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Quantitative Finance
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Volatility behavior, information efficiency and risk in
the S&P 500 index markets
Shu-Mei Chiang a , Huimin Chung b & Chien-Ming Huang c a
Department of Finance , Lunghwa University of Science and Technology , 300 Wanshou Road, Section 1, Gueishan Shiang , Taoyuan County 333 , Taiwan
b
Graduate Institute of Finance , National Chiao Tung University , Hsinchu , Taiwan c
Department of Banking and Finance , Tamkang University , Taipei , Taiwan Published online: 15 Apr 2011.
To cite this article: Shu-Mei Chiang , Huimin Chung & Chien-Ming Huang (2012) Volatility behavior, information efficiency
and risk in the S&P 500 index markets, Quantitative Finance, 12:9, 1421-1437, DOI: 10.1080/14697688.2010.547512 To link to this article: http://dx.doi.org/10.1080/14697688.2010.547512
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Volatility behavior, information efficiency and risk in
the S&P 500 index markets
SHU-MEI CHIANG*y, HUIMIN CHUNGz and CHIEN-MING HUANGx
yDepartment of Finance, Lunghwa University of Science and Technology, 300 Wanshou Road, Section 1,Gueishan Shiang, Taoyuan County 333, Taiwan
zGraduate Institute of Finance, National Chiao Tung University, Hsinchu, Taiwan xDepartment of Banking and Finance, Tamkang University, Taipei, Taiwan
(Received 4 June 2008; revised 5 December 2010; in final form 9 December 2010)
We propose an ARJI-Trend model—a combination of the ARJI and component models—to capture the distinguishing features of US index returns, with the results indicating that our model has a good fit for the volatility dynamics of spot, floor- traded and E-mini index futures in US markets. Although certain analogous characteristics are discernible amongst the three indices (such as the responses by the transitory components to innovations, the high persistence in the trends, and the relative importance of jump variance), the reaction to news is found to be heterogeneous amongst the S&P 500 indices. Furthermore, the out-of-sample forecasting performances of both the ARJI-Trend model and the GARCH model are found to have general equivalence for the S&P 500 indices. Our analyses further show that the mini-sized index market is the most efficient with regard to the transmission of information in both the short and long run. This suggests that, following the introduction of E-mini futures, these instruments have come to play a dominant role in price discovery. Overall, our empirical results are very encouraging, insofar as the proposed ARJI-Trend model is found to be a useful tool for helping practitioners to gain a better understanding of the differential attributes between spot, general and mini-sized products in US stock markets.
Keywords: ARJI-Trend model; Components; Jumps; S&P 500 index 1. Introduction
Despite a general worldwide trend which has witnessed the transfer of trading systems from open outcry to electronic trading, both types of trading systems are simultaneously operated during normal trading hours in the index futures markets of the US. Given that
E-mini and floor-traded index futures are the
counterparts of the spot index, significant linkages may be found to exist between the spot and the corresponding index futures.ô
Furthermore, it is quite reasonable to assume that both spot and index futures should be driven by, and result in, similar reactions to the same news arrivals; nevertheless, several prior studies have demonstrated that there are certain properties of the index markets that are not
necessarily found to be the same (Chiu et al. 2006, Chung and Chiang 2006).?
Asset volatility refers to fluctuations in stock returns irrespective of direction, with volatility having been recognized for a considerable period of time as playing an important and predominant role in the field of finance. Andersen (1996) notes that changes in return volatility are directly related to the rate of news flows entering into the market. There are also many examples of other studies within which various approaches have been applied to the measurement of volatility and hedging.k Hence, it is essential for investors, speculators and hedgers to have a firm understanding of the features of volatility so as to avoid investment risk and try to defeat the market.
Structural change refers to long-term changes in the parameters of a structure generating a time series, and, *Corresponding author. Email: shumei@mail.lhu.edu.tw
ôRefer to Chung and Chiang (2006) for a detailed description of the differences between the contract specifications for floor-traded and E-mini futures indices.
?Chiu et al. (2006) demonstrate that there are divergent responses to news arrivals for the S&P 500 index spot and futures, whilst Chung and Chiang (2006) also note that there are different degrees of price clustering between the index spot and futures. kSee Giot and Laurent (2004), Chiu et al. (2005b) and Hung et al. (2006).
Quantitative Finance
ISSN 1469–7688 print/ISSN 1469–7696 onlineß 2012 Taylor & Francis http://www.tandfonline.com
http://dx.doi.org/10.1080/14697688.2010.547512
indeed, as stated by Perron (1997), if such structural changes are not taken into consideration with the occurrence of certain events, the outcomes may differ markedly from expectations. Accordingly, in a highly uncertain and volatile economy, policymakers must have a firm understanding of any potential transition. Of particular importance is the need to explore potential occurrences of structural changes in markets character-ized by a rapid pace of change, a prime example of which is the financial markets.
There are certain characteristics of volatility which have been shown to exist in financial stock returns, such as time-varying conditional volatility (Jacquier et al. 1994), time-varying conditional jumps (Maheu and McCurdy 2004, Daal et al. 2007) and permanent and transitory components (Engle and Lee 1999, Ane´ 2006).y Despite this, there is a distinct lack of studies with a focus on empirically exploring the existence of these three features. Although the results reported by Chen and Shen (2004) did provide some evidence on these three features with regard to the Taiwanese exchange rate, their analysis was confined to the consideration of constant jumps. According to Maheu and McCurdy (2004), it is of crucial importance to allow for both time variations and cluster-ing in the process governcluster-ing jumps. Nevertheless, despite the successful development of E-mini index futures in the US, very few studies have set out to examine the potential co-existence of time-varying GARCH and jump compo-nents, and the related responses to the arrival of news in the index markets.z
It has already become clear that E-mini futures provide traders with greater advantages than those provided by floor-traded index futures, in terms of lower margins, operational efficiency, quotation transparency and ano-nymity. Chung and Chiang (2006) also note that the differences between E-mini and open-outcry futures, with regard to contract size and tick size, are quite distinct. Furthermore, the absence of human intervention and the almost 24-hour trading period for E-mini futures may well have succeeded in attracting more day traders or speculators.
An issue of particular importance is the fact that, according to the Chicago Mercantile Exchange (CME) trading rules, during those periods when unexpected events occur but the floor-traded futures markets are closed, investors in floor-traded futures can write off their positions through the corresponding E-mini futures.
