forecasting accuracy of stochastic volatility, garch and ewma models under different volatility...

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Forecasting accuracy of stochastic volatility, GARCHand EWMA models under different volatility scenariosJie Ding a & Nigel Meade aa Imperial College London, Tanaka Business School, South Kensington , London SW7 2AZ, UKPublished online: 17 May 2010.

To cite this article: Jie Ding & Nigel Meade (2010) Forecasting accuracy of stochastic volatility, GARCH and EWMA modelsunder different volatility scenarios, Applied Financial Economics, 20:10, 771-783, DOI: 10.1080/09603101003636188

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Applied Financial Economics, 2010, 20, 771–783

Forecasting accuracy of stochastic

volatility, GARCH and EWMA

models under different volatility

scenarios

Jie Ding and Nigel Meade*

Imperial College London, Tanaka Business School, South Kensington,

London SW7 2AZ, UK

The forecasting of the volatility of asset returns is a prerequisite for many

risk management tasks in finance. The objective here is to identify the

volatility scenarios that favour either Generalized Autoregressive

Conditional Heteroscedasticity (GARCH) or Stochastic Volatility (SV)

models. Scenarios are defined by the persistence of volatility (its robustness

to shocks) and the volatility of volatility. A simulation experiment

generates return series using both volatility models for a range of volatility

scenarios representative of that observed in real assets. Forecasts are

generated from SV, GARCH and Exponentially Weighted Moving

Average (EWMA) volatility models. SV model forecasts are only

noticeably more accurate than GARCH in scenarios with very high

volatility of volatility and a stochastic volatility generating process. For

scenarios with medium volatility of volatility, there is little penalty for

using EWMA regardless of the volatility generating process. A set of

return time series selected from FX rates, equity indices, equities and

commodities is used to validate the simulation-based results. Broadly

speaking, the real series come from the medium volatility of volatility

scenarios where EWMA forecasts are reliably accurate. The robust

structure of EWMA appears to contribute to its greater forecasting

accuracy than more flexible GARCH model.

I. Introduction

The forecasting of the volatility of asset returns is

required for many risk management tasks in finance.

The use of Value-at-Risk (VaR) is ubiquitous in

financial institutions and is increasingly widespreadas a risk management tool in corporate institutions.

Future volatility is the crucial input to VaR calcula-

tions; an overestimate of volatility leads to an

opportunity loss due to capital being tied up unne-

cessarily and an underestimate of volatility leads to

risks being under protected. Risk control strategies,

such as delta hedging, are reliant on the estimates offuture volatility for the creation of riskless portfoliosof options and the underlying asset. Mean-variance

portfolio selection relies on the estimates of futurevolatility. Poon and Granger (2003) review the

extensive literature on forecasting the volatility offinancial markets.

Our contribution to this literature is to explorethe comparative effectiveness of the main two

volatility modelling methodologies, Generalized

*Corresponding author. E-mail: [email protected]

Applied Financial Economics ISSN 0960–3107 print/ISSN 1466–4305 online � 2010 Taylor & Francis 771http://www.informaworld.com

DOI: 10.1080/09603101003636188

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Autoregressive Conditional Heteroscedasticity(GARCH) and Stochastic Volatility (SV) in thecontext of possible volatility scenarios. We use twodimensions, the persistence of volatility (its robust-ness to shocks) and the volatility of volatility, todefine the space of volatility scenarios. We investigateif there are regions where either modelling approachis dominant using the accuracy of volatility forecastsas our measure of dominance. The investigation is intwo parts. First, a simulation experiment is per-formed where data from known volatility models aregenerated to cover the space of volatility scenariosand the accuracy of forecasts prepared from thecompeting models is compared. Second, data fromdifferent financial markets are located within thespace of volatility scenarios and a similar comparisonof forecasting accuracy is carried out.

In addition to forecasts using SV and GARCHmodels, we compute forecasts using ExponentiallyWeighted Moving Average (EWMA) volatility, thespecial case of the GARCH model favoured byRiskMetrics. For the simulated data, we contrast theaccuracy of volatility estimates in-sample and out-of-sample using a range of error measures. Weconclude from this experiment that, for most scenariosand regardless of the data generating process, usinga GARCH volatility forecast involves little penaltycompared to using a SV forecast. SV model forecastsare only noticeably more accurate than GARCH inscenarios with very high volatility of volatility and astochastic volatility generating process. For scenarioswith medium volatility of volatility, there is littlepenalty for using EWMA regardless of the volatilitygenerating process.

In order to validate these simulation-based results,a set of return time series selected from a ForeignExchange (FX) rates, equity indices, equities andcommodities is used. The real series are related to thepersistence and volatility of volatility scenarios andout-of-sample forecasts are generated from the threemethods discussed. We find that, broadly speaking,the real assets come from the medium volatility ofvolatility scenarios where EWMA forecasts are reli-ably accurate.

This article is structured as follows. In Section II,we discuss volatility models and we describe thesimulation experiment in Section III. The validationof our findings using real data is given in Section IVand we give our conclusions in Section V.

II. Volatility Models

A stylized fact of time series of returns on financialassets is the clustering behaviour of volatility.

Two modelling approaches have been used to capture

this behaviour. The GARCH model represents con-

ditional variance as a function of lagged squared

residuals and lagged conditional variance. The sto-

chastic variance model (as implied by its name)

assumes that the variance follows a stochastic pro-

cess. Both approaches will be described below. Note

that we will focus on the basic formulation of each

model, we wish to facilitate comparisons between the

modelling approaches rather than be distracted by the

differences of models within each approach. Our

notation is rt, which is the log-return in period t, �̂tþhis the estimated volatility in period tþ h, given data

up to period t.

Generalised autoregressive conditionalheteroscedasticity models

The GARCH model was proposed by Bollerslev

(1986), generalizing Engle’s (1982) Autoregressive

Conditional Heteroscedasticity (ARCH) model. For

surveys of the extensive literature on these models, see

Bollerslev et al. (1992, 1994), and Li et al. (2002). The

GARCH( p, q) model is defined as follows:

rt ¼ �þ "t ð1Þ

where "t � zt�t and E(zt)¼ 0 and V(zt)¼ 1. The

variance, �2t , obeys this process

�2t ¼ !þPqi¼1

�i"2t�i þ

Ppj¼1

�j�2t�j ð2Þ

such that !4 0, �i � 0, �j � 0,P�iþ

P�j 5 1.

