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TIME OF THE DAY EFFECT IN THE MALAYSIAN STOCK MARKET
Labuan Bulletin OF INTERNATIONAL BUSINESS & FINANCE Labuan Bulletin of International Business & Finance 2(1), 2004, 31-49 ISSN 1675-7262 TIME-OF-THE-DAY EFFECT IN THE MALAYSIAN STOCK MARKET Goh Kim Leng∗a and Kok Kim Lianb a Faculty of Economics and Administration, University of Malaya b Taylor’s Business School Abstract This paper explores if ten-minute stock returns depend systematically on time periods of a trading day, and days of a week. Systematic pattern that is associated with the time of a trading day is discovered, but the returns do not vary significantly according to different days of the week. The return is typically found to be negative in the first ten minutes after the open of the day and positive in the last ten minutes of the trading session. The intra-day volatility of returns follows an inverse-J curve, where the highest variance is recorded in the first ten minutes of the trading day. When used for forecasting stock prices ten minutes ahead, the model incorporating the time-of-theday effect produces average forecast error as low as 0.15 per cent, while the average absolute deviation of the forecast from the actual value is no higher than 1.08 points. 1. Introduction Recent technological advances have enabled stock markets to record intraday stock prices and indices at 15-minute and 1-minute intervals. The Kuala Lumpur Stock Exchange (KLSE) started recording the KLSE Composite Index at 15-minute intervals on 3 February 1990 and 1-minute intervals since April 1995. This, in turn, has permitted intraday stock price behaviour to be studied. Chang et al. (1994) used the KLSE Composite Index at 15-minute interval over a 2year period, from 3 February 1990 to 10 February 1992, and found that the mean intraday returns were large at the beginning and at the end of each of the morning and afternoon trading sessions, while recording its lowest level during the trading period. Thus, the intraday returns generated roughly a U-shaped curve in the morning and the ∗ Corresponding author: Goh Kim Leng, Faculty of Economics and Administration, University of Malaya, 50603 Kuala Lumpur, Malaysia. E-mail: klgoh@um.edu.my Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 32 afternoon trading sessions. These results are generally consistent with those obtained by Wood et al. (1995), Lockwood and Linn (1990) and Gerety and Mulherin (1992) on the New York Stock Exchange, and those obtained by Chang et al. (1993) on the Tokyo Stock Exchange. Chang et al. (1994) also found that intraday volatility in the KLSE was the highest at the beginning of the morning session and declined after that but rose again at the end of the morning session. The intraday volatility rose to the highest level sometime during the afternoon session and also at the close of the market. Thus, a U-shaped curve was observed only in the morning session. Using the KLSE Composite Index at 15-minute interval over the period March 1992April 1993, Lim (1996) obtained results similar to those of Chang et al. (1994). Lim (1996) also showed that there was generally no day-of-the-week effect for any 15minute time interval but there was time-of-the-day effect for any weekday. The research gap on studies of time-of-the-day effect has yet to be bridged since the availability of minute-to-minute data. Given this increased frequency of information dissemination, traders have more information in hand and their trading patterns may change accordingly. While these changes can introduce ascertainable price movements, the external shocks of the recent financial turmoil experienced by many East Asian economies can cause the market to behave differently. It is the objective of this paper to revisit the subject matter in the light of such changes, and also to shed additional light using new tools of analysis emerging from the recent development in financial econometrics specifically for modelling of the time-varying volatility or risks. To keep the study manageable, the intraday behaviour for returns over tenminute intervals is examined, but not that for minute by minute data. This paper is also motivated to ascertain systematic patterns in intraday returns to document evidence on the microstructure of the trading in KLSE, which in turn provides simple trading rules in terms of strategies for timing the trading in the stock market. Although not anticipated by theory, evidence of seasonal regularities in stock returns also has important bearing on the empirical validity of the efficient market hypothesis. This notable paradigm suggests that stock prices should follow a martingale process and returns should not exhibit any systematic patterns. Further, this paper differs from other works in that it shows how such systematic patterns can be used for forecasting purposes. Also, a comparison of the behaviour of intraday returns before and after the implementation of capital controls is provided. Following this introduction, Section 2 of this paper describes the framework of analysis. Association of intraday returns to days of a trading week and time periods within a trading day is examined in Section 3. This section also presents the regression models that capture the time-of-the-day effect, and evaluates how well the performance of these models is when used for forecasting the KLSE Composite Index. Section 4 summarizes the findings and concludes with some discussions. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 33 2. Data and Methodology This study uses ten-minute interval data on the KLSE Composite Index for the period December 1997 to April 1999. This sample period consists of 334 trading days and 36 ten-minute intervals in each weekday, and has a total of 12,024 data points. Intraday ten-minute returns are calculated as: rt = ln (pt/pt-1) x 100 (1) where pt is the Composite Index observed at time t and pt-1 is the index observed ten minutes before. The over-night returns are not included in the analysis. Five return data points outside the bound of ten standard errors were excluded to avoid the unusual impact of excessively extreme outliers. A sub-period analysis is also conducted to compare the intraday behaviour for the period before and after the implementation of capital controls. The two sub-periods are from December 1997 to August 1998 and September 1998 to April 1999. Mansor (1997) and the studies therein surveyed found significant day-of-the-week effect in the Malaysian stock market. Different from these studies, the motivation here is to establish whether the ten-minute returns are affected by the effect of days of the week in order to decide if this effect must be accordingly controlled in the analysis of time-of-the-day effect. The one-way ANOVA (see Chapter 14, Keller and Warrack, 2000) is used for this purpose. The null hypothesis involved is: H0: µ1 = µ2 = µ3 = µ4 = µ5 (2) where µi denotes the mean of the ten-minute returns for day-i, i = 1 for Monday, i = 2 for Tuesday, ..., i = 5 for Friday. The ANOVA assumes equality of variance and the Levene test (see Chapter 2, Madansky, 1988) is employed to check the validity of this assumption. The null hypothesis to be tested is: H0: σ21 = σ22 = σ23 = σ24 = σ25 (3) where σ2i denotes the variance of the intraday returns for day-i. If the null hypothesis is rejected, a non-parametric approach is more suitable for examining H0 in (2) and the Kruskal-Wallis test (see Chapter 16, Keller and Warrack, 2000) is used in this study. The dependence of the behaviour of mean return on the time of the day is tested via a one-way ANOVA. The null hypothesis is: H0: µ1 = µ2 = ... = µ36 (4) where µi signifies the mean of the ten-minute returns for time-i, i = 1 for 9:10 am, i = 2 for 9:20 am, ..., i = 36 for 5:00 pm. Possible heteroscedastic behaviour is tested using the Levene test where the null hypothesis is: Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 34 H0: σ21 = σ22 = ... = σ236 (5) The previous analyses consider the time-of-the-day and day-of-the-week effects separately. A two-way ANOVA (see Chapter 14, Keller and Warrack, 2000) is used to test the significance of one effect while keeping the other controlled and also the interaction between the two effects. If returns are dependent on the time of the day, it would be of interest to know which time period witnessed significant negative or positive returns. A regression model as follows is fitted: rt = α1 D1t + α2 D2t + ... + α36 D36,t + θ Ct + ut (6) where Dit = 1 for time-i and zero otherwise, i is as defined in (4), Ct = 1 for the trading days beginning from September 1998 and zero otherwise, and ut is the error term. The variable Ct is included to capture the effect of the implementation of capital controls. The t-test for the significance of αi (which measures the mean return for time-i) has to take the possible heteroscedastic behaviour into account. The White’s (1980) heteroscedastic consistent standard errors are used in the computation of the tstatistics. It must be ascertained if model (6) is sufficient to capture the time-varying volatility of the returns. The Lagrange multiplier (LM) test for presence of ARCH (autoregressive conditional heteroscedasticity) effect at 5 and 10 lags are performed (see Engle (1982)). A generalized ARCH or GARCH model is considered to examine if the day-of-the-week effect is affected when the time-varying volatility is modelled (see Bollerslev et al. (1992) for a survey of ARCH models). The following model, which is based on a GARCH(1,1) specification, is adopted: rt = α1 D1t + α2 D2t + ... + α36 D36,t + θ Ct + ut ut = z t h t ht = β0 + β1 u2t-1 + β2 ht-1 where ut ~ (0, ht) and zt ~ i.i.d.