Therefore, if such investors were capable of recognizing the information transmission efficiency, risk and volatility dynamics of each market, they would be in a position to adopt appropriate hedging and arbitrage policies. Given the peculiarity of the market trading mechanism in the US, we are provided with an ideal environment to directly and empirically compare the volatility dynamics, infor-mation efficiency and risk amongst spot, regular and E-mini index futures.
In an attempt to capture the stochastic process of volatility for S&P 500 index spot and futures, we combine the Engle and Lee (1999) ‘component’ model with the Chan and Maheu (2002) ‘autoregressive jump intensity’ (ARJI) model, creating an ARJI-Trend model to under-take this challenging task.x Furthermore, the structural change analysis of Bai and Perron (2003) is also applied in advance to enable us to determine whether the market reverses; if this is the case, then we must consider the existence of structural changes for the three indices. Finally, we go on to compare the responses to news by the S&P 500 spot, regular and E-mini index futures markets. Despite the patterns of public information arrival to financial markets having been documented by Berry and Howe (1994) as non-constant, distinct, displaying season-ality and having an insignificant relationship with price volatility, their study did not consider potential jumps, time-varying volatility and components. They may, therefore, have failed to completely capture the actual characteristics of the volatility. In the present study, we set out to investigate volatility behavior and to carry out a risk comparison with the impacts of news arrivals on the S&P 500 index in an attempt to resolve this issue.
It is anticipated that such contrasts amongst the spot, open-outcry and E-mini futures indices will be particu-larly informative, not only because the three financial products are simultaneously traded during regular trading hours, but also because their prices are highly correlated; consequently, investors can readily achieve a complete understanding of the microstructure in the S&P 500 index markets. Thus, based upon our results, S&P 500 index market participants would be in a position to make appropriate investment and financial allocation decisions. This could also result in making market supervision much easier for government regulators.
This study makes several contributions to the literature, as follows. Firstly, the results show that a new model, incorporating long- and short-run GARCH dynamics yAlthough the ‘long memory’ phenomenon is also one of the characteristics of volatility in stock returns, the models generally applied to test this long-memory phenomenon include ARFIMA, FIGARCH and so forth; however, it seems that, to date, the empirical models are still unable to deal with a combination of ARJI-Trend plus the long-memory phenomenon.
zThere have been many studies exploring the price efficiency resulting from the changes from regular to electronic trading (Naidu and Rozeff 1994, Blennerhassett and Bowman 1998), the effects of electronic trading on the spot or futures market (Aitken et al. 2004, Chung and Chiang 2005) and the intraday price discovery process between floor-traded and E-mini index futures (Ates and Wang 2005); however, seldom has there been any focus on this particular topic.
xThere are several reasons for our examination of the dynamic process in the S&P 500 index spot and futures markets. Firstly, since both S&P 500 regular and E-mini index futures were the first commodities to be traded in the futures markets, they play a leading role in these markets. Secondly, the S&P 500 index is a market-value-weighted index of 500 stocks traded on the NYSE, AMEX and NASDAQ. The weights ensure that the influence of each firm on the performance of the index is proportional to the market value of the firm; hence, the index is an accurate substitute for a fully diversified equity portfolio. Thirdly, the S&P 500 stock index has long been the investment industry standard for measuring performance; thus, it is a comparable gauge for studying the reaction, in terms of volatility, to specific events.
and autoregressive conditional jump dynamics, provides a good fit for the S&P 500 index markets. Secondly, we find that there are certain similarities in some of the charac-teristics of volatility within the S&P 500 spot and futures; that is, it would seem that, in the long run, the risks in the S&P 500 index markets may be controllable. Thirdly, despite the fact that open-outcry and mini-sized futures are counterparts of the corresponding spot index, their reactions to news are found to be heterogeneous. Fourthly, E-mini index futures are the most efficient instrument in terms of the transference of information; hence, they appear to play a dominant role in price discovery within the index markets.
The remainder of this paper is organized as follows. Descriptions of the data and the empirical methodology adopted for this study are provided in section 2, followed in section 3 by presentation and discussion of the empirical results. Finally, the conclusions drawn from this study are summarized in the closing section.
2. Data and methodology 2.1. Data description
Our analysis of volatility dynamics is based upon the daily closing prices of the S&P 500 spot, futures and E-mini futures indices; the details are obtained from Tickdata Inc. Nearby floor-traded and E-mini futures contracts are selected for our discussion since these are the most actively traded futures contracts in their classifications. Given that the E-mini index futures contracts were introduced on the S&P 500 index of the CME on 9 September 1997, in order to unify the trading period for the three indices, the sample period used for our analysis covers the 10 years from 11 September 1997 to 31 August 2006. All of the analyses undertaken in this study are carried out on returns data.
2.2. Methodology
The primary aim of the present study is to determine whether the permanent and transitory components, in conjunction with the dynamic jump process, can explain volatility behavior in the S&P 500 spot and futures indices. In order to gain a complete understanding of whether these three singular features coexist in the three indices, we combine the Engle and Lee (1999) component model with the Chan and Maheu (2002) ARJI model to decompose the GARCH conditional variance into per-manent and transitory components. We hypothesize that jump intensity will follow an ARMA process, and also incorporate the GARCH effect of the return series. 2.2.1. ARJI model
The ARJI model of returns can be expressed asy:
Rt¼ þ Xp i¼1 iRtiþ ffiffiffiffi ht p Ztþ Xnt k¼1 t,k, ð1Þ ht¼! þ Xq i¼1 i"2tiþ Xp i¼1 ihti, ð2Þ t¼0þ Xr i¼1 itiþ Xs i¼1 iti, ð3Þ Zt NIDð0, 1Þ, t,k Nð , 2Þ,
where Rt is the return series of the S&P 500 spot and
futures. The conditional jump size, t,k, is assumed to be
independent and normally distributed with mean and
variance 2; htdenotes the conditional volatility dynamics
for the return, following a GARCH(p, q) process with
"t¼Rt Ppi¼1iRti; nt is the discrete counting
process governing the number of jumps arriving between
t 1 and t, which is distributed as a Poisson random
variable with a time-varying conditional intensity
param-eter, t.
