on the Taiwan Futures Exchange (TAIFEX) and Singapore Exchange Derivatives Trading Division (SGX-DT). Two different methodologies, the OLS-based and GARCH-based models, are used to test the robustness of the result and to obtain a sensitivity check. The major findings of this investigation are as follows. First, the estimates of the conditional mean function of the two futures markets are consistent with weak-form efficiency. Second, the evidence suggests that volatility is higher during periods of high futures trading volume for the TAIFEX and SGX-DT futures markets, supporting the mixture of distribution hypothesis by Clark (1973). Lastly, inconsistent with market depth theories, this study demonstrates that existing market depth does not mitigate volatility in the SGX-DT and TAIFEX futures markets. This result is noteworthy because it provides evidence that the relation between price volatility and market depth may vary with the market maturity. Keywords: Price volatility; Trading activity; Market depth; Stock index futures; GARCH 1. Introduction Numerous works have examined the relation between trading volume and price volatility for equities and futures. Considerable evidence exists a positive contemporaneous correlation between price volatility and trading volume. Karpoff (1987) extensively reviews previous theoretical and empirical research on the price-volume relation, and finds 18 studies documenting the positive relation. Regarding the theoretical aspect, the two leading models, namely the sequential information model of Copeland (1976) and the mixture of distribution hypothesis of Clark (1973), explain for the positive relationship between volume and price volatility. As for the empirical aspect, various studies such as Crouch (1970), Epps and Epps (1976), Cornell (1981), Harris (1986), Tauchen and Pitts (1983), Gallant, Rossi and Tauchen (1992), Chen, Firth and Rui (2001), and Ciner (2002) demonstrate the positive contemporaneous correlation between volatility and volume in equities and futures markets. The most recent model developed by Blume, Easley, and O’Hara (1994) describes the informational role of volume, and documents in which volume conveys information to the market and then improves the accuracy of price movement forecasts. Furthermore, this model demonstrates how volume can affect market be- havior, rather than simply describing the corre- This study documents the relations among price volatility, trading activity and market depth for selected futures contracts traded on the Taiwan Futures Exchange (TAIFEX) and Singapore Exchange Derivatives Trading Division (SGX-DT) Taiwan Stock Index Futures markets. This subject has been extensively investigated for U.S. capital markets, and also for some developed international markets. However, this issue has been little examined in less-developed markets. The reason for our particular interest in empirical work on the Taiwan Stock Index Futures markets is that it is a rapidly expanding emerging market1 and is characterized by high volatility in Asia. Therefore, this study provides additional empirical evidence by exploring whether the volatility patterns and contribution of market depth of emerging markets differ from those of developed markets. That is, we will test the mixture of distribution hypothesis that there is a positive contemporaneous correlation between price volatility and trading volume, and the market depth theory that price volatility is affected by existing market depth. The main contribution of this work is to present empirical comparisons to fill this gap in the literature. * Corresponding author. E-mail: [email protected] 1 See Hsu and Lin (2002), Lin and Hsu (2003) and Lin, Hsu and Chiang (2004). 131 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 lation between trading volume and price movements. Therefore, the literature widely uses volume as a proxy for information arrival. appears to serve as a good proxy for the order flow concept that relates directly to market depth. Therefore, Bessembinder and Seguin (1993) use expected open interest as a proxy for market depth since it reflects order flow of the futures transactions and willingness of traders to risk their capital. Lamoureux and Lastrapes (1990) first apply Bollerslev’s (1986) GARCH methodology to investigate the price volatility-volume relationship in equity markets, employing daily trading volume as a measure of the information flow (the rate of information arrivals) based on Clark’s (1973) MDH.2 Lamoureux and Lastrapes find that autoregressive conditional heteroscedasticity (ARCH) effects disappear when trading volume is introduced into the conditional variance equation, suggesting that using trading volume as a proxy for the information variable causes price volatility and largely explains the GARCH effects. Jones, Kaul and Lipson (1994) also obtain findings similar to those of Lamoureux and Lastrapes (1990). Unlike the study by Lamoureux and Lastrapes (1990), Najand and Yung (1991) and Foster (1995) show that GARCH effects remain when current volume is included in the conditional variance equation. These authors find a positive relation between trading volume and price volatility, which is consistent with the theoretical models provided by Clark (1973), Copeland (1976) and Blume, Easley, and O’Hara (1994). Consistent with earlier studies, Bessembinder and Seguin (1993) identify a strong positive relation between contemporaneous price volatility and trading volume, but unexpected volume shocks influence price volatility more than expected volume shocks. Additionally, Bessembinder and Seguin find that expected open interest is negatively related to price volatility in all eight futures markets, indicating that increased depth (larger expected open interest) mitigates price volatility and supporting market depth theory. Following the method developed by Bessembinder and Seguin (1993), Ragunathan and Peker (1997), Watanabe (2001) and Fung and Patterson (1999) all investigate the relationships between price volatility, trading activity and market depth in the Australian futures market, the Japanese stock index futures market and the USA futures market, respectively. Their empirical results resemble those of Bessembinder and Seguin (1993) despite differences in sample data . The volume-volatility relationship frequently has been examined, but few works include market depth in the volume-volatility analysis. Kyle (1985) develops a theoretical model of market depth, and defines market depth as the volume of order flows required to move prices by one unit. His model suggests that market depth changes with trading activity and a deep market helps create market conditions that reduce price pressures when trading provides new information. Consequently, market depth theory says that lower price volatility frequently may exist in deeper markets that facilitate trading activity. That