Asymmetric volatility dynamics in high frequency FTSE-100 stock index futures
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Applied Financial Economics, 2003, 13, 599–607
Asymmetric volatility dynamics in high
frequency FTSE-100 stock index futures
D A V I D G. Mc M I L L A N and A L A N E . H . SP E I G H T {*
Department of Economics, University of St Andrews, Scotland, UK and { Department
of Economics, University of Wales, Swansea, SA2 8PP, UK
This paper examines whether variants of the GARCH class of model with the
capacity to accommodate volatility asymmetries and volatility feedback are able
to provide an adequate representation of non-linear dependency in intraday
FTSE-100 stock index futures returns at the quarter-hour and hourly frequency.
Significant variance asymmetry is identified, and such that negative shocks induce
a greater response in volatility than equivalent positive shocks, but with the addi-
tional effect of subsequently depressing volatility at the 15-minute frequency. In the
absence of financial leverage arguments in the market considered, and the absence of
a statistically significant volatility feedback effect, such asymmetry is interpreted as
indirect evidence for the presence of noise traders, attracted to such markets by low
transaction costs and margin requirements. In contrast with previous results using
intraday data, a notable absence of remaining structure in asymmetric GARCH
models at the hourly frequency is found, but neither symmetric nor asymmetric
models are able to fully account for nonlinear dependence at the higher intraday
frequency.
I. INTRODUCTION As first noted by Black (1976), a potential explanation
for this negative returns-volatility correlation, or ‘predic-
Over the past decade and a half, the genre of models of tive asymmetry’, in equity markets is offered by the ‘lever-
generalized autoregressive conditional heteroscedasticity age effect’. That is, where a firm with debt and equity
(GARCH: Engle, 1982; Bollerslev, 1986) have provided outstanding is exposed to large negative returns which
the dominant means for modelling nonlinear dependence decrease the market value of the firm, raising the debt-to-
in financial data.1 A significant issue that has arisen in the equity ratio and increasing the risk associated with the
empirical application of GARCH models to interdaily claim of equity, so increasing returns volatility. An alter-
equity data concerns the potentially asymmetric behaviour native explanation for volatility asymmetry and the nega-
of volatility. Whilst a symmetric effect of positive and nega- tive correlation between stock returns and future volatility
tive shocks on conditional variance is imposed by the linear of the kind implied by the leverage effect is offered by
dependence of the GARCH conditional variance on the ‘volatility feedback’ (eg. Campbell and Hentschel, 1992).
squares of past shocks and past variances, an asymmetric That is, where large items of ‘news’ increase expected
relationship is often argued to hold between current vari- future volatility, so increasing the required rate of return
ance and past shocks, negative shocks typically increasing and depressing the current asset price, thereby magnifying
conditional variance by a greater amount than equally the negative price effects of negative news and mitigating
sized positive shocks. Alternatively put, volatility is typi- the positive price impact of positive news. As a conse-
cally higher following stock market falls than rises, such quence, returns are characterized by negative skewness,
that returns are negatively correlated with future volatility. large negative returns being more common than large posi-
* Corresponding author. E-mail: a.speight@swan.ac.uk
1
For reviews, see Bollerslev, Chou and Kroner (1992), Bera and Higgins (1993), and Bollerslev et al. (1994).
Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online # 2003 Taylor & Francis Ltd 599
http://www.tandf.co.uk/journals
DOI: 10.1080/0960310022000040715600 D. G. McMillan and A. E. H. Speight
tive returns, and asset price movements are correlated with II. MODELS
future volatility. However, predictive asymmetry may also
reflect the role of market dynamics. In a market model With the exception of preliminary models of conditional
comprising insiders, uninformed investors and market mean structure alone, all estimation is conducted jointly in
makers, differences in investors’ expectations may take conditional mean and conditional variance. Conditional
time to be eliminated as the information held by insiders mean models are of linear autoregressive ARðmÞ class,
takes time to be disseminated, so accounting for volatility while conditional variance structure is modelled, at least
clustering (Kyle, 1985). In particular, as demonstrated by initially, by GARCHðp; qÞ processes for conditional vola-
Sentana and Wadhwani (1992) in an extension of this tility defined in terms of the errors of the mean models,
model to accommodate feedback (or ‘noise’) traders pos- expressed in general form as:
sessing less information than their informed counterparts
and who follow market trends and trade on price move- X
m
rt ¼ þ ai rti þ "t ð1Þ
ments, responses to bad news (negative price shocks) lead i¼1
to greater volatility than do responses to good news, so
offering an alternative explanation for predictive asymme- "t ¼ t ht ð2Þ
try.
