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Forecasting UK stock market volatilityDavid McMillan

a, Alan Speight

b& Owain Apgwilym

c

aDepartment of Economics, University of St Andrews, Fife, KY16 9AL

bDepartment of Economics, University of Wales, Swansea SA2 8PP, UK

cDepartment of Management, University of Southampton, Southampton, SO171BJ

Version of record first published: 07 Oct 2010.

To cite this article: David McMillan , Alan Speight & Owain Apgwilym (2000): Forecasting UK stock market volatility

Applied Financial Economics, 10:4, 435-448

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Forecasting UK stock market volatility

D A V I D MCM I L L A N , A L A N S P E I G H T *{ and OWAIN A PG W I L Y M}

Department of Economics, University of St Andrews, Fife, KY 16 9AL {Department ofEconomics, University of W ales, Swansea SA2 8PP, UKand }Department of Manage-ment, University of Southampton, Southampton, SO17 1BJ

The paper analyses the forecasting performance of a variety of statistical and econo-metric models of UK FTA All Share and FTSE100 stock index volatility at themonthly, weekly and daily frequencies under both symmetric and asymmetric lossfunctions. Under symmetric loss, results suggest that the random walk model pro-vides vastly superior monthly volatility forecasts, while random walk, moving aver-

age, and recursive smoothing models provide moderately superior weekly volatilityforecasts, and GARCH, moving average and exponential smoothing models providemarginally superior daily volatility forecasts. If attention is restricted to one fore-casting method for all frequencies, the most consistent forecasting performance isprovided by moving average and GARCH models. More generally, results suggestthat previous results reporting that the class of GARCH models provide relativelypoor volatility forecasts may not be robust at higher frequencies, failing to hold herefor the crash-adjusted FTSE100 index in particular.

I . I N T R O D U C T IO N

Since the stock market crash of 1987, stock price volatilityhas been the focus of both empirical academic research and

regulatory concern. This attention reects three importantconsiderations. First, the well-noted observation that stockmarket volatility has been higher in more recent than his-

torical periods, and the perception that such increasedvolatility is due to institutional changes such as automatedtrading and the introduction of trading in derivative

futures and options contracts which may have enhancedthe likelihood of large swings in mean stock returns(Edwards, 1988a, b; Schwert, 1990; Robinson, 1994;

Antoniou and Holmes, 1995). Second, although the ten-

dency for stock market volatility to exhibit `clusteringhas long been recognized (Mandelbrot, 1963; Fama,

1965), it is only since introduction of the autoregressiveconditional heteroscedasticity (ARC H) model by Engle(1982) and its subsequent generalization (GARCH) by

Bollerslev (1986) that researchers have formally modelledthe second and higher moments of nancial time series

using econometric techniques. Third, increased recognitihas been paid to the practical importance of accurate voltility estimates and forecasts in asset and option prici

models, and portfolio selection and market timing desions (Gemmill, 1993; Vasilellis and Meade, 1996). Theconsiderations have led to examinations of the stationariand persistence of volatility over time and the accuracy

volatility forecasting techniques. In particular, the comparative forecasting performance of statistical procedurassociated with `technical analysis1, other more na

methods, and forecasts generated from extended mod

of the GARCH class (Bollerslev et al., 1992).While the modelling and forecasting of US sto

market conditional volatility has been extensively inves

gated, the analysis of conditional volatility in othinternational stock markets has only recently beundertaken, with conicting results.2 For example, in forcasting monthly US stock market volatility, Akgiray (198nds that a GARCH(1, 1) model outperforms mo

traditional technical analysis. Tse (1991) and Tse aTung (1992) examine the stock markets of Japan a

Applied Financial Economics ISSN 09603107 print/ISSN 14664305 online # 2000 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

Applied Financial Economics, 2000, 10, 435448

4

* Corresponding author. E-mail: a.speight@s wan.ac.uk1 For extended discussions of technical analysis and associated `chartism, and the formal underpinnings of such analysis, see Stew(1966) and Plummer (1989) respectively.2 Also see de Jong et al. (1992), and Kearns and Pagan ( 1993).


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Singapore respectively, and nd that an exponentiallyweighted moving average method is superior to theGARCH model in both cases. Franses and van Dijk(1996 ) compare the volatility f orecasting performance ofthe GARCH, quadratic-GARCH (QGARCH, Engle andNg, 1993), and threshold-GARCH (TGARCH, Glosten etal., 1992) models to the random walk model using weekly

German, Dutch, Italian, Spanish and Swedish stock indexreturns over the period 1986 to 1994. They report that therandom walk model performs particularly well when thecrash of 1987 is included in the estimation sample, whilethe QGARCH model performs better upon its exclusion.For the monthly volatility of Australian stock market data,Brailsford and Fa (1996) compare the predictiveperformance of several statistical methods with GARCHand TGARCH models. Using several loss functions, theyare unable to identify a clearly superior model and suggestthat the `best forecasting model depends upon the sub-sequent application, as the ranking of models is sensitive

to the choice of loss function. However, to date, little evi-dence is available for the UK. Dimson and Marsh (1990)examine various technical methods of predicting the vola-tility of UK stock market returns over the period 1955

1989 and nd that exponential smoothing and simpleregression models perform best according to their criteria,but do not consider the genre of ARCH models. Indeed, noanalysis of the volatility forecasting performance ofGARCH models for the UK stock market has yet beenreported.

Several issues therefore arise from the existing empiricalliterature. First, there is no clear consensus across inter-

national stock markets as to the superior conditional volati-lity forecasting method. Second, that these conicting resultsin part reect dierences in the choice of classes of model forinclusion in the forecasting competition. Third, such resultsare obtained for dierent frequencies of observation of stockmarket prices, and with diering treatments of the 1987crash. Fourth, that the identication of a superior volatilityforecasting method may not be invariant to the choice of lossfunction used in forecast appraisal.

This paper therefore provides a comparative evaluationof the volatility forecasting ability of GARCH models,asymmetric TGARCH and exponential-GARCH

(EGARCH, Nelson, 1991) models, and a wide range ofmore traditional methods associated with technical analy-sis, for the Financial TimesStock Exchange 100 index andthe Financial TimesActuaries All Share index at theLondon Stock Exchange. A primary aim of the paper isto attempt to clarify the usefulness of GARCH type modelsin predicting volatility while using a dierent stock marketsetting from previous research, and to identify the `bestforecasting models for the volatility of these indices ofreturns on the London stock exchange in particular.Additionally, and in extension of previous research inves-tigating such models, we also examine the component-

GARCH model introduced by Engle and Lee (199which permits the separation of volatility into long-terand short-term eects and has not previously been consiered in the literature. In total, ten models are considerethese being the historical mean, moving average, randowalk, exponential smoothing, exponentially weighted moing average, simple regression, GARCH, TGARC

EGARCH, and component-GARCH models. Further, forecast evaluations are performed for the monthly, weekand daily data frequencies in order that the sensitivity our results to the choice of sample frequency may assessed. This exercise also has a practical motivation that while tactical short-term investment decisions avaluations of short-lived derivatives frequently f ocus oshort-term predictions of volatility, strategic decisioand the valuation of long-lived assets require longer-terpredictions of volatility. The consideration of volatilforecasting performance over dierent frequencies athereby dierent horizons, provides one means of accom

modating such considerations and is an issue likely to be interest to such market participants as managers of indtracking funds who need to be aware of the volatility these indices at dierent frequencies. All forecasts are alconducted on the basis of in-sample conditional memodels both with and without adjustment of the 19crash. Finally, forecast evaluation is conducted wirespect to a variety of loss functions. At a practical levit is unlikely that all market participants will attach equimportance to overpredictions and underpredictions volatility, as in the pricing of call options on the buyiand selling sides of the market for example, and both sym

metric and asymmetric loss function results are thereforeported.The remainder of the paper is structured as follow

Section II describes the data series, source, sample periand methodology. Section III describes the volatility forcasting models employed and characteristics of their isample estimation. Section IV discusses the criteria which the forecast performance of these models is assessand Section V reports the outcomes of the comparatiforecast exercise. Section VI summarizes our ndings anconcludes.

I I . D A T A

The Financial TimesStock Exchange 100 index ( FTSE10was launched in January 1984 and represents the stoprices of the 100 largest listed UK companies weighted market capitalization. One of the main aims of launchinthe index was to enable the introduction of derivativbased on a single indicator of the equity market. It therefoconstitutes the rst real-time index for the UK equmarket which is small enough to be updated regularly anaccurately but also comprehensive enough to act as

436 D. McMillan et a


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eective proxy for the UK equity market, and is now awidely used indicator of UK equity market conditions.3

The FTSE100 is thus a blue-chip subset of the FinancialTimesActuaries (FTA) All Share index introduced in1962 which covers about 93% of the market capitalizationof the UK equity market and is the long-standing bench-mark index for the UK market. While the FTSE100 and

FTA All Share index are closely correlated, the dierentcomposition of the two indices means there is alwayssome scope for divergence between them and both aretherefore analysed here.