is, observed price volatility, conditional on contemporaneous volume, is expected to decrease with increasing market depth. The purpose of this work extends previous research and further examines the relationships between price volatility, trading activity and market depth in TAIFEX and SGX-DT Taiwan Stock Index Futures markets using two different methodologies for testing the robustness of our results. This paper provides insights into the different methodologies specified. First, this work adopts the methodology of Bessembinder and Seguin (1993), in which the conditional mean and volatility equations are estimated sequentially using OLS. Second, rather than selecting a OLS-based volatility estimation framework, we also generalize the methodology of Bessembinder and Seguin (1992, 1993) to a GARCH-based framework following Gulen and Mayhew (2000). The conditional mean and volatility equations are jointly estimated numerically by maximizing the likelihood function (MLE) in GARCH-based model. Earlier studies measure trading activity in futures markets based on volume alone. Recently, various studies3 have examined the relation between open interest and price volatility in developed futures markets. Particularly, Bessembinder and Seguin (1993) first investigate the relations between price volatility, trading activity variables (trading volume and open interest) and market depth for eight futures markets in the U.S. As the level of open interest is a measure of trading activity and reflects the current willingness of futures traders to risk their capital in the futures position in the presence of price volatility, open interest Overall, the findings of this study indicate that, with the exception of slight differences in the estimation results of Monday effect, expected volume and unexpected open interest variables, the estimation results of the remaining variables for the two models are similar for both the TAIFEX and SGX-DT futures markets. Further comparisons with developed markets are as follows. First, like Bessembinder and Seguin (1993), Ragunathan and Peker (1997), Gulen and Mayhew (2000) and Watanabe (2001), the results of this study show that the TAIFEX and SGX-DT futures markets are consistent with weak-form efficiency and support the mixture of distribution hypothesis by Clark (1973). Second, this study tests the market 2 The framework that the MDH can be represented as GARCH model is refered to Lamoureux and Lastrapes (1990). 3 See Bessembinder and Seguin (1992, 1993), Harris and Raviv (1993), Bessembinder, Chan and Seguin (1996), Ragunathan and Peker (1997), Watanabe (2001), Fung and Patterson (1999), Ferris, Park and Park (2002) and Girma and Mougoue (2002). 132 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 equation based on Clark’s (1973) MDH.4 The conclusion indicates that trading volume used as a proxy for the information variable explains much of the GARCH effects, supporting the information-based variance structure. depth theory and finds that existing market depth does not mitigate price volatility in the TAIFEX and SGX-DT futures markets. That is, the finding of this investigation is inconsistent with market depth theory. This evidence contrasts with the findings of developed markets by Bessembinder and Seguin (1993) and Watanabe (2001), who find that large expected open interest (e.g. market depth) mitigates volatility. Therefore, this study concludes that market maturity affects price volatility. Recently, some researchers have studied the relation between price volatility and trading activity measured by trading volume and open interest in futures markets. Using open interest as a proxy for market depth, Bessembinder and Seguin (1993) first investigate the relations between price volatility, trading activity variables (trading volume and open interest) and market depth in eight futures markets in the USA. Bessembinder and Seguin find a strong positive relation between contemporaneous price volatility and trading volume, supporting the mixture of distribution hypothesis by Clark (1973). Besides, the findings of Bessembinder and Seguin also support the market depth hypothesis, namely that expected open interest is positively related to number of traders or total capital dedicated to a market at the beginning of a trading session, and that an increase in the number of traders or amount of capital involved in a market enhances market depth and thus lessens volatility. The remainder of this paper is organized as follows. Section 2 reviews the literature on the relationship between price volatility, trading activity and market depth. Section 3 then describes our data set and the empirical methodology. Subsequently, section 4 presents the empirical results. Finally, section 5 makes concluding remarks. 2. Literature Review The “mixture of distributions hypothesis” (MDH) by Clark (1973) postulates that price volatility and volume jointly and simultaneously respond to the common (mixing) directing variable, which is interpreted as the rate of information flow to the market. Based on the MDH, price volatility and trading volume should be positively correlated. Ragunathan and Peker (1997) investigate the nature of the relationship between volumes, price variability and market depth for four futures contracts traded on the Sydney Futures Exchange, and obtain analytical results that confirm the empirical findings of Bessembinder and Seguin (1993). Copeland (1976) constructs a “sequential arrival of information” model in which the sequential arrival of new information to the market causes both volume and price movements through numerous information shocks and a final equilibrium is established. The model of sequential information arrival indicates a positive relationship between volume and price volatility and suggests that price volatility potentially is predictable based on trading volume knowledge. Watanabe (2001) examines the relation between price volatility, trading volume and open interest for the Nikkei 225 stock index futures traded on the Osaka Securities Exchange (OSE). The OSE gradually changes regulations such as margin requirements, price range and time interval in updating quotaion several times. The regulations were relaxed beginning February 14, 1994. Therefore, the samples prior to and beginning February 14, 1994 are examined separately. The results obtained by Watanabe for the period beginning February 14, 1994 resemble the findings of Bessembinder and Seguin (1993). However, no relationship between price volatility, volume and open interest is found for the period before February 14, 1994. This analytical result shows that the relation between price volatility, volume and open interest may vary with the regulations. Blume, Easley and O’Hara (1994) develop a model that traders can use to learn valuable information regarding a security by observing past price and volume information. The model