Whilst the issue of predictive asymmetry in volatility has X
q X
p
h2t ¼ ! þ i "2ti þ i h2t1 ð3Þ
been extensively investigated using interdaily and lower i¼1 i¼1
frequency equity spot market data, it has received far less
attention in the context of higher frequency data, and where ð; ai ; !; i ; i Þ are real constants, the unexpected
equity futures markets in particular. This is perhaps sur- return "t rt Eðrt jOt1 Þ is serially uncorrelated with
prising given that futures markets are characterized by low zero mean and conditional variance h2t varðrt jOt1 Þ
transaction costs and low margin requirements, and that where Ot1 represents the information set containing rea-
the consequent attraction of noise and feedback traders to lized values of r up to t 1, and the standardized error t is
such markets is likely to engender significant asymmetries identically and independently distributed (iid) with zero
in futures market volatility.2 This paper examines these mean and unit variance. For the GARCH model expressed
issues in the context of intraday UK FTSE-100 index in Equation 3, where i , i and ! are non-negative par-
futures contract returns. A particular feature of this data ameters,
P it P is necessary and sufficient that the sum
is that it comprises actual transaction prices, rather than ¼ i i þ i < 1 in order for a finite unconditional
the notional prices based on bid-ask spread midpoints that variance to exist, that sum also providing a measure of
are commonly used in the high frequency analysis of spot the persistence of shocks to h2i , with half-life given by % ¼
market returns. This has the further advantage of permit- ½ lnð2Þ= lnðÞ . Those measures also define the limiting
ting the interpretation of our empirical results in terms of integrated-GARCH (IGARCH) case under ¼ 1, % ¼ 1,
asset price adjustment. Specifically, conditional volatility in such that current shocks persist indefinitely in conditioning
hourly and quarter-hourly FTSE-100 futures transaction future variances (Engle and Bollerslev, 1986; Nelson,
returns is modelled using the GARCH model and its 1990). However, whilst ! > 0 and i , i 5 0 may be
threshold-GARCH (TGARCH) and quadratic-GARCH imposed to ensure non-negativity of the conditional vari-
(QGARCH) extensions which permit investigation of the ance, Nelson and Cao (1992) have shown that these
potentially asymmetric response of volatility to past inequalities need not hold to ensure a positive variance.
shocks, as well as a QGARCH-in-mean (QGARCH-M) For example, in the GARCH(1,2) case which holds empiri-
form with the ability to accommodate volatility feedback.3 cally below, where 2 is negative, 1 1 5 2 is sufficient to
The remainder of the paper proceeds as follows. Section ensure h2t > 0.
II introduces the model specifications and their properties. In order to capture potential volatility asymmetry, two
Section III outlines the specification tests employed. alternatives to Equation 3 are considered which permit the
Section IV describes the data. Section V reports model asymmetric response of conditional volatility to past
estimates, test results and residual diagnostics. Section VI shocks. First, the threshold-GARCH (TGARCH)
summarizes the findings and conclusions. model of Glosten et al. (1993) (also closely related to the
2
A notable exception is the recent work of Antoniou et al. (1998) which reports estimated asymmetric volatility models for a variety of
daily futures markets in the context of an examination of the impact of futures trading on asymmetric volatility in equity spot markets.
3
For related studies of financial futures markets at lower, typically interday, frequencies, see for example, Praschnik (1991), Yang and
Brorsen (1994), Fujihara and Mougoué (1997) and Robinson (1998). On the efficiency and hedging effectiveness of the FTSE-100 stock
index futures contract, see Antoniou and Holmes (1996) and Butterworth and Holmes (2001) respectively. On the interaction between the
FTSE-100 spot and futures markets, see Abhyankar (1995), Antoniou and Holmes (1995), and Antoniou et al. (1998).Asymmetric volatility dynamics 601
6
threshold-ARCH representations proposed by volatility risk. However, whilst the GARCH-M model
Rabemananjara and Zakoian, 1993, and Zakoian, 1994): formed by the conjunction of Equations 3 and 6 allows
X
q X
p the conditional mean to depend on the conditional vari-
h2t ¼ ! þ i "2ti þ i h2t1 þ 2
1 D1 "t1 ð4Þ ance, that model imposes zero correlation between returns
i¼1 i¼1 and future volatility, and therefore does not capture the
mechanism underlying volatility feedback. That is, as
where potential asymmetry, restricted to a first-order effect
described in the Introduction, where changes in volatility
only, is captured by the use of the dummy variable D1 such
have important effects on required returns and thus on the
that D1 ¼ 1 if "t1 < 0 and D1 ¼ 0 otherwise. This
current level of asset prices. Following Campbell and
TGARCHðp; q; 1Þ specification thus allows negative shocks
Hentschel (1992), the extension of the QGARCHðp; q; rÞ
to have a greater impact on subsequent volatility if the real
model to the QGARCH-M form given by Equations 6
constant 1 P > 0, whilst
P overall shock persistence is quanti- and 5 is utilized in order to capture this potential effect.
fied by ¼ i i þ i i þ ð 1 =2Þ with half-life calculated
In contrast to the GARCH or TGARCH model, the
as above. An alternative to the asymmetric TGARCH
QGARCH model permits a non-zero correlation between
model is provided by the quadratric-GARCH
returns and future volatility through the last term in
(QGARCH) model (Engle, 1990; Engle and Ng, 1993;
Equation 5, and the QGARCH-M model is therefore
Sentana, 1995), the diagonal variant of which (excluding
able to capture the mechanism underlying volatility feed-
shock cross-products, see Sentana, 1995) is given by:4
back. By amplifying the predictive asymmetry of the basic
X
q X
p X
r QGARCH model in this manner, the QGARCH-M model
h2t ¼ ! þ i "2ti þ i h2ti þ i "ti ð5Þ is also able to accommodate the negative skewness and
i¼1 i¼1 i¼1
excess kurtosis implied by volatility feedback without
with shock persistence and half-life calculated as for the recourse to alternative statistical models characterized by
basic GARCH model (Sentana, 1995, p. 646) whilst nonnormal return innovations (Nelson 1991, Engle and
i 6¼ 0 yields a direct measure of potentially higher-order González-Rivera, 1991).