Although there is no consensus about the ideal

GARCH specification (see, e.g. Brailsford and Faff,

1996), there is little doubt that GARCH models

capture the most important stylized facts describing

the asset return volatility; excess kurtosis and vola-

tility clustering. For our purposes, we will use the

above ‘classic’ GARCH model with p¼ q¼ 1. The

range of possible models has been investigated by

several authors; for example, McMillan et al. (2000)

compare the forecast accuracy of a variety of

GARCH formulations; Chuang et al. (2007) investi-

gate the effect on the forecasting accuracy of

GARCH models using different density

functions for zt.A feature of the GARCH model is that forecast

volatility reverts to an equilibrium level. For the

widely used GARCH(1,1), the one-step variance

forecast is

�̂2tþ1 ¼ !̂þ �̂"2t þ �̂�̂

2t ð3Þ

772 J. Ding and N. Meade

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by successive substitution, the h-step forecast can bederived as

�̂2tþh ¼ !̂Xh�2i¼0

ð�̂þ �̂Þi þ ð�̂þ �̂Þh�1�̂2tþ1 ð4Þ

We can see that as h gets larger, the forecast tendstowards its long-term mean of != 1� �� ð Þ. Thisproperty is consistent with volatility mean-revertingpatterns observed in the markets; however, it meansthat extreme values of volatility, events of greatpractical interest, cannot be predicted. It can be seenfrom this formulation that as (�þ �) approachesunity, the long-run variance approaches nonstatio-narity. For this reason, we use (�þ �) as a measureof the persistence of the GARCH volatility process.

A simple model that is a special case of theGARCH model is the EWMA volatility estimator;here volatility is implicitly assumed to follow arandom walk process. This implies that any shockin the market price introduces a permanent impact onvolatility, such that volatility does not have along-run mean level to revert to. This is denoted asIntegrated GARCH (IGARCH). For a sample sizeof nþ 1 time periods, the EWMA estimate is

�̂2tþh ¼1� �

1� �nþ1

Xni¼0

�ir2t�i ð5Þ

and 0 � � � 1 is the decay factor, which typicallytakes a value between 0.94 and 0.97. Note that asingle volatility prediction applies to all future timehorizons. This volatility estimation approach is sup-ported by RiskMetrics (2001).

Stochastic volatility models

Unlike the GARCH model, where conditional vola-tility is determined by lagged residuals and laggedconditional volatility, the SV model, introduced byTaylor (1982), considers volatility as a stochasticprocess. We consider an SV model where the instan-taneous variance follows an AR(1) process, the SV(1)model is

rt ¼ ��e0:5ht�t where �t�N 0, 1ð Þ

ht ¼ �ht�1 þ t where t�N�0, �2

�and E �ttð Þ ¼ � ð6Þ

where rt is the return of an asset and �� is the averagelevel of volatility, our formulation of the modelfollows Ghysels et al. (1996). Taking the logarithm ofsquared returns gives a linear model where

ln r2t ¼ ln ��2 þ ht þ "t where "t ¼ ln �2t� �

� ln �21� �

ht ¼ �ht�1 þ t t � N�0, �2

�but E "ttð Þ ¼ 0:

ð7Þ

After the log transformation, the information

regarding the correlation coefficient is lost

(Harvey et al., 1994). However, this can be recovered

by conditioning on the signs of the original returns

(Harvey and Shephard, 1996).This model captures some financial time series

properties, such as volatility clustering and high

persistence. Ghysels et al. (1996) assert that the SV

model describes financial time series better than the

ARCH models because the extra noise process tmakes it more flexible. The price of this flexibility is

that, it is no longer possible to specify an analytic

expression for the Log-Likelihood Function (LLF).

To date, methods used to estimate the SV model are

computationally expensive and, in some cases, the

theoretical properties of the estimators are still

unknown. Examples include: simulated maximum

likelihood suggested by Danielsson (1994); a Markov

Chain Monte Carlo (MCMC) approach suggested

by Jacquier et al. (1994); a Generalized Method of

Moments (GMM) approach used by Melino and

Turnbull (1990) and Andersen and Sorensen (1996).

Ruiz (1994) shows that for the kind of parameter

values likely to arise in practice, GMM is less efficient

than Quasi-Maximum Likelihood (QML) together

with the Kalman filter. The Kalman filter QML

procedure for SV model estimation can also be found

in Harvey et al. (1994), Heynen and Kat (1994), Yu

(2002) and Javaheri et al. (2003). Finally, there is the

empirical characteristic function method of Knight

et al. (2002). Of these methods, QML stands out due

to its consistency and relative ease of implementation.

Sandmann and Koopman (1998) propose a Monte

Carlo Likelihood (MCL) approach, which is a

significant improvement over QML. MCL uses the

idea of importance sampling to eliminate the need to

approximate a log�2 distribution with a normal one

as under QML.Compared to other volatility forecasting techni-

ques, the SV model is favoured by Heynen (1998) for

stock indices in the US, UK, Hong Kong, Japan and

Australia, and by Yu (2002) for the New Zealand

stock market. It is also preferred by Heynen and Kat

(1994) for stock indices, but less so compared with

GARCH(1,1) for currencies. Dunis et al. (2001)

compare GARCH, the SV model and other models

using a group of six FX rates. They find that

GARCH is more accurate (using Root Mean

Squared Error (RMSE)) than SV for five out of the

six rates over a 1-month horizon and for four out of

the six rates for a 3-month horizon. Sadorsky (2005)

uses a range-based SV model to forecast volatility

1 day ahead. For a variety of assets, an equity index,

crude oil, bonds and FX, he found that SV forecasts

Forecasting accuracy under different volatility scenarios 773

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were less accurate than EWMA and other naivemodels.

We refer to our implementation of MCL estima-tion of the SV model as SV(MCL). The full details ofthis complex estimation method are given inSandmann and Koopman (1998). Our objective isto estimate the parameter vector ð�,�, �Þ where� ¼ lnð ��2Þ (Equations 6 and 7). For this model, theparameter � is used as a measure of persistence, when� ¼ 1, the volatility process has a unit root.