demonstrates how volume affects market behavior and argues that volume provides data on the quality or precision of about past price movement information. Numerous studies such as Crouch (1970), Epps and Epps (1976), Cornell (1981), Harris (1986), Gallant, Rossi and Tauchen (1992), Chen, Firth and Rui (2001), and Ciner (2002) use equity and futures market data to investigate the price-volume relationship. These empirical results support an almost positive relationship between price volatility and volume, consistent with the theoretical models developed by Clark (1973), Copeland (1976) and Blume, Easley, and O’Hara (1994). Fung and Patterson (1999) also investigate the relationship between price volatility, trading volume and market depth for seven U.S. futures markets.5 Fung and Patterson identify a strong, positive relation between volume and price volatility. Furthermore, Fung and Patterson also find an negative relationship between market depth and volatility, suggesting that increased depth reduces price Lamoureux and Lastrapes (1990) investigate the efficacy of the GARCH effects in GARCH specification in equity markets when the information variable (e.g., trading volume) is incorporated into the conditional variance 4 The framework in which the MDH can be represented as GARCH model is refered to Lamoureux and Lastrapes (1990). 5 The futures data were obtained from the Futures Industry Institute and consist of five currency futures prices and two interest-rate futures prices. 133 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 volatility. Table 1. Features of Sampled Futures Contracts 1. Futures Exchange 2. Underlying asset 3.Contract size 4.Contract month 5.Trading Hours 6.Last Trading Day 7.Tick size TX TF TE Mini-TX TAIFEX Taiwan Capitalization Weighed Index TWD 200 × TX index Spot month, the next calendar month, and the next three quarter months 8:45am-1: 45pm TAIFEX Taiwan Banking & Insurance Sector Index TWD 1000× TF index Spot month, the next calendar month, and the next three quarter months 8:45am-1: 45pm TAIFEX Taiwan Electronic Sector Index TWD 4000× TE index Spot month, the next calendar month, and the next three quarter months TAIFEX Taiwan Capitalization Weighed Index TWD 50 × TX index Spot month, the next calendar month, and the next three quarter months 8:45am-1: 45pm 8:45am-1: 45pm SGX-DT MSCI Taiwan Index USD100 × TiMSCI index March, June, September, December and two nearest serial months 8:45am-12: 15pm 14:45pm-19:00pm The third Wednesday of The third Wednesday of The third Wednesday of the The third Wednesday of Second last business day of the delivery month of the delivery month of delivery month of each the delivery month of the contract month each contract each contract contract each contract 0.1 index point 1 index point 0.2 index point 0.05 index point 1 index point The findings of these studies suggest that an inverse relationship exists between market depth and the price volatility. However, previous studies have neglected the relations between price volatility, trading activity and market depth in emerging futures markets. This work addresses this deficiency and provides an additional evidence by empirically assessing the TAIFEX and SGX-DT Taiwan Stock Index futures markets. from January 2001 to March 2003. Since the Mini-TX is not introduced until April 9, 2001, its sample period runs from April 9, 2001 to March 2003. Following Bessembinder and Seguin (1993), the index futures prices are taken from the nearby futures contracts to obtain a representative futures price series, since the nearby futures contracts usually are most actively traded. The return Rt is computed using the natural logarithmic difference in the price levels. That is, Rt = ln( Pt / Pt −1 ) where Pt denotes the closing price of the nearby contract on day t. Additionally, trading vol- 3. Data and Methodology 3.1 Data ume and open interest are summed across all outstanding maturities to obtain an aggregate measure of activity for each futures contract. Table 1 details the specifications of the five futures contracts investigated here. The Singapore Exchange Derivatives Trading Limited Division (SGX-DT) and Taiwan Futures Exchange (TAIFEX) introduce futures contracts of the Taiwan stock index on January 9, 1997 and July 21, 1998, respectively. The SGX-DT offers Morgan Stanley Capital International Taiwan Stock Index Futures (MSCITX), which contains 77 large capitalization stocks on the Taiwan stock exchange. Moreover, the TAIFEX offers the Taiwan Stock Exchange Capitalization Weighted Stock Index Futures (TX), which includes all of the stocks6 listed on the Taiwan Stock Exchange (TSE). Subsequently, TAIFEX also successively launched Taiwan Stock Exchange Electronic Sector Index Futures (TE), Taiwan Stock Exchange Banking and Insurance Sector Index Futures (TF) and Mini Taiwan Stock Exchange Capitalization Weighted Stock Index Futures (Mini-TX). Table 2 lists some basic descriptive statistics of the returns, absolute returns, trading volume and open interest for each futures contract examined here. The most volatile futures contract is TE, with daily returns standard deviation of approximately 2.598% per day. In contrast, the least volatile futures contract is TX, which has daily standard deviation in returns of 2.115%. The partial autocorrelations of returns at all lags are statistically insignificant, implying that no evidence exists predictability of returns. MSCITX is the most active futures contract, measured in terms of volume and open interest, while the least active is TF. First order autocorrelation coefficients are significant for every trading volume and open interest, showing that high autocorrelation exists in volumes and open interests. The presence of unit roots in volume and open series is tested using the Augmented Dickey-Fuller (ADF) tests. The empirical findings of the tests firmly reject the null hypothesis of a unit root at the 5% significance level for all volume series and open interest series, implying that the series need not be differenced to achieve stationarity, a condition necessary to The daily closing prices, trading volumes, and open interests of TX, TF, TE, Mini-TX and MSCITX are obtained from the TAIFEX and the SGX-DT, respectively. The sample period for TX, TF, TE, and MSCITX extends 6 MSCITX The TSE Weighted Index does not include full-delivery stocks, preferred stocks, and newly listed stock, which are listed for less than one calendar month. 