Let tE[nt|t1] be the conditional expectation of the
counting process, which is assumed to follow an
ARMA(r, s) process, where tis related to r past lags of
the conditional jump intensity plus lags of t. Thus, t1is
the innovation to t1which is measured, ex post, by the
econometrician. The jump intensity residual is then computed as
ti E n½ tijti ti¼
X1
i¼0
jPðnti¼j jtiÞ ti,
ð4Þ
and the conditional density of nt, following a Poisson
distribution, is
Pðnt¼j jt1Þ ¼
etj
t
j! , j ¼0, 1, 2, . . . : ð5Þ
Maheu and McCurdy (2004) propose a further ex-post
distribution for the number of jumps, nt, within which the
filter is contracted as Pðn ¼ j jtÞ ¼ f ðRtjnt¼j, t1ÞPðnt ¼j jt1Þ PðRtjt1Þ , j ¼0, 1, 2 . . . : ð6Þ
After integrating out all of the jumps during a one-unit interval, the conditional probability density function of the returns can be expressed as
PðRtjt1Þ ¼
X1
j¼1
f ðRtjnt¼j, t1ÞPðnt¼j jt1Þ: ð7Þ
On the condition that the j jumps which occur during the conditional density of returns are normal, the likeli-hood function can be constructed as
f Rð tjnt ¼j, t1Þ ¼ 2ðhtþj 2Þ 1=2 exp Rt Pp i¼1iRti j 2 2ðhtþj 2Þ ! : ð8Þ yRefer to Chiu et al. (2005a) for a detailed description.
The log-likelihood function can then be written as
LðÞ ¼X
T
t¼1
log f Rð tjt1; Þ, ð9Þ
where ¼ (, i, !, i, i, , , 0, i, i) are the parameters
to be estimated.
Given that the log-likelihood function in equation (9) involves an infinite summation, in order to ensure that our estimations are feasible, as the truncation point for the distribution determining the number of jumps, we select a large value of five as the maximum number. The reason for this is that, in practice, we find that within our model estimates, the conditional Poisson distribution of equation (5) has a zero probability in the tail for all values
of nt45; thus, in the present study, we set the maximum
number of jumps (nt) as five.
2.2.2. Component model. The component model, which
was devised by Engle and Lee (1999) to describe both long- and short-run volatility, is stated as
ht¼qtþ "2t1qt1 þðht1qt1Þ, ð10Þ qt¼! þ qt1þ "2t1ht1 , ð11Þ
where qtis the permanent (or trend) component within the
GARCH conditional variance which captures the time-varying long-term volatility, with the mean reversion speed being determined by .
It should be noted that equation (2.4) in the GARCH(1,1) model of Engle and Lee (1999) can also be represented as
ht ¼ ð1 Þ 2þ"2t1þht1¼ 2
þ " 2t1 2þðht1 2Þ:
It is therefore quite easy to recognize that equation (10) in the present study is simply a reinterpretation of equation (2.4) in the Engle and Lee (1999) study, albeit
with the constant long-run volatility, 2, being replaced
by the time-varying trend, qt, and its past value. The
difference between ht and qt is simply the transitory
component of the conditional variance which dies out with time.
The forecasting error term, "2
t1ht1, which is
zero-mean and serially uncorrelated, drives the evolution of the
permanent component, whilst þ refers to the
persistence of the short-run component. The term is an autoregressive root, for which a typical value is
between 0.9 and 1; therefore, qt will approach the
unconditional variance very slowly. The stability terms of are 51 and þ 51. When 4 þ , the transitory component will converge more rapidly than the perma-nent compoperma-nent.
2.2.3. ARJI-Trend model. In addition to replacing
equation (3) t1with equation (4), we also combine the
ARJI model with the component model and rewrite equation (2) as equations (10) and (11). The resultant equations can be written as followsy:
ht¼qtþð"2t1qt1Þ þðht1qt1Þ, ð12Þ
qt¼! þ qt1þ "2t1ht1
, ð13Þ
t ¼0þ ð1 1Þt1þ 1E n½ t1jt1: ð14Þ
The log-likelihood function of equation (9) is used to
estimate the parameters in the present study.z
Furthermore, in the present study, we carry out the Bai and Perron (2003) procedures in advance, so as to ensure that we can effectively identify the potential structural change within the conditional mean of the returns for the spot and futures markets; this enables us to determine the possible break points in the three series.x
If such structural changes are found to exist, then the dummy variables are added into the mean equation to indicate the existence of a structural transition; that is, equation (1) is subsequently rewritten as
Rt ¼0þ Xn i¼1 iDiþ Xp i¼1 i0þ Xn j¼1 ijDj ! Rti þ ffiffiffiffiht p Ztþ Xn i¼1 i ffiffiffiffiffiffiffiffi hti p Ztiþ Xnt k¼1 tk, ð15Þ Zt NIDð0, 1Þ, t,k Nð , 2Þ, nt PoissonðtdtÞ,
where Di represents the structural change dummy
variables captured by the Bai and Perron (2003) procedure.
yIf ||51, and when ¼ s, equation (3) can be rewritten as equation (14).
zThe distribution of the error terms for standardized GARCH-type models is usually non-normal. In order to resolve the potential problem of the residuals not having a conditional normal distribution, we apply the quasi-maximum likelihood estimation (QMLE) approach for our empirical examination. According to Tsai and Chan (2005), the QMLE is asymptotically normal; therefore, our results are robust, producing consistent estimates of the parameters.
xBai and Perron (2003) note that a total of five breaks should prove to be efficient for empirical applications. In any event, the critical values for choices greater than five are very small. Accordingly, in the present study, the maximum permitted number of breaks allowed is set at five. In addition, to match the maximum number of breaks, we use a trimming of " ¼ 0.15; thus, the minimal distances between each break are 338 for the spot index, 340 for floor-traded futures and 341 for E-mini index futures. Adhering to the Bai and Perron (2003) approach for determining the potential break points in the spot, floor-traded and E-mini futures, we execute the following steps. Firstly, we construct the supF, UDmax and WDmax tests for the three indices, and then also construct the supF(l þ 1 | l) tests. Secondly, we refer to the procedure for selecting the number of breaks using information criteria as the BIC, LWZ and sequential procedures. Finally, we obtain the optimal number of breaks for spot, floor-traded and E-mini futures.
3. Estimation results
In order to illustrate the dynamic properties of S&P 500 spot and futures, we estimate the ARJI-Trend model in conjunction with a structural change test. A description of the data adopted for this study and subsequent analysis of the estimation results based upon the ARJI-Trend model are provided in the following subsections.
3.1. Basic analysis of the data
The summary descriptive statistics for the daily logarith-mic returns of the S&P 500 spot, regular and E-mini futures indices over the 10-year sample period are presented in table 1, from which we can see that the mean daily return is found to be the highest (lowest) for the spot (futures) index.
Amongst the three indices, the standard deviation of the returns is found to be the highest for regular futures, with all of the indices exhibiting large kurtosis, and the Jarque–Bera tests for normality revealing that the stock returns have a non-normal distribution.y Therefore, according to the preliminary results of the skewness, kurtosis and Jarque–Bera tests, the return distribution is more fat-tailed and high-peaked than a normal distribu-tion, which is consistent with the ARCH effect.