All data are obtained from the Datastream market infor-mation service. For the FTSE100 index we consider dailyclosing price data from 2 January 1984 to 31 July 1996,while for the FTA All Share index we study daily closingprice data from 1 January 1969 to 31 July 1996. These priceindices p are converted to returns r by the standardmethod of calculating the log-dierence, rt

log pt=pt1. Further, in order to examine the performance

of the various volatility estimators over dierent forecasthorizons, the data are re-sampled at a weekly and monthlyreturns frequency. Additionally, given the data periodextends over the crash of October 1987, volatility estima-tion is conducted both including and excluding a dummyvariable for this event in order that the sensitivity of ourresults to this extreme volatility event may be assessed.Finally, for each frequency the data are partitioned intothe in-sample estimation periods 19691994 for the FTAdata and 19841994 for the FTSE data, the out-of-sampleforecast periods covering the remaining period 19951996for both data sources. This separation provides in-sample

(out-of-sample) observations of 312 ( 19), 1357 (82) a6783 (413) for the monthly, weekly and daily FTA serieand 132 (20), 574 (83) and 2869 (413) for the monthlweekly and daily FTSE series.

In order to generate an historical `actual volatility serion the basis of which volatility forecasts may be generatusing statistical methods, we follow Pagan and Schwe

(1990) in representing true past volatility by the squarresiduals from a conditional mean equation for returestimated over the in-sample period. Examination of tin-sample returns data reveals signicant serial correlatiin the levels of the weekly and daily but not monthseries. 4 The in-sample conditional mean is therefore esmated using the autoregressive (AR) process:

Lrt ut; L 1 1L L;

t 1 ;2; . . . ;T

where L is the orthodox lag operator Lkxtk xtk and

denotes the number of in-sample observations.5

With esmates of the parameters of L and determined minimization of the in-sample sum of squared residuals Equation 1, historical volatility,

2t using the convention

notation in the literature, is formally generated using:

2t rt fT ;TL 1rtg2

;

t 1 ;2; . . . ;T

where denotes the number of out-of-sample observationand the subscript T on a coecient indicates that itestimated conditional on the in-sample information se

Forecasting UK stock market volatility 4

3 The base level of the index was set at 1000.0 on 31 December 1983, and it is now calculated on a rm basis by the London StoExchange every 60 seconds during market hours, currently 08301630 GMT.4 For example, the LjungBox Q-statistics for up to twelfth-order autocorrelation for the FTSE (FTA) series are 36.43 (71.74) for weekreturns and 47.04 (325.45) for daily returns.5 Empirically, the lag length is determined by the Akaike and Schwarz information criterion. Additionally, dummies for Monday anholiday eects are included in the AR model for daily data, while a January dummy is included for monthly data. Typically, low-ordAR models are determined. In particular, for the FTSE, an AR(0) model is appropriate for monthly data, AR(2) for weekly data, ARfor daily data including a dummy f or the 1987 crash, and AR(4 ) for daily data without a crash dummy. For the FTA series, an ARmodel holds for monthly data, an AR(3) for weekly data and AR(10) for daily data. The AR(10) in daily FTA data is a result of the twweek settlement-of-account period operating prior to reform in 1995. Thus an investor could have bought at the beginning of tsettlement period but held both cash and the asset, and returns thus compensate for this (see Theobald and Price, 1984). In 1995 this wchanged to a rolling settlement system.6 Thus, historical volatility is generated by the application of a conditional mean model determined on in-sample data to both the i

sample and out-of-sample returns data. While not discussed in the text, for several of the fo recasting methods described in Section III, have also considered f ully recursive variants of Equations 1 and 2 entailing re-estimation o f the parameters L; as the data sampis extended over the out-of-sample horizon. Indeed, this approach is required in the recursive estimation and forecasting of the GAR Cclass of models. However, this alternative, fully recursive, method of generating 2t imposes severe computational b urdens in the contof the moving average, smoothing and simple regression frameworks, since at each point in time over the out-of-sample forecast horizthe entire history of volatility must then be reassessed and revised. Given that one objective of the present study is to ascertain the relatibenets of alternative fo recasting mechanisms in a practical context entailing repeated forecasting exercises, it is likely that simplicity idesirable f eature of any recommended forecasting method, and the f ully recursive approach may not be favoured under these methods such grounds. Moreover, the conditional mean functions estimated exhibit inherent stability. For example, comparison of the in-sampconditional mean functions and f ull-sample conditional mean f unctions (the l atter being employed in generating forecast error statistisee Section IV), reveal only minor dierences in coecient estimates beyond the fourth decimal place. Thus, for those statistical methowe have investigated using the f ully recursive approach to the generation of historic volatility, whether that approach is employed or nyields little or no quantitative dierence with the forecast error statistics reported in section V, and no qualitative dierence in termsrelative forecasting performance across models.


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This notation is employed throughout the remainder of thepaper.

I I I . V OL A T I L I T Y F O R E C A S T I N G M O D EL S

Historical meanExtrapolation of the historical mean in volatility providesperhaps the most obvious means of forecasting future vola-tility. Moreover, if the distribution of volatility has a sta-tionary mean, all variation in estimated volatility isattributable to measurement error and the historicalmean, -, computed as the unweighted average of volatilityobserved in-sample, then gives the optimal forecast of vola-tility, h, for all future periods:

ht1 -

2 1

T XT

j 1

2j; t T ; T 1 ; . . . ;T 1

3

Forecasts based on this mean also provide a benchmark forthe comparative evaluation of the alternative forecastingmodels outlined below. In addition to this in-sample histor-ical mean, we also consider the recursive assessment of thehistorical mean, iteratively updated with each incrementalobservation on volatility over the out-of-sample period:

ht1 -

2t

1

t

Xtj 1

2j ; t T ;T 1 ; . . . ; T 1 4

such that the mean of historic volatility and forecast offuture volatility at any point in time during the out-of-sample period is based on all information on actual vola-tility available at that point in time.

Moving average

Under the moving average method volatility is forecastby an unweighted average of past observed volatilitiesover a particular historical time interval of xedlength:

ht1 -

2

t;T

1

TXt

j TT 2

j ; t T ; T 1 ; . . . ; T 1

5

where T is the moving average period or `rolling window.The choice of this interval is essentially arbitrary and twolengths are considered here for each frequency. Thesearbitrary choices are ten years and ve years for monthly

data, two-and-one-half years and one-and-one-quarte r yeafor weekly data, and six months and three months for dadata, corresponding to T 120 and T 60 data points fthe longer and shorter window lengths in each case. 7

Random walk

The preceding models presume reversion to a stable gradually shifting trend in volatility. However, if volatiliuctuates randomly the optimal forecast of next periovolatility is simply current actual volatility:

ht1 2t ; t T ; T 1; . . . ;T 1

This `random walk model thus suggests that toptimal forecast of volatility is for no change since tlast true observation. This model also provides us wian alternative benchmark for appraising the relative forcasting performance of methods considered here, beingstandard comparative method in econometric foreca

appraisal.

Exponential smoothing

Under exponential smoothing the one-step-aheaforecast of volatility is a weighted function of timmediately preceding volatility forecast and curreactual volatility:

ht1 Tht 1 T2t ; t T ; T 1; . . . ;T

where the `smoothing parameter is constrained such th0 1. For 0 (or ht 2t , an exactly correct pa

forecast) the exponential smoothing model collapses to trandom walk, while as ! 1 major weight is given to tprior period forecast. As the subscripts indicate, the valof T is determined empirically as that which minimizes tin-sample sum of squared prediction errors.8

Exponentially weighted moving average (EW MA)

The exponentially weighted moving average model similar to the exponential smoothing model, but whe

past observed volatility in Equation 7 is replaced withmoving average forecast as in Equation 5:

ht1 T ht 1 T1

T

Xtj TT

2j ;

t T ;T 1 ; . . . ; T 1


7 Thus, approximately the same number of observations are used in constructing the moving averages over the three sample f requenci8 The values estimated here lie in the range 0.0010.1 for monthly and weekly data, and 0.140.83 for daily data. This contrasts wiresults reported elsewhere. For example, Dimson and Marsh (1990) using quarterly volatilities report a value of 0.66 in their preferrmodel, while Brailsford and Fa (1996 ) report values in the range 0.51 to 0.98 f or monthly data.


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with T specied as for the longer of two horizons consideredfor each frequency in the moving average model above. 9

Simple (mean) regression

The simple regression model provides one-step-ahead fore-casts generated from the application of an in-sample esti-

mated ordinary least squares regression of observed actualvolatility upon immediately preceding actual volatility toout-of-sample data:

ht1 T T2t1; t T ;T 1 ; . . . ; T 1 9

Following Dimson and Marsh (1990), again assuming thestationarity of volatility over the longer term, if suchforecasts are to be unbiased then the simple regressionimplicitly forecasts volatility as a weighted sum of recentvolatility and long-run mean volatility, and such thatvolatility will regress from its most recent level, 2t1,towards its long-run mean

-h, with determining the

speed of regression towards that mean.10;11

Generalized autoregressive conditional heteroscedasticity

(GARCH)

The GARCH model of Engle (1982) and Bollerslev (1986)requires joint estimation of the conditional mean and vari-ance process, the former being represented by an auto-regressive process for stock index returns as above inEquation 1. On the assumption that the resulting con-ditional mean stochastic error, "t, is normally distributedwith zero mean and time-varying conditional variance

quantifying volatility, out-of-sample forecasts of volatiliare generated by the GARCHp; q model:12

F0

;TLrt 0T "t;