134 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 avoid spurious regressions. Table 2. Summarized Statistics of Futures Returns, Volume and Open Interest Contract Mean TX Return (%) Absolute Return (%) Volume Open Interest -0.0242 1.585 14406 11699 TF Return (%) Absolute Return (%) Volume Open Interest Standard Deviation Partial Autocorrelation at Lag 1 2 2.115 1.399 5294 3405 -0.060 -0.016 0.705** 0.968** 0.052 0.098** 0.360** -0.108** -0.0058 1.6138 1792 2177 2.156 1.4283 1330 1172 -0.024 0.115** 0.86** 0.981** TE Return (%) Absolute Return (%) Volume Open Interest -0.0502 1.9411 3175 3151 2.5977 1.7251 1286 1126 Mini-TX Return (%) Absolute Return (%) Volume Open Interest -0.0489 1.5881 3459 3721 MSCITX Return (%) Absolute Return (%) Volume Open Interest -0.0463 1.8299 15542 41414 Unit Root Test Statistic 4 5 0.054 0.103** 0.173** 0.000 0.069 0.061 0.238** -0.030 -0.037 0.034 0.200** -0.027 -4.11** -4.69** 0.051 0.169** 0.253** -0.203** 0.066 0.057 0.199** 0.042 -0.004 0.061 0.057 -0.045 -0.042 0.091 0.024 0.052 -4.13** -3.71** -0.06 -0.007 0.654** 0.975** 0.082 0.072 0.330** 0.012 0.017 0.119** 0.127** -0.035 0.027 0.014 0.275** -0.043 -0.008 0.039 0.226** 0.037 -3.81** -3.81** 2.1522 1.4516 1504 1392 -0.079 -0.013 0.72** 0.966** 0.062 0.101** 0.369** -0.062 0.056 0.106** 0.206** -0.035 0.082 0.098** 0.244** 0.031 -0.049 0.042 0.169** 0.026 -3.77** -5.00** 2.4072 1.5624 5234 7071 -0.025 -0.056 0.245** 0.740** -0.018 0.075 0.109 0.064 -6.65** -8.57** 0.072 0.060 0.089 0.009 3 0.073 0.009 0.019 0.127** -0.035 -0.065 0.022 0.079 1. To test for the stationarity of the volume and open interest series, we use the augmented Dickey and Fuller (1981) test to perform the unit root test. We employ the Akaike Information Criterion (AIC) to determine the appropriate lag structure in ADF test. The critical value for ADF test at the 5% is –3.42. H0 : unit root, HA : no unit root. 2.** indicate statistically significant at 5% level. Additionally, to capture the evidence8 that past volatilities can predict volumes, the forecast errors are regressed against lags in volatility, volume, and open interest. Finally, the following equation is used to estimate the expected and unexpected trading activity variables: 3.2 Measurement of Trading Activity Variables and Market Depth Following previous studies,7 this work uses trading volume and open interest as proxies for trading activity in futures markets. Employing a technique similar to that of Bessembinder and Seguin (1992, 1993), the trading activity series are decomposed into expected and unexpected components. First, the expected trading activity series is the fitted value from the univariate ARMA model, while the unexpected trading activity series is defined as εˆit , which is the actual trading activity series minus expected trading activity. This step yields ten series of forecast errors, εˆit , { εˆit = activity it − E activity it activity i ,t −τ , τ = 1,..., n } εˆit = ϕ + j =1 ρijσˆ i ,t − j + n k =1 λikVoli ,t − k + n µmOIi ,t − m + ν it (2) m =1 where Voli and OIi denote the trading volume and open interest of futures contract i, respectively. This study utilizes the Akaike information criterion (AIC) to determine the appropriate lag structure in Eqs. (1) and (2). Therefore, this study employs a different model for each time series. The residuals νˆit from the Eq. (2) serve as the unexpected component of each trading volume and open interest series, while the expected component is defined as the difference between the actual trading activity series and the unexpected component, activityit-νˆit . (1) for activity = trading volume or open interest i = TX, TF, TE, Mini-TX, MSCITX 7 n Bessembinder and Seguin (1992, 1993), Ragunathan and Peker (1997), Watanabe (2001) and Fung and Patterson (1999). 8 2 See Schwert (1990), Gallant, Rossi and Tauchen (1992). Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 According to market depth theory by Kyle (1985), Bessembinder and Seguin (1993) argue that market depth varies with recent trading activity, which is proxied by endogenously determined open interest. As the level of open interest is a measure of trading activity and reflects the current willingness of futures traders to risk their capital in the futures position in the presence of price volatility, expected open interest is a good proxy for market depth. Therefore, this study also uses expected open interest as a proxy for market depth. We will test the market depth theory that observed volatility, conditional on contemporaneous volume, is expected to decrease with increasing market depth (larger expected open interest) in TAIFEX and SGX-DT Taiwan stock index futures markets. lagged volatility variables are included in Eq. (4) to measure and accommodate the effect of persistence in price volatility, a phenomenon called volatility-clustering. According to Bessembinder and Seguin (1992, 1993), estimates of daily standard deviations are obtained using the following transformation. σˆ t = Uˆ t To improve study robustness, two commonly used price volatility measures, the OLS-based and GARCHbased models, are employed to examine the relationship between price volatility, trading activity and market depth for selected futures contracts traded on the TAIFEX and the SGX-DT. To determine the existence of an asymmetry effect, namely, the effects of unexpected changes in trading activity variables on price volatility varying according to shock sign, dummy variables are defined as 0 for a negative shock (lower than expected trading activities) and 1 for a positive shock (higher than expected trading activities). The product of the dummy variable and the unexpected trading activity variable thus is created. Consequently, the estimated equation is expressed as follows, 3.3.1 Model 1(OLS-Based Volatility Estimate) First, this study measures the unbiased estimate of daily futures price volatility using the same procedure as Davidian and Carroll (1987) and Bessembinder and Seguin (1992, 1993). The procedure involves iteration between the following two equations: a conditional mean Eq. (3) and a conditional volatility Eq. (4), Rt = α + γ j Rt− j + j =1 σˆ t = δ + n j =1 ω j Uˆ t − j + 4 ρidi + i =1 4 i =1 ηi di + n π j σˆ t − j + U t σˆ t = δ 0 + k =1 µ j Ak + n β jσˆ t − j + et ω j Uˆ t − j + 4 ηidi + i =1 m k =1 µ j Ak + n β jσˆ t − j j =1 (6) + δ 1Un exp Voldum t + δ 2Un exp OIdum t + et where UnexpVoldumt represents the product term of the unexpected volume and the dummy variable, which is 1 for a positive shock on day t and 0 otherwise, and UnexpOIdumt represents the product term of the unexpected open interest and the dummy variable, which is 1 for a positive shock and 0 otherwise. (3) (4) j =1 where Rt denotes futures return on day t, di (i=1,..., 4) represents four dummy variables9 which captures differences in mean and volatility for the ith day of the week and Ut (residuals) is unexpected returns from Eq. (3). σ̂ t is the conditional standard deviation, which is called price volatility in this paper. The method for estimating σ̂ t will be explained below. Ak are the trading activity variables, which are divided into expected and unexpected components, and indicate how trading activity variables affect price volatility in Eq. (4). Lagged unexpected returns are included in Eq. (4) to capture this possible asymmetry (e.g., leverage effect10) in the relation between returns and volatility. That is, negative unexpected return shocks impact volatility more than positive unexpected return shocks. The 9 n j =1 j =1 m (5) Davidian and Carroll (1987) recommend that Eqs. (3) and (4) are estimated sequentially. Equation (3) first is estimated without the lagged volatility estimates by using OLS. The obtained residuals then are transformed into the volatility estimates using Eq. (5), and subsequently Eq. (4) is estimated. Fitted values calculated from Eq. (4) are used as regressors in reestimating Eq. (3). Equation (4) is finally reestimated using the residuals from Eq. (3). 