3.2. Empirical results for the S&P 500 index markets
3.2.1. Structural break points. The daily stock price
trends of the S&P 500 spot index, regular and E-mini futures markets from 11 September 1997 to 31 August 2006 are illustrated in figures 1(a)–(c). We apply the Bai and Perron (2003) procedure to extract the structural break point so as to explore the possible structural break in these indices during the sample period.
The extracted results indicate that the break date in the S&P 500 is 13 August 2002 for the spot index and 30 September 2002 for the open-outcry and E-mini futures indices. At first glance, such a difference in the timing of the structural changes in the two markets implies that, on the one hand, there are divergent responses to news in the spot and futures markets; that is, the S&P 500 spot and futures markets may well have different microstructures. On the other hand, however, the identical date for the structural change in the open-outcry and E-mini index futures markets seems to confirm high homogeneity between the two markets, despite the differences in their trading hours and trading systems.
In order to confirm that the inclusion of such breaks is essential, we first perform likelihood ratio tests to examine the validity of the ARJI-Trend model with structural breaks. The results are presented in tables 2–4, from which we can see that the ARJI-Trend model with structural breaks is an appropriate model for the S&P 500 spot index, floor-traded and E-mini futures markets.
In the following subsections, we go on to further examine the null hypotheses on whether the coefficients with break dummies are significantly different from zero
(that is, H0: 1¼0 and H0: 11¼0). The F-test results,
presented in table 5, reveal that a structural break must be considered when undertaking empirical research using the ARJI-Trend model. An evaluation of the existence of structural change is therefore seen as important when setting out to determine the presence of time-varying,
continuous-state GARCH and discrete jump
components.
3.2.2. Comparison between ARJI-Trend and
GARCH-type models. The ARJI-Trend model proposed in the
present study nests the widely adopted classical GARCH
structures: GARCH(1,1), component-GARCH and
GARCH- Jump with constant jump intensity and size. In order to determine whether it is the ARJI-Trend model which actually provides the best goodness-of-fit, we apply the likelihood ratio (LR) tests to examine the validity of each of the models. The LR results on the S&P 500 spot index, floor-traded and E-mini index futures (tables 2–4) indicate that, for these three markets, the GARCH, component-GARCH and GARCH-Jump models are rejected at the 1% level of significance.
3.3. Empirical results for the ARJI-Trend model
3.3.1. GARCH components. Table 5 also presents the
related results for the ARJI-Trend model, from which we can see that, for the three indices, all of the related parameters characterizing the dynamics in the models are significant. Furthermore, almost all of the parameters
pass the Ljung–Box Q(26) and Q2(26) tests, thereby
indicating the superior performance of the ARJI-Trend model in describing the behavior of S&P 500 spot and futures indices.
Table 5 further reveals that the permanent component of the GARCH conditional variance exhibits a high degree of persistence, with the autoregressive parameters Table 1. Summary descriptive statistics of daily returns for the S&P 500 spot index, floor- traded and E-mini index futures
markets. Variable Spot index Floor-traded futures E-mini futures Mean 0.0143 0.0137 0.0137 Std. dev. 1.1720 1.2021 1.2004 Max. 5.3080 5.8141 5.9128 Min. 7.1127 7.7058 7.8793 Skewness 0.0620 0.0958* 0.0820 Kurtosis 2.9755*** 3.5493*** 3.5098*** Jarque–Bera 832.9338*** 1192.8664*** 1170.2833*** No. of observations 2254 2266 2275
*Significant at the 10% level. **Significant at the 5% level. ***Significant at the 1% level.
yThe kurtosis examined here is excess kurtosis, which deviates from the respective means.
1600 1500 1400 1300 1200 1100 1000 900 800 700
11 Sep 30 Jun 19 Apr 20 Feb 16 Nov 6 Sep 1 Jul 17 Apr 4 Feb 19 Nov 8 Sep 27 Jun 1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006 (a) (b) 1600 1500 1400 1300 1200 1100 1000 900 800 700 16 Feb 30 Nov
11 Sep 1 Jul 21 Apr 7 Feb 22 Nov 19 Sep 10 Jul 29 Apr 13 Sep 27 Jun 1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006 1600 (c) 1500 1400 1300 1200 1100 1000 900 800 700 16 Feb 30 Nov
11 Sep 1 Jul 21 Apr 7 Feb 22 Nov 19 Sep 10 Jul 29 Apr 13 Sep 27 Jun 1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006
Figure 1. Daily stock price trend on the S&P500 (a) spot index, (b) futures index, and (c) E-mini futures index, September 1997 to August 2006.
Table 2. Parameter estimates of the GARCH-type model for the S&P 500 spot index. ARJI-Trend GARCH(1,1) Component GARCH GARCH Jump ARJI-Trend (no breaks) ARJI-Trend ( ¼ 1) Variable Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. 0 0.3504*** 0.0193 0.0396* 0.0215 0.0437** 0.0176 0.0478*** 0.0177 0.1743*** 0.0126 0.2646*** 0.0001 1 0.1394*** 0.0240 – 0.0744*** 0.0126 10 0.0022 0.0226 0.0265 0.0227 0.0347* 0.0187 0.0372** 0.0182 0.0576*** 0.0075 0.0148*** 0.0000 11 0.0900** 0.0418 0.1589*** 0.0213 v1 0.0233*** 0.0000 ! 0.0000 0.0558 0.0107** 0.0047 0.4414** 0.0872 0.0186*** 0.0024 0.0000 0.0558 1.0717*** 0.2965 0.0849*** 0.0062 0.0795*** 0.0140 0.0209*** 0.0027 0.0772** 0.0034 0.0790*** 0.0014 0.1094*** 0.0145 0.8608*** 0.0121 0.9150*** 0.0148 0.9762*** 0.0036 0.9030*** 0.0024 0.8848*** 0.0063 0.8515*** 0.0172 0.9932*** 0.0013 0.9364** 0.0228 0.9951*** 0.0012 1.0000*** 0.0000 0.0086** 0.0021 0.0933*** 0.0147 0.0108*** 0.0001 0.0280*** 0.0012 0.8056*** 0.0456 3.3365*** 0.8431 0.7535*** 0.0715 0.5852*** 0.0001 0.1915** 0.0739 1.8788*** 0.6852 0.5155*** 0.0224 0.0000** 0.0000 0 0.0130*** 0.0015 0.0068** 0.0000 0.1101*** 0.0062 0.2631*** 0.0126 1 0.9628*** 0.0044 0.4606*** 0.0152 0.1944*** 0.0379 1 0.0852*** 0.0289 0.4447*** 0.0225 0.5582*** 0.0022 Log L 3262.1331 3304.8791 3295.8603 3271.4754 3269.6633 3258.6476 H0 : GARCH 85.4920*** H0 : Component-GARCH 67.4544*** H0 : GARCH-Jump 18.6652*** H0 : ARJI-Trend (no breaks) 15.0604*** H0 :( ¼ 1) 6.9710 *Significant at the 10% level. **Significant at the 5% level. ***Significant at the 1% level.