F0

;TL 1 F0

1;TL F;TL; t 1; 2; . . . ;T 1

"t N0 ;ht 1

ht1 !T Xqi 1

i;T"2

ti1 Xpi 1

-i;Thti1;

t T ; T 1 ; . . . ;T 1 1

where ! > 0, i, -i 0, andP

i i P

i -i < 1, the latsum quantifying the persistence of shocks to volatilityThe GARCH(1, 1) model, for example, thus generates onstep-ahead forecasts of volatility, ht1, as a weighted averaof the constant long-run or average variance, !, the prviously forecast variance for the current period, ht, and cu

rent volatility reecting squared `news about the indreturn, "2t . In particular, volatility forecasts are increasfollowing a large positive or negative index return, tGARCH specication thus capturing the well-documentvolatility clustering evident in nancial returns data.14

Threshold-GARCH (TGARCH)

The GARCH model, although non-linear in the coditional mean error term, "t, postulates a linear relatioship between forecast volatility, previous forecasts


9 As the subscript notation indicates, the value of is again determined empirically by minimization of the in-sample residual suof squares and found to be 0.999 for both the FTSE and FTA series at all three frequencies, with the exception of the monthsampled FTSE where the value is 0.4340 for data adjusted for the 1987 stock market crash and 0.5060 without such adjustment. Thcompares with a range of values reported by Brailsford and Fa (1996) of between 0.0 and 0.9. Additionally, it should be noted thmodels were also estimated for the shorter moving average intervals discussed, but with no qualitative dierence from the resureported here.10 To see this, note that the expected value of both sides of Equation 9 must equal the long-run mean,

-h, for unbiasedness to hold. Hen

may be replaced by 1 -

h by virtue of E2t1 -

h, yielding:

ht1 1 T-

h T2t1; t T ; T 1; . . . ; T 1 9

11 In-sample estimates of are typically low and positive in the lower frequency data, at 0.05 (0.11) and 0.08 (0.12) in monthly and weekFTA data (crash adjusted values in brackets), and 0.04 (70.06) and 0.01 (0.03) in monthly and weekly FTSE data. Only for the adjustmonthly FTA and unadjusted weekly FTSE series are these values statistically signicantly dierent from zero. At the higher dai

frequency all estimated coecients are positive and statistically signicant with values of 0.46 (0.35) and 0.63 (0.27) for the FTA aFTSE respectively.12 Note that in the GARCH model, "2t is the analogue of historic volatility,

2t , in the statistical forecasting models described above, t

only essential dierence being that while 2t is generated as the squared residuals from the conditional mean function estimated alone,is generated as the squared residuals from the conditional mean function estimated jointly with the conditional variance process Equations 1012. While alternative notation could be employed in clarication of this connection, we retain the representation emploing "2t in keeping with conventions in the GARCH literature.13 As Nelson and Cao (1992) have shown, the rst two inequalities stated need not necessarily hold in order to ensure non-negativitythe conditional variance. In the GARCH(1, 2) case for example, 2 < 0 with -11 2 is sucient to ensure h

2t > 0. However, it

necessary and sucient thatP

i i P

i -i < 1 in order for a nite unconditional variance to exist, that sum also dening the limitincase of the integrated-GARCH (IGARCH) model for

Pi i

Pi -i 1, such that current news has a permanent eect on foreca

volatility (Engle and Bollerslev, 1986).14 Preliminary tests for the presence of ARCH eects in in-sample conditional mean errors using Engles (1982) LM t est indicsignicant conditional heteroscedasticity in all cases other than the weekly FTSE series unadjusted for the 1987 crash, that eve


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volatility, and current and lagged measured volatility inresponse to news. However, it has been observed that posi-tive and negative shocks of equal magnitude have a dier-ential impact upon stock price volatility, which may beattributable to a `leverage eect (Black, 1976). Further,stock market returns series have been noted to display sig-nicant negative skewness, possibly due to the fact that

market crashes are greater in absolute size and occurmore frequently and more quickly than booms (Fransesand van Dijk, 1996). One model that is able to capturethese features is the TGARCH model (Glosten et al.,1993) which, for a rst-order threshold, is expressed as(in conjunction with Equations 10 and 11):

ht1 !T Xqi 1

i;T"2ti1 TIt"

2t

Xpi 1

-i;Thti1;

t T ;T 1 ; . . . ; T 1 13

where It 1 if "t < 0, and It 0 if "t > 0. Thus, in theTGARCH(1, 1) case, for example, positive `news has animpact of 1 on volatility while negative news has animpact of 1 , positive (negative) news therefore havingthe greater impact on subsequent volatility for < 0

> 0, while shock persistence is quantied byPi i

Pi -i =2.

15

Exponential-GARCH (EGARCH)

The EGARCH model (Nelson, 1991) provides an alterna-

tive asymmetric model, the conditional variance beingexpressed as an asymmetric function of past errors (againin conjunction with Equations 10 and 11):

log ht1 !T Xqi 1

i;T 1;Tj"ti1j

j

hti1jp 2 ;T "ti1

hti1p

Xpi 1

-i;T log hti1 1

t T ; . . . ;T 1, where the 1 coecient captures tvolatility clustering eect noted above and the coecientmeasures the asymmetric eect, which if negative indicatthat negative shocks have a greater impact upon coditional volatility than positive shocks of equal magnitudAdditionally, the use of logs allows the parameters i an-i to be negative without the conditional variance becoming negative, while the persistence of shocks to conditionvariance is given by

Pi -i.

16

Component-GARCH (CGARCH)

The CGARCH model of Engle and Lee (1993) attempts

separate long-run and short-run volatility eects in a simlar manner to the BeveridgeNelson (1981) decompositiof conditional mean AR MA models for economic timseries. Thus, while the GAR CH model and its asymmetrTGARCH and EGARCH extensions exhibit mean revesion to !, the CGARCH model allows mean reversion totime-varying level, !t. The CGARCH specication is:

ht1 !t1 1"2t !t -1ht !t;

t T ;T 1; . . . ; T 1 1

!t1 ! !t "2t ht

where !t represents a time-varying trend or permanecomponent in volatility which is integrated if 1. Tvolatility prediction error "2t1 ht1 serves as the drivi


swamping all other conditional variance behaviour in that s eries. For all remaining series, in-sample A R-GARCH model orders ;pare initially determined by application of the Akaike and Schwarz information criterion. However, as noted by Nelson ( 1991, p. 356) , tasymptotic properties of the Schwarz criterion in the context of ARCH are unknown, while the A kaike criterion tends to prefer long lmodels. Thus, examination of the residual diagnostics and maximization of the log-likelihood are also used to guide order selectioEstimation of GARCH models and extended GARCH models is performed jointly with the AR model throughout, and dummies fpossible Monday and holiday eects are again included in the AR model for daily data, while a January dummy is again included monthly data. The resulting jointly estimated AR models orders remain the same as reported above under AR estimation alone, wh

joint GAR CH(1, 1) models hold for all series other than FTA daily data where a GARCH(3, 1) model holds. Persistence measures fthese models applied to FTA and FTSE data are found to be 0.900.93 for monthly data, 0.930.94 for weekly data and 0.950.97 daily data, with the exception of the monthly FTSE data unadjusted for the 1987 crash where a persistence measure of 1.00 identies

IGARCH process. Thus, GARCH volatility f orecast mean reversion is absent for the latter series, and generally sluggish for tremaining series in view of their relatively high shock persistence measures.15 Estimation and forecasting of the TGARCH model is conducted along the same lines as for the GARCH model, the AR-GARCmodel orders reported above continuing to hold for jointly estimated AR-TGARCH models with rst-order thresholds imposed. Testimated asymmetric coecient, , is everywhere positive and statistically signicant in-sample for the 1987 crash-adjusted FTA seriesall frequencies, the non-adjusted daily FTA series, the non-adjusted monthly FTSE series and the crash-adjusted weekly FTSE seriePersistence measures remain virtually unchanged relative to those for in-sample GARCH models, except for the IGAR CH craunadjusted monthly FTSE case where persistence falls to 0.50, and the crash-unadjusted monthly FTA case where persistence falls 0.84.16 Estimation and forecasting of the EGARCH model is conducted as for GARCH and TGARCH models, model orders continuing hold for jointly estimated AR-EGARCH models. The asymmetric coecient, 2, is consistently negative and statistically signicant isample for the 1987 crash-adjusted FTA series at all frequencies, the non-adjusted FTA series at the daily frequency, and the FTScrash-adjusted series at both the monthly and weekly frequencies. Persistence measures conrm those obtained for GARCH aTGARCH models, with the exception of the monthly FTSE series where the previously integrated crash-unadjusted GARCH ca


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force for the time-dependent movement of the trend, andthe dierence between the conditional variance and itstrend ht qt denes the transitory component of the con-ditional variance. The transitory component then con-verges to zero with powers of - while the long-runcomponent converges on ! with powers of .17

Recursively estimated models

In addition to recursive appraisal of the historical mean(see Equation 4 above) we also consider forecasts generatedby recursive variants of the above models which involveparameters estimated using in-sample data. That is, thosemodels involving parameters with a T subscript. Thesealternative recursively generated one-step-ahead forecastsare based on re-estimation of the underlying parameters ateach data point over the out-of-sample period, that re-esti-mation utilizing all information available at that point intime. Thus, while not specied in full, the set of parameters

T T ;T ;T ;T ;F;T ; T ;! T ;i;T ; -i;T ; T ;i;T ;

2;T ;T

in Equations 710 and 1215 are correspondingly replacedby the set t t ; t ; . . . ;t for t T ; T 1; . . . ;T 1.18 The criteria on which the forecast perform-ance of the resulting and the foregoing models are assessedis discussed in the following section.