3.3 Measurement of Price Volatility-Empirical Model Specification n π /2 3.3.2 Model 2(GARCH-Based Volatility Estimate) Second, using a procedure inspired by Gulen and Mayhew (2000), this study transforms Eqs. (3) and (4) to the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model proposed by Bollerslev (1986). In contrast to the regression Eqs. (3) and (4) estimated by OLS, the GARCH model empirically has been demonstrated to capture reasonably well the time-varying volatility of financial data returns.11 The estimated GARCH(p, q) specification can be expressed as below, Rt = a + n j =1 d1t takes one when day t is Monday and otherwise takes zero; d2t takes one when day t is Tuesday and otherwise takes zero, and so on. See Black (1976), Christie (1982), Nelson (1991). The leverage effect means a reduction in stock price increases the debt/equity ratio (or leverage) and thus raises equity returns volatility. 10 11 136 bj Rt − j + 4 ci di + εt , εt Ωt −1 ~ N ( 0, ht ) (7) i =1 See Bollerslev (1986), French,Schwert and Stambaugh (1987), Nelson (1991), Bollerslev ,Chou and Kroner (1992). Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 p ht = α 0 + α i ε t2− i + i =1 q 4 β j ht − j + j =1 η i d i + ω 1 ExpVol i =1 + ω 2 Un exp Vol + ω 3 ExpOI different from 0, then negative unexpected return shocks impact volatility more than positive unexpected return shocks. Trading activity variables also are decomposed into expected and unexpected components, and then are inserted into the EGARCH conditional volatility equation as additional explanatory variables. (8) + ω 4 Un exp OI t If the asymmetric term, , is negative and statistically t where Rt denotes futures return on day t, di (i=1,…, 4) represents four dummy variables12 which captures differences in mean and volatility for the ith day of the week, ε t (residuals) is unexpected returns from Eq. (7), ExpVolt and UnexpVolt represent the expected and unexpected volume, and ExpOIt and UnexpOIt are the expected and unexpected open interest, respectively. The conditional mean equation provided in Eq. (7) can be expressed as a function of exogenous variables with an error term. The ht is the one- period ahead forecast variance based on past information (Ω t −1 ) , and is called the conditional variance, which denotes price volatility in this paper. The conditional variance equation specification in Eq. (8) is a linear function of lagged squared residuals and lagged residual conditional variance. Furthermore, Equation (8) can be extended to permit the inclusion of exogenous independent variables such as daily dummies and trading activities in this investigation. 4. Empirical Results Table 3 presents the estimated results of the conditional mean Eq. (3) for each of the five futures contracts. The findings of this study are consistent with those of Bessembinder and Seguin (1993) and Ragunathan and Peker (1997). The largest adjusted R2 is 1.39% for TF. Moreover, none of the day-of-the-week dummies are significant across all contracts. Lagged returns do not have significant explanatory power for all contracts. This evidence can be attributed to weak-form efficiency, indicating that past returns are of limited use in predicting present returns. Nevertheless, lagged volatilities for TE, Mini-TX and MSCITX are positive and statistically significant at the 10% level. Additionally, to determine whether an asymmetry in volume and open interest shocks exists, the UnexpVoldumt and UnexpOIdumt terms are included in Eq. (8). Equation (9), given below, shows that the allowed variation in the effects of unexpected changes in volume and open interest with shock sign is reestimated. p ht = α 0 + q α i ε t2−i + 4 β j ht − j + i =1 Table 4 reports the estimation results of the conditional volatility in Eq. (4). Monday dummy exhibits significant day-of-the-week effect across all contracts. Moreover, all ten coefficient estimates for expected and unexpected volume are positive. However, magnitudes and statistical significance levels show that expected and unexpected volume has heterogeneous effects on volatility. The estimated coefficient on expected volume is significant at the 10% level for two of five contracts, while the estimated coefficient on unexpected volume is significant at the 5 % level for all five contracts. Furthermore, the coefficients of unexpected volume are larger than expected volume. The ratio of unexpected volume to expected volume varied from 2 for the TF to 111 for the Mini-TX, indicating that unexpected volume impacts volatility more than expected volume. This analytical result is consistent with the findings of Bessembinder and Seguin (1993), Ragunathan and Peker (1997) and Watanabe (2001). j =1 η i d i + ω1ExpVolt + ω2Un exp Volt i =1 (9) + ω3 ExpOIt + ω4Un exp OI + ω5Un exp Voldumt + ω6Un exp OIdumt However, the GARCH is a symmetric volatility model and its conditional variance depends on the magnitude rather than the sign of the unexpected disturbance term. Thus, GARCH fails to capture the “leverage effect”. The EGARCH model proposed by Nelson (1991) overcomes this problem. The conditional variance in the model depends on both the sign and magnitude of the unexpected returns, and thus is asymmetric in its response to positive and negative unexpected returns. This study designs a more rigorous model that is used to investigate the relationship between price volatility, trading activity and market depth for the TAIFEX and SGX-DT Taiwan Stock Index Futures under the framework of EGARCH specification. The EGARCH (1,1) specification replaces Eq. (9) with the following Eq. (10). log h t = ω 0 + φ log h t − 1 + α + ω 1 ExpVol t + ω 2 Un exp Vol + ω 5Un exp Voldum 12 t ε t −1 ht −1 t − 2 π + ω 3 ExpOI + ω 6Un exp OIdum +γ t t ε t −1 ht − 1 4 Except for TF, the coefficient estimates for expected open interest are negative and statistically significant. This measurement result contrasts with the findings of Bessembinder and Seguin (1993) and Watanabe (2001), who find that all coefficients relating expected open interest to volatility are negative and significant. Bessembinder and Seguin (1993) note that a significant negative coefficient for the effect of expected open interest on volatility is consistent with the market depth hypothesis that expected open interest