Definition of the volatility scenario space

The space is defined by the persistence of volatilityand the volatility of volatility. Here, we relate thesetwo dimensions to both modelling approaches. Thepersistence of volatility reflects its robustness toshocks; the higher the measure, the longer the effectof a shock persists. For a GARCH model, persistenceis measured by (�þ�), and for an SV model, it ismeasured by �. These measures behave similarly inthat ð�þ �Þ ¼ � ¼ 0 implies a constant variance andno persistence, whereas ð�þ �Þ ¼ � ¼ 1 implies thevariance process is integrated and has high persis-tence. The volatility of volatility is determined asfollows. The Coefficient of Variation (CV) is a widelyused statistical measure of dispersion of a variablerelative to its mean, given by

CV ¼SD

mean

Here, we shall use CV2, the square of CV, forthe purpose of computational simplicity. If a returnseries follows a standard GARCH(1,1) process(Equation 2), the CV2 of its variance is

CV2GARCH ¼

2�2

1� 2�2 � �þ �ð Þ2

ð8Þ

where 1� 2�2 � �þ �ð Þ2 4 0

Note that the above condition for a finite secondmoment of the variance process encompasses thecondition for a finite first moment, �þ�<1. Theconsequence is that for a given value of �þ�, there isan upper limit on the value of � such that theGARCH(1,1) variance has a finite second moment.For example, if �þ�¼ 0.9, then �<0.31.

It can be shown that if a return series follows anSV(1) process (Equation 6), the CV2 of its variance is

CV2SV ¼ exp

�21� �2

!� 1 ð9Þ

Once the expected variance of the process is fixedto some arbitrary constant, the position of a GARCH

model in the volatility scenario space is determined bythe triplet of coefficients ð!,�,�Þ. Similarly, theposition of an SV model is determined by the tripletof coefficients ð�,�, �Þ.

III. Description of Simulation Experiment

This experiment involves the estimation of EWMA,GARCH and SV models using simulated data setsrepresenting different scenarios. These scenarios aredefined by the persistence of volatility and thevolatility of volatility; for each scenario two volatilitygenerating processes are used, GARCH and SV.

Simulation of time series with given CV2

The first step in the simulation process is to investi-gate how accurately the empirical CV2 calculatedfrom the simulated series reflects the CV2 used in thesimulation. We conduct experiments covering a rangeof parameter values commonly found in practice.Most empirical studies find values of persistence closeto unity, ranging between 0.9 and 0.995.

The range of parameter values ð!,�,�Þ for theGARCH model is chosen by first setting the persis-tence to realistic values, �þ� are set to be 0.90, 0.95and 0.98. Then for each value of �þ �, the values of �are chosen such that the CV2, defined in Equation 8,takes the value of 10, 1 and 0.1. A high value of CV2

indicates a high volatility of volatility, whereas alow value suggests that the model tends to constantvolatility. Lastly, values of ! are selected such thatthe expected variance !=ð1� �� Þ equals (1%)2,taking the simulated data as daily returns, thiscorresponds to 16% annualized volatility. Thevalues of these nine parameter triplets ð!,�,�Þi fori ¼ 1, . . . , 9 can be found in Table 1.

The range of parameter values ð�,�, �Þ for the SVmodel is selected similarly. First, values of theautoregressive parameter � are set to 0.90, 0.95 and0.98; then, for each value of �, three values of � areselected such that the CV2 takes the value of 10, 1 and0.1, respectively. Lastly, values of the locationparameter, � (recall that � ¼ ln ��2), are chosen tofix the expected variance at (1%)2. This proceduregives nine parameter triplets, ð�,�, �Þi fori ¼ 1, . . . , 9, whose values can be found in Table 1.

We generate 100 samples of length T¼ 250, 1000and 4000 for each parameter triplet and compute thesample CV2 based on the realized variance process.The results are summarized in Table 2, for each tripletthe average and SD of CV2 are given, the latter initalics. For both volatility models, the theoretical CV2


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is based on an assumption of an infinite series, thusdealing with a finite series leading to imperfectestimation of CV2.

The GARCH sample CV2s converge to theirtheoretical value as the sample length increases.However, for the three cases where the theoreticalvalue of CV2 is 10, the convergence is extremely slow.An exceedingly long GARCH series would be neededfor the sample CV2 to match its high theoreticalvalue. The results for the SV model follow a similarpattern although the sample CV2s converge morequickly than the GARCH model.

The implications for our study are as follows. Fora typical length of series, say 1000 observations, thesample CV2 underestimates the theoretical CV2. It ismore noticeable for high values of CV2 than lowvalues, and it is more pronounced for GARCHmodels than for SV models. If a set of data shows ahigh value of CV2, say 10, other considerations aside,it is more likely that the underlying data generatingprocess follows SV rather than GARCH.

Forecasting experiment

To assess the performance of SV and GARCH inmodelling data with different volatility properties, thefollowing simulation experiment is conducted. Foreach of the 18 parameter triplets defined in Section‘Simulation of time series with given CV2’, 100sample realizations with length T¼ 1050 are simu-lated. Using the simulated return data, these threemodels are used to estimate volatility: an SV model,a GARCH model and an EWMA model with 0.94(the decay factor suggested by RiskMetrics, 2001).The first 1000 data points are used for in-sampleestimation and the remaining 50 are reserved forout-of-sample forecasting. To measure the differencesbetween the observed and estimated volatilities foreach of the three models (Brailsford and Faff, 1996),three measures of forecasting accuracy are calculated:one RMSE

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

T

XTi¼1

�i � �̂ið Þ2

vuut ð10Þ

and two Mean Mixed Error (MME) measures

MMEðUÞ ¼1

T

Xi2U

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�i � �̂ij j

pþXj2O

�j � �̂j�� !

ð11Þ

MME Oð Þ ¼1

T

Xi2U

�i � �̂ij j þXj2O

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�j � �̂j�� q !

i 2 U if �i 4 �̂i and j 2 O if �i 5 �̂i ð12Þ

Note that in this simulated environment, thevolatility at a particular time period, �i, is known,this is not the case with real data. MME(U) penalizesunderestimations more heavily than overestimationssince (

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�i � �̂ij j

p4 �i � �̂ij j when �i � �̂ij j5 100%),

MME(O) does the opposite and RMSE applies equalpenalty regardless of the direction of errors.