I V . F O R E C A S T E V A L U A T I O N

In order to provide a measure of `true volatility againstwhich to assess the forecast performance of the volatilityestimators, we follow Pagan and Schwert (1990) in usingthe squared error term from a conditional mean model forreturns estimated over the full data set comprising both thein-sample and out-of-sample data. That is, `true volatilitygenerated by:

s2t rt f ;L 1rtg2 16

where the subscript on a coecient indicates that it isestimated over the entire data sample. The ability of the

above models to adequately forecast true volatility someasured in the FTSE and FTA stock index markets isevaluated using the mean error (ME), root mean squared

error (RMSE) and mean absolute error ( MAE), dened follows:

ME 1

XTtT1

ht s2t 1

MAE 1

XT

tT1

jht s2t j 1

RMSE

1

XTtT1

ht s2t

vuut 1where is the number of forecast data points and s2t`true volatility as dened above. The ME statistic is ushere as a general guide to the direction of over- or undeprediction on average. The MAE is an orthodox forecaappraisal criterion which does not permit the osettieects of overprediction and underprediction as in M

while the RMSE is a conventional criterion which clearweights greater forecast errors more heavily in the averaforecast error penalty.

These error statistics assume the underlying loss functito be symmetric. However, as noted in the introduction,is probable that as a practical matter not all investors wattach equal weight to similar sized overpredictions aunderpredictions of volatility.19 Following previoresearch (Pagan and Schwert, 1990; Brailsford and Fa1996) we therefore also consider error statistics designto account for potential asymmetry in the loss functioThat is, mean mixed error statistics which penalize, rst

underpredictions more heavily:

MMEU 1

XOi 1

jht s2t j

XUi 1

jht s

2t j

q" #2

and secondly, overpredictions more heavily:

MMEO 1

XOi 1

jht s

2t j

qXUi 1

jht s2t j

" #1

where O denotes the number of overpredictions and U tnumber of underpredictions among the out-of-sample for

casts. Finally, and again f ollowing previous research, walso report standardized values for all error statistics usinthe error statistic for the historical mean benchmark f


now obtains a persistence measure of 0:94, suggesting oscillation in conditional variance, and the corresponding crash-adjustmeasure falls to 0.43.17 In-sample estimates of generally conrm earlier persistence measures, at 0.83 (0.64), 0.92 (0.91), and 0.99 (0.99) in crash-adjust(unadjusted) FTA monthly, weekly and daily series respectively, and 0.91 (0.68), 0.94 and 0.94 (0.92) in corresponding FTSE adjust(unadjusted) series. Persistence in the transitory component, -, is everywhere lower than in the permanent component, though onmarginally f or daily FTA data, and negative for the monthly crash-adjusted and weekly unadjusted FTA series and adjusted daily FTseries, suggesting the oscillation of volatility about the long-run trend during convergence in those cases.18 Note that, as mentioned in footnote 6, for the class of GARCH models such recursive estimation is `fully recursive in the sense thnot only are the volatility forecasting equation parameters iteratively updated with the acquisition of out-of-sample information, but t


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each series. This has the advantage of allowing the errorstatistics to be more easily interpreted in a relative context.

V . F OR E C A S T R E S U LT S

Symm etric forecast error results

Tables 1A and 1B report forecast ME, RMSE and MAEstatistics for the FTSE and FTA indices sampled at themonthly frequency, unadjusted and adjusted for the 1987crash respectively. The ME statistic indicates that allmodels overpredict volatility for both series, with the soleexception of the random walk, which also provides theforecast with the smallest absolute ME for both series.More substantively, the random walk model also domi-nates on both the RMSE and MAE statistics for bothseries, and both including and excluding the crash period.Additionally, as indicated by the standardized statistics, the

gain in performance of the random walk model over allother models is considerable. The simple regression andhistorical mean perform generally poorly, particularly inforecasting FTA monthly volatility, providing the worstforecasts for that series. Among those other models, theperformance of the GARCH and smoothing models(moving average, exponential smoothing, and EWMA) issimilar across measures and series, with some notableexceptions: the good performance of recursive exponentialsmoothing for the FTA series; the poor performance of thenon-recursive GARCH and TGARCH for the FTSEseries, particularly evident where poorer relative to the his-

torical mean benchmark; likewise the poor performance ofthe exponential smoothing models for the crash-unadjustedFTSE series. Further, the rst of these exceptions apart, theve-year moving average (and equally the recursiveEWMA model for FTSE data) consistently marginally out-performs other models.20

Tables 2A and 2B present ME, R MSE and MAE statis-tics for forecasts of weekly sampled FTA and FTSEreturns volatility, again crash-unadjusted and crash-adjusted respectively. As for the monthly frequency, allmodels again generate overpredictions of volatility withthe exception of the random walk, although minimum

ME statistics are now obtained by the recursive exponen-tial smoothing model for FTA data and the 1.25 year mov-

ing average for FTSE data. Nevertheless, on the basis the MAE forecast error statistic, the random walk modcontinues to dominate the forecasting performance fboth indices, followed by the exponential smoothimodel for the FTA series while the moving average anrecursive EWMA models provide the next best forecastiperformance on broadly similar forecast error statistics f

both sets of FTA and FTSE data, followed by the GARCand remaining smoothing models with broadly simiforecast error statistics. However, under the RMSforecast error statistic, both moving average models athe recursive exponential smoothing and exponentEWMA models now provide equivalently superior f orcasts for the FTA series, both including and excludithe crash period. Similarly, both moving average modeand the recursive EWMA model provide equivalensuperior forecasts for crash-unadjusted FTSE data, whthe 1.25 year moving average is singularly superior for tcrash-adjusted FTSE. On these criteria the GARCH mo

els perform comparably with the remaining smoothimethods, and more favourably in some instances, notabthe recursive GARCH and EGARCH models. Finally, oboth the MAE and RMSE criteria, the simple regressioand historical mean models again provide the pooreforecasts for the FTA series, while the exponential smooting model provides the poorest forecasts for the FTseries, although the historical mean and simple regressimodels again perform poorly.

Tables 3A and 3B report forecast error statistics for returvolatility on daily sampled unadjusted and crash-adjustFTSE and FTA data. ME statistics indicate overpredicti

for all models except the random walk and recursive expnential smoothing in crash-adjusted series and, additional(non-recursive) exponential smoothing in the crasunadjusted series. Recursive exponential smoothing alnow gives the minimum ME statistic across all series oththan the crash-adjusted FTA, where the 3 month moviaverage model is preferred. For the FTA series, the exponetial smoothing and three-month moving average models prvide the best forecasts on the MAE statistic for crasunadjusted and crash-adjusted data respectively, followby the remaining smoothing models (excepting the non-recusive EWMA) and GARCH models thereafter. This orderi

also holds for crash-unadjusted FTSE data on the MAE cterion, but not the crash-adjusted FTSE series, for which t


conditional mean function for stock index returns, which is used to generate historic volatility (i.e. "2t in the GARCH context), is al

iteratively updated. On the implementation of similar procedures for the class of statistical forecasting models, see footnote 6.19 For example, consider the positive relationship between the volatility of underlying stock prices and call option prices previously notby Brailsford and Fa (1996) . A n underprediction of volatility will impart a downward bias to estimates of the call option price, whichlikely to be of more concern to a seller than a buyer, while the reverse is true of overpredictions of stock price volatility.20 As noted in the Introduction, Dimson and Marsh (1990) reject the random walk model for quarterly volatility forecasting of the FTpreferring the simple regression and exponential smoothing models. One possible explanation for the divergence between their resuand ours is that Dimson and Marsh compute actual volatility as annualized standard deviations of daily returns over non-overlappincalendar quarters, and generate one-step-ahead f orecasts of that quarterly series. Nevertheless, their ndings receive some support herethe monthly frequency in terms of our recursive exponential smoothing model results for FTA data, which prove `second best to trandom walk model.


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three-month moving average is marginally superior to therandom walk and other (particularly recursive) smoothingmodels. On the basis of the RMSE criterion there is far lessdivergence in error statistics f or both series for the majority ofmodels considered, although the 3-month moving averagemodel is marginally favoured for both FTA and FTSE datacrash-unadjusted, exponential smoothing for the FTAadjusted series, and the GARCH model for the crash-adjusted FTSE series. For all daily series, all of these models

are far superior (using RMSE) to the historical mean ansimple regression models which again provide the poorforecasts. Moreover, the random walk model performs pooron the RMSE measure, both relative to the results for lowfrequencies noted above and to other models for daily daand for (particularly crash-adjusted) FTSE data especially

In sum, on the basis of symmetric forecast error statitics, while the random walk model provides the most accrate volatility forecasts at the monthly and week


Table 1. Forecast error statistics; monthly frequency

FTA-All FTSE

Model ME MAE RMSE ME MAE RMSE

A. Excluding dummy for 1987 crashHistorical mean 0. 003 5 (1. 00 ) 0. 0035 ( 1. 00 ) 0. 003 5 ( 1. 00 ) 0. 002 5 ( 1. 00 ) 0. 0026 ( 1. 00 ) 0. 0026 ( 1. 0Recursive historical mean 0.0034 (0.97) 0.0034 (0.97) 0.0034 (0.97) 0.0024 (0.96) 0.0024 (0.92) 0.0024 (0.92