Table 3. Parameter estimates of the GARCH-type model for S&P 500 floor-traded index futures. ARJI-Trend GARCH(1,1) Component GARCH GARCH Jump ARJI-Trend (no breaks) ARJI-Trend ( ¼ 1) Variable Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. 0 0.2383*** 0.0166 0.0400** 0.0195 0.0435** 0.0190 0.0507*** 0.0180 0.2523*** 0.0208 0.0547*** 0.0000 1 0.0766*** 0.0206 0.0377 0.0240 10 0.0109 0.0224 0.0268 0.0223 0.0272 0.0233 0.0284 0.0220 0.1228*** 0.0113 0.0507*** 0.0000 11 0.0427 0.0399 0.0288 0.0432 v1 0.0518*** 0.0222 v2 0.0576*** 0.0198 ! 0.0000 0.0528 0.0108*** 0.0035 0.4335*** 0.0645 0.0176*** 0.0026 0.0002 0.0003 0.0000*** 0.0000 0.0735*** 0.0066 0.0868*** 0.0126 0.0210*** 0.0010 0.0811*** 0.0058 0.0093*** 0.0066 0.0886*** 0.0077 0.8737*** 0.0117 0.9088*** 0.0128 0.9764*** 0.0010 0.9004*** 0.0035 0.9823*** 0.0093 0.8701*** 0.0136 0.9937*** 0.0011 0.9423** 0.0112 0.9471*** 0.0179 1.0000*** 0.0000 0.0081*** 0.0019 0.0959*** 0.0079 0.0797*** 0.0123 0.0215*** 0.0052 0.6927*** 0.0620 2.7810*** 0.6354 0.6956*** 0.0863 0.5739*** 0.1020 0.5710*** 0.0508 2.0560*** 0.5792 0.4559*** 0.0663 0.7909*** 0.0817 0 0.0246*** 0.0031 0.0095*** 0.0000 0.2490*** 0.0729 0.0404*** 0.0088 1 0.9259*** 0.0099 0.2950* 0.1650 0.6872*** 0.0647 1 0.1939*** 0.0576 0.4262*** 0.1171 0.5861*** 0.2261 Log L 3303.7758 3352.6116 3342.5027 3321.1007 3303.7319 3309.5355 H0 : GARCH 97.6716*** H0 : Component-GARCH 77.4538*** H0 : GARCH-Jump 34.6498*** H0 : ARJI-Trend (no breaks) 0.0878 H0 :( ¼ 1) 11.5194*** *Significant at the 10% level. **Significant at the 5% level. ***Significant at the 1% level.
Table 4. Parameter estimates of the GARCH-type model for S&P 500 E-mini index futures. ARJI-Trend GARCH(1,1) Component GARCH GARCH Jump ARJI-Trend (no breaks) ARJI-Trend ( ¼ 1) Variable Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. Coeff. S.E. 0 0.3331*** 0.0000 0.0394** 0.0194 0.0429** 0.0178 0.0500*** 0.0182 0.2268*** 0.0122 0.2203*** 0.0280 1 0.1024*** 0.0247 0.0590* 0.0342 10 0.1270*** 0.0000 0.0290 0.0227 0.0254 0.0221 0.0271 0.0226 0.0840*** 0.0184 0.0122 0.0211 11 0.0763*** 0.0390 0.0896*** 0.0282 v1 0.1030*** 0.0223 ! 0.0000*** 0.0000 0.0108*** 0.0041 0.4359*** 0.1131 0.0175*** 0.0020 0.0005*** 0.0002 0.0107*** 0.0011 0.0745*** 0.0071 0.0853*** 0.0132 0.0213* 0.0110 0.0821*** 0.0030 0.0818*** 0.0075 0.0574*** 0.0196 0.8708*** 0.0142 0.9101*** 0.0133 0.9761*** 0.0112 0.9001*** 0.0027 0.8592*** 0.0143 0.9114*** 0.0287 0.9924*** 0.0014 0.9411*** 0.0228 0.9896*** 0.0017 1.0000*** 0.0000 0.0142*** 0.0036 0.0961*** 0.0161 0.0143*** 0.0037 0.0656*** 0.0192 0.7103*** 0.0470 2.8126*** 0.7682 0.6825*** 0.0481 0.5536*** 0.1004 0.4752*** 0.0492 2.1382*** 0.6459 0.4925*** 0.0503 0.4284*** 0.0744 0 0.1305*** 0.0118 0.0083*** 0.0032 0.2082*** 0.0199 0.2253** 0.0913 1 0.6107*** 0.0358 0.3542*** 0.0603 0.2389 0.2039 1 0.2233*** 0.0911 0.3632*** 0.1283 0.4330*** 0.1465 Log L 3302.0651 3364.0375 3353.7131 3331.6032 3314.6898 3327.6124 H0 : GARCH 123.9448*** H0 : Component-GARCH 103.2960*** H0 : GARCH-Jump 59.0762*** H0 : ARJI-Trend (no breaks) 25.2494*** H0 :( ¼ 1) 51.0946*** *Significant at the 10% level. **Significant at the 5% level. ***Significant at the 1% level.
() in the trend equation being very close to 1 (0.9932 for spot, 0.9937 for regular futures, and 0.9924 for E-mini futures). In order to determine whether the permanent component of volatility is integrated, we carry out the statistical test by letting ¼ 1. In the subsequent proce-dure, we follow Chen and Shen (2004) to re-estimate the model and calculate the restricted log likelihood function. As shown in tables 2 to 4, the log-likelihood ratio tests are 6.9710 for spot, 11.5194 for floor-traded futures, and 36.1116 for E-mini index futures, results which reveal no rejection of the restricted ARJI-Trend model for the S&P 500 spot; that is, the permanent component of the spot volatility is a random walk, whereas those of the regular and E-mini futures are not. Furthermore, the convergence speed to long-run innovation appears to be most rapid in the E-mini futures market, with the slowest convergence speed being found for floor-traded futures.
