Forecasting accuracy of stochastic volatility, GARCH and EWMA models under different volatility scenarios

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<ul><li><p>This article was downloaded by: [University of North Texas]On: 26 November 2014, At: 17:25Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK</p><p>Applied Financial EconomicsPublication details, including instructions for authors and subscription information:</p><p>Forecasting accuracy of stochastic volatility, GARCHand EWMA models under different volatility scenariosJie Ding a &amp; Nigel Meade aa Imperial College London, Tanaka Business School, South Kensington , London SW7 2AZ, UKPublished online: 17 May 2010.</p><p>To cite this article: Jie Ding &amp; 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</p><p>To link to this article:</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. 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Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms &amp; Conditions of access and use can be found at</p><p></p></li><li><p>Applied Financial Economics, 2010, 20, 771783</p><p>Forecasting accuracy of stochastic</p><p>volatility, GARCH and EWMA</p><p>models under different volatility</p><p>scenarios</p><p>Jie Ding and Nigel Meade*</p><p>Imperial College London, Tanaka Business School, South Kensington,</p><p>London SW7 2AZ, UK</p><p>The forecasting of the volatility of asset returns is a prerequisite for many</p><p>risk management tasks in finance. The objective here is to identify the</p><p>volatility scenarios that favour either Generalized Autoregressive</p><p>Conditional Heteroscedasticity (GARCH) or Stochastic Volatility (SV)</p><p>models. Scenarios are defined by the persistence of volatility (its robustness</p><p>to shocks) and the volatility of volatility. A simulation experiment</p><p>generates return series using both volatility models for a range of volatility</p><p>scenarios representative of that observed in real assets. Forecasts are</p><p>generated from SV, GARCH and Exponentially Weighted Moving</p><p>Average (EWMA) volatility models. SV model forecasts are only</p><p>noticeably more accurate than GARCH in scenarios with very high</p><p>volatility of volatility and a stochastic volatility generating process. For</p><p>scenarios with medium volatility of volatility, there is little penalty for</p><p>using EWMA regardless of the volatility generating process. A set of</p><p>return time series selected from FX rates, equity indices, equities and</p><p>commodities is used to validate the simulation-based results. Broadly</p><p>speaking, the real series come from the medium volatility of volatility</p><p>scenarios where EWMA forecasts are reliably accurate. The robust</p><p>structure of EWMA appears to contribute to its greater forecasting</p><p>accuracy than more flexible GARCH model.</p><p>I. Introduction</p><p>The forecasting of the volatility of asset returns is</p><p>required for many risk management tasks in finance.</p><p>The use of Value-at-Risk (VaR) is ubiquitous in</p><p>financial institutions and is increasingly widespreadas a risk management tool in corporate institutions.</p><p>Future volatility is the crucial input to VaR calcula-</p><p>tions; an overestimate of volatility leads to an</p><p>opportunity loss due to capital being tied up unne-</p><p>cessarily and an underestimate of volatility leads to</p><p>risks being under protected. Risk control strategies,</p><p>such as delta hedging, are reliant on the estimates offuture volatility for the creation of riskless portfoliosof options and the underlying asset. Mean-variance</p><p>portfolio selection relies on the estimates of futurevolatility. Poon and Granger (2003) review the</p><p>extensive literature on forecasting the volatility offinancial markets.</p><p>Our contribution to this literature is to explorethe comparative effectiveness of the main two</p><p>volatility modelling methodologies, Generalized</p><p>*Corresponding author. E-mail:</p><p>Applied Financial Economics ISSN 09603107 print/ISSN 14664305 online 2010 Taylor &amp; Francis 771</p><p>DOI: 10.1080/09603101003636188</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f N</p><p>orth</p><p> Tex</p><p>as] </p><p>at 1</p><p>7:25</p><p> 26 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>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.</p><p>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.</p><p>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.</p><p>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.</p><p>II. Volatility Models</p><p>A stylized fact of time series of returns on financialassets is the clustering behaviour of volatility.</p><p>Two modelling approaches have been used to capture</p><p>this behaviour. The GARCH model represents con-</p><p>ditional variance as a function of lagged squared</p><p>residuals and lagged conditional variance. The sto-</p><p>chastic variance model (as implied by its name)</p><p>assumes that the variance follows a stochastic pro-</p><p>cess. Both approaches will be described below. Note</p><p>that we will focus on the basic formulation of each</p><p>model, we wish to facilitate comparisons between the</p><p>modelling approaches rather than be distracted by the</p><p>differences of models within each approach. Our</p><p>notation is rt, which is the log-return in period t, this the estimated volatility in period t h, given dataup to period t.