Moving average 5 years 0.0014 (0.40) 0.0012 ( 0.34) 0.0013 ( 0.37) 0.0013 ( 0.55) 0.0013 ( 0.50) 0.0014 ( 0.54Moving average 10 ye ar s 0.0026 (0.74) 0.0015 ( 0.43) 0.0015 ( 0.43) 0.0025 ( 1.03) 0.0025 ( 0.96) 0.0026 ( 1.0Random walk 70.0005 (0.13)* 0.0004 (0.11)* 0.0006 (0.17)* 70.0004 (0.16)* 0.0004 (0.15)* 0.0005 (0.1Expo nential s mo othing 0.0016 (0.46) 0.0016 ( 0.46) 0.0017 ( 0.49) 0.0032 ( 1.30) 0.0032 ( 1.23) 0.0032 ( 1.2Recursive exponential smoothing 0.0009 (0.26) 0.0010 (0.29) 0.0011 (0.31) 0.0030 (1.20) 0.0030 (1.15) 0.0030 (1.1EWMA 0.0027 (0.76 ) 0.0019 (0.54) 0.0019 (0.54) 0.0017 (0.71) 0.0026 (1. 00) 0.0026 (1.0R ecursive EWMA 0. 0026 (0. 74 ) 0. 0026 ( 0. 74 ) 0. 0026 ( 0. 74 ) 0. 001 4 ( 0. 56 ) 0. 001 4 ( 0. 54 ) 0. 0014 ( 0. 54Simple regression 0. 003 3 (0. 96 ) 0. 003 3 ( 0. 94 ) 0. 003 4 ( 0. 97 ) 0. 002 5 ( 1. 00 ) 0. 002 5 ( 0. 96 ) 0. 0025 ( 0. 96Recursive simple regression 0.0027 (0.76) 0.0027 (0.76) 0.0027 (0.76) 0.0023 (0.92) 0.0023 (0.88) 0.0023 (0.8GARCH 0.0020 (0.56 ) 0.0020 (0.57) 0.0020 (0.57) 0.0021 (0.84) 0.0021 (0. 81) 0.0021 (0.8R ecursive GA RCH 0. 001 8 (0. 51 ) 0. 001 8 ( 0. 51 ) 0. 001 9 ( 0. 54 ) 0. 001 7 ( 0.68 ) 0. 001 7 ( 0. 65 ) 0. 0019 ( 0. 7TGARCH 0.0022 (0.62) 0. 0022 (0.63 ) 0.0022 (0.63) 0.0030 (1.22) 0. 0028 (1. 08) 0.0030 (1.1R ecursive TGA RCH 0. 002 1 (0.60 ) 0. 002 1 ( 0.60 ) 0. 002 1 ( 0.60 ) 0. 002 7 ( 1. 08 ) 0. 002 7 ( 1. 04 ) 0. 0027 ( 1. 04EGARCH 0.0016 (0.47) 0. 0016 (0.46 ) 0.0018 (0.51) 0.0024 (0.96 ) 0. 0024 (0. 92) 0.0026 (1.0

R ecursive EGA RCH 0. 001 5 (0. 43 ) 0. 001 5 ( 0. 43 ) 0. 001 7 ( 0. 49 ) 0. 002 2 ( 0. 88 ) 0. 002 2 ( 0. 85 ) 0. 0024 ( 0. 92C-GARCH 0.0022 (0.62) 0. 0022 (0.63 ) 0.0022 (0.63) 0.0028 (1.15) 0. 0028 (1. 08) 0.0029 (1.12R ec ur sive C-GAR CH 0.0021 (0.60) 0.0021 ( 0.60) 0.0022 ( 0.63) 0.0025 ( 1.00) 0.0026 ( 1.00) 0.0026 ( 1.0

B. Including dummy for 1987 crashHistorical mean 0. 003 1 (1. 00 ) 0. 0031 ( 1. 00 ) 0. 003 1 ( 1. 00 ) 0. 001 8 ( 1. 00 ) 0. 0018 ( 1. 00 ) 0. 0018 ( 1. 0Recursive historical mean 0.0030 (0.97) 0.0030 (0.97) 0.0030 (0.97) 0.0016 (0.89) 0.0016 (0.89) 0.0017 (0.94Moving average 5 years 0.0014 (0.47) 0.0012 ( 0.39) 0.0013 ( 0.42) 0.0014 ( 0.78) 0.0014 ( 0.78) 0.0014 ( 0.7Moving average 10 ye ar s 0.0017 (0.56) 0.0015 ( 0.48) 0.0016 ( 0.52) 0.0016 ( 0.89) 0.0016 ( 0.89) 0.0017 ( 0.94Random walk 70.0005 (0.16)* 0.0004 (0.13)* 0.0006 (0.19)* 70.0004 (0.22)* 0.0004 (0.22)* 0.0005 (0.2Expo nential s mo othing 0.0016 (0.53) 0.0016 ( 0.52) 0.0017 ( 0.55) 0.0017 ( 0.94) 0.0017 ( 0.94) 0.0017 ( 0.94Recursive exponential smoothing 0.0008 (0.26) 0.0008 (0.26) 0.0010 (0.32) 0.0017 (0.94) 0.0017 (0.94) 0.0017 (0.94EWMA 0.0018 (0.58) 0.0020 (0.65) 0.0019 (0.61) 0.0018 (1.00) 0.0017 (0. 94) 0.0017 (0.94R ecursive EWMA 0. 001 7 (0. 55 ) 0. 001 7 ( 0. 55 ) 0. 001 8 ( 0. 58 ) 0. 001 4 ( 0. 78 ) 0. 001 4 ( 0. 78 ) 0. 0015 ( 0. 8Simple regression 0. 002 8 (0. 89 ) 0. 002 8 ( 0. 90 ) 0. 002 8 ( 0. 90 ) 0. 001 9 ( 1. 06 ) 0. 001 9 ( 1. 06 ) 0. 0019 ( 1. 06Recursive simple regression 0.0027 (0.87) 0.0027 (0.87) 0.0027 (0.87) 0.0017 (0.94) 0.0017 (0.94) 0.0017 (0.94GARCH 0.0015 (0.47) 0.0015 (0.48) 0.0015 (0.48) 0.0029 (1.64) 0.0029 (1.64) 0.0030 (1.6R ecursive GA RCH 0. 001 3 (0. 42 ) 0. 001 3 ( 0. 42 ) 0. 001 4 ( 0. 47 ) 0. 0016 ( 0. 89 ) 0. 0016 ( 0. 89 ) 0. 0018 ( 1. 0TGARCH 0.0017 (0.56 ) 0. 0017 (0.55 ) 0.0018 (0.58) 0.006 5 (3.72) 0. 006 5 (3. 72) 0.0067 (3.72R ecursive TGA RCH 0. 0016 (0. 51 ) 0. 0016 ( 0. 51 ) 0. 001 8 ( 0. 58 ) 0. 002 2 ( 1. 22 ) 0. 002 2 ( 1. 22 ) 0. 0024 ( 1. 3EGARCH 0.0020 (0.64) 0. 0020 (0.65 ) 0.0021 (0.68) 0.0023 (1.28) 0. 0023 (1. 28) 0.0024 (1.3R ecursive EGA RCH 0. 001 7 (0. 55 ) 0. 001 7 ( 0. 55 ) 0. 001 9 ( 0.61 ) 0. 001 9 ( 1. 06 ) 0. 001 9 ( 1. 06 ) 0. 0019 ( 1. 06C-GARCH 0.0016 (0.51) 0. 0016 (0.52 ) 0.0016 (0.52) 0.0022 (1.22) 0. 0022 (1. 22) 0.0022 (1.22R ec ur sive C-GAR CH 0.0015 (0.47) 0.0015 ( 0.47) 0.0016 ( 0.51) 0.0018 ( 1.00) 0.0018 ( 1.00) 0.0019 ( 1.06

Notes : FTA-ALL refers to the Financial Times-Actuaries All Share Index, and FTSE to the Financial TimesStock Exchange 100 indME is the mean error statistic dened in (17). MAE is the mean absolute error statistic dened in (18). RMSE is the root mean squareerror statistic dened in (19). Relative error statistics obtained by expressing the actual statistic for each model as a ratio to tcorresponding error statistic for the historical mean are provided in parentheses, minimum values for which are indicated by asteris

All forecast error statistics relate to the period 1/1/199531/7/1996. For f orecasting model descriptions and denitions see Section IIIthe text, expressions 315.


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frequencies (especially monthly, where it dominates all

others by a sizeable margin), it performs poorly in forecast-

ing the volatility of daily returns. The exponential smooth-ing and moving average models provide some of the most

accurate weekly and daily volatility forecasts, but poorer

monthly volatility forecasts, the moving average models

providing a consistently good relative forecasting perform-

ance. Similarly, the GARCH genre of models provide a

consistently fair relative forecasting performance,

GARCH and EGARCH models outperforming the

TGARCH and CGARCH models. The performance of

the remaining models, and especially the simple regression

and historical mean models, ranks poorly compared to tabove models.