is positively related to the number of traders or amount of capital dedicated to a market at the beginning of a trading session, and moreover increasing the number of traders or amount of capital involved in a market improves market depth and thus reduces volatility. Accordingly, inconsistent with theories of market depth, this study does not η idi + i =1 + ω 4 Un exp OI t (10) d1t takes one when day t is Monday and otherwise takes zero, d2t takes one when day t is Tuesday and otherwise takes zero, and so on. 137 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 Table 3. Estimation Results of the Autoregressive Model for Daily Futures Returns Rt = α + Futures contracts Intercept n j =1 γ j Rt− j + 4 i =1 TX ρ id i + TF n j =1 π j σˆ t − j + U t TE Mini-TX MSCITX -0.00364 (-0.81585) 0.00065 (0.17174) -0.00604 (-1.36766) -0.00400 (-1.45935) -0.01538 (-1.89779) -0.00189 (-0.63259) -0.00256 (-0.86081) 0.00071 (0.24139) -0.00019 (-0.06338) 0.20228 (1.00145) 0.05631 (0.33818) -0.00234 (-0.78531) -0.00434 (-1.44190) -0.00259 (-0.85164) 0.00337 (1.12764) 0.00342 (0.00045) 0.03443 (0.13016) -0.00232 (-0.66441) -0.00175 (-0.49276) 0.00114 (0.32781) -0.00153 (-0.43640) 0.25482 (2.81211)* 0.02035 (0.09741) 0.00009 (0.02902) -0.00213 (-0.67522) 0.00157 (0.50080) -0.00063 (-0.19907) 0.29254 (3.30876)* 0.01681 (0.03786) -0.00382 (-1.02495) 0.00034 (0.09468) 0.00069 (0.18327) 0.00247 (0.66704) 0.64834 (3.60275)* 0.03271 (0.11005) Daily dummies Monday Tuesday Wednesday Thursday Sum of lagged volatilities Sum of lagged returns Ljung-Box Q (12) Ljung-Box Q2 (12) Adjusted R2 Regression F-statistic 4.2827 3.8209 7.4793 8.3059 6.4976 38.997** -0.00232 0.91257 121.39** 0.01391 1.53094 45.607** 0.01271 1.85095* 41.845** 0.00475 1.22647 20.160* 0.008753 1.30275 1. Test statistics in parentheses for individual coefficients are t–statistics. Test statistics in parentheses for lagged coefficients are F–statistics for the hypothesis that the sum of the lagged coefficients is zero. 2. Ljung-Box Q (k) statistic tests the joint significance of the autocorrelations of the daily return series up to the k-th order. Ljung-Box Q2 (k) statistic tests the joint significance of the autocorrelations of the squared daily return series up to the k-th order. 3. * and ** indicate statistically significant at the 10% and 5% level, respectively. 4. The appropriate lag-length specification of each equation is determined using Akaike’s Information Criterion (AIC). unexpected volume shocks influence volatility more than negative unexpected volume shocks. find any evidence that higher levels of expected open interest significantly mitigate volatility. Four of the five estimated coefficients relating unexpected open interest to volatility are negative and significant. This analytical result resembles that of Bessembinder and Seguin (1993), implying that an increase in open interest during the trading day lessens the influence of a volume shock on volatility. The sum of the estimated coefficients on the lags of the volatility series is positive and significant for all contracts, displaying significant persistence in volatility in financial markets. The sum of estimated coefficients on lagged unexpected returns is negative and significant for TE and MSCITX, indicating that negative return shocks have a larger effect on subsequent volatility than positive return shocks do, otherwise known as the “leverage effect”. This study reports Ljung-Box test statistics for the 12th order serial correlation including both the residuals and squared residuals in Table 3. The Ljung-Box test statistics for the residual levels are insignificant at the 5% level, revealing that no serial autocorrelation remains in the five futures contracts. However, the Ljung-Box test statistics for the squared residuals are highly significant for all contracts, indicating the existence of time-varying index returns volatility. Furthermore, Tables 4 and 5 also demonstrate that the sum of estimated coefficients on the lags of the volatility series is positive and significant for all contracts. Therefore, these analytical results indicate the volatility-clustering phenomenon observed in the present volatility series. This study Table 5 reports the estimation results of Eq. (6) that allow the effects of unexpected changes in volume and open interest to vary with the shock sign. The coefficients associated with the UnexpVoldumt and UnexpOIdumt are not statistically significant for all contracts, indicating no asymmetries for unexpected volume and open interest. This finding is inconsistent with the findings by Bessembinder and Seguin (1993) and Watanabe (2001), who find positive Additionally, the 1 and 1 coefficients for the persistence of volatility measures are statistically positive for all contracts when current volume and open interest are included in the conditional variance equation simultaneously. Therefore, current volume and open interest do not remove the ARCH effects in the SGX-DT and TAIFEX futures. This finding is consistent with the findings of Najand and Yung (1991) and Foster (1995), but differs from those of 138 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 Table 4. Regressions of Futures Returns Volatility on Expected and Unexpected Trading Activity σˆ t = δ + Futures contracts Intercept Daily dummies Monday Tuesday Wednesday Thursday Expected volume Unexpected volume Expected open interest Unexpected open interest Sum of lagged volatilities Sum of lagged unexpected returns Adjusted R2 Regression F-statistic n j =1 ω j Uˆ t − j + 4 i =1 ηid i + m k =1 µ j Ak + n j =1 β j σˆ t − j + e t TX 0.01186 (3.34560)** TF 0.018215 (7.35168)** TE 0.00981 (2.5436)** Mini-TX 0.01271 (4.03618)** MSCITX 0.03360 (3.11148)** 0.00752 (3.49979)** 0.00004 (0.02049) 0.00313 (1.46045) -0.00056 (-0.26353) 4.57E-07 (1.69090) 1.91E-06 (8.54184)** -2.61E-07 (-0.79992) -4.94E-06 (-5.6621)** 0.21238 (3.67422)** -0.10259 (1.93938) 0.22044 9.18490** 0.00389 (1.83411)* -0.00287 (-1.35707) -0.00140 (-0.65788) -0.00341 (-1.58038) 5.21E-06 (3.80526)** 1.05E-05 (9.52388)** -4.54E-06 (-3.54286)** -8.60E-06 (-2.591493)** 0.14487 (2.94785)* -0.04863 (0.71218) 0.20179 10.46253** 0.00771 (2.90466)** 0.00266 (1.01856) 0.00333 (1.26180) -0.00105 (-0.39339) 3.21E-06 (1.88732)* 7.85E-06 (7.47906)** -1.24E-06 (-0.99024) -2.26E-05 (-6.66454)** 0.35699 (4.18475)** -0.264952 (6.13189)** 0.20803 5.86893** 0.00814 (3.59289)** 0.000667 (0.29810) 0.00215 (0.95247) -0.00103 (-0.45849) 9.49E-08 (0.08660) 1.06E-05 (12.2295)** -1.69E-07 (-0.17595) -4.88E-06 (-2.07135)** 0.27227 (8.66916)** -0.04985 (0.58856) 0.30943 14.21835** -0.00706 (-2.66275)** 0.00202 (0.77245) -0.00820 (-3.04590)** -0.00474 (-1.80079)* 3.07E-08 (0.05643) 1.08E-06 (6.48034)** -2.24E-07 (-1.44620) -1.07E-07 (-0.56401) 0.22482 (2.75632)* -0.17657 (4.58891)** 0.15236 4.55372 1. * and ** indicate statistically significant at the 10% and 5% levels, respectively. 