The ratio of computation time taken by each modelis approximately 1:7:300 for EWMA, GARCH andSV(MCL), respectively. The results are presented inTables 3–6. The format of each table is the same. Thepersistence increases from left to right and volatility

Table 2. Sample versus theoretical CV2values for the SV

and GARCH models

SV GARCH

TheoreticalCV2 Triplet 250 1000 4000 250 1000 4000

10 1 4.48 6.91 8.60 0.83 1.44 2.073.92 5.72 5.04 0.53 1.34 2.27

2 3.07 5.58 8.00 0.62 1.13 1.822.20 4.48 7.66 0.42 0.89 1.60

3 1.78 3.64 6.33 0.42 0.84 1.531.11 2.36 5.02 0.36 0.76 1.41

1 4 0.82 0.94 0.98 0.44 0.68 0.820.47 0.31 0.15 0.25 0.50 0.49

5 0.68 0.89 0.97 0.36 0.57 0.770.42 0.36 0.25 0.23 0.38 0.48

6 0.46 0.76 0.94 0.25 0.46 0.700.26 0.35 0.30 0.21 0.36 0.52

0.1 7 0.09 0.10 0.10 0.08 0.10 0.100.03 0.02 0.01 0.03 0.04 0.02

8 0.08 0.09 0.10 0.07 0.09 0.100.04 0.02 0.01 0.04 0.04 0.03

9 0.06 0.09 0.10 0.05 0.08 0.100.03 0.03 0.02 0.04 0.04 0.03

Note: Mean values and SDs (in italics) from 100 runs.

Table 1. Parameter triplets for the SV and GARCH model

SV GARCH

CV2 � � � ! � �

10 0.675 0.900 �1.199 0.100 0.294 0.6060.484 0.950 �1.199 0.050 0.211 0.7390.308 0.980 �1.199 0.020 0.134 0.846

1 0.363 0.900 �0.347 0.100 0.218 0.6820.260 0.950 �0.347 0.050 0.156 0.7940.166 0.980 �0.347 0.020 0.099 0.881

0.1 0.135 0.900 �0.048 0.100 0.093 0.8070.096 0.950 �0.048 0.050 0.067 0.8830.061 0.980 �0.048 0.020 0.042 0.938

Note: For each value of CV2, there are three triplets withpersistence values of 0.90, 0.95 and 0.98.


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of volatility increases upwards. Specifically, eachtable is divided horizontally into three panels inaccordance with the CV2 value; the results aregrouped vertically into three categories dependingon the parameters, � for the SV model and �þ � for

the GARCH model, giving nine segments corre-sponding to the nine parameter triplets.

First, we look at volatility generated by an SVprocess, from Table 3, summarizing in-sample esti-mation results, we see the following.

Table 4. Out-of-sample accuracy (%) error measures for SV-simulated samples

� � � � � � � � �0.675 0.900 �1.199 0.484 0.950 �1.199 0.308 0.980 �1.199

EWMA GARCH SV EWMA GARCH SV EWMA GARCH SV

RMSE 0.145 0.208 0.131 0.134 0.149 0.114 0.125 0.116 0.095MME(U) 0.101 0.090 0.113 0.093 0.100 0.119 0.084 0.107 0.128MME(O) 0.273 0.378 0.227 0.274 0.282 0.196 0.275 0.221 0.155

� � � � � � � � �0.363 0.900 �0.347 0.260 0.950 �0.347 0.166 0.980 �0.347


RMSE 0.088 0.089 0.090 0.082 0.085 0.082 0.075 0.076 0.071MME(U) 0.108 0.098 0.098 0.102 0.096 0.111 0.096 0.094 0.122MME(O) 0.172 0.186 0.190 0.172 0.184 0.161 0.170 0.170 0.131

� � � � � � � � �0.135 0.900 �0.048 0.096 0.950 �0.048 0.061 0.980 �0.048


RMSE 0.042 0.035 0.034 0.040 0.033 0.032 0.037 0.030 0.028MME(U) 0.096 0.077 0.075 0.093 0.077 0.072 0.091 0.071 0.067MME(O) 0.096 0.096 0.095 0.095 0.094 0.094 0.094 0.092 0.091

Table 3. In-sample accuracy (%) error measures for SV-simulated samples

� � � � � � � � �0.675 0.900 �1.199 0.484 0.950 �1.199 0.308 0.980 �1.199


RMSE 0.73 0.53 0.39 0.72 0.44 0.33 0.71 0.34 0.26MME(U) 2.08 1.88 2.64 2.10 1.86 2.43 2.14 1.76 2.15MME(O) 5.49 3.75 1.64 5.46 3.10 1.52 5.42 2.52 1.39

� � � � � � � � �0.363 0.900 �0.347 0.260 0.950 �0.347 0.166 0.980 �0.347


RMSE 0.43 0.33 0.28 0.42 0.29 0.24 0.42 0.24 0.20MME(U) 2.17 1.87 2.62 2.19 1.78 2.44 2.24 1.67 2.18MME(O) 3.50 3.00 1.60 3.48 2.73 1.50 3.45 2.38 1.40

� � � � � � � � �0.135 0.900 �0.048 0.096 0.950 �0.048 0.061 0.980 �0.048


RMSE 0.20 0.15 0.15 0.20 0.15 0.14 0.20 0.13 0.14MME(U) 1.86 1.51 1.88 1.87 1.48 1.89 1.89 1.42 1.85MME(O) 1.94 1.81 1.38 1.93 1.76 1.31 1.91 1.68 1.26


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While � is kept constant, the accuracy of all threemodels improves as CV2 decreases, i.e. as the truevolatility becomes less stochastic. The relativeimprovement for EWMA is the most obvious

whereas that for SV(MCL) is the most modest.While CV2 is kept constant, the accuracyof GARCH and SV(MCL) improves as thevalue of � increases, i.e. the true volatility

Table 6. Out-of-sample accuracy (%) error measures for GARCH-simulated samples

! � � ! � � ! � �0.100 0.294 0.606 0.050 0.211 0.739 0.020 0.134 0.846


RMSE 0.077 0.076 0.076 0.076 0.073 0.070 0.076 0.063 0.062MME(U) 0.082 0.079 0.092 0.082 0.078 0.106 0.077 0.075 0.116MME(O) 0.179 0.179 0.160 0.182 0.175 0.136 0.189 0.159 0.111

! � � ! � � ! � �0.100 0.218 0.682 0.050 0.156 0.794 0.020 0.099 0.881


RMSE 0.063 0.061 0.067 0.060 0.058 0.062 0.058 0.050 0.054MME(U) 0.083 0.077 0.074 0.082 0.076 0.086 0.075 0.073 0.105MME(O) 0.150 0.151 0.173 0.152 0.149 0.151 0.155 0.135 0.114