dynamic asymmetries in conditional variance with respect
to past shocks, h2t being greater for negative "ti when
i < 0.5
I I I . S P E C I FI C A T I O N T E S T S
Further extension of the model in Equation 1 allows the
conditional mean to be an explicit function, in part, of the
A variety of specification tests are employed in order to
conditional variance process under the GARCH-in-mean
ensure that the estimated models obtained offer appropri-
(GARCH-M) specification (Engle et al. 1987):
ate characterizations of market conditional volatility.
X
m These include the ARCHðqÞ Lagrange Multiplier (LM)
rt ¼ þ ai rti þ h2t þ "t ð6Þ test due to Engle (1982), computed as TR2 in the OLS
i¼1
regression of "2t on its first q lags and a constant, the result-
with h2t defined as in Equation 3. Following Merton (1980), ing test statistic, denoted here by Aq , being distributed as
the parameter in such models may be interpreted as the 2q under the null of no ARCH effects. The extension of this
coefficient of relative risk aversion, and h2t as a time- test proposed by Sentana (1995) is also implemented which
varying risk premium in the sense of the increased expected involves augmenting the test equation by the inclusion of
rate of return required in response to an increase in the the first q lags of "t as regressors, so providing an LM test
predictable variance of the return, such that profitable for diagonal-variant QARCHðqÞ effects as described by
trading opportunities may go unexploited due to perceived Equation 5 under the restriction p ¼ 0, denoted here by
4
For details of more general QGARCH specifications and their interpretation, see Sentana (1995).
5
An alternative and more direct illustration of the asymmetry effect afforded by the QGARCH model is given by noting that the
restricted first-order QGARCH(0,1,1) case may also be expressed as:
þ ð"t1 cÞ2
h2t ¼ !
Further, when i < 0 the derivative of h2t with respect to "ti is also greater, implying greater ‘steepness’ in the conditional variance
function (Nelson, 1991; Sentana, 1995). The contrast between the QGARCH and TGARCH specifications may also be expressed in
terms of their differing ‘news impact curve’ representation, which depicts the relationship between "t1 and h2t (Engle and Ng, 1993). In
the TGARCH case the news impact curve is centred on "t1 ¼ 0 but with asymmetric slopes for "t1 < 0 and "t1 > 0. In the QGARCH
case the news impact curve is centred on "t1 ¼ c with symmetric slopes for "t1 6¼ c.
6
For details of the conditions under which Equation 6 can be derived in general equilibrium, and under which Equation 6 holds as an
approximation, see Campbell (1993). For general surveys of the (G)ARCH literature discussed in this section see Engle and Bollerslev
(1986), Bollerslev et al. (1992), Bera and Higgins (1993), Bollerslev et al. (1994), and for a review and comparison of a variety of related
symmetric and asymmetric GARCH models, Hentschel (1995).602 D. G. McMillan and A. E. H. Speight
QAq , and distributed as 22qunder the null of homoscedas- tlements based on the Exchange Delivery Settlement Price
ticity. Furthermore, a variety of tests developed by Engle (EDSP).8 Thus, several contracts are traded simultaneously
and Ng (1993) are employed to test for the predictability of and given the need for a continuous series a decision must
the squared standardized error, t2 ¼ "2t =h2t . The sign bias be made as to which contract price to utilize in constructing
test examines whether positive and negative innovations the returns series. The criteria employed here is that the
have a differential impact on subsequent volatility over contract most heavily traded at each point in time is
and above that allowed for by the estimated model, as used. Throughout the sample period, heaviest trading is
indicated by the statistical significance of the dummy vari- found to occur in the contract closest to expiry up to the
able D1 as defined in Equation 4 in a regression of t2 on D1 day preceding its expiry at which time trading then switches
and a constant. The negative size bias and positive size bias to the immediately succeeding contract. The continuous
tests examine whether the magnitude of negative and posi- contract price data thus constructed is sampled at both
tive innovations respectively have any effect on subsequent 15-minute and 60-minute intervals in order to provide
volatility that is not captured by the estimated model. alternative quarter-hour and hourly frequency returns
These tests are again tests of significance in the context series using the conventional transformation rt ¼
of regressions of t2 on a constant and either D1 "t1 or logðPt =Pt1 Þ where P is the price of the security. With a
ð1 D1 Þ"t1 respectively. The joint LM test of both sign
floor trading time for FTSE-100 contracts of 8:35 a.m.–
and size bias provides a further general test for variance
4.10 p.m. and 846 trading days in the sample, and with the
specification, computed as TR2 in the regression of t2 on a
overnight return excluded so as to ensure a consistent
constant, D1 , D1 "t1 and ð1 D1 Þ"t1 , and distributed as
returns series, this yields 25 380 observations at the 15-
23 . Finally, the Brock et al. (1987) test for departures from
minute frequency, and 6345 observations at the hourly
iid is considered as a portmanteau test for remaining resi-
frequency.9
dual structure, the BDSðm; Þ test statistic being defined