2. Test statistics in parentheses for individual coefficients are t–statistics. Test statistics in parentheses for lagged coefficients are F–statistics for the hypothesis that the sum of the lagged coefficients is zero. 3. The appropriate lag-length specification of each equation is determined using Akaike’s Information Criterion (AIC). Lamoureux and Lastrapes (1990) and Jones, Kaul and Lipson (1994), who find that ARCH effects on price volatility are insignificant when volume is included in the conditional variance equation. and Patterson (1999), few researchers have examined the same relations for emerging markets. This study aims to further extend the methodology used by Bessembinder and Seguin (1993) and improve understanding of the relations among price volatility, trading activity and market depth for selected futures contracts in the Taiwan Futures Exchange (TAIFEX) and Singapore Exchange Derivatives Trading Division (SGX-DT) Taiwan Stock Index Futures markets. Table 7 reports the estimation results of a standard GARCH model [Eqs. (7) and (9)] that allows the effects of unexpected volume and open interest shocks on volatility to vary with shock sign by introducing the UnexpVoldumt and UnexpOIdumt terms. The estimation results are almost identical regardless of the presence of dummy variables, except that the coefficients on the expected volume variable for TE and MSCITX become negative and the coefficient on unexpected open interest for TE becomes insignificant. The coefficients associated with UnexpVoldumt and UnexpOIdumt are not statistically significant for all contracts, indicating that neither of the unexpected trading activity shocks are asymmetric in SGX-DT and TAIFEX. This result also is consistent with the Schwert volatility estimates listed in Table 5. This study investigates this issue using two different modeling methodologies, the OLS-based and GARCHbased models, to test the robustness of the results and to obtain a sensitivity check. Essentially, the findings of this study regarding the relationship among volatility, volume and open interest are robust for different measures of price volatility. Besides slight differences in the estimation results for Monday dummy, expected volume and unexpected open interest variables, the estimation results of the remaining variables are similar across the two models for the TAIFEX and SGX-DT futures markets. Further comparison with developed markets is presented below. 5. Conclusion First, the results of conditional mean function of the two models in the TAIFEX and SGX-DT futures markets are consistent with weak-form efficiency. The findings resemble those found for other developed markets in studies by Bessembinder and Seguin (1993), Ragunathan and Peker Despite extensive research on the relations among trading activity, volatility and market depth in futures markets, including that by Bessembinder and Seguin (1993), Ragunathan and Peker (1997), Watanabe (2001) and Fung 139 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 Table 5. Regressions of Futures Returns Volatility on Trading Activity (Allowing for Asymmetries) σˆ t = δ 0 + Futures contracts Intercept Daily dummies Monday Tuesday Wednesday Thursday Expected volume Unexpected volume UnexpVoldum Expected open interest Unexpected open interest UnexpOIdum Sum of lagged volatilities Sum of lagged unexpected returns Adjusted R2 Regression F-statistic n j =1 ω j Uˆ t − j + 4 i =1 ηi d i + m k =1 µ j Ak + n j =1 β j σˆ t − j + δ 1Un exp Voldumt + δ 2Un exp OIdumt + et TX 0.01177 (3.3156)** TF 0.01798 (7.1292)** TE 0.00972 (2.51604)** Mini-TX 0.01124 (3.5929)** MSCITX 0.03391 (3.0363)** 0.00749 (3.48117)** 0.00003 (0.01829) 0.00326 (1.51685) -0.00051 (-0.23957) 4.28E-07 (1.5662) 1.80E-06 (4.0393)** 1.77E-07 (0.24913) -2.91E-07 (-0.8881) -6.00E-06 (-4.0320)** 2.43E-06 (0.87776) 0.38125 (4.28517)** -0.10332 (1.96408) 0.21866 8.29627** 0.004087 (1.9098)* -0.00279 (-1.31371) -0.00133 (-0.62479) -0.00333 (-1.5456) 5.00E-06 (3.4711)** 1.12E-05 (4.3101)** -1.39E-06 (-0.36328) -4.48E-06 (-3.4843)** -1.27E-05 (-2.1228)** 8.67E-06 (0.85692) 0.15024 (3.18855)* -0.05585 (0.92808) 0.20723 9.56114** 0.00765 (2.8762)** 0.00249 (0.95571) 0.00316 (1.20244) -0.00066 (-0.24971) 2.88E-06 (1.6506) 6.11E-06 (2.8104)** 2.87E-06 (0.84442) -1.36E-06 (-1.0784) -2.58E-05 (-4.2724)** 7.13E-06 (0.67820) 0.38746 (5.09621)** -0.25911 (5.90118)** 0.21372 5.70246** 0.008054 (3.60693)** 0.00058 (0.2662) 0.00144 (0.6468) -0.00099 (-0.44852) -6.30E-07 (-0.5699) 7.71E-06 (4.5213)** 4.09E-06 (1.48395) -2.91E-07 (-0.3072) -1.48E-05 (-3.9279)** 2.61E-06 (1.46801) 0.27185 (8.72858)** -0.04916 (0.57127) 0.31817 13.23680** -0.00701 (-2.6456)** 0.00199 (0.76674) -0.00848 (-3.14368)** -0.00503 (-1.90519)* -3.42E-07 (0.5892) 1.79E-06 (3.6902)** -1.06E-06 (-1.55929) -2.83E-07 (-1.6481) -7.14E-08 (-0.2348) -8.76E-08 (-0.16226) 0.31854 (3.18756)** -0.17741 (4.6917)** 0.15691 4.22498** 1. * and ** indicate statistically significant at the 10% and 5% levels, respectively. 2. Test statistics in parentheses for individual coefficients are t–statistics. Test statistics in parentheses for lagged coefficients are F–statistics for the hypothesis that the sum of the lagged coefficients is zero. 3. The appropriate lag-length specification of each equation is determined using Akaike’s Information Criterion (AIC). (1997), Gulen and Mayhew (2000) and Watanabe (2001). the findings of Bessembinder and Seguin (1993) and Watanabe (2001), namely that positive unexpected volume shocks impact volatility more than negative unexpected volume shocks. Second, as found for the developed markets, both models also show that unexpected volume influences volatility more than expected volume for the TAIFEX and SGX-DT futures markets. This finding indicates that volatility increases with increasing futures volume, but this behavior is driven by the unexpected volume component, rather than the expected component. This finding seems to support the mixture of distribution hypothesis by Clark (1973). Finally, empirical tests of the market depth theory demonstrate that increased levels of expected open interest (viewed as a proxy for market depth) do not mitigate volatility significantly. This finding suggests that market depth does not have an effect on volatility in the SGX-DT and TAIFEX futures markets, implying that existing market depth does not mitigate price volatility. This analytical result contrasts with the findings of the developed markets examined by Bessembinder and Seguin (1993) and Watanabe (2001). Because the Taiwan Stock Index Futures market is a emerging market and characterized by high volatility, this difference can be attributed to market maturity. This result is noteworthy because it provides evidence that the relation between price volatility and market depth may vary with the market maturity. It can be concluded that market maturity affects price volatility. Thus, the findings of this