! � � ! � � ! � �0.100 0.093 0.807 0.050 0.067 0.883 0.020 0.042 0.938


RMSE 0.036 0.030 0.031 0.032 0.030 0.031 0.027 0.027 0.028MME(U) 0.080 0.064 0.064 0.075 0.064 0.069 0.069 0.059 0.068MME(O) 0.095 0.096 0.101 0.092 0.096 0.097 0.086 0.094 0.091

Table 5. In-sample accuracy (%) error measures for GARCH-simulated samples

! � � ! � � ! � �0.100 0.294 0.606 0.050 0.211 0.739 0.020 0.134 0.846


RMSE 0.31 0.04 0.24 0.30 0.04 0.21 0.30 0.05 0.18MME(U) 1.24 0.80 2.88 1.21 0.81 2.67 1.32 0.80 2.35MME(O) 3.29 0.78 1.09 3.37 0.77 1.07 3.48 0.77 1.05

! � � ! � � ! � �0.100 0.218 0.682 0.050 0.156 0.794 0.020 0.099 0.881


RMSE 0.22 0.04 0.21 0.21 0.04 0.19 0.22 0.05 0.16MME(U) 1.15 0.82 2.83 1.12 0.83 2.65 1.27 0.84 2.30MME(O) 2.61 0.79 1.00 2.68 0.79 1.01 2.78 0.81 1.02

! � � ! � � ! � �0.100 0.093 0.807 0.050 0.067 0.883 0.020 0.042 0.938


RMSE 0.07 0.04 0.13 0.06 0.05 0.13 0.09 0.06 0.12MME(U) 0.98 0.87 2.16 0.81 0.89 2.06 1.10 0.94 1.95MME(O) 1.06 0.83 0.86 0.90 0.88 0.91 1.28 0.96 0.95


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becomes more persistent and less vulnerable toshocks.

The accuracy of EWMA seems independent of �for any given value of CV2, whereas we might expectbetter results when � ¼ 0:94. Since EWMA is arestricted version of GARCH, the accuracy ofGARCH dominates that of EWMA.

According to the RMSE, SV(MCL) holds a cleardominance when CV2 is high. As CV2 decreases,EWMA and GARCH start to catch up, withGARCH matching the accuracy of SV(MCL) whenCV2 is 0.1.

From MME(U) and MME(O) we see that, onaverage, EWMA and GARCH produce more over-estimations of volatility than underestimations.The opposite is true for SV(MCL). In terms of theMME(U) statistic, GARCH dominates SV(MCL) inall cases. Thus, if underestimation of volatility is ofgreater concern than overestimation, it is better to usea GARCHmodel even if the underlying process is SV.

From the summary of the accuracy of theout-of-sample predictions of this SV simulated dataset in Table 4, the following observations can be made.

Overall, the differences between the accuracy of thethree models are less marked than in-sample.The accuracy of out-of-sample forecasts improves asthe value of CV2 decreases for a constant �, similarly,accuracy improves as persistence (�) increases for aconstant CV2. The accuracy of EWMA behaves moreintuitively reasonably, i.e. greater accuracy for higher�. The greater accuracy of GARCH over EWMAnoticed in-sample does not persist out-of-sample.Considering RMSE, SV(MCL) produces more accu-rate forecasts for CV2

¼ 10; however for CV2<1,there is little difference between the accuracy of thethree models, i.e. little penalty for using ‘incorrect’GARCH or EWMA forecasts.

Considering the MME measures, all three methodstend to produce more overestimations than under-estimations. Using MME(U), GARCH tends to bemore accurate; using MME(O), SV(MCL) tends todominate.

In general, for this SV simulated data set, we seethat when the degree of stochasticity in the realvolatility is high, the true model SV(MCL) generatesmore accurate (using RMSE) forecasts. However, asthe degree of stochasticity (measured by CV2) falls,there is little difference in forecasting accuracybetween the three models.

Considering the case where the data are generatedby a GARCH process, we first consider the in-sampleaccuracy of the three models and then theout-of-sample accuracy. From Table 5, which sum-marizes the in-sample accuracy, we observe thefollowing.

The accuracy of EWMA and SV(MCL) improvesas the value of CV2 decreases (as � increases and� decreases) for constant �þ�. The accuracy ofGARCH remains virtually the same. As �þ�increases, with CV2 constant, SV(MCL) tends tobecome more accurate, whereas GARCH becomesless accurate. EWMA is most accurate for�þ �¼ 0.95, which is consistent with the chosenparameter value of 0.94.

The ‘correct’ model GARCH is the most accuratemodel for all accuracy measures and parametercombinations. The GARCH model has almost iden-tical MME(U) and MME(O) statistics, suggestingthat in addition to producing more accurate estimatesthan the other two models, GARCH also producesunbiased estimates. Although SV(MCL) is moreaccurate than EWMA for high CV2, this dominancedisappears as CV2 decreases.

The out-of-sample accuracy of the three modelsforecasting the volatility of the GARCH generateddata is summarized in Table 6. We note the following.

The accuracy of all three models improves as CV2

decreases for constant �þ � and for constant CV2 aspersistence (�þ �) increases. GARCH tends to be amore accurate forecast for most cases but only by asmall margin. For CV2

� 1, there is a small penaltyfor using the ‘incorrect’ SV(MCL) forecast. Thepenalty for using ‘incorrect’ EWMA forecasts isgreatest for the high CV2 and high persistence case.For all models, MME(O) is larger than MME(U)showing a general tendency to overestimate volatility.

From this experiment, the more important pointsare these. The only scenario where choosingSV(MCL) as a volatility forecasting model leads tonoticeably greater accuracy is where the underlyingvolatility model is SV and CV2 is high. As CV2

decreases (as volatility becomes less stochastic), thepenalty of incorrect model choice becomes lessserious.

IV. Estimation Using Real Data

We estimate the EWMA, GARCH and SV(MCL)models for 12 financial data sets. The data are chosento give a cross-section across the main financialmarkets with the objective of capturing examples thatcover as much as possible of the volatility scenariospace examined in the simulation experiment. Withineach market, we choose a small number of exampleswhere the market for asset is very liquid; thesecomprise three FX series, three equity indices, threeequities and three commodities. All the sets of dataare collected from 1 January 2001 to 29 December


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2006, giving 1564 observations for each return series.Returns are calculated by taking the difference oflogarithms of consecutive closing prices. The iden-tity of the assets and the descriptive statistics for thetime series of returns are given in Table 7.