High frequency intraday data is strongly characterized
over ‘embedding dimension’, m, and sample residual stan-
dard deviation, , and asymptotically distributed as by high-frequency periodicity often corresponding to
Nð0; 1Þ.7 proximity in time to market opening and closing, as well
as macroeconomic news and other systematic information
IV . DA T A releases. As has been noted elsewhere, the strength of these
intraday effects is such that failing to adjust for them can
The data analysed here relate to FTSE-100 stock index result in misleading analysis of the dynamic dependencies
futures contracts traded at the London International in the data (Goodhart et al. 1993; Andersen and Bollerslev,
Financial Futures and Options Exchange (LIFFE) over 1997b; Guillaume et al. 1997; Goodhart and O’Hara,
the period January 1992 to June 1995. This contract is of 1997). In order to more clearly identify the relevant
particular interest since it constitutes a major investment dynamic dependencies in the data, the recommendations
and hedging instrument that is heavily traded, and the of Andersen and Bollerslev (1997b) are followed in
underlying security represents the principal index for the standardizing the raw returns data by the sample mean
London Stock Exchange. absolute value for each intraday time interval, at the 15-
LIFFE futures contracts have four delivery months minute and hourly frequencies respectively, prior to
(March, June, September and December) with all cash set- estimation.10
7
More formally, the BDS statistic tests the null that the series in question are iid against an unspecified alternative, and has power against
a variety of non-linear processes including ARCH (Brock et al. 1991). The statistic is based upon a measure of spatial correlation in m-
dimensional space known as the ‘correlation integral’ (Grassberger and Procaccia, 1983) defined as:
Wm;T ðdÞ ¼ T 0:5 ½Cm:T ðdÞ C1;T ðdÞm =m;T ðdÞ
where is the sample standard deviation of the data, and Cm;T ðkÞ the sample correlation integral given ‘embedding dimension’, m, and
distance, d. In applications to iid series the BDS statistic is asymptotically distributed as a standard normal, W Nð0; 1Þ, and the
asymptotic distribution of the BDS statistic remains a valid approximation when testing the residuals from a linear autoregressive model.
For further details see Brock et al. (1987), Brock et al. (1991) and Brock and Potter (1993).
8
The FTSE-100 index futures contracts deliver on the third Friday in the delivery months (or last preceding working day if the third
Friday is not a working day), with EDSP based on the average level of the FTSE-100 index between 10:10 a.m. and 10.30 a.m. on the last
trading day.
9
This data period also extends and updates the three-month September–November 1991 sample analysed by Abhyankar et al. (1997).
10
Various alternative adjustments for systematic intraday effects have been proposed in the literature, including the use of interval
dummies (Baillie and Bollerslev, 1990, 1991), time-scaling (Dacorogna et al., 1993). Fourier transforms (Anderson and Bollerslev,
1997a). In order not to compound the dependencies tested for, the latter approaches are forgone in favour of the methodology described
in the text. For a more detailed discussion of the intraday deterministic patterns in LIFFE futures returns data, including that analysed
here, see Gwilym et al. (1999).Asymmetric volatility dynamics 603
V . R E SU L T S lidity of the normality assumption, with robust standard
errors computed using the method of Bollerslev and
Preliminary models of conditional mean autoregressive Wooldrige (1992). Model lag lengths at each frequency
structure alone are determined by reference to the are again determined by reference to the Schwarz (1978)
Schwarz (1978) criterion and the estimated log-likelihood. criterion and the estimated log-likelihood, as well as par-
The coefficients of resulting AR(1) and AR(0) models at ameter significance and residual diagnostics including the
the fifteen-minute and sixty-minute frequencies respectively residual correlogram and correlogram of squared residuals.
are negative throughout, but only the autoregressive coeffi- As reported in Table 2, GARCH(1,2) models are conse-
cient at the quarter-hour frequency is statistically signif- quently determined for both frequencies analysed.
icant.11 Indeed, the lack of autoregressive structure and Coefficients are significant throughout, and all positive
insignificant constant for the sixty-minute data suggests with the exception of the second lag of the squared error
that FTSE-100 futures prices follow a pure random walk term which is negative. Nevertheless, the conditional vari-
in mean at the hourly frequency. Residual summary statis- ance remains positive given satisfaction of the restriction
tics for these simple preliminary models, reported in 1 1 > 2 noted in Section II. However, as reported in
Table 1, exhibit similar characteristics to the raw returns Table 3, these simple GARCH models appear unable to
series in terms of skewness and kurtosis, the latter in adequately capture the non-linear dependence in the data
particular giving rise to highly non-normal conditional at the lower frequency of one-hour on the basis of the QA
distributions on the basis of Jarque–Bera statistics. tests, and on both QA and sign and size bias tests at the
Further, Engle (1982) ARCH-LM test results indicate the higher frequency of 15-minutes.