Accordingly, this also implies that, in the spot market, almost 87.24% of the news shock on the permanent components still persists after 20 trading days, whilst 88.13% (85.85%) of the news shock persists in the open-outcry (E-mini) futures market over the same horizon.y Conversely, the respective sums of þ , which represent the continuance level of the transitory component, are 0.9457 for spot, 0.9472 for open-outcry, and 0.9453 for E-mini futures. These figures correspond to their respective half-lives of 18.5, 19 and 18 trading days.
These results reveal that the trend has a high level of persistence, whereas the arrival of new information has only minor impacts on the markets. The outcome also reveals that the deviations in GARCH conditional variance from the general trend are temporary, thereby providing support for the findings of the prior studies.z
The shock effects on the permanent and transitory components are respectively given by and , with a comparison of the parameter estimates showing that, for the three markets, the arrival of news has a much greater influence on the transitory component than on the permanent component. However, combined with the previous results, despite the shock effect on the transitory component being stronger, as in Engle and Lee (1999) and Chen and Shen (2004), the effect is found to be short-lived. Furthermore, as the relationships between the parameters are ( þ )551, the permanent component will tend to dominate the forecasting of GARCH.
To briefly summarize, when certain information arrives into the market, the more rapid convergence speed of the permanent and transitory components of the E-mini futures indicates that the E-mini market is better at transmitting the information. Conversely, floor-traded futures appear to be the least efficient in dealing with the news. This suggests that the introduction of E-mini futures may well have led to them playing a governing role in price discovery. Indeed, since the open-outcry Table 5. Estimation results of the ARJI-Trend model for the S&P 500 indices.
Variablea
Spot indexb Floor-traded futuresb E-mini futuresb
Coeff. S.E. Coeff. S.E. Coeff. S.E.
0 0.3504*** 0.0193 0.2383*** 0.0166 0.3331*** 0.0000 1 0.1394*** 0.0240 0.0766*** 0.0206 0.1024*** 0.0247 10 0.0022 0.0226 0.0109 0.0224 0.1270*** 0.0000 11 0.0900** 0.0418 0.0427 0.0399 0.0763* 0.0390 v1 0.0233*** 0.0000 0.0518** 0.0222 0.1030*** 0.0223 v2 0.0576*** 0.0198 ! 0.0000 0.0558 0.0000 0.0528 0.0000*** 0.0000 0.0849*** 0.0062 0.0735*** 0.0066 0.0745*** 0.0071 0.8608*** 0.0121 0.8737*** 0.0117 0.8708*** 0.0142 0.9932*** 0.0013 0.9937*** 0.0011 0.9924*** 0.0014 0.0086** 0.0021 0.0081*** 0.0019 0.0142*** 0.0036 0.8056*** 0.0456 0.6927*** 0.0620 0.7103*** 0.0470 0.1915** 0.0739 0.5710*** 0.0508 0.4752*** 0.0492 0 0.0130*** 0.0015 0.0246*** 0.0031 0.1305*** 0.0118 1 0.9628*** 0.0044 0.9259*** 0.0099 0.6107*** 0.0358 1 0.0852*** 0.0289 0.1939*** 0.0576 0.2233** 0.0911 Q(26) 25.6189 28.2439 29.7274 Q2(26) 22.6970 38.2480* 32.7562 Log L 3262.1331 3303.7758 3302.0651 H0: 1¼0 and H0: 11¼0 31.0737*** 14.7318*** 21.1511*** a
The Q(26) and Q2(26), distributed as 2
26, are the Ljung–Box Q-statistics with a lag of 26 for standardized residuals and standardized squared
residuals for the purpose of examining whether linear and nonlinear dependence have been removed.
b
Significant at the 10% level. **Significant at the 5% level. ***Significant at the 1% level.
yGiven that the daily impact of news shocks on the permanent components for each market are 0.9932 for the spot market, 0.9937 for the open-outcry market, and 0.9924 for the E-mini market, their effects after 20 trading days are, respectively, computed as 0.993220, 0.993720and 0.992420. The resultant percentages are 87.24%, 88.13% and 85.85%.
zSee Engle and Lee (1999), Speight et al. (2000) and Ane´ (2006).
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
11 Sep 30 Jun 19 Apr 2 Feb 16 Nov 6 Sep 1 Jul 17 Apr 4 Feb 19 Nov 8 Sep 27 Jun 1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006 (a) 1.2 1.0 0.8 0.6 0.4 0.2 0
11 Sep 1 Jul 21 Apr 7 Feb 22 Nov 19 Sep 10 Jul 29 Apr 16 Feb 30 Nov 13 Sep 27 Jun
1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006 (b) 1.2 1.0 0.8 0.6 0.4 0.2 0
11 Sep 2 Jul 22 Apr 8 Feb 24 Nov 21 Sep 12 Jul 1 May 17 Feb 1 Dec 13 Sep 27 Jun 1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006 (c)
Figure 2. Conditional time-varying jump intensity on the S&P500 (a) spot index, (b) index futures, and (c) E-mini futures, September 1997 to August 2006.
trading mechanism involves greater human participation within the overall trading process (Christie et al. 1994), this is likely to result in a reduction in information efficiency within the market.
3.3.2. Jump intensity. The jump-size mean is found to
be significant at the 1% level for all three indices, ranging from 0.8056 to 0.6927, with such significance implying that there is abnormal information resulting in a discon-tinuous jump in the index spot and futures market which, in the long run, will be significantly different from zero.
The jump parameters 0are also found to be significant
at the 1% level, thereby implying the existence of jump behavior for the S&P 500 index spot and futures whenever there is abnormal information flowing into the markets.
Furthermore, the parameters 1and 1are also found
to be significant at the 1% level, thereby indicating that the probability of jumps incited by abnormal information would change over time, leading to clustering, as noted by Maheu and McCurdy (2004). The unconditional jump intensity is 0.3495 for spot, 0.3320 for regular futures, and 0.3352 for E-mini index futures, which suggests that jumps to returns will occur, on average, once every 1.537, 1.497 and 1.504 business days for the corresponding markets.y Thus, the abnormal innovations occurring in the S&P 500 index markets are frequent; however, since the jumps will almost completely decay in about 4–5 trading days in all three markets, their impacts will rapidly disappear.
The conditional jump intensities for the S&P 500 spot, open-outcry and E-mini index futures markets are illus-trated in figures 2(a)–(c), which clearly reveal that jump intensity is time-varying. Thus, although some studies (such as Chen and Shen (2004)) hypothesize that jump intensity is constant, it will vary when different events occur over various time spans.