The three commodities show a positive averagedaily return (in excess of 10% pa), Microsoftincreases and the Euro/US$ rate decreases by morethan 5% pa, and for the other series the averagereturns are close to zero. FX rates have lowervolatilities than equity indices, which in turn havelower volatilities than individual stocks. The volatilityof commodities varies, depending on the commodityin question. The volatility of gold is between that ofFX rates and of equity indices; the volatility of oilis similar to that of individual stocks while thevolatility of electricity is more than 10 times that ofindividual stocks. The Jarque–Bera (JB) statistics for

all return series indicate significant nonnormality.

The main cause of nonnormality is excess kurtosis

rather than skewness. The more skewed series, HSBC

and electricity are also the ones with the highest

excess kurtosis. We would expect the series with high

kurtosis to show relatively high values of CV2.The SV(MCL) and GARCH parameter estimates

are given in Table 8. In addition, model-based

estimates of CV2 and long-term volatility are given,

and they are calculated from these formulae:

Expected annualized GARCH volatility

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!

1� ��

r�ffiffiffiffiffiffiffiffi250p

Expected annualized SV

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiexp

�22 1� �2ð Þ

þ �

� �s�ffiffiffiffiffiffiffiffi250p

Table 8. Parameter estimates for SV and GARCH models of financial data

SV model GARCH model

� � � CV2Annualizedvolatility ! � � �þ � CV2

Annualizedvolatility

Euro/US$ 0.14 0.946 �1.01 0.09 10.0 0.005 0.016 0.970 0.986 0.03 9.4JPY/US$ 0.18 0.889 �1.04 0.06 9.8 0.030 0.018 0.897 0.916 0.01 9.4UK£/US$ 0.05 0.989 �1.45 0.09 7.9 0.004 0.023 0.963 0.986 0.05 8.1FTSE 100 0.15 0.988 �0.25 1.52 17.7 0.012 0.111 0.880 0.991 1.56 18.4Nikkei 225 0.11 0.988 0.52 0.31 22.9 0.016 0.077 0.918 0.995 0.37 28.4S&P 500 0.09 0.993 �0.14 0.83 17.4 0.007 0.057 0.937 0.993 0.81 15.7Advantest 0.08 0.994 2.01 0.43 50.1 0.021 0.042 0.956 0.998 0.41 50.4HSBC 0.17 0.986 0.22 1.25 22.9 0.008 0.070 0.929 0.999 1.32 52.1Microsoft 0.24 0.977 0.78 1.10 32.3 0.064 0.055 0.926 0.981 0.52 28.7Electricity 0.65 0.905 6.16 2.83 611.5 4.050 0.115 0.886 1.000 1.46 n/aGold 0.22 0.953 �0.32 0.58 15.4 0.027 0.042 0.929 0.972 0.23 15.3Oil 0.07 1.000 �0.83 0.24 47.6 0.278 0.066 0.878 0.944 0.21 35.3

Note: Using the returns from 2 January 2001 to 11 August 2006.

Table 7. Descriptive statistics of returns on actual financial data

Data Mean (%) Minimum (%) Maximum (%)Annualizedvolatility Skewness

Excesskurtosis JB statistic

FX Euro/US$ �0.0219 �1.88 2.35 9.4 0.23 0.65 42JPY/US$ 0.0026 �2.93 2.45 9.2 �0.29 1.66 202UK£/US$ �0.0172 �2.05 1.84 8.1 0.02 0.57 21

Indices FTSE 100 0.0000 �5.89 5.90 17.7 �0.22 4.17 1148Nikkei 225 0.0142 �6.86 7.22 21.5 �0.08 1.78 209S&P 500 0.0046 �5.05 5.57 16.7 0.16 3.06 617

Stocks Advantest 0.0155 �11.27 13.30 47.8 0.29 1.66 201HSBC �0.0036 �14.33 8.67 22.1 �0.50 11.27 8337Microsoft 0.0204 �12.08 10.54 29.8 0.06 5.13 1713

Commodities Electricity 0.1770 �266.38 287.75 580.5 0.45 10.89 7787Gold 0.0539 �5.54 5.95 15.5 �0.28 3.54 836Oil 0.0646 �16.97 10.98 35.6 �0.23 3.08 633

Note: All series are derived from daily price data from 1 January 2001 to 29 December 2006.


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From Table 8, we can see the similarity of theestimates of � and �þ � for each series. Both describethe persistence of the volatility process; the values areall close to unity indicating that the volatility processis close to a unit root. The largest discrepancybetween the estimated persistence is with electricity,where GARCH suggests a unit root, but SV(MCL)suggests lower persistence than the other seriesconsidered.

In most cases, the model-based annualizedlong-term volatilities are similar to each other andthat based on sample SDs shown in Table 7. Thisconfirms that both SV and GARCH models captureproperties of the data and produce sensible parameterestimates. Two exceptions are electricity and HSBC;for the former GARCH, estimation suggests that thevolatility process is integrated; for the latter GARCHestimates, a long-term volatility twice that ofSV(MCL) or that observed in Table 7. Note thatthese two exceptions exhibit high CV2 (for eithermodel). High values of excess kurtosis (Table 7) tendto be associated with high values of CV2 (theSpearman’s rank correlation coefficient is 0.84 for

SV(MCL), 0.74 for GARCH). Intuitively, fatter tailsimply more frequent extreme values, which will causelarger changes in conditional volatility. Note also thatwe cannot infer the level of volatility from the CV2 orvice versa, all combinations are possible. Forinstance, the FTSE 100 has a comparatively lowvolatility but high CV2; oil has a high volatility butlow CV2; electricity has both a high volatility andhigh CV2 and all three FX rates have both a lowvolatility and low CV2.