Asymmetric forecast error results

Finally, Tables 4 and 5 report MME(U) and MME(statistics for all series at the weekly and daily frequencirespectively. No such statistics are reported for monthseries, where all models other than the random walk cosistently overpredict volatility throughout the out-osample data, (the MME(U) statistic in particular collapsito the MAE statistic reported above in such circumstances). At the daily frequency, for both the FTA an


Table 2. Forecast error statistics; weekly frequency

FTA-All FTSE


A. Excluding dummy for 1987 crashH istor ical m ean 0.00042 (1.00) 0.00050 ( 1.00) 0.00053 ( 1.00) 0.00037 ( 1.00) 0.00040 ( 1.00) 0 .00044 ( 1.00Recursive historical mean 0.00045 (1.07) 0.00049 (0.98) 0.00051 (0.96) 0.00034 (0.92) 0.00038 (0.95) 0.00042 (0.95

Moving average 1.25 years 0.00005 (0.11) 0.00020 (0.40) 0.00025 (0.47)* 0.00010 (0.28)* 0.00020 (0.50) 0.00027 (0.6Moving average 2.5 years 0.00005 (0.13) 0.00020 (0.40) 0.00025 (0.47)* 0.00012 (0.33) 0.00021 (0.53) 0.00027 (0.61Random walk 70.00016 (0.37) 0.00016 (0.32)* 0.00029 (0.55) 70.00016 (0.45) 0.00016 (0.40)* 0.00029 (0.66Exponential smoothing 0.00015 (0.36) 0.00026 (0.52) 0.00029 (0.55) 0.00043 (1.17) 0.00046 (1.15) 0.00049 (1.11Recursive exponential smoothing 0.00002 (0.06)* 0.00019 (0.38) 0.00025 (0.47)* 0.00036 (0.97) 0.00040 (1.00) 0.00044 (1.00EWMA 0. 00005 (0. 13) 0. 0002 5 (0. 50) 0. 0002 9 ( 0. 55 ) 0. 0001 7 ( 0. 47 ) 0. 00024 (0.60) 0. 0003 0 (0.68Recursive EWMA 0.00005 (0.13) 0.00020 (0.40) 0.00025 (0.47)* 0.00012 (0.32) 0.00021 (0.53) 0.00027 (0.61Simple regression 0.00055 (1.09) 0.00047 (0.94) 0.00049 (0.93) 0.00036 (0.99) 0.00041 (1.03) 0.00044 (1.00Recursive simple regression 0.00042 (1.00) 0.00046 (0.92) 0.00048 (0.91) 0.00034 (0.92) 0.00038 (0.95) 0.00042 (0.95GARCH 0.00021 (0.50) 0.00029 (0.58) 0.00033 (0.62) NA NA NARecursive GARCH 0.00020 (0.48) 0.00028 (0.56) 0.00032 (0.60) NA NA NATGARCH 0.00021 (0.49) 0.00029 (0.58) 0.00032 (0.60) NA NA NARecursive TGARCH 0.00019 (0.42) 0.00028 (0.56) 0.00031 (0.58) NA NA NAEGARCH 0.00013 (0.31) 0.00024 (0.48) 0.00029 (0.55) NA NA NA

Recursive EGARCH 0.00012 (0.45) 0.00024 (0.48) 0.00028 (0.53) NA NA NAC-GARCH 0.00021 (0.50) 0.00029 (0.58) 0.00033 (0.62) NA NA NARecursive C-GARCH 0.00020 (0.42) 0.00028 (0.56) 0.00032 (0.60) NA NA NA

B. Including dummy for 1987 crashH istor ical m ean 0.00040 (1.00) 0.00047 ( 1.00) 0.00049 ( 1.00) 0.00027 ( 1.00) 0.00032 ( 1.00) 0 .00036 ( 1.00Recursive historical mean 0.00042 (1.05) 0.00046 (0.98) 0.00048 (0.98) 0.00026 (0.96) 0.00030 (0.94) 0.00034 (0.94Moving average 1.25 years 0.00005 (0.12) 0.00020 (0.43) 0.00025 (0.51)* 0.00010 (0.38)* 0.00020 (0.63) 0.00025 (0.69Moving average 2.5 years 0.00005 (0.13) 0.00019 (0.40) 0.00025 (0.51)* 0.00013 (0.46) 0.00021 (0.66) 0.00026 (0.72Random walk 70.00016 (0.38) 0.00016 (0.34)* 0.00029 (0.59) 70.00016 (0.57) 0.00016 (0.50)* 0.00027 (0.75Exponential smoothing 0.00017 (0.42) 0.00026 (0.55) 0.00030 (0.61) 0.00028 (1.03) 0.00033 (1.03) 0.00036 (1.00Recursive exponential smoothing 0.00001 (0.03)* 0.00018 (0.38) 0.00025 (0.51)* 0.00012 (0.44) 0.00022 (0.69) 0.00027 (0.75EWMA 0. 00014 (0. 35) 0. 0002 5 (0. 53) 0. 0002 8 ( 0. 57 ) 0. 0001 8 ( 0. 66 ) 0. 00025 (0. 78) 0. 0002 9 (0. 81Recursive EWMA 0.00006 (0.15) 0.00020 (0.43) 0.00025 (0.51)* 0.00013 (0.48) 0.00021 (0.66) 0.00026 (0.72Simple regression 0.00037 (0.92) 0.00042 (0.89) 0.00045 (0.92) 0.00027 (0.97) 0.00031 (0.97) 0.00035 (0.97Recursive simple regression 0.00037 (0.93) 0.00041 (0.87) 0.00044 (0.90) 0.00025 (0.93) 0.00030 (0.94) 0.00034 (0.94GARCH 0. 00014 (0. 34) 0. 0002 5 (0. 53) 0. 0002 9 ( 0. 59 ) 0. 00016 ( 0. 59 ) 0. 00024 (0. 75) 0. 0002 8 (0. 78Recursive GARCH 0.00013 (0.33) 0.00024 (0.51) 0.00028 (0.57) 0.00013 (0.48) 0.00022 (0.69) 0.00026 (0.72TGA RCH 0. 0001 3 (0. 31) 0. 0002 4 (0. 51 ) 0. 0002 8 ( 0. 57 ) 0. 0001 8 ( 0. 67 ) 0. 0002 5 (0. 78) 0. 0002 9 (0. 81Recursive TGARCH 0.00011 (0.28) 0.00023 (0.49) 0.00027 (0.55) 0.00015 (0.56) 0.00023 (0.72) 0.00027 (0.75EGA RCH 0. 0000 9 (0. 22) 0. 0002 2 (0. 47 ) 0. 0002 7 ( 0. 55 ) 0. 0001 7 ( 0. 62 ) 0. 0002 4 (0. 75) 0. 0002 9 (0. 81Recursive EGARCH 0.00008 (0.20) 0.00021 (0.45) 0.00027 (0.55) 0.00014 (0.52) 0.00022 (0.69) 0.00027 (0.75C-GA RCH 0. 0001 4 (0. 35) 0. 0002 5 (0. 53 ) 0. 0002 9 ( 0. 59 ) 0. 0001 5 ( 0. 53 ) 0. 0002 3 (0. 72) 0. 0002 8 (0. 78Recursive C-GARCH 0.00012 (0.30) 0.00024 (0.51) 0.00028 (0.57) 0.00013 (0.48) 0.00022 (0.69) 0.00027 (0.75

Note : As Table 1.


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FTSE series adjusted or unadjusted for the 1987 crash, thehistorical (or recursive historical) mean is the best volatilityforecasting model if underpredictions are penalized moreheavily, while the random walk model is massively pre-ferred if overpredictions are penalized more heavily.These conclusions also hold for the FTA at the weeklyfrequency, although the simple regression model is also apreferred model on the MME(U) criterion. For the weeklyFTSE series, the random walk continues to be heavilypreferred on the MME(O) criterion penalizing overpredic-tions more heavily, while the exponential smoothing modelis preferred on the MME(U) criterion.

V I . C O N C L U S I O N

This paper provides a comparative evaluation of the abiliof a variety of statistical and econometric models to forcast the volatility of the UK FTA All Share and FTSE1stock indices, motivated by recognition of the practicneed for accurate volatility forecasts in areas such as optipricing and the limited empirical evidence available to dafor the UK. A total of ten volatility forecasting models aconsidered, including the historical mean, moving averagrandom walk, exponential smoothing, exponentialweighted moving average, simple regression, GARC


Table 3. Forecast error statistics; daily frequency

FTA-All FTSE


A. Excluding dummy for 1987 crash

Histo rical m ean 7. 34e -05 ( 1. 00 ) 7. 87e-05 ( 1. 00 ) 8. 32e -0 5 ( 1. 00 ) 5. 41e -05 ( 1. 00 ) 6. 74e-0 5 ( 1. 00 ) 7. 48e-05 ( 1. 00Recursive historical mean 7.13e-05 (0.97) 7.67e-05 (0.97) 8.14e-05 (0.98) 5.06e-05 (0.94) 6.45e-05 (0.96) 7.23e-05 (0.97

Moving average 3 months 1.58e-07 (0.02) 2.45e-05 (0.31) 3.93e-05 (0.47)* 7.96e-07 (0.01) 3.45e-05 (0.51) 5.19e-05 (0.69Moving average 6 months 2.88e-06 (0.04) 2.56e-05 (0.33) 3.98e-05 (0.48) 4.29e-06 (0.08) 3.57e-05 (0.53) 5.24e-05 (0.70

Random walk 72.54e-05 (0.35) 2.56e-05 (0.33) 4.68e-05 (0.56) 73.57e-05 (0.66) 3.59e-05 (0.53) 6.30e-05 (0.84Exponential smoothing 73.56e-05 (0.05) 2.34e-05 (0.30)* 3.94e-05 (0.47) 71.98e-05 (0.36) 3.09e-05 (0.46)* 5.54e-05 (0.74

Recursive exponential smoothing 71.17e-07 (0.002)* 2.65e-05 (0.34) 4.17e-05 (0.50) 71.42e-07 (0.003)* 4.31e-05 (0.64) 6.74e-05 (0.90