(0, 1). To further examine the robustness of findings on time-of-the-day effect, an extended model using the ARCH-in-mean or ARCH-M specification (see Engle et al., 1987) is fitted: rt = α1 D1t + α2 D2t + ... + α36 D36,t + γ1 ht1/2 + γ2 rt-1 + θ Ct + ut ut = z t h t ht = β0 + Σφi Dit + β1 u2t-1 + β2 ht-1 The ht term in the mean equation takes into consideration any risk-return tradeoff. Equation (8) includes the time-of-the-day effect not only in the mean equation, but (8) (7) Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 35 also the variance equation. The purpose of this is to investigate if the time-of-the-day effect in mean returns is caused by the intraday seasonal behaviour in the volatility.1 In addition, a lagged return is included in the mean equation of (9) to account for firstorder serial correlation in return. The practical usefulness of the time-of-the-day seasonal patterns is not limited to only timing of trading. The models that capture this effect can also be employed to forecast the Composite Index for the 36 time periods in any given trading day. The models in (6), (7) and (8) estimated for the whole sample period are utilized to forecast the Composite Index at the end of each ten-minute interval for the first two trading days of May 1999 (3rd and 4th). To evaluate the forecast performance, the actual and the forecast values are compared. The criteria for measuring this performance includes: Root mean squared error (RMSE) = t = n +1 ∑ (p n +s t ˆ − pt )2 (9) s n +s Mean absolute deviation (MAD) = t = n +1 ∑p t ˆ − pt s n +s (10) Mean absolute percent error (MAPE) = t = n +1 ∑ (p t ˆ − pt ) / pt s (11) ˆ where s is the number of observations in the out-of-sample forecast period and p t is the forecast value of pt. In all cases, a one-period (ten-minute) ahead forecast is performed. 3. Results Day-of-the-week Effect The mean and variance of the ten-minute returns for the five trading days are given in Table 1. The returns are averaged across all the time intervals for each trading day to obtain the mean. The variance is computed based on the dispersion of returns across all the time intervals for each of the trading days. The mean and variance of the tenminute returns are plotted against the trading days in Figures 1 and 2, respectively. The mean return is the lowest and negative on Monday. It turns positive and peaks on Wednesday but slowly tapers off. A similar pattern is observed for the first sub1 In the study of the day-of-the-week effect, for instance, Davidson and Peker (1996) found that this effect in the mean equation is no longer significant after it was incorporated in the variance equation. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 36 period, except that the mean is negative for all five trading days. The second subperiod witnessed a rather different behaviour, with a start of an almost zero mean. It dips before turning positive on Wednesday, after which the mean increases until the peak on Friday. The behaviour of the variance for the whole period seems similar to that of the second sub-period. The highest volatility is found on Monday, and this decreases over the remaining trading days. For the first sub-period, the variance starts low on Monday and Tuesday but increases after that to Thursday and takes a small dip before the close of the week. Table 1 Summary Statistics for Ten-Minute Returns by Days of the Trading Week December 1997April 1999 Mean (%) Variance Monday Tuesday Wednesday Thursday Friday All days -0.0166 -0.0132 0.0046 0.0028 0.0029 -0.0036 0.1348 0.1343 0.1224 0.1189 0.0921 0.1202 December 1997August 1998 Mean (%) Variance -0.0317 -0.0069 -0.0004 -0.0071 -0.0157 -0.0123 0.0892 0.0813 0.1037 0.1205 0.1076 0.1005 September 1998April 1999 Mean (%) Variance 0.0018 -0.0197 0.0101 0.0122 0.0227 0.0056 0.1898 0.1889 0.1429 0.1173 0.0749 0.1411 n 2230 2411 2482 2448 2448 12019 n 1224 1223 1295 1188 1260 6190 n 1006 1188 1187 1260 1188 5829 Figure 1 Mean (%) of the Ten-Minute Returns by Days of the Trading Week 0.04 0.02 September 1998 – April 1999 December 1997 – April 1999 0.00 December 1997 – August 1998 -0.02 -0.04 Monday Tuesday Wednesday Thursday Friday Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 37 Figure 2 Variance of the Ten-Minute Returns by Days of the Trading Week 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 Monday Tuesday Wednesday Thursday Friday December 97 – April 1999 September 1998 – April 1999 December 1997 – August 1998 The results of the ANOVA test presented in Table 2 show that the null hypothesis of equal mean returns for the five trading days cannot be rejected at the 5 per cent significance level for the whole period and both sub-periods. The null hypothesis under the Levene test is strongly rejected in all the cases, indicating violation of the constant variance assumption underlying the ANOVA. The results of the KruskalWallis test in Table 2 provide no evidence to support significance of the day-of-theweek effect either. This is consistent with the findings of Lim (1996). Overall, the findings are that the ten-minute mean returns for different trading days do not differ significantly from one another, but the volatility behaviour is different across different days of the week. It must be borne in mind, however, this analysis does not control for possible differences across time periods within a trading day. Table 2 Results of Tests of Equality of Mean and Variance for Different Days of the Week ANOVAa Levene Testb Kruskal-Wallis Testa December 1997-April 1999 December 1997-August 1998 September 1998-April 1999 Notes: 2.023 1.809 2.137 3.145* 6.510** 6.387** 9.139 6.753 4.939 a Both tests if the mean returns for every trading day of the week are equal. bTests if the variances of return for every trading day of the week are equal. *Significant at 5 per cent. **Significant at 1 per cent. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 38 Time-of-the-day Effect The non-significance of the day-of-the-week effect on mean returns reported in the previous section suggests that the analysis on time-of-the-day effect can be performed without the need to control for differences in the trading day. For every trading day, 36 ten-minute returns are computed for 9:10 am, 9:20 am, ..., 12:30 pm, 2:40 pm, 2:50 pm, ..., and 5 pm. Table 3 shows the mean and variance of these returns for the 36 time intervals. The mean is computed by averaging the returns across all the trading days for each of the 36 time intervals, while the variance is computed based on the dispersion of returns across all the trading days for each of the 36 time intervals. The plots of the mean and variance for every time interval are given in Figures 3 and 4, respectively. Table 3 Summary Statistics for Ten-Minute Returns by Time of the Trading Day December 1997-April 1999 Mean (%) Variance n -0.1063 0.0300 0.0143 -0.0036 0.0125 -0.0355 -0.0092 0.0129 -0.0188 -0.0089 -0.0190 -0.0274 0.0143 0.0209 -0.0092 -0.0176 -0.0231 -0.0191 -0.0044 -0.0242 0.0080 0.0029 -0.0252 -0.0181 0.0295 0.0346 0.0208 0.0222 0.0170 -0.0066 0.0007 -0.0212 -0.0168 0.0033 -0.0334 0.0720 0.5894 0.3475 0.2506 0.1592 0.1241 0.1306 0.1165 0.1178 0.0955 0.0937 0.0657 0.0703 0.0686 0.0663 0.0666 0.0684 0.0560 0.0712 0.0551 0.0397 0.0945 0.1362 0.0907 0.0883 0.0716 0.0641 0.1320 0.0667 0.0795 0.0821 0.1330 0.1637 0.0920 0.1199 0.1099 0.1414 330 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 334 333 334 December 1997-August 1998 Mean (%) Variance n -0.1978 0.0513 -0.0417 -0.0159 -0.0037 -0.0191 0.0132 0.0107 -0.0069 -0.0316 -0.0357 -0.0559 -0.0059 0.0163 -0.0258 -0.0189 -0.0067 -0.0153 -0.0166 -0.0139 -0.0206 -0.0150 -0.0398 -0.0242 0.0047 0.0158 0.0627 0.0166 -0.0049 -0.0039 0.0070 0.0140 -0.0257 -0.0009 -0.0314 0.0205 0.6148 0.3618 0.1938 0.1547 0.1387 0.1131 0.0883 0.1345 0.1015 0.0942 0.0686 0.0553 0.0471 0.0653 0.0537 0.0453 0.0427 0.0508 0.0408 0.0386 0.0524 0.1817 0.0720 0.0603 0.0554 0.0569 0.0440 0.0454 0.0693 0.0472 0.0622 0.0626 0.0540 0.0526 0.0573 0.1103 170 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 September 1998-April 1999 Mean (%) Variance n -0.0091 0.0075 0.0739 0.0094 0.0296 -0.0529 -0.0329 0.0153 -0.0314 0.0152 -0.0013 0.0029 0.0359 0.0258 0.0086 -0.0163 -0.0405 -0.0232 0.0086 -0.0351 0.0383 0.0220 -0.0096 -0.0115 0.0559 0.0546 -0.0238 0.0282 0.0402 -0.0096 -0.0060 -0.0585 -0.0073 0.0076 -0.0356 0.1267 0.5477 0.3334 0.3054 0.1647 0.1087 0.1493 0.1460 0.1008 0.0894 0.0927 0.0623 0.0849 0.0910 0.0677 0.0802 0.0933 0.0699 0.0931 0.0703 0.0408 0.1379 0.0880 0.1106 0.1185 0.0878 0.0714 0.2223 0.0898 0.0897 0.1197 0.2089 0.2693 0.1327 0.1920 0.1668 0.1694 160 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 161 162 Time 9:10 9:20 9:30 9:40 9:50 10:00 10:10 10:20 10:30 10:40 10:50 11:00 11:10 11:20 11:30 11:40 11:50 12:00 12:10 12:20 12:30 2:40 2:50 3:00 3:10 3:20 3:30 3:40 3:50 4:00 4:10 4:20 4:30 4:40 4:50 5:00 Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 39 Figure 3 Mean (%) of the Ten-Minute Returns by Time of the Trading Day (a) December 1997-April 1999 0.08 0.04 0.00 -0.04 -0.08 -0.12 9:10 9:50 10:30 11:10 11:50 12:30 3:10 3:50 4:30 (b) December 1997-August 1998 0.04 -0.02 -0.08 -0.14 -0.20 9:10 0.12 0.06 0.00 -0.06 9:10 9:50 10:30 11:10 11:50 12:30 3:10 3:50 4:30 9:50 10:30 11:10 11:50 12:30 3:10 3:50 4:30 (c) September 1998-April 1999 Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 40 Figure 4 Variance of the Ten-Minute Returns by Time of the Trading Day (a) December 1997-April 1999 0.6 0.4 0.2 0.0 9:10 0.6 0.4 0.2 0.0 9:10 9:30 9:50 10:10 10:30 10:50 11:10 11:30 11:50 12:10 12:30 2:50 3:10 3:30 3:50 4:10 4:30 4:50 (b) December 1997-August 1998 (c) September 1998-April 1999 0.6 9:30 9:50 10:10 10:30 10:50 11:10 11:30 11:50 12:10 12:30 2:50 3:10 3:30 3:50 4:10 4:30 4:50 0.4 0.2 0.0 9:10 9:30 9:50 10:10 10:30 10:50 11:10 11:30 11:50 12:10 12:30 2:50 3:10 3:30 3:50 4:10 4:30 4:50 Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 41 For the whole period, the lowest return (which is negative) is found in the first ten minutes after the open of the market, and the highest return (positive) is recorded over the last ten-minute interval before the close of the trading day. The lowest return at the first 10 minutes is observed for the first sub-period, and the highest return at the last 10 minutes is observed for the second sub-period. It is interesting to note that in the first sub-period most of the positive returns are found in the afternoon session. The variance exhibits an inverse-J curve, which is particularly obvious for the