The SV(MCL) estimated CV2 is generally greaterthan the corresponding GARCH estimate (except forthe FTSE 100, the Nikkei 225 and HSBC), thereasons for this are discussed in Section ‘Simulationof time series with given CV2’. From the evidence ofthe underestimation of the theoretical CV2, we caninfer that the theoretical value for electricity could bearound 10. Using average estimates of persistence andaverage estimates of CV2 from both models, the datasets are plotted in Fig. 1. Note that the volatilityscenario space covered by this plot corresponds withthat used in the simulation experiment, confirmingthe experiment’s relevance across markets. Most

Fig. 1. Data sets plotted according to volatility persistence ðAverageð/, aþ bÞÞ and volatility of volatility (CV2)


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assets cluster to the right of the plot where persis-tence, � or �þ �, is high. This effect is emphasized bythe low CV2 and persistence of JPY/US$. Electricitywould be in the top right-hand corner using only theGARCH persistence measure, it is brought into themiddle of the figure by averaging with the lowSV(MCL) persistence measure. From the simulationexperiment, we know that the lower CV2 (CV2

� 1)region strongly favours the GARCH model: whensamples are generated by SV, GARCH estimationsare as good as SV(MCL); whereas when samples aregenerated by GARCH, GARCH estimations arebetter than SV(MCL). Since the majority of financialassets we consider exhibit lower CV2, we mayconjecture that, for these data sets, the GARCHmodel is a better choice than SV(MCL). However,when CV2 is high, as in electricity, GARCH andSV(MCL) models may lead to different results.

A forecasting exercise was carried out to test theseconjectures using out-of-sample data. The threeforecasting methods compared are GARCH,SV(MCL) and EWMA. Five different origins wereused for forecasting (the first origin, denoted Tm is11 August 2006); the last 100 observations of the dataset were withheld from the estimation set andvolatility forecasts prepared for up to 100 daysahead; the last 80 days were withheld and forecastsprepared for up to 80 days ahead; etc. This processwas repeated so that five sets of 1 month aheadvolatility forecasts were available, four sets of 2months ahead forecasts and so on. The accuracy ofthe forecasts is summarized by the measures usedpreviously, RMSE, MME(U) and MME(O) forhorizons of 1–5 months ahead. However, whenusing real data, an unambiguous means of observingdaily volatility is not available. In this case, theobserved volatility is the SD of returns over the20-day period and the estimated volatility is averagedover the month.

RMSE h months aheadð Þ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

6�h

X6�hm¼1

SD rtjt2A,Bð Þ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMean �̂2t jt2A,B

� �q� �2

vuutwhere A ¼ Tm þ 20 h� 1ð Þ þ 1 and B ¼ Tm þ 20h.The accuracy measures are given in Table 9. Lookingfirst at the symmetric measure of accuracy, RMSE,we see that for FX rates, equity indices and equities,EWMA provides the most accurate forecasts for allfive horizons and GARCH provides the second mostaccurate. For the commodities, GARCH providesthe most accurate forecasts for horizons of at least3 months. The magnitude of the RMSE is closelyrelated to the volatility of the return series

(Spearman’s rank correlation of 0.9). For the asym-metric measures of accuracy, the model rankingsaccording to MME(O) (which penalizes overestima-tion) are very similar to those for RMSE, apart fromelectricity and oil where EWMA dominates GARCH.For MME(U), EWMA is penalized for underestima-tion for the 1 month horizon for all 12 assets. For theFTSE 100, Advantest, Gold and Oil, SV(MCL) givesthe lowest MME(U) for horizons of 1 or 2 months. Itis only under this measure of accuracy that SV(MCL)forecasts are not least accurate.

In the region where an SV process was most likelyto be recognized, the high CV2 region inhabited byelectricity, FTSE 100 and HSBC, it was not detected.This indicates that the underlying volatility processesare closer to a GARCH process than SV. Overall, theconvincing domination of the SV(MCL) forecasts bythe GARCH forecasts supports the finding of thesimulation experiment that the use of GARCH is asafe course of action. The generally greater accuracyof the EWMA model suggests that this forecastingprocedure is more robust to departures fromGARCHness experienced in real data, but notcaptured in the simulation experiments.

V. Summary and Conclusions

The simulation experiment based on a range ofvolatility scenarios defined by persistence and vola-tility of volatility acted as a ‘laboratory’ representa-tion of scenarios encountered in real market data.Under these ‘laboratory’ conditions, the main con-clusion was that very little difference was foundbetween the forecasting accuracy of the three models,EWMA, GARCH and SV. The axes of the scenarioshad different effects on forecasting accuracy. Aspersistence increased, forecasting accuracy improvedfor all three methods. As volatility of volatilityincreased, forecasting accuracy deteriorated and thepenalty for using the ‘incorrect’ method became morenoticeable. In particular, the GARCH model is poorat forecasting volatility where the underlying processwas SV with high CV2.

Away from ‘laboratory’ conditions, using real datawhere the volatility process is unknown, the evidenceindicated that the data were better forecast by aGARCH process than by SV(MCL). In addition, inmost cases, the EWMA forecast was more accuratethan GARCH.

In summary, fitting GARCH or SV models to realdata undoubtedly increases one’s insight into thenature of the volatility process. However, if theobjective is to achieve forecasting accuracy, then


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forecasting volatility using EWMA is a reliable policythat will only come unstuck if the series exhibits

SV with a high volatility of volatility. The robust

structure of EWMA appears to contribute to itsgreater forecasting accuracy compared to GARCH.

References

Andersen, T. and Sorensen, B. (1996) GMM estimation ofa stochastic volatility model: a Monte Carlo study,Journal of Business and Economic Statistics, 14,326–52.

Bollerslev, T. (1986) Generalized autoregressive conditionalheteroscedasticity, Journal of Econometrics, 31,307–27.

Bollerslev, T., Chou, R. and Kroner, K. (1992) ARCHmodelling in finance: a review of the theoryand empirical evidence, Journal of Econometrics, 52,5–59.

Bollerslev, T., Engle, R. and Nelson, D. (1994) ARCHmodels, in The Handbook of Econometrics (Eds)R. Engle and D. McFadden, North-Holland,Amsterdam, pp. 2959–3038.

Brailsford, T. and Faff, R. (1996) An evaluation ofvolatility forecasting techniques, Journal of Bankingand Finance, 20, 419–38.