presence of GARCH effects in the data at all frequencies, Further model estimates confirm the significance of con-
test statistics being highly significant at all lags examined. ditional variance asymmetries at both frequencies consid-
The unanimous significance of Brock et al. (1987) BDS test ered. In the first-order TGARCH(1,2,1) models, the
values confirms the presence of residual nonlinear structure positive coefficients obtained for 1 indicate that negative
and further motivates this examination of models of the shocks increase volatility by a greater magnitude than posi-
GARCH class. The preceding autoregressive model orders tive shocks of equal size. However, QA and sign and size
are maintained in all estimated models reported below but bias tests again reveal remaining asymmetric structure at
coefficient estimates are suppressed in tabulation and the 15 minute frequency. Investigation of the QGARCH
discussion. form suggests that predictive asymmetry is of second-order
Joint conditional mean and conditional volatility estima- form, though this effect is statistically insignificant in the
tion is conducted by quasimaximum likelihood given inva- hourly returns data. The resulting QGARCH(1,2,2) model
Table 1. Summary statistics and preliminary diagnostics
BDS
Data/model Mean S.D. Sk. Ku. JB A1 A4 A8 A12 ð2; Þ ð3; Þ ð4; Þ
Fifteeen-minute returns
Unadjusted 74.43e706 0.0015 70.003 8.80 35 777* 1411* 1903* 2124* 2185* – – –
Adjusted 0.0097 2.4285 0.017 10.15 54 127* 2105* 2240* 2353* 2428* – – –
AR 0.0000 2.4300 0.020 9.96 51 298* 1996* 2143* 2262* 2340* 20.42* 25.52* 29.71*
Sixty-minute returns
Unadjusted 71.48e705 0.0026 0.090 6.94 4932* 184.7* 352.9* 462.4* 497.0* – – –
Adjusted 70.032 4.3200 0.050 6.30 3452* 215.7* 399.2* 497.5* 546.5* – – –
AR 0.000 4.3200 0.050 6.30 3452* 215.7* 399.2* 497.5* 546.5* 8.03* 10.30* 12.32*
Notes: Under ‘Data/Model’, ‘Unadjusted’ refers to the raw returns data, and ‘Adjusted’ to returns deflated by the relevant intraday
interval average absolute return in order to account for the intraday periodicity present in the data, whilst AR denotes the application of
an autoregressive conditional mean model, of orders AR(1) and AR(0) in 15-minute and 60-minute returns respectively; Sk. and Ku.
denote measure of the second and third moments of skewness and kurtosis, on the basis of which the Jarque–Bera test for normality is
calculated, JB 22 ; Ai denotes the ith order Engle (1982) ARCH-LM test, Aq 2q ; ‘BDS’ denotes the Brock et al. (1987) test for
departures from iid defined over ðm; dÞ where ‘m’ denotes embedding dimension and ‘d’ distance (determined with reference to the
sample residual standard deviation, ), asymptotically distributed as Nð0; 1Þ. For further details of specification tests, see text, Section
III. An asterisk denotes significance in a test statistic at the 5% level or higher.
11
The resulting coefficient estimates, intercept first (and standard errors adjusted by the method of White, 1980) are: at the quarter-hour
frequency, 70.0101(0.0152), 0.0383(0.0118); and at the hourly frequency, 70.323(0.0495). The absence of remaining linear structure in
the residuals of these autoregressive model residuals is confirmed using the robust version of the standard LM test (Wooldridge, 1990).604
Table 2. Model estimates
Model ! 1 2 1 1 2
Fifteen-minute returns
GARCH 0.0804 (0.0108)* 0.0917 (0.0114)* 70.0472 (0.0121)* 0.9420 (0.0045)* – – –
TGARCH 0.0757 (0.0103)* 0.0833 (0.0115)* 70.0477 (0.0121)* 0.9448 (0.0043)* 0.0134 (0.0056)* – –
QGARCH 0.0759 (0.0102)* 0.0912 (0.0111)* 70.0489 (0.0118)* 0.9449 (0.0043)* 70.1313 (0.0340)* 0.1059 (0.0348)* –
QGARCH-M 0.0761 (0.0102)* 0.0909 (0.0111)* 70.485 (0.0117)* 0.9447 (0.0043)* 70.1310 (0.0340)* 0.1056 (0.0348)* 0.0044 (0.0056)
Sixty-minute returns
GARCH 0.1042 (0.0314)* 0.0776 (0.0214)* 70.0546 (0.0216)* 0.9713 (0.0047)* – – –
TGARCH 0.1004 (0.0291)* 0.0628 (0.0206)* 70.0515 (0.0206)* 0.9729 (0.0046)* 0.0207 (0.0047)* – –
QGARCH 0.1301 (0.0243)* 0.0754 (0.0123)* 70.0549 (0.0124)* 0.9736 (0.0028)* 70.1130 (0.0137)* – –
QGARCH-M 0.1299 (0.0376)* 0.0758 (0.0170)* 70.0553 (0.0172)* 0.9735 (0.0043)* 70.1131 (0.0302)* – 0.0099 (0.0066)
Notes: For equation specifications, see text, Section II, Equations 2–6. Conditional mean estimates suppressed. Standard errors, in parentheses, adjusted by the method of
Bollerslev and Wooldridge (1992). An asterisk denotes asymptotic coefficient significance at the 5% level.