3.3.3. Comparison of the three features of total conditional
variance (risk). Total conditional variance comprises the
permanent and transitory components, along with jump variance; however, an interesting point from the results shown in table 6 is that volatility arises mainly from the transitory component (48.50% for the spot market, 47.40% for floor-traded futures, and 53.50% for
E-mini futures). Although GARCH variance provides the main contribution to total variance (about 83%), jump-induced variance similarly explains about 17% of the volatility for the three index markets.
The results indicate that jump behavior is important, and must therefore be taken into account when consid-ering investing in the S&P 500 index. However, responses to events seem to be less dramatic in the US, as compared to the Taiwan index market (Chiu et al. 2008). Furthermore, since the total variance is greatest in the spot index (1.468), followed by the floor-traded index futures (1.462), and the E-mini index futures being the smallest (1.224), this suggests that trading is riskier in the spot and regular index futures markets than in the E-mini futures market.
The permanent and transitory components and the jump variance for the three markets are illustrated in figures 3(a)–(c), which show that the permanent compo-nent moves in a rather even manner, whilst the transitory component is driven up and down by news arrivals. Furthermore, despite the permanent component lasting for a longer period, it is not found to be particularly large. Overall, the transitory component is found to be far greater than either the permanent component or the jump-induced variance.
For investors, although the effects of transitory and jump components are significant, they are merely tempo-rary. Conversely, despite the more prolonged impacts of permanent components, in the long run, these are relatively smooth. From a perspective of long-term investment, the investment risks appear to be under control, essentially because the market will ultimately return to its long-run state; however, what such investors do need to be aware of is fundamental risk (economics, politics, and so on). Conversely, speculators in the market can make use of the sizable temporary and jump component to engage in arbitrage activities.
3.4. Out-of-sample analysis
For our evaluation of the out-of-sample one-period-ahead variance forecasting ability of the GARCH(1,1) and ARJI-Trend models, we select the first nine years as our in-sample period, and set the final year as our out-of-sample period. The loss functions, mean absolute error (MAE) and root mean square error (RMSE) are
consid-ered as the criteria for assessing the forecasting
Table 6. Basic statistics for the total conditional variance combination.
Variance combination Value % Value % Value %
GARCH conditional variance
Transitory component 0.712 48.50 0.693 47.40 0.655 53.50
Permanent component 0.411 28.00 0.444 30.37 0.489 39.95
Jump conditional variance 0.344 23.43 0.326 22.30 0.081 6.62
Total conditional variance 1.468 100.00 1.463 100.00 1.224 100.00
yWhen the ARJI specification is stationary (||51), then the unconditional jump intensity will be equal to E [t] ¼ 0/(1 1).
7 6 5 4 3 2 1 0
11 Sep 30 Jun 19 Apr 2 Feb 16 Nov 6 Sep 1 Jul 17 Apr 4 Feb 19 Nov 8 Sep 27 Jun
1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006
Permanent component Transitory component Jump variance
(a)
8 7 6 5 4 3 2 1 011 Sep 1 Jul 21 Apr 2 Feb 22 Nov 19 Sep 10 Jul 29 Apr 16 Feb 30 Nov 13 Sep 27 Jun
1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006
Permanent component Transitory component Jump variance
(b)
7 6 5 4 3 2 1 011 Sep 2 Jul 22 Apr 8 Feb 24 Nov 21 Sep 12 Jul 1 May 17 Feb 1 Dec 13 Sep 27 Jun 1997 1998 1999 2000 2000 2001 2002 2003 2004 2004 2005 2006
Permanent component Transitory component Jump variance
(c)
Figure 3. Jump variance and permanent and transitory components for the S&P 500 (a) spot index, (b) index futures, and (c) E-mini index futures, September 1997 to August 2006.
performance of the ARJI-Trend model relative to that of the GARCH(1,1) model.y The formulae for computing the MAE and RMSE are as follows:
MAE ¼ PT t¼1jYt ^Ytj T , RMSE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PT t¼1 Yt ^Yt 2 T v u u t ,
where Ytis the observed value, and ^Yt is the estimator of
the estimated parameter Y. The forecasting performance will be better with a smaller value of MAE or RMSE.
The model parameters are initially set at the values shown in table 5; we then use a fixed-sized rolling window of 250 observations, and also adopt the Diebold and Mariano (1995) ‘DM-test’ (the forecast comparison test) to compute the MAE and RMSE, comparing the forecasting efficiency between the ARJI-Trend and GARCH (1,1) models during the out-of-sample period.
Although the outcomes of the DM test appear to reveal that the ARJI-Trend model performs better than the GARCH model in the S&P 500 spot index, there are only very small differences in the values of MAE (1.3847 vs. 1.3826) and RMSE (3.3517 vs. 3.3516). Therefore, the DM-test results reported in table 7 indicate that, for the two models, the out-of-sample forecasting performance is virtually the same for the S&P 500 spot, floor-traded and E-mini indices.
3.5. News effects
As shown in figures 2(a)–(c) and 3(a)–(c), during the event periods in our sampling span, there are increases in the permanent and temporary components, jump intensity and jump variance for all three indices. To explain this in a concise way, we simply display the jump intensity and transitory components for the related events in table 8. Both jump intensity and the transitory components are generally larger than their means during these periods; however, the degree of priority is not consistent amongst the three indices. We surmise that the different responses to news arrivals during such spans stem mainly from the divergent trading systems and clientele characteristics. These results provide some support for the dissimilarities
in structural fundamentals in the index markets, as noted by both Chiu et al. (2006) and Chung and Chiang (2006). To summarize, the results indicate the coexistence of time-varying jumps, as well as both permanent and transitory components in the volatility behavior of the S&P 500 spot and index futures markets. Furthermore, the diverse responses to news arrivals amongst the S&P 500 indices may be attributable to certain characteristics relating to market structure. Indeed, this also suggests that, under the new form of economic generation—within which greater uncertainty is the norm—aside from traditional GARCH volatility, jumps leading to infre-quent large moves in returns must be taken into account when considering investing in the financial markets. Although the ARJI-Trend model does provide us with a heightened perception of the reaction to events and volatility behavior within the S&P 500 markets, when faced with shocks and innovations within these markets, we suggest that investors should treat them as a normal condition, since, in the long run, the impacts will ultimately disappear.
4. Conclusions
We propose a model structure, comprising a combination of time-varying continuous-state GARCH and discrete jump components, which we suggest is appropriate for describing the volatility features of the S&P 500 index markets. Our results reveal the coexistence of trend and transitory components, together with time-varying jumps, and although we find that the trend has a high level of persistency, the arrival of new information appears to have only minor impacts on the markets.