Chuang, I. Y., Lu, J.-R. and Lee, P.-H. (2007) Forecastingvolatility in the financial markets: a comparison of

Table 9. Measures of forecasting accuracy for SV and GARCH models of financial data

RMSE� 100 MME(O)� 100 MME(U)� 100

Horizon (months) 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

Euro/US$ GARCH 0.15 0.17 0.20 0.21 0.25 3.80 4.12 4.44 4.62 5.00 0.14 0.17 0.20 0.21 0.25SV 0.33 0.46 0.48 0.49 0.51 5.75 6.75 6.90 6.96 7.15 0.33 0.46 0.48 0.48 0.51EWMA 0.07 0.06 0.09 0.09 0.16 1.99 2.24 2.63 3.04 3.98 0.16 0.05 0.08 0.09 0.16

JPY/US$ GARCH 0.18 0.20 0.20 0.23 0.29 4.14 4.36 4.33 4.61 5.40 0.17 0.19 0.19 0.22 0.29SV 0.32 0.38 0.37 0.37 0.43 5.61 6.04 5.91 5.98 6.58 0.32 0.37 0.36 0.36 0.43EWMA 0.09 0.11 0.10 0.14 0.24 2.17 2.43 2.87 3.69 4.95 0.30 0.37 0.09 0.14 0.24

UK£/US$ GARCH 0.10 0.09 0.11 0.10 0.13 2.69 2.86 3.09 3.05 3.62 0.09 0.09 0.10 0.09 0.13SV 0.17 0.26 0.34 0.39 0.46 3.96 5.03 5.80 6.23 6.80 0.16 0.25 0.34 0.39 0.46EWMA 0.09 0.06 0.08 0.05 0.12 2.07 1.35 1.68 2.09 3.42 0.59 0.80 0.60 0.04 0.12

FTSE 100 GARCH 0.24 0.28 0.43 0.48 0.64 3.92 5.23 6.40 6.84 7.97 0.62 0.28 0.42 0.47 0.64SV 0.58 1.24 2.07 2.61 3.33 7.14 11.00 14.32 16.12 18.26 0.54 1.22 2.06 2.60 3.33EWMA 0.21 0.19 0.30 0.30 0.52 3.40 3.79 4.88 5.50 7.21 0.81 0.16 0.26 0.30 0.52

Nikkei 225 GARCH 0.34 0.42 0.64 0.68 1.00 4.49 6.29 7.67 8.13 9.98 0.59 0.40 0.61 0.67 1.00SV 0.70 1.37 2.09 2.49 3.00 8.01 11.67 14.40 15.75 17.31 0.66 1.36 2.08 2.48 3.00EWMA 0.30 0.30 0.51 0.50 0.82 4.06 5.23 6.49 6.92 9.05 0.93 0.28 0.46 0.48 0.82

S&P 500 GARCH 0.17 0.24 0.31 0.33 0.44 3.80 4.71 5.43 5.78 6.64 0.16 0.23 0.30 0.33 0.44SV 0.35 0.77 1.28 1.78 2.48 5.64 8.63 11.23 13.30 15.75 0.33 0.75 1.27 1.77 2.48EWMA 0.12 0.13 0.18 0.16 0.29 2.74 2.77 3.53 4.01 5.37 0.63 0.10 0.15 0.16 0.29

Advantest GARCH 0.53 0.62 0.53 0.80 1.33 4.99 4.62 5.26 8.14 11.54 1.55 2.20 0.37 0.71 1.33SV 0.97 1.95 3.01 4.71 6.99 8.48 13.39 17.30 21.71 26.44 0.83 1.85 3.00 4.71 6.99EWMA 0.47 0.56 0.40 0.52 1.05 4.06 4.27 2.80 5.92 10.26 2.21 3.28 2.72 0.42 1.05

HSBC GARCH 0.31 0.36 0.48 0.54 0.71 4.98 5.80 6.79 7.32 8.43 0.28 0.35 0.47 0.54 0.71SV 0.80 1.69 2.60 3.15 3.93 8.67 12.93 16.05 17.67 19.83 0.77 1.68 2.58 3.13 3.93EWMA 0.21 0.16 0.21 0.24 0.40 3.39 3.19 4.09 4.62 6.29 0.60 0.40 0.18 0.22 0.40

Microsoft GARCH 0.50 0.79 0.94 0.98 0.94 6.91 8.80 9.68 9.89 9.68 0.49 0.78 0.94 0.98 0.94SV 0.91 2.44 3.67 3.95 4.42 9.19 15.37 19.11 19.85 21.03 0.87 2.39 3.66 3.94 4.42EWMA 0.19 0.35 0.45 0.50 0.50 3.28 4.70 6.16 6.69 7.04 0.85 1.06 0.41 0.46 0.50

Electricity GARCH 9.80 8.58 5.94 5.98 10.34 16.21 17.87 14.75 15.58 32.15 11.62 15.43 11.45 12.83 10.34SV 25.81 28.32 27.29 30.16 28.85 50.36 52.44 51.76 54.78 53.71 25.51 27.78 26.96 30.06 28.85EWMA 10.90 8.78 6.81 6.02 7.59 14.91 13.96 13.42 15.41 27.55 12.01 16.39 19.30 15.13 7.59

Gold GARCH 0.37 0.42 0.31 0.30 0.16 4.02 4.29 5.30 5.15 3.98 1.62 2.03 0.29 0.28 0.16

SV 0.70 0.96 0.98 0.85 0.71 6.67 7.69 9.73 9.17 8.43 0.75 0.92 0.96 0.84 0.71EWMA 0.47 0.58 0.77 0.84 0.86 5.11 6.01 8.78 8.91 9.25 1.40 0.47 0.77 0.81 0.86

Oil GARCH 0.53 0.50 0.57 0.64 0.91 4.98 4.88 6.39 6.78 9.52 1.76 1.84 0.47 0.53 0.91SV 0.88 1.54 2.88 3.16 6.75 7.60 11.48 15.80 17.38 25.99 1.60 1.40 2.63 3.07 6.75EWMA 0.56 0.57 0.66 0.37 0.34 3.94 3.66 3.62 3.14 5.81 2.79 3.72 3.97 3.31 0.34

Notes: The volatility forecasts are for 20-day periods, ‘months’ up to 5 months ahead (Tþ 1,Tþ 20), (Tþ 21,Tþ 40),(Tþ 41,Tþ 60), (Tþ 61,Tþ 80), (Tþ 81,Tþ 100). The forecast origins, T, are: 14 August 2006, 11 September 2006,9 October 2006, 6 November 2006 and 4 December 2006. In each comparison, the lowest (most accurate) error measure isshown in bold type; the second most accurate is shown in italics.


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alternative distributional assumptions, AppliedFinancial Economics, 17, 1051–60.

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