D. G. McMillan and A. E. H. SpeightTable 3. Specification tests
Asymmetric volatility dynamics
BDS
Model A1 A4 A8 A12 QA1 QA4 QA8 QA12 St St "t1 St "t1 S1 ð2; Þ ð3; Þ ð4; Þ
Fifteen-minute returns
GARCH 2.46 7.24 11.04 15.88 11.90* 18.14* 24.48 31.00 0.126 (4.13)* 70.024 (72.69)* 70.024 (72.32)* 12.94* 0.22 1.26 2.56*
TGARCH 3.37 8.91 12.47 16.98 8.65* 19.36* 21.39 27.09 0.117 (3.78)* 70.019 (72.12)* 70.017 (71.62) 10.30* 0.18 1.26 2.59*
QGARCH 2.68 8.82 12.31 17.08 2.72 11.14 15.37 21.43 0.047 (1.54) 0.000 (0.04) 0.011 (0.14) 2.12 0.07 1.16 2.52*
QGARCH-M 2.95 8.92 12.40 17.24 2.97 11.29 5.50 21.64 0.028 (0.91) 0.001 (0.05) 0.002 (0.17) 2.23 0.09 1.17 2.52*
Sixty-minute returns
GARCH 0.13 8.08 9.15 11.74 4.58 16.99* 27.66* 38.84 0.098 (1.75) 70.014 (71.37) 70.010 (71.12) 3.21 71.41 71.01 0.03
TGARCH 0.13 8.27 9.26 12.04 1.82 14.67 20.05 24.49 0.079 (1.43) 70.010 (71.03) 70.004 (70.44) 2.40 71.33 70.93 0.12
QGARCH 0.02 8.28 9.42 11.54 1.39 12.69 14.28 16.78 0.065 (1.19) 70.010 (70.98) 70.003 (70.30) 1.49 71.10 70.71 0.32
QGARCH-M 0.05 9.05 10.49 13.32 1.42 12.69 14.30 16.85 0.064 (1.17) 70.009 (70.92) 70.003 (70.31) 1.73 71.21 70.70 0.45
Notes: Under ‘Model’, GARCH denotes the standardized residuals from a generalized autoregressive conditional heteroscedasticity model, TGARCH the standardized
residuals from a threshold-GARCH model, QGARCH the standardized residuals from a quadratic-GARCH model, and QGARCH-M the standardized residuals from the
QGARCH extension of a GARCH-in-mean model. For equation specifications, see text, Section II, Equations 2–6. Conditional mean estimates suppressed. Sk. and Ku.
denote measures of the second and third moments of skewness and kurtosis, on the basis of which the Jarque–Bera test for normality is calculated, JB 22 ; Aq denotes the
qth order Engle (1982) ARCH-LM test, Ai 2q ; QAq denotes the qth order Sentana (1995) QARCH-LM test, Ai 22q ; St , St "t1 , and Stþ "t1 denote the negative sign bias
negative size bias test, and positive size bias tests of Engle and Ng (1993), whilst S1 denotes the Engle and Ng (1993) joint LM test of sign and size bias, distributed as 23
under the null of symmetry; ‘BDS’ denotes the Brock et al. (1987) test for departures from iid defined over ðm; dÞ where‘m’ denotes embedding dimension and ‘d’ distance
(determined with reference to the sample residual standard deviation, ), asymptotically distributed as Nð0; 1Þ. For further details of specification tests, see text, Section III.
An asterisk denotes significance in a test statistic at the 5% level or higher.
605606 D. G. McMillan and A. E. H. Speight
estimates qualitatively confirm the TGARCH results in significant volatility feedback, and due to inherent difficul-
that the first-order effect of a negative shock is again to ties in appealing to leverage effects as an explanation for
increase conditional volatility relative to the effect of an such predictive asymmetry in the context of futures mar-
equivalent size positive shock. However, the additional kets, the view advanced is that such asymmetry in volatility
second-order asymmetric QGARCH effect of that negative reflects the activities of noise traders in the market, in keep-
shock is to depress subsequent conditional volatility. Thus, ing with the theoretical framework of Sentana and
QGARCH conditional volatility is first raised then Wadhwani (1992), who are attracted by low margin
depressed by a negative shock or ‘bad news’. This is in requirements and raise volatility by trading on price move-
contrast to the asymmetric effects of a positive shock, ments induced by the sequential dissemination of informa-
which serve to mitigate rather than amplify the usual tion, with subsequent depression of volatility following the
GARCH impact on conditional volatility operating convergence of prices on a new market equilibrium and the
through the squared error terms parameterized by i . abatement of noise trading. Finally, and in further contrast
However, as revealed by the higher dimension BDS tests with previous results, it is found that all nonlinear depen-
reported in Table 3, this asymmetry is unable to account dence in the data at the hourly frequency is satisfactorily
for all of the remaining GARCH residual non-linear struc- represented by GARCH models of asymmetric form.
ture at the quarter-hour frequency noted above. The final However, there is evidence of remaining nonlinear residual
extension of the model to QGARCH-M form yields no structure at the intrahourly frequency, and further research
evidence of a statistically significant effect of volatility on might seek to resolve those aspects of non-linear depen-
returns at either the hourly or quarter-hourly frequency, dence in quarter-hour returns that are not fully captured
and therefore no evidence of any volatility feedback by the class of models considered here.
through the interaction of predictive asymmetry and a
risk premium. Neither is this extended form capable of
further removing the residual nonlinear dependence in REFERENCES
the quarter-hour data.12;13 Abhyankar, A. (1995) Return and volatility dynamics in the
FTSE-100 stock index and stock index futures market,
Journal of Futures Markets, 15, 457–88.