whole period and the first sub-period. The variance is the highest in the first ten minutes, and declines gradually before a slight increase again before the close of the morning session. This is observed for all the three different sample periods considered. For the first sub-period, a smaller inverse-J curve is found in the afternoon session. The variance rises in the first 10 minutes in the afternoon session, fluctuates within a narrow range after that and increases slightly during the last 10 minutes. The variance seems to behave more erratically in the afternoon session of the second sub-period, with few jumps at 3:30 pm, 4:20 pm and 4:40 pm. This is reflected to a smaller extent in the results for the whole period. These results on the mean and variance of intraday returns are rather different from the U-curve phenomenon reported by Chang et al. (1994) and Lim (1996). The striking feature is the evidence contrary to the high riskhigh return tradeoff, particularly at the first and last ten minutes of the trading day. Investors acting on information received overnight may have led to the high variance at the open of the trading session, but it is peculiar that high positive returns are associated with low variance at the end of the day. The results of the ANOVA test for equality of the mean ten-minute returns across the 36 time intervals are reported in Table 4. The null hypothesis stated in (4) is rejected for the whole period and the two sub-periods, showing strong evidence of time-of-theday effect. As is observed earlier, the variance behaves differently for different time period of the day. The Levene test rejects strongly the null hypothesis of (5), thus showing the problem of heteroscedasticity. The Kruskal-Wallis test is relied on to check the robustness of the results of ANOVA. Similarly, this test rejects the equality of mean return for different time periods of the day stated in the null hypothesis of (4). This rejection concurs with the findings of Lim (1996). Thus far, the finding is that the mean and variance of the returns exhibit significantly different behaviour for different time of a trading day. Table 4 Results of Tests of Equality of Mean and Variance for Different Time Periods of the Day ANOVAa Levene Testb Kruskal-Wallis Testa December 1997-April 1999 December 1997-August 1998 September 1998-April 1999 Notes: a 2.308** 2.750** 1.606* 30.823** 28.462** 9.266** 118.632** 75.168** 106.979** Both tests if the mean returns for every ten-minute interval of the trading day are equal. bTests if the variances of return for every ten-minute interval of the trading day are equal. *Significant at 5 per cent. **Significant at 1 per cent. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 42 The results of the two-way ANOVA are given in Table 5. It can be seen that the timeof-the-day effect is significant, but not the day-of-the-week effect at the 5 per cent level. This reaffirms our earlier findings. Interestingly, the interaction between these two effects is significant only for the first sub-period, but not the whole period and the second sub-period. Table 5 Results of Two-Way ANOVA for Day-of-the-Week and Time-of-the-Day Effects Time-of-the-day effect December 1997-April 1999 December 1997-August 1998 September 1998-April 1999 Notes: 2.294** 2.722** 1.649** Day-of-the-week effect 2.031 1.845 2.136 Interactions: time & day 0.959 1.421** 0.847 *Significant at 5 per cent. **Significant at 1 per cent. Modelling the Time-of-the-day Effect Table 6 reports the estimated results for regression model (6). The coefficients reflect the mean returns after controlling for the effects of capital controls. For the overall sample period, the return for the first ten minutes is significantly negative and that for the last ten minutes is significantly positive. The return for the penultimate ten-minute interval before the close of the morning session is also significantly negative. The coefficient of Ct is significantly positive, showing that the average return is higher for the period after the implementation of capital controls, when the time-of-the-day effect is controlled. The significance of this variable further justifies the sub-period analysis. The first sub-period sees a significant negative return in the first ten minutes but this does not prevail in the second sub-period. While the return for the last ten minutes is not significant in the first sub-period, it is significantly positive in the second sub-period. The return for the second last ten-minute interval before lunch break is found to be significantly negative in the second sub-period. The model, however, is statistically inadequate as the results of the LM test at both 5 and 10 lags indicate strongly the presence of ARCH effects. The GARCH(1,1) model defined in (7) is estimated and the results are given in Table 7.2 As the residuals are not normally distributed,3 the quasi-maximum likelihood standard errors suggested by Bollerslev and Wooldridge (1992) are used. The estimated coefficients of β1 and β2 are statistically significant. The