The variance in the transitory and jump components is capable of explaining about 83% of the total conditional variance in the US markets; nevertheless, such variance will decay in the long run, which suggests that following the occurrence of trading noise and events, there may be some inherent force stabilizing the market. Rational investors must therefore treat such news arrivals and noise trading as normal occurrences, since the market will eventually return to its long-run state; that is, in addition to the fundamental risk and trading noise, investors in the Table 7. Out-of-sample forecasting performance of conditional volatility for S&P 500 spot, floor-traded and E-mini index futures.
GARCH(1,1) ARJI-trend DM statisticsa,b
Index market MAE RMSE MAE RMSE MAE RMSE
Spot 1.3847 3.3517 1.3826 3.3516 6.8586*** 2.7203***
Futures 1.4559 3.6901 1.5131 4.0618 1.6234 1.2487
E-mini 1.4516 3.6692 1.4803 3.7406 1.9153 1.4194
a
DM refers to the Diebold and Mariano (1995) t-statistics. The null hypothesis of the DM-test is of equal predictive ability for the two models, whereas a significantly negative t-statistic would indicate that the GARCH(1,1) model was dominated by the ARJI-Trend model.
b
Significant at the 1% level.
yThe MAE and RMSE are frequently used measures for determining the differences between values predicted by a model and the values actually observed from the issue being modeled or estimated.
Table 8. Conditional jump intensity and transitory components of the S&P 500 spot, floor-traded and E-mini index futures during specific events. Spot market Floor-traded futures E-mini futures Event Date Jump intensity Transitory component Jump intensity Transitory component Jump intensity Transitory component Southeast Asian financial crisis hampers both the economic growth and profitability of US industries, leading to economic decay 3 August 1998 0.4723 1.0389 0.5779 0.9805 0.4697 0.7906 4 August 1998 0.4758 0.9665 0.5523 0.8653 0.4233 0.7013 5 August 1998 0.5854 2.2185 0.8100 2.1023 0.7157 1.8967 6 August 1998 0.5616 1.8989 0.7254 1.8476 0.5111 1.6650 The Fed reduces interest rates by 0.25%, lower than expected, leading to general disappointment in the financial markets 1 October 1998 0.4873 2.4006 0.4262 2.5474 0.3979 1.8455 2 October 1998 0.5166 3.0905 0.4722 2.9974 0.4393 2.3386 9 November 2000 0.4032 0.7789 0.3830 0.9357 0.4551 0.6777 Dell announces lower than expected sales growth for the coming year 10 November 2000 0.4079 0.7127 0.3899 0.8637 0.4187 0.5975 13 November 2000 0.4715 1.2598 0.4842 1.2228 0.5051 0.9368 The Fed reduces interest rates by only 0.5%, again not in line with the expectations of investors 21 March 2001 0.5726 2.4824 0.6212 2.2789 0.3988 1.8306 22 March 2001 0.5802 2.4685 0.6299 2.2618 0.4051 1.8129 18 September 2001 0.6422 3.4222 0.8429 3.3676 0.5657 1.9141 The ‘911’ terrorist attacks on the World Trade Centre in New York 19 September 2001 0.6312 2.9404 0.7861 2.9423 0.4631 1.5960 20 September 2001 0.6326 2.7944 0.7956 3.0785 0.4687 1.7498 21 September 2001 0.6561 3.3968 0.8313 3.6244 0.5181 2.3882 Fall in Boeing stock prices leads to concern amongst investors of an economic downturn 30 October 2001 0.4244 0.9519 0.4071 1.0315 0.5076 0.7630 31 October 2001 0.4502 1.1398 0.4315 1.0529 0.4834 0.7946 19 July 2002 0.5377 2.2893 0.6008 2.6014 0.4407 2.0848 US Congress passes the industrial refor-mation law (Sarbanes–Oxley Act) 22 July 2002 0.5832 3.5011 0.6703 3.3989 0.4962 2.8976 23 July 2002 0.6024 4.0996 0.7012 3.8201 0.4918 3.2848 24 July 2002 0.6089 4.1282 0.7221 4.1515 0.4757 3.4864 Period of structural change 13 August 2002 0.4482 3.0241 14 August 2002 0.4589 3.0588 30 September 2002 0.5054 1.9901 0.4279 1.5786 1 October 2002 0.5062 1.8341 0.4116 1.4499 Concerns amongst investors over the possible outbreak of war between the US and Iraq 22 January 2003 0.3539 0.8240 0.3639 0.7071 0.4541 0.7722 23 January 2003 0.3667 0.8276 0.3983 0.7835 0.4546 0.7593 27 January 2003 0.4088 1.3862 0.4717 1.1304 0.4837 0.6364 28 January 2003 0.4270 1.5218 0.4964 1.1813 0.4798 0.7411 Sample average 0.3471 0.7002 0.3272 0.6913 0.3363 0.4545
S&P 500 markets must also take into account the potential occurrence of such events when considering engaging in short-run investment activities. In the long run, however, what investors need to do is to allocate assets within their portfolios to earn better returns.
The out-of-sample forecast performance evaluated by the DM-test reveals that the ARJI-Trend model has superior performance in the S&P 500 spot market, a result which is consistent with the finding of Chan and Maheu (2002), that the ARJI-Trend model adequately captures the time variation in the conditional jump intensity and improves both the in-sample fit and out-of-sample fore-casting in the S&P 500 spot market.
Of the three indices, the mini-sized index futures market exhibits more rapid convergence to normality, as well as less trading noise, implying that this market deals with information more efficiently. This result concurs with the findings of Hasbrouck (2003) and Kurov and Lasser (2004), that the introduction of E-mini futures has led to these instruments playing a dominant role in price discovery within the index markets. Furthermore, as noted by Chiu et al. (2006) and Chung and Chiang (2006), the different responses to news arrivals in the S&P 500 indices may stem from market-specific characteristics associated with the market microstructure.
In conclusion, with the coexistence of trend, transitory and jump components in the volatility behavior of the S&P 500 index markets, we suggest that governments should strive to gain a firm understanding of the overall effects of changes in trading systems and policies. This would undoubtedly lead to improvements in information efficiency, thereby enabling such governments to better manage both the financial markets, and market risk. Failure to take these issues into consideration may well result in mistakes or the likelihood of erroneous decision making.
Acknowledgements
The authors would like to take this opportunity to express our sincere appreciation for the financial support pro-vided by the National Science Council. Huimin Chung would like to gratefully acknowledge the financial sup-port provided by the MoE ATU Plan of the National Chiao Tung University.
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