study have implications for financial market regulators concerned with how to enhance market Third, both models demonstrate the volatility-clustering phenomenon observed in the present volatility series for the TAIFEX and SGX-DT futures markets when current volume and open interest are included in the conditional variance equation. This analytical result is consistent with those of Bessembinder and Seguin (1993), Ragunathan and Peker (1997), Watanabe (2001), but is inconsistent with the findings of Lamoureux and Lastrapes (1990). Fourth, no asymmetric relationship is found between unexpected trading activity and volatility in the TAIFEX and SGX-DT futures markets. This finding contrasts with 140 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 depth in the SGX-DT and TAIFEX futures markets. Table 6. Effect of Futures Trading Activity on Volatility-GARCH Model Rt = a + ht = α 0 + p i =1 α i ε t2− i + Panel A Conditional Mean Equation Futures contracts Intercept Daily dummies Monday Tuesday Wednesday Thursday Sum of lagged returns q j =1 n j =1 β j ht − j + b j Rt− j + 4 i =1 4 i =1 c i d i + ε t , ε t Ω t −1 ~ N ( 0 , h t ) η i d i + ω 1 ExpVol t + ω 2Un exp Vol t + ω 3 ExpOI t + ω 4Un exp OI TX -0.00048 (0.7855) TF -0.00069 (0.6989) TE -0.00073 (0.8363) Mini-TX -0.00076 (0.7309) TiMSCI -0.00185 (0.3880) -0.00109 (0.6773) -0.00193 (0.4672) 0.00136 (0.6042) 0.00074 (0.7763) 0.02196 (0.8679) -0.00172 (0.4997) -0.00311 (0.2141) -0.00064 (0.8023) 0.00082 (0.7206) 0.06446 (0.5066) -0.00168 (0.7039) 4.21E-05 (0.9931) 0.00080 (0.8570) -8.96E-05 (0.9858) 0.02192 (0.8710) 0.00019 (0.9548) -0.00047 (0.8812) 0.00113 (0.6871) 0.00039 (0.9151) -0.0033 (0.9743) -0.00139 (0.6470) 0.00266 (0.4389) 0.00188 (0.5715) 0.00266 (0.3328) 0.0720 (0.5439) TE GARCH (2,2) 0.00051 (0.0228)** 0.17565 (0.0102)** 0.04000 (0.6282) 0.47999 (0.0771)* 0.03999 (0.8975) -1.80E-05 (0.9323) -0.00011 (0.5244) -0.00014 (0.4205) -3.25E-05 (0.8535) 3.38E-08 (0.5678) 1.24E-07 (0.0018)** -2.11E-08 (0.6611) -4.48E-07 (0.0001)** Mini-TX GARCH (2,2) 0.00036 (0.0088)** 0.15000 (0.0713)* 0.04000 (0.6468) 0.48000 (0.0078)** 0.04000 (0.8557) -4.78E-05 (0.7762) -0.00016 (0.1922) -8.68E-05 (0.5034) -6.41E-05 (0.6330) 2.10E-08 (0.3700) 1.06E-07 (0.0000)** -1.52E-08 (0.4908) -4.32E-08 (0.4206) TiMSCI GARCH (2,1) 0.00062 (0.0018)** 0.13332 (0.1071) 0.04444 (0.5939) 0.53333 (0.0000)** Panel B Conditional Variance Equation Intercept 1 TX GARCH (1,2) 1.17E-05 (0.8524) 0.13332 (0.0415)** TF GARCH (1,1) 9.01E-05 (0.0673)* 0.14994 (0.0040)** 0.53333 (0.0334)** 0.04444 (0.8423) 9.37E-05 (0.3083) -2.51E-05 (0.7374) 4.17E-05 (0.5884) -1.61E-05 (0.8249) 1.67E-09 (0.7949) 2.97E-08 (0.0000)** -9.70E-10 (0.8767) -7.84E-08 (0.0012)** 0.59996 (0.0000)** 2 1 2 Monday Tuesday Wednesday Thursday Expected volume Unexpected volume Expected open interest Unexpected open interest 5.78E-06 (0.9404) -0.00013 (0.1194) -2.77E-05 (0.6158) -0.00016 (0.0067)** 4.07E-09 (0.8974) 2.65E-07 (0.0000)** -4.21E-08 (0.0285) -2.12E-07 (0.0379)** -0.00018 (0.0321)** 0.00010 (0.2570) -0.00013 (0.2029) -0.00027 (0.0014)** 2.84E-08 (0.0069)** 3.39E-08 (0.0000)** -2.74E-09 (0.3471) 1.81E-09 (0.7094) Model Diagnostics Test on Standardized Residuals Ljung-Box Q (12) Ljung-Box Q2 (12) ARCH (12) 5.8042 (0.926) 16.728 (0.160) 15.4364 (0.2184) 4.0047 (0.9830) 8.0298 (0.783) 7.8705 (0.7951) 9.4180 (0.667) 8.3470 (0.106) 17.0041 (0.118) 12.828 (0.382) 24.231 (0.029)** 25.050 (0.034)** 9.8115 (0.632) 7.3806 (0.112) 15.9548 (0.1933) 1. The number in parentheses are the p-values. 2. LB Q (12) and LB Q2 (12) are the Ljung-Box statistics applied on the standardized and squared standardized residuals, respectively. ARCH (12) is the statistics used to test whether standardized residuals exists ARCH effect up to the order 12. 3. * and ** indicate statistically significant at the 10% and 5% levels, respectively. 141 Wen-Hsiu Kuo et al./Asia Pacific Management Review (2005) 10(1), 131-143 Table 7. Effect of Futures Trading Activity on Volatility-GARCH Model with Dummy Variables Rt = a + ht = α 0 + p i =1 αiε t2−i + q j =1 β j ht − j + 4 i =1 n j =1 b j Rt− j + 4 i =1 c i d i + ε t , ε t Ωt −1 ~ N ( 0 , h t ) ηi di + ω1ExpVolt + ω 2Un expVolt + ω3 ExpOIt + ω 4Un exp OI + ω5Un expVoldumt + ω6Un exp OIdumt Panel A Conditional Mean Equation Futures contracts Intercept TX -0.00054 (0.7602) TF -0.00097 (0.6143) TE -0.00073 (0.8319) Mini-TX -0.00075 (0.8271) TiMSCI -0.00180 (0.3732) -0.00032 (0.9172) -0.00057 (0.8673) 0.00042 (0.8908) 0.00022 (0.9402) 0.0219 (0.871) -0.00151 (0.5730) -0.00309 (0.2283) -0.00063 (0.8123) 0.00074 (0.7537) 0.06439 (0.5074) -0.00168 (0.6943) 4.25E-05 (0.9929) 0.00080 (0.8587) -8.94E-05 (0.9856) 0.03508 (0.7804) 0.00022 (0.9577) -0.00047 (0.9142) 0.00113 (0.7898) 0.00039 (0.9315) -0.00692 (0.9525) -0.00138 (0.6647) 0.00267 (0.3782) 0.00190 (0.5774) 0.00266 (0.4202) 0.08536 (0.7158) Daily dummies Monday Tuesday Wednesday Thursday Sum of lagged returns Panel B Conditional Variance Equation Intercept 1 TX GARCH (1,2) 0.00010 (0.1212) 0.13333 (0.0215)** TF GARCH (1,1) 0.00010 (0.0549) 0.14985 (0.0015)** 8.07E-06 (0.9233) -0.00015 (0.0047)** -3.82E-05 (0.4683) -0.00016 (0.0139)** 1.15E-08 (0.7395) 2.44E-07 (0.0001)** -1.89E-09 (0.9869) -2.97E-08 (0.2320) -3.81E-07 (0.0499)** 4.20E-07 (0.1843) TE GARCH (2,2) 0.00050 (0.0335)** 0.12000 (0.1118) 0.04000 (0.6148) 0.67999 (0.0195)** 0.04000 (0.8949) -4.09E-05 (0.8381) -7.96E-05 (0.6163) -9.72E-05 (0.5798) -2.67E-05 (0.8770) -3.00E-08 (0.5031) 1.46E-07 (0.0441)** -2.11E-08 (0.8562) -1.77E-08 (0.6525) -1.54E-07 (0.5019) -3.79E-07 (0.30900) Mini-TX GARCH (2,2) 0.00037 (0.0001)** 0.12000 (0.1974) 0.04000 (0.6196) 0.48000 (0.0412)** 0.04000 (0.8512) -5.49E-05 (0.6658) -0.00014 (0.1296) -6.23E-05 (0.5889) -7.74E-05 (0.4825) 1.92E-08 (0.3386) 1.28E-07 (0.0000)** 5.56E-08 (0.4333) -1.53E-08 (0.3910) -2.18E-08 (0.3812) -2.83E-07 (0.2169) TiMSCI GARCH (2,1) 0.00055 (0.0039) 0.13333 (0.1092) 0.04444 (0.6191) 0.53333 (0.0002)** 0.0720 (0.5522) -2.47E-05 (0.7713) 5.66E-05 (0.5407) -6.77E-05 (0.4408) -5.08E-05 (0.6865) -1.06E-08 (0.5216) 3.78E-08 (0.0037)** -1.58E-08 (0.5105) -3.41E-09 (0.4985) 1.14E-09 (0.8786) -1.63E-08 (0.1012) 3.9760 (0.9840 7.3295 (0.835) 7.1379 (0.8483) 8.6695 (0.731) 19.772 (0.072) 20.3916 (0.060) 12.940 (0.373) 22.575 (0.032)** 23.3324 (0.025)** 8.1982 (0.769) 20.962 (0.051) 17.505 (0.1315) 2 0.53333 (0.0093)** 0.04444 2 (0.8027) 7.30E-05 Monday (0.4570) -8.44E-05 Tuesday (0.2937) -8.02E-05 Wednesday (0.2812) -5.96E-05 Thursday (0.4376) 9.89E-10 Expected volume (0.8732) 2.05E-08 Unexpected volume (0.0053)** 2.44E-08 UnexpVoldum (0.2123) -9.87E-10 Expected open interest (0.7378) -1.24E-07 Unexpected open interest (0.0118)** 9.58E-08 UnexpOIdum (0.1927) Model Diagnostics Test on Standardized Residuals 8.9643 Ljung-Box Q (12) (0.706) 11.563 Ljung-Box Q2 (12) (0.481) 11.4361 ARCH (12) (0.4919) 1 0.59992 (0.0000)** 1. 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