Abhyankar, A., Copeland, L. S. and Wong, W. (1997)
VI. SUMMARY AND CONCLUSIONS Uncovering nonlinear structure in real-time stock market
indices, Journal of Business and Economic Statistics, 15, 1–14.
In summary of the findings described above, evidence is Andersen, T. G. and Bollerslev, T. (1997a) Intra-day periodicity
and volatility persistence in financial markets, Journal of
reported of significant predictive asymmetry at both the Empirical Finance, 4, 115–58.
quarter-hour and hourly intra-day data frequencies for Andersen, T. G. and Bollerslev, T. (1997b) Heterogeneous infor-
FTSE-100 index futures returns, but no evidence is found mation arrivals and return volatility dynamics: uncovering
of a significant volatility feedback effect induced by the the long-run in high frequency returns, Journal of Finance,
interaction of such asymmetry and a risk premium. The 52, 975–1005.
Antoniou, A. and Holmes, P. (1995) Futures trading, information
asymmetry detected is of a form reported widely elsewhere and spot price volatility: evidence for the FTSE-100 stock
for spot markets. Namely, where negative shocks to the index futures contract using GARCH, Journal of Banking
returns process, usually associated with ‘bad news’ events, & Finance, 19, 117–29.
have a greater impact on subsequent volatility than positive Antoniou, A. and Holmes, P. (1996) Futures market efficiency,
‘good new’ shocks of equivalent magnitude. However, in the unbiasedness hypothesis and variance-bounds tests: the
case of the FTSE-100 futures contract, Bulletin of Economic
contrast to previous studies, which have typically focused Research, 48, 115–28.
on the application of low-order conditional variance pro- Antoniou, A., Holmes, P. and Priestley, R. (1998) The effects of
cesses to interdaily or lower frequency data, it is found that stock index futures trading on stock index volatility: an ana-
a higher order model is warranted at the higher intradaily lysis of the asymmetric response of volatility to news, Journal
frequency. This asymmetry is such that negative shocks of Futures Markets, 18, 151–66.
Baillie, R. T. and Bollerslev, T. (1990) A multivariate generalised
lead to increased volatility but, additionally, to sub- ARCH approach to modelling risk premia in forward foreign
sequently decreased volatility, again relative to the impact exchange rate markets, Journal of International Money and
of an equivalent positive shock. In the absence of any Finance, 9, 309–24.
12
Note that whilst the significance of the BDS statistics reported in Table 2 is appraised relative to the asymptotic Nð0; 1Þ value of 1:96,
the asymptotic distribution does not well approximate the finite distribution of the BDS statistic applied to the standardized residuals of
GARCH models. Therefore the simulation quantiles reported by BHL are used as guidance to the actual sizes of the BDS statistics
applied to hourly GARCH model residuals. The significance of test statistics indicated in Table 3 and the inferences made in the text
continue to hold on these alternative critical values.
13
Final tests for the effects of contract rollover and time-to-maturity confirm the robustness of the empirical results and the insignificance
of these additional considerations. These further test results are therefore suppressed here but are available from the authors on request.Asymmetric volatility dynamics 607
Baillie, R. T. and Bollerslev, T. (1991) Intra-day and inter-market nominal excess return on stocks, Journal of Finance, 48,
volatility in foreign exchange markets, Review of Economic 1779–801.
Studies, 58, 565–85. Goodhart, C. A. E. and O’Hara, M. (1997) High frequency data
Bera, A. K. and Higgins, M. L. (1993) ARCH models: properties, in financial markets: issues and applications, Journal of
estimation and testing, Journal of Economic Surveys, 7, 305– Empirical Finance, 4, 74–114.
62. Goodhart, C. A. E., Hall, S. G., Henry, S. G. B. and Pesaran, B.
Black, F. (1976) Studies in stock price volatility changes, (1993) News effects in a High Frequency Model of the
Proceedings of the 1976 Business Meeting of the Business Sterling-dollar Exchange Rate, Journal of Applied
and Economics Statistics Section, American Statistical Econometrics, 8, 1–13.
Association, New York, 177–81. Grassberger, P. and Procaccia, I. (1983) Measuring the strange-
Bollerslev, T. (1986) Generalised autoregressive heteroscedasti- ness of strange attractors, Physica D, 9, 189–208.
city, Journal of Econometrics, 31, 307–27. Guillaume, D. M., Dacorogna, M. M., Davé, R. R., Müller, U. A.,
Bollerslev, T. and Wooldridge, J. M. (1992) Quasi maximum like- Olsen, R. B. and Pictet, O. (1997) From the bird’s eye to the
lihood estimation in inference in dynamic models with time microscope: a survey of new stylized facts of the intra-daily
varying covariances, Econometric Reviews, 11, 143–79. foreign exchange markets, Finance and Stochastics, 1, 61–76.