sum of the two coefficients is close to one, suggesting that the time-varying volatility is rather 2 Note that the days of the week and the interaction between days of the week and time periods of the day are not included in the current analysis. This is because the earlier results show that the day-of-theweek effect is not significant. By and large, this is also true of the interaction effect, with the exception of the first sub-period. Since the day-of-the-week effect is not considered, the interaction effect is treated the same way. From the computational viewpoint, this is essential as the GARCH model is estimated using the maximum likelihood procedure which is iterative in nature, and convergence is a problem when the number of parameters in the model increases. 3 The results are not reported but available on request. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 43 persistent. For the entire sample period, the results are rather similar to that reported in Table 6 with the exception that more coefficients in the mean equation are found to be significant, and the returns associated with the last ten minutes is no longer significant. The dummy variable for capital controls is significant as before. Table 6 Regression Results for Time-of-the-Day Effect Without Volatility Modeling December 1997-April 1999 Coefficient Std. Error -0.1150** 0.0214 0.0057 -0.0123 0.0038 -0.0442* -0.0179 0.0042 -0.0274 -0.0176 -0.0277 -0.0361* 0.0056 0.0123 -0.0178 -0.0263 -0.0318* -0.0278 -0.0131 -0.0329** -0.0007 -0.0058 -0.0338* -0.0267 0.0209 0.0259 0.0121 0.0136 0.0083 -0.0153 -0.0080 -0.0299 -0.0255 -0.0054 -0.0421* 0.0633** 0.0179** 0.0424 0.0325 0.0273 0.0220 0.0196 0.0200 0.0188 0.0191 0.0172 0.0170 0.0143 0.0147 0.0145 0.0144 0.0144 0.0145 0.0133 0.0148 0.0131 0.0114 0.0169 0.0206 0.0167 0.0164 0.0148 0.0141 0.0198 0.0143 0.0156 0.0158 0.0199 0.0220 0.0167 0.0189 0.0182 0.0206 0.0063 December 1997-August 1998 Coefficient Std. Error -0.1978** 0.0513 -0.0417 -0.0159 -0.0037 -0.0191 0.0132 0.0107 -0.0069 -0.0316 -0.0357 -0.0559** -0.0059 0.0163 -0.0258 -0.0189 -0.0067 -0.0153 -0.0166 -0.0139 -0.0206 -0.0150 -0.0398 -0.0242 0.0047 0.0158 0.0627** 0.0166 -0.0049 -0.0039 0.0070 0.0140 -0.0257 -0.0009 -0.0314 0.0205 0.0601 0.0459 0.0336 0.0300 0.0284 0.0256 0.0227 0.0280 0.0243 0.0234 0.0200 0.0179 0.0165 0.0195 0.0177 0.0162 0.0158 0.0172 0.0154 0.0150 0.0175 0.0325 0.0205 0.0187 0.0180 0.0182 0.0160 0.0162 0.0201 0.0166 0.0190 0.0191 0.0177 0.0175 0.0183 0.0253 September 1998-April 1999 Coefficient Std. Error -0.0091 0.0075 0.0739 0.0094 0.0296 -0.0529 -0.0329 0.0153 -0.0314 0.0152 -0.0013 0.0029 0.0359 0.0258 0.0086 -0.0163 -0.0405 -0.0232 0.0086 -0.0351* 0.0383 0.0220 -0.0096 -0.0115 0.0559* 0.0546** -0.0238 0.0282 0.0402 -0.0096 -0.0060 -0.0585 -0.0073 0.0076 -0.0356 0.1267** 0.0585 0.0454 0.0434 0.0319 0.0259 0.0304 0.0300 0.0250 0.0235 0.0239 0.0196 0.0229 0.0237 0.0204 0.0222 0.0240 0.0208 0.0240 0.0208 0.0159 0.0292 0.0233 0.0261 0.0270 0.0233 0.0210 0.0370 0.0235 0.0235 0.0272 0.0359 0.0408 0.0286 0.0344 0.0322 0.0323 Variable 9:10 9:20 9:30 9:40 9:50 10:00 10:10 10:20 10:30 10:40 10:50 11:00 11:10 11:20 11:30 11:40 11:50 12:00 12:10 12:20 12:30 2:40 2:50 3:00 3:10 3:20 3:30 3:40 3:50 4:00 4:10 4:20 4:30 4:40 4:50 5:00 Capital control ARCH LM Test 5 lags 2903.32** 406.82** 1784.43** 10 lags 2989.60** 454.88** 1820.36** Notes: The White’s (1980) heteroscedastic consistent standard errors are reported. *Significant at 5 per cent. **Significant at 1 per cent. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 44 Following this, the model is reestimated for the two sub-periods. All the coefficients in the variance equation appear to be significant. More time-of-the-day effect is found to be significant compared to the model without the variance equation. The significantly negative return associated with the first ten minutes of the trading days in the first sub-period and the significantly positive return for the last ten minutes for the second sub-period are also observed. The use of a GARCH(1,1)-based model is sufficient to capture the time-varying volatility. This is shown by the ARCH LM test that is generally not significant. Table 8 reports the results for the estimation of the ARCH-M model stated in (8). As the aim here is to investigate if the day-of-the-week effect is spuriously induced by time-varying volatility, only the time interval reported to have significant mean returns in Table 6 is included. We see that the model is sufficient to account for the time-varying volatility as none of the ARCH LM test is significant. The estimated coefficient on the expected risk (ht) is significant for the whole period and the first sub-period, but the negative sign is contrary to expectation. It could be that in the period immediately after the financial crisis, not only there is no payoff for risk takers, volatile market conditions were associated with poor returns. In the second subperiod, however, the same is not observed where the ht coefficient, although negative, is not significant. The lagged