EWMA 2. 30e-05 (0. 31) 3.70e-05 (0. 47) 4. 95e-05 (0.60 ) 3. 02e-05 (0. 56 ) 4.96e-05 (0. 74) 5. 99e-05 (0.80Rec ursive EWMA 2.94e -06 (0.04 ) 2.58e -05 (0.33 ) 4.01e -0 5 (0.48 ) 4.36 e-06 (0.08 ) 3.60e -0 5 (0.53 ) 5.27e-0 5 (0.70Simple re gression 3.91e -05 (0.53 ) 4.96e -05 (0.63 ) 5.81e -0 5 (0.70 ) 1.97e -05 (0.36 ) 4.65e -0 5 (0.69 ) 6 .47e-0 5 (0.87

Recursive simple regression 3.79e-05 (0.52) 4.86e-05 (0.62) 5.74e-05 (0.69) 1.85e-05 (0.34) 4.58e-05 (0.68) 6.43e-05 (0.86

GA RCH 1. 36e-05 (0. 19) 3. 13e-05 (0. 40 ) 4. 20e-05 (0. 51 ) 2. 08e-05 (0. 38) 4. 42e-05 (0.66 ) 5.63e-05 (0. 75Rec ursive GARCH 1.31e -05 (0.18 ) 3.10e -05 (0.39 ) 4.19e -0 5 (0.50 ) 1.90e -05 (0.35 ) 4.31e -0 5 (0.64 ) 5.56 e-0 5 (0.74

TGAR CH 1. 46e-05 (0. 20) 3. 19e-05 (0. 41 ) 4. 23e-05 (0. 51 ) 2. 07e-05 (0. 38) 4. 39e-05 (0.65 ) 5. 59e-05 (0. 75Rec ursive TGARCH 1.36 e-05 (0.19 ) 3.12e -05 (0.40 ) 4.20e -0 5 (0.50 ) 1.89e -05 (0.35 ) 4.29e -0 5 (0.64 ) 5.53e-0 5 (0.74

EGAR CH 8. 40e-06 (0. 11) 2. 86e-05 (0. 36 ) 4. 07e-05 (0. 49 ) 1. 97e-05 (0. 36 ) 4. 35e-05 (0.65 ) 5. 59e-05 (0. 75Rec ursive EGARCH 9.03e -06 (0.12 ) 2.89e -05 (0.36 ) 4.07e -0 5 (0.49 ) 1.76 e-05 (0.33 ) 4.26e -0 5 (0.63 ) 5.53e-0 5 (0.74

C-GA RCH 1. 15e-05 (0. 16 ) 3. 02e-05 (0. 38 ) 4. 15e-05 (0. 50 ) 2. 02e-05 (0. 37) 4. 39e-05 (0.65 ) 5.61e-05 (0. 75Rec ursive C-GARCH 1.02e -05 (0.14 ) 2.93e -05 (0.37 ) 4.12e -0 5 (0.50 ) 1.67e -05 (0.31 ) 4.19e -0 5 (0.62 ) 5.49e-0 5 (0.73

B. Including dummy f or 1987 crash

Histo rical m ean 7. 05e -05 ( 1. 00 ) 7.6 0e-05 ( 1. 00 ) 8. 07e -0 5 ( 1. 00 ) 4. 46e -05 ( 1. 00 ) 5. 98e-0 5 ( 1. 00 ) 6. 84e-05 ( 1. 00Recursive historical mean 6.84e-05 (0.97) 7.41e-05 (0.98) 7.89e-05 (0.98) 4.17e-05 (0.93) 5.76e-05 (0.96) 6.69e-05 (0.97

Moving average 3 months 1.88e-07 (0.003)* 2.45e-05 (0.32)* 3.94e-05 (0.49) 9.14e-07 (0.02) 3.45e-05 (0.58)* 5.25e-05 (0.77Moving average 6 months 2.88e-06 (0.04) 2.55e-05 (0.34) 3.98e-05 (0.49) 4.40e-06 (0.10) 3.56e-05 (0.60) 5.24e-05 (0.77

Random walk 72.53e-05 (0.36) 2.55e-05 (0.34) 4.68e-05 (0.58) 73.52e-05 (0.79) 3.54e-05 (0.59) 6.32e-05 (0.92

Exponential smoothing 1.80e-06 (0.03) 2.51e-05 (0.33) 3.93e-05 (0.49)* 3.33e-06 (0.07) 3.54e-05 (0.59) 5.24e-05 (0.77Recursive exponential smoothing 72.53e-07 (0.004) 2.57e-05 (0.34) 4.06e-05 (0.50) 71.57e-07 (0.004)* 3.58e-05 (0.60) 5.41e-05 (0.79EWMA 2. 29e-05 (0. 32) 3.69e-05 (0. 49) 4. 55e-05 (0. 56 ) 3. 00e-05 (0.67) 4.92e-05 (0. 82) 5. 98e-05 (0.87

Rec ursive EWMA 2.94e -06 (0.04 ) 2.57e -05 (0.34 ) 4.01e -0 5 (0.50 ) 4.47e -06 (0.10 ) 3.60e -0 5 (0.60 ) 5.33e-0 5 (0.78

Simple re gression 4.61e -05 (0.65 ) 5.50e -05 (0.72 ) 6.20e -0 5 (0.77 ) 3.22e -05 (0.72 ) 5.16e -0 5 (0.86 ) 6 .34e-0 5 (0.93Recursive simple regression 4.46e-05 (0.63) 5.38e-05 (0.71) 6.09e-05 (0.75) 3.02e-05 (0.68) 5.01e-05 (0.84) 6.24e-05 (0.91

GA RCH 1. 27e-05 (0. 18) 3. 08e-05 (0. 41 ) 4. 19e-05 (0. 52 ) 1. 59e-05 (0. 36 ) 4. 19e-05 (0. 70 ) 5. 23e-05 (0. 77rec ursive GARCH 1.19e -05 (0.17 ) 3.04e -05 (0.40 ) 4.17e -0 5 (0.52 ) 1.58e -05 (0.35 ) 4.16e -0 5 (0.70 ) 5.52e-0 5 (0.80

TGAR CH 1. 21e-05 (0. 17) 3. 04e-05 (0. 40 ) 4. 15e-05 (0. 51 ) 1. 74e-05 (0. 39) 4. 22e-05 (0. 71 ) 5. 54e-05 (0. 81Rec ursive TGARCH 1.14e -05 (0.16 ) 3.00e -05 (0.39 ) 4.14e -0 5 (0.51 ) 1.56 e-05 (0.35 ) 4.14e -0 5 (0.69 ) 5.50e-0 5 (0.80

EGAR CH 8. 81e-06 (0. 12) 2. 88e-05 (0. 38 ) 4. 09e-05 (0. 51 ) 1. 59e-05 (0. 36 ) 4. 19e-05 (0. 70 ) 5. 54e-05 (0. 81Rec ursive EGARCH 8.34e -06 (0.12 ) 2.86e -05 (0.38 ) 4.07e -0 5 (0.50 ) 1.43e -05 (0.32 ) 4.11e -0 5 (0.69 ) 5.50e-0 5 (0.80

C-GA RCH 9. 90e-06 (0. 14) 2. 94e-05 (0. 39 ) 4. 11e-05 (0. 51 ) 1. 73e-05 (0. 39) 4. 21e-05 (0. 70 ) 5. 50e-05 (0. 80Rec ursive C-GARCH 8.70e -06 (0.00 ) 2.88e -05 (0.38 ) 4.09e -0 5 (0.51 ) 1.53e -05 (0.34 ) 4.13e -0 5 (0.69 ) 5.50e-0 5 (0.80

Note : As Table 1.


13/15


Table4.Meanmixedforecasterrorstatisticsweeklyfrequency

FTA-All

FTSE

Excluding1987crashdummy

Including1987crashdummy


Including1987crashdummy

Model

MM

E(U)

MME(O)

MM

E(U)

MME(O)

MME

(U)

MME(O)

MME(U)

MME(O)

Historicalmean

0.0014(1.00)*

0.0211(1.00)

0.0014(1.00)*

0.0200(1.00)

0.0012

(1.00)

0.0188(1.00)

0.0014

(1.00)

0.0161(1.00)

Recursivehistoricalmean

0.0014(1.00)*

0.0208(0.99)

0.0014(1.00)*

0.0200(1.00)

0.0012

(1.00)

0.0181(0.96)

0.0015

(1.07)

0.0154(0.96)

Movingaverage1.25years

0.0042(3.00)

0.0094(0.45)

0.0041(2.93)

0.0093(0.47)

0.0030

(2.50)

0.0105(0.56)

0.0028

(2.00)

0.0105(0.65)

Movingaverage2.5years

0.0040(2.86)

0.0095(0.45)

0.0039(2.79)

0.0094(0.47)

0.0028

(2.33)

0.0113(0.60)

0.0026

(1.86)

0.0114(0.71)

Randomwalk

0.0096(6.86)

0.0002(0.01)*

0.0094(6.71)

0.0002(0.01)*

0.0105

(8.75)

0.0002(0.01)*

0.0102

(7.29)

0.0002(0.01)*

Exponentialsmoothing

0.0030(2.14)

0.0126(0.60)

0.0027(1.93)

0.0129(0.65)

0.0010

(0.83)*

0.0205(1.09)

0.0013

(0.93)*

0.0163(1.01)

Recursiveexponential

0.0045(3.21)