Bollerslev, T., Chou, R. Y. and Kroner, K. F. (1992) ARCH Gwilym, ap, O., McMillan, D. G. and Speight, A. E. H. (1999)
modelling in finance, Journal of Econometrics, 52, 5–59. The intra-day relationship between volume and volatility in
Bollerslev, T., Engle, R. F. and Nelson, D. B. (1994) ARCH LIFFE futures markets, Applied Financial Economics, 9, 593–
models, in Handbook of Econometrics (Eds) R. F. Engle 604.
and D. L. McFadden, Elsevier Science, Amsterdam, Hentschel, L. (1995) All in the family: nesting symmetric and
pp. 2959–3038. asymmetric GARCH models, Journal of Financial
Brock, W. A. and Potter, S. A. (1993) Nonlinear time series and Economics, 39, 71–104.
macroeconomics, Handbook of Statistics, 11, 195–229. Kyle, A. S. (1985) Continuous auctions and insider trading,
Brock, W., Dechert, W. D. and Scheinkman, J. (1987) A test for Econometrica, 53, 1315–35.
independence based on the correlation dimension, University Merton, R. C. (1980) On estimating the expected return on the
of Wisconsin, Economics Working Paper No: SSRI-8702. market: an exploratory analysis, Journal of Financial
Brock, W., Hsieh, D. A. and LeBaron, L. (1991) Nonlinear Economics, 8, 323–61.
Dynamics, Chaos and Instability, MIT Press, Cambridge.
Nelson, D. B. (1990) Stationarity and persistence in the
Butterworth, D. and Holmes, P. (2001) The hedging effectiveness
GARCH(1,1) model, Econometric Theory, 6, 318–34.
of stock index futures; evidence for the FTSE-100 and FTSE-
Nelson, D. B. (1991) Conditional heteroscedasticity in asset
MID250 indexes, Applied Financial Economics, 11, 57–68.
returns: a new approach, Econometrica, 59, 347–70.
Campbell, J. Y. (1993) Intertemporal asset pricing without con-
Nelson, D. B. and Cao, C. Q. (1992) Inequality constraints in the
sumption data, American Economic Review, 83, 487–512.
univariate GARCH model, Journal of Business and Economic
Campbell, J. Y. and Hentschel, L. (1992) No news is good news:
an asymmetric model of changing volatility in stock returns, Statistics, 10, 229–35.
Journal of Financial Economics, 31, 281–318. Praschnik, J. (1991) Nonlinear dependence? A look at the treasury
Dacorogna, M. M., Müller, U. A., Nagler, R. J., Olsen, R. B. and bill futures market, Applied Financial Economics, 1, 165–73.
Pictet, O. V. (1993) A geographical model for the daily and Rabemananjara, R. and Zakoian, J. (1993) Threshold ARCH
weekly seasonal volatility in the FX market, Journal of models and asymmetries in volatility, Journal of Applied
International Money and Finance, 12, 413–38. Econometrics, 8, 31–49.
Engle, R. F. (1982) Autoregressive conditional heteroscedasticity Robinson, D. M. (1998) Nonlinear dependence, asymmetry and
with estimates of the variance of United Kingdom inflation, threshold in Australian futures markets, University of
Econometrica, 50, 987–1007. Melbourne, Department of Accounting and Finance, mimeo.
Engle, R. F. (1990) Discussion: stock market volatility and the Schwarz, G. (1978) Estimating the dimension of a model, Annals
crash of ’87, Review of Financial Studies, 3, 103–6. of Statistics, 4, 461–64.
Engle, R. F. and Bollerslev, T. (1986) Modelling the persistence of Sentana, E. (1995) Quadratic ARCH models, Review of Economic
conditional variances, Econometric Reviews, 5, 1–87. Studies, 62, 639–61.
Engle, R. F. and González-Rivera, G. (1991) Semiparametric Sentana, E. and Wadhwani, S. (1992) Feedback traders and stock
ARCH models, Journal of Business and Economic Statistics, return autocorrelations: evidence from a century of daily
9, 345–60. data, Economic Journal, 102, 415–25.
Engle, R. F. and Ng, V. K. (1993) Measuring and testing the White, H. (1980) A heteroscadasticity-consistent covariance
impact of news on volatility, Journal of Finance, 48, 1749–78. matrix estimator and a direct test for heteroscedasticity,
Engle, R. F., Lilien, D. M. and Robins, R. P. (1987) Estimating Econometrica, 48, 817–38.
time varying risk premia in the term structure: the ARCH-M Wooldridge, J. M. (1990) A unified approach to robust, regres-
model, Econometrica, 55, 391–407. sion-based specification tests, Econometric Theory, 6, 17–43.
Fujihara, R. A. and Mougoué, M. (1997) Linear dependence, Yang, S. and Brorsen, B. W. (1994) Daily futures price changes
nonlinear dependence and petroleum futures market effi- and nonlinear dynamics, Structural Change and Economic
ciency, Journal of Futures Markets, 17, 75–99. Dynamics, 5, 111–32.
Glosten, L. R., Jagannathan, R. and Runkle, D. E. (1993) On the Zakoian, J. (1994) Threshold heteroscedastic models, Journal of
relation between the expected value and the volatility of the Economic Dynamics and Control, 18, 931–55.You can also read