return variable is significant in all cases. Also, most of the dummy variables are significant in the variance equation, suggesting that the timevarying volatility exhibits time-of-the-day effect. Almost all but a few of the time-ofthe-day effects reported earlier continue to be significant, and thus the results are rather robust. The results for the evaluation of forecast performance of models (6), (7) and (8) are shown in Table 9. These models are estimated using observations from the entire sample period, and also the second sub-period. All models have forecast errors of between 0.15 to 0.16 per cent, and the average absolute deviation from the actual value is between 1.05 to 1.08 points. For the entire sample period, the GARCH(1,1) model performs better than the OLS model without the variance equation based on RMSE but the reverse is true based on MAD and MAPE. The ARCH-M model is better than the other two models based on all the three criteria. For the second subperiod, the GARCH(1,1) model is better than the OLS model, but it is outperformed by the ARCH-M model on all accounts. This suggests that the ARCH-M model is consistently the best model. All the models estimated for the second sub-period have lower forecasts errors compared to the models estimated for the whole period. The results suggest that the market information after the implementation of capital controls is more relevant for forecasting the more recent index. The actual and forecast values of the Composite Index based on the ARCH-M model estimated for the second sub-period are plotted in Figure 5. The forecast values track the actual values very closely. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 45 Table 7 Regression Results for Time-of-the-Day Effect With Volatility Modeling Using GARCH(1,1) December 1997-April 1999 Coefficient Std. Error -0.0635* -0.0250 -0.0218 0.0021 -0.0300 -0.0413** -0.0180 -0.0262* -0.0283* -0.0256* -0.0301** -0.0438** -0.0010 -0.0146 -0.0093 -0.0446** -0.0388** -0.0398** -0.0321** -0.0286** -0.0188 -0.0099 -0.0242* -0.0094 0.0045 0.0072 0.0296* 0.0099 -0.0004 0.0007 -0.0136 -0.0080 -0.0569** 0.0099 -0.0546 -0.0480 0.0206** 0.0322 0.0228 0.0211 0.0174 0.0167 0.0153 0.0121 0.0132 0.0127 0.0113 0.0113 0.0103 0.0112 0.0106 0.0105 0.0098 0.0094 0.0112 0.0106 0.0106 0.0138 0.0160 0.0119 0.0111 0.0112 0.0115 0.0122 0.0115 0.0113 0.0115 0.0125 0.0184 0.0188 0.0186 0.0297 0.0412 0.0051 0.0011 0.0245 0.0231 0.0158** 0.4308** 0.4973** 10.54 11.91 0.0020 0.0524 0.0396 0.0102** 0.3082** 0.6083** 6.18 7.33 0.0014 0.0324 0.0289 December 1997-August 1998 Coefficient Std. Error -0.1427** 0.0139 -0.0306 -0.0078 -0.0322 -0.0317 0.0139 -0.0131 -0.0187 -0.0341* -0.0257 -0.0648** -0.0136 -0.0031 -0.0107 -0.0367** -0.0235 -0.0359* -0.0198 0.0161 -0.0426 -0.0223 -0.0292 -0.0220 -0.0074 -0.0042 0.0465** 0.0135 -0.0023 0.0082 -0.0036 0.0068 -0.0495** 0.0395 0.0955** -0.1444 0.0457 0.0338 0.0313 0.0261 0.0263 0.0205 0.0188 0.0204 0.0190 0.0164 0.0163 0.0140 0.0155 0.0148 0.0137 0.0123 0.0122 0.0165 0.0150 0.0204 0.0225 0.0260 0.0169 0.0146 0.0127 0.0138 0.0153 0.0152 0.0152 0.0132 0.0153 0.0154 0.0179 0.0209 0.0355 0.0753 September 1998-April 1999 Coefficient Std. Error 0.0306 -0.0302 0.0077 0.0222 -0.0058 -0.0275 -0.0222 -0.0125 -0.0173 0.0005 -0.0142 -0.0014 0.0290 -0.0063 0.0103 -0.0343* -0.0340** -0.0234 -0.0240 -0.0334** 0.0092 0.0256 0.0004 0.0250 0.0398* 0.0441* 0.0348 0.0287 0.0192 0.0154 -0.0022 0.0198 -0.0355 -0.0424 -0.1376** 0.1243** 0.0415 0.0323 0.0262 0.0225 0.0201 0.0217 0.0152 0.0166 0.0174 0.0153 0.0140 0.0138 0.0160 0.0141 0.0146 0.0141 0.0131 0.0124 0.0129 0.0125 0.0152 0.0155 0.0155 0.0158 0.0170 0.0178 0.0184 0.0170 0.0156 0.0185 0.0200 0.0361 0.0250 0.0237 0.0335 0.0434 Variable 9:10 9:20 9:30 9:40 9:50 10:00 10:10 10:20 10:30 10:40 10:50 11:00 11:10 11:20 11:30 11:40 11:50 12:00 12:10 12:20 12:30 2:40 2:50 3:00 3:10 3:20 3:30 3:40 3:50 4:00 4:10 4:20 4:30 4:40 4:50 5:00 Capital control Variance Equation constant 0.0119** u2t-1 0.3228** ht-1 0.5977** ARCH LM Test 5 lags 10 lags Notes: 13.41* 14.89 The quasi-maximum likelihood standard errors suggested by Bollerslev and Wooldridge (1992) are reported. *Significant at 5 per cent. **Significant at 1 per cent. Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 46 Table 8 Regression Results for Time-of-the-Day Effect With Volatility Modeling Using ARCH-M December 1997-April 1999 Coefficient Std. Error -0.0808* -0.0227 -0.0277** -0.0246** -0.0145 -0.0116 0.0349 0.0159 0.0106 0.0089 0.0095 0.0119 December 1997-August 1998 Coefficient Std. Error -0.1448** -0.0509** 0.0557 0.0148 -0.0233 0.0388* 0.0355* 0.0553** -0.0218 0.0952** -0.0304* 0.2117** 0.0114** 0.0124 0.0175 0.0127 0.0113 0.0041 0.0009 0.0245 0.0077 0.0043 0.0029 0.0061 0.0054 -0.0343* 0.1769** 0.0154 0.0133 0.0167 0.1606** -0.0094 0.2520** 0.0351 0.0149 0.0164 0.0120 0.0168 0.0180 September 1998-April 1999 Coefficient Std. Error Variable 9:10 10:00 11:00 11:50 12:20 2:50 3:10 3:20 3:30 4:50 5:00 ht1/2 Returnt-1 Capital control Variance Equation Constant 0.0059** 9:10 0.1725** 10:00 -0.0432** 11:00 -0.0100* 11:50 -0.0116** 12:20 0.0153* 2:50 -0.0028 3:10 3:20 3:30 4:50 0.0056 5:00 0.0472** u2t-1 0.1699** ht-1 0.7154** ARCH LM Test 5 lags 10 lags Notes: 4.72 6.77 0.0125** 0.3004** -0.0176** 0.0014 0.0495 0.0054 0.0059** 0.0013 -0.0053 0.0078 0.0049 -0.0109* 0.0053 0.0210 0.0266 0.1756** 0.2343** 0.6515** 1.39 4.09 0.0043 0.0081 0.0073 0.0276 0.0288 0.0277 0.0055 0.0129 0.0135 0.0168 0.2088** 0.5872** 3.78 9.36 The quasi-maximum likelihood standard errors suggested by Bollerslev and Wooldridge (1992) are reported. *Significant at 5 per cent. **Significant at 1 per cent. Table 9 Forecast Performance Based on Time-of-the-Day Effect Estimation period: December 1997-April 1999 OLS GARCH ARCH-M Root mean squared error (RMSE) Mean absolute deviation (MAD) Mean absolute percent error (MAPE) (%) 1.3933 1.0814 0.1583 1.3886 1.0853 0.1588 1.3532 1.0665 0.1560 Estimation period: September 1998-April 1999 OLS GARCH ARCH-M 1.3755 1.0615 0.1554 1.3542 1.0611 0.1552 1.3395 1.0538 0.1542 Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 47 Figure 5 Forecast Values of the KLSE Composite Index Using an ARCH-M Model Estimated for the Period September 1998-April 1999 700 690 680 actual forecast 670 9:10 9:50 10:30 11:10 11:50 12:30 3:10 3:50 4:30 9:10 9:50 10:30 11:10 11:50 12:30 3:10 3:50 4:30 3 May 1999 4 May 1999 4. Conclusion The intra-day analysis conducted in this study found systematic pattern in the tenminute returns that are associated with the time of a trading day. Interestingly, the average of these returns does not vary significantly according to different days of the week. There is also no strong evidence that suggests presence of interaction between the time-of-the day behaviour and the effect of days of the week. The return is typically found to be negative in the first ten minutes after the open of the day and positive in the last ten minutes of the trading session. The former is particularly true for the period before the implementation of capital control, while the latter for the period after the implementation of capital controls. The volatility of the ten-minute returns exhibits different behaviour across different time periods of the day. The volatility is the highest in the first ten minutes of the trading day and gradually declines after that for the morning session. Although not as high as that at the open of the market, the volatility increases at the start of the afternoon trading session, but the pattern is less clear for the rest of the session. Nevertheless, the volatility for the last ten minutes of the trading day, conditioned on information of the penultimate ten minutes, can be high particularly for the period after the implementation of capital controls. On average, the implication is clear that trading risk is the highest at the start of a trading day, with high possibility of negative returns. This is especially true for the period after the start of the financial crisis (the first sub-period of this study). The Goh and Kok / Labuan Bulletin of International Business & Finance, 2(1), 2004, 31-49 48 results suggest that risk takers are not rewarded a premium but rather have to pay a penalty in depressed market conditions. This phenomenon, however, disappears when the market condition improves (the second sub-period of this study). Risk is found to be more moderate at the end of the trading day, with better likelihood of positive returns. The high positive increase in the KLSE Composite Index in the last ten minutes is particularly prominent in the second sub-period. The strong evidence of seasonal regularities in the ten-minute returns is not induced by the time-of-the-day seasonality in volatility or any possible risk-return tradeoff. The revelation of these unsuspecting regularities poses considerable challenge to the efficient market hypothesis. The implication is that market information is not fully reflected in prices at ten-minute intervals, thus systematic patterns remain in successive price changes arising from the unexploited informational content. It would be interesting for future research to investigate if period-specific events can be identified for explaining the seasonal patterns reported in this study in order to understand if they are data-driven, occurrence by chance, or outcome of these events. Consequently, the information content in the time-of-the-day effect is of use for forecasting stock prices at different time of a trading day. The average error for tenminute ahead forecasts can be as low as 0.15 per cent, while the average absolute deviation of the forecast from the actual value is no higher than 1.08 points. For the more recent market index, the findings further suggest that information after the implementation of capital controls produces better forecast accuracy than a combination of information before and after the control. Forecast performance is also better when time-varying volatility of the returns is taken into account apart from relying only on the time-of-the-day effect of mean returns. 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