0.0086(0.41)

0.0046(3.29)

0.0082(0.41)

0.0012

(1.00)

0.0186(0.99)

0.0028

(2.00)

0.0111(0.69)

smoothing

EWMA

0.0040(2.86)

0.0095(0.45)

0.0031(2.21)

0.0122(0.61)

0.0023

(1.92)

0.0129(0.69)

0.0021

(1.50)

0.0131(0.81)

RecursiveEWMA

0.0040(2.86)

0.0095(0.45)

0.0040(2.86)

0.0096(0.48)

0.0028

(2.33)

0.0113(0.60)

0.0026

(1.86)

0.0114(0.71)

Simpleregression

0.0014(1.00)*

0.0202(0.96)

0.0014(1.00)*

0.0187(0.94)

0.0012

(1.00)

0.0188(1.00)

0.0014

(1.00)

0.0158(0.98)

Recursivesimpleregression

0.0014(1.00)*

0.0199(0.94)

0.0015(1.07)

0.0186(0.93)

0.0012

(1.00)

0.0181(0.96)

0.0016

(1.14)

0.0151(0.94)

GARCH

0.0024(1.71)

0.0142(0.67)

0.0031(2.21)

0.0121(0.61)

N

A

NA

0.0023

(1.64)

0.0126(0.78)

RecursiveGARCH

0.0025(1.86)

0.0138(0.65)

0.0033(2.36)

0.0119(0.60)

N

A

NA

0.0026

(1.86)

0.0115(0.71)

TGARCH

0.0024(1.71)

0.0141(0.67)

0.0031(2.21)

0.0117(0.59)

N

A

NA

0.0022

(1.57)

0.0131(0.81)

RecursiveTGARCH

0.0026(1.86)

0.0136(0.64)

0.0033(2.36)

0.0115(0.58)

N

A

NA

0.0024

(1.71)

0.0120(0.75)

EGARCH

0.0032(2.29)

0.0118(0.56)

0.0036(2.57)

0.0104(0.52)

N

A

NA

0.0023

(1.64)

0.0128(0.80)

RecursiveEGARCH

0.0033(2.36)

0.0115(0.55)

0.0037(2.64)

0.0103(0.52)

N

A

NA

0.0026

(1.86)

0.0118(0.73)

C-GARCH

0.0024(1.71)

0.0142(0.67)

0.0031(2.21)

0.0121(0.61)

N

A

NA

0.0025

(1.79)

0.0120(0.75)

RecursiveC-GARCH

0.0025(1.79)

0.0138(0.65)

0.0033(2.36)

0.0118(0.59)

N

A

NA

0.0027

(1.93)

0.0114(0.71)

Notes:AsTable1additionally,MM

E(U)istheasymmetricmeanerrors

tatisticdenedinEquation20whichpenalizesunderpredictionsmoreheav

ilythanoverpredictions;

andMME(O)istheasymmetricm

eanerrorstatisticdenedinEquatio

n21whichpenalizesoverpredictions

moreheavily.


14/15


Table5.

Meanmixedforecasterrorstatisticsdailyfrequency

FT

A-All

FTSE


Includ

ing1987crashdummy


Includ

ing1987crashdummy

Model

MM

E(U)

MME(O)

MM

E(U)

MME(O)

MME

(U)

MME(O)

MME(

U)

MME(O)

Historicalmean

0.0004(1.00)*

0.0084(1.00)

0.0004(1.00)*

0.0082(1.00)

0.0008

(1.00)*

0.0072(1.00)

0.0009

(1.00)*

0.0066(1.00)

Recursivehistoricalmean

0.0004(1.00)*

0.0083(0.99)

0.0004(1.00)*

0.0081(0.99)

0.0008

(1.00)*

0.0070(0.97)

0.0009

(1.00)*

0.0064(0.97)

Movingaverage3months

0.0017(4.25)

0.0028(0.33)

0.0017(4.25)

0.0028(0.34)

0.0021

(2.63)

0.0033(0.46)

0.0021

(2.33)

0.0033(0.50)

Movingaverage6months

0.0016(4.00)

0.0030(0.36)

0.0016(4.00)

0.0030(0.37)

0.0019

(2.38)

0.0036(0.50)

0.0019

(2.11)

0.0036(0.55)

Eandomwalk

0.0040(10.0)

0.00002(0.002)*

0.0040(10.0)

0.00002(0.002)*

0.0048

(6.00)

0.00004(0.005)*

0.0047

(5.22)

0.0001(0.02)*

Exponentialsmoothing

0.0019(4.75)

0.0025(0.30)

0.0016(4.00)

0.0030(0.37)

0.0032

(4.00)

0.0016(0.22)

0.0019

(2.11)

0.0036(0.55)

Recursiveexponential

0.0019(4.75)

0.0028(0.33)

0.0018(4.50)

0.0028(0.34)

0.0027

(3.38)

0.0030(0.42)

0.0022

(2.44)

0.0033(0.50)

smoothing

EWMA

0.0009(2.25)

0.0049(0.58)

0.0009(2.25)

0.0049(0.60)

0.0011

(1.38)

0.0056(0.78)

0.0011

(1.22)

0.0056(0.85)

RecursieEWMA

0.0016(4.00)

0.0030(0.36)

0.0016(4.00)

0.0030(0.37)

0.0019

(2.38)

0.0036(0.50)

0.0019

(2.11)

0.0036(0.55)

Simpleregression

0.0007(1.75)

0.0061(0.73)

0.0006(1.50)

0.0066(0.80)

0.0016

(2.00)

0.0046(0.64)

0.0011

(1.22)

0.0057(0.86)

Recursivesimpleregression

0.0007(1.75)

0.0060(0.71)

0.0006(1.50)

0.0065(0.79)

0.0017

(2.13)

0.0045(0.63)

0.0012

(1.33)

0.0055(0.83)

GARCH

0.0012(3.00)

0.0041(0.49)

0.0012(3.00)

0.0040(0.49)

0.0014

(1.75)

0.0049(0.68)

0.0016

(1.78)

0.0045(0.68)

RecursiveGARCH

0.0012(3.00)

0.0040(0.48)

0.0013(3.25)

0.0039(0.48)

0.0015

(1.88)

0.0047(0.65)

0.0016

(1.78)

0.0045(0.68)

TGARCH

0.0012(3.00)

0.0042(0.50)

0.0013(3.25)

0.0039(0.48)

0.0014

(1.75)

0.0049(0.68)

0.0015

(1.67)

0.0046(0.70)

RecursiveTGARCH

0.0012(3.00)

0.0041(0.49)

0.0013(3.25)

0.0039(0.48)

0.0015

(1.88)

0.0047(0.65)

0.0016

(1.78)

0.0045(0.68)

EGARCH

0.0014(3.50)

0.0036(0.43)

0.0014(3.50)

0.0036(0.44)

0.0015

(1.88)

0.0048(0.67)

0.0016

(1.78)

0.0045(0.68)

RecursiveEGARCH

0.0014(3.50)

0.0036(0.43)

0.0014(3.50)

0.0036(0.44)

0.0015

(1.88)

0.0047(0.65)

0.0016

(1.78)

0.0044(0.67)

C-GARCH

0.0013(3.25)

0.0039(0.46)

0.0014(3.50)

0.0037(0.45)

0.0014

(1.75)

0.0048(0.67)

0.0015

(1.67)

0.0046(0.70)

RecursiveC-GARCH

0.0013(3.25)

0.0037(0.44)

0.0014(3.50)

0.0036(0.44)

0.0015

(1.88)

0.0046(0.64)

0.0016

(1.78)

0.0045(0.68)

Note:asTable4.


15/15

TGARCH, EGARCH, and component-GARCH models,and, additionally, recursive variants of these models whereappropriate. Forecast evaluations are performed formonthly, weekly and daily data frequencies, both withand without adjustment for the 1987 crash, and withrespect to both symmetric and asymmetric loss functions.

In summary of our results, when asymmetric loss is con-

sidered, the ranking of forecasting methods is dependenton the series, frequency and direction of that asymmetry, inkeeping with the previous results of Brailsford and Fa(1996). If overpredictions are penalized more heavily thanunder predictions, the random walk model is favoured.However, if underpredictions are penalized more heavilythan overpredictions, then the historical mean is favouredfor the forecasting of daily FTA and FTSE volatility, whilethe historical mean and simple regression are jointlyfavoured for weekly FTA volatility, and exponentialsmoothing is favoured for forecasting weekly FTSE vola-tility. For the symmetric loss case, the random walk model

is found to provide vastly superior monthly volatility fore-casts, while the random walk, moving average, and recur-sive smoothing models provide moderately superior weeklyvolatility forecasts, and GARCH, moving average, andexponential smoothing models provide marginally superiordaily volatility forecasts. Our ndings also lend some sup-port to the results of Franses and van Dijk (1996) concern-ing the dominance of the random walk model when thecrash of 1987 is included in the estimation sample, and theimprovement in GARCH model forecasts obtained follow-ing its exclusion. Indeed, if attention is restricted to sym-metric loss and the proposal of one forecasting method for

all f requencies, the most consistent forecasting performanceis provided by moving average and GARCH models. Moregenerally, our results suggest that previous results reportingthat the class of GARCH models provide relatively poorvolatility f orecasts, so limiting their practical usefulness,may be data specic, holding more robustly at lower fre-quencies but failing to hold at higher frequencies, as herefor the crash-adjusted daily FTSE100 index in particular.

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