</p><p>Generalised autoregressive conditionalheteroscedasticity models</p><p>The GARCH model was proposed by Bollerslev</p><p>(1986), generalizing Engles (1982) Autoregressive</p><p>Conditional Heteroscedasticity (ARCH) model. For</p><p>surveys of the extensive literature on these models, see</p><p>Bollerslev et al. (1992, 1994), and Li et al. (2002). The</p><p>GARCH( p, q) model is defined as follows:</p><p>rt "t 1</p><p>where "t ztt and E(zt) 0 and V(zt) 1. Thevariance, 2t , obeys this process</p><p>2t !Pqi1</p><p>i"2ti </p><p>Ppj1</p><p>j2tj 2</p><p>such that !4 0, i 0, j 0,P</p><p>iP</p><p>j 5 1.Although there is no consensus about the ideal</p><p>GARCH specification (see, e.g. Brailsford and Faff,</p><p>1996), there is little doubt that GARCH models</p><p>capture the most important stylized facts describing</p><p>the asset return volatility; excess kurtosis and vola-</p><p>tility clustering. For our purposes, we will use the</p><p>above classic GARCH model with p q 1. Therange of possible models has been investigated by</p><p>several authors; for example, McMillan et al. (2000)</p><p>compare the forecast accuracy of a variety of</p><p>GARCH formulations; Chuang et al. (2007) investi-</p><p>gate the effect on the forecasting accuracy of</p><p>GARCH models using different density</p><p>functions for zt.A feature of the GARCH model is that forecast</p><p>volatility reverts to an equilibrium level. For the</p><p>widely used GARCH(1,1), the one-step variance</p><p>forecast is</p><p>2t1 ! "2t 2t 3</p><p>772 J. Ding and N. Meade</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f N</p><p>orth</p><p> Tex</p><p>as] </p><p>at 1</p><p>7:25</p><p> 26 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>by successive substitution, the h-step forecast can bederived as</p><p>2th !Xh2i0</p><p> i h12t1 4</p><p>We can see that as h gets larger, the forecast tendstowards its long-term mean of != 1 . 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 ( ) approachesunity, the long-run variance approaches nonstatio-narity. For this reason, we use ( ) as a measureof the persistence of the GARCH volatility process.</p><p>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 n 1 time periods, the EWMA estimate is</p><p>2th 1 </p><p>1 n1Xni0</p><p>ir2ti 5</p><p>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).</p><p>Stochastic volatility models</p><p>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</p><p>rt e0:5htt where tN 0, 1 ht ht1 t where tN</p><p>0, 2</p><p>and E tt 6</p><p>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</p><p>ln r2t ln 2 ht "t where "t ln 2t </p><p> ln 21 </p><p>ht ht1 t t N0, 2</p><p>but E "tt 0:</p><p>7</p><p>After the log transformation, the information</p><p>regarding the correlation coefficient is lost(Harvey et al., 1994). However, this can be recovered</p><p>by conditioning on the signs of the original returns</p><p>(Harvey and Shephard, 1996).This model captures some financial time series</p><p>properties, such as volatility clustering and high</p><p>persistence. Ghysels et al. (1996) assert that the SV</p><p>model describes financial time series better than the</p><p>ARCH models because the extra noise process tmakes it more flexible. The price of this flexibility is</p><p>that, it is no longer possible to specify an analytic</p><p>expression for the Log-Likelihood Function (LLF).</p><p>To date, methods used to estimate the SV model are</p><p>computationally expensive and, in some cases, the</p><p>theoretical properties of the estimators are still</p><p>unknown. Examples include: simulated maximum</p><p>likelihood suggested by Danielsson (1994); a Markov</p><p>Chain Monte Carlo (MCMC) approach suggested</p><p>by Jacquier et al. (1994); a Generalized Method of</p><p>Moments (GMM) approach used by Melino and</p><p>Turnbull (1990) and Andersen and Sorensen (1996).</p><p>Ruiz (1994) shows that for the kind of parameter</p><p>values likely to arise in practice, GMM is less efficient</p><p>than Quasi-Maximum Likelihood (QML) together</p><p>with the Kalman filter. The Kalman filter QML</p><p>procedure for SV model estimation can also be found</p><p>in Harvey et al. (1994), Heynen and Kat (1994), Yu</p><p>(2002) and Javaheri et al. (2003). Finally, there is the</p><p>empirical characteristic function method of Knight</p><p>et al. (2002). Of these methods, QML stands out due</p><p>to its consistency and relative ease of implementation.</p><p>Sandmann and Koopman (1998) propose a Monte</p><p>Carlo Likelihood (MCL) approach, which is a</p><p>significant improvement over QML. MCL uses the</p><p>idea of importance sampling to eliminate the need to</p><p>approximate a log2 distribution with a normal oneas under QML.</p><p>Compared to other volatility forecasting techni-</p><p>ques, the SV model is favoured by Heynen (1998) for</p><p>stock indices in the US, UK, Hong Kong, Japan and</p><p>Australia, and by Yu (2002) for the New Zealand</p><p>stock market. It is also preferred by Heynen and Kat</p><p>(1994) for stock indices, but less so compared with</p><p>GARCH(1,1) for currencies. Dunis et al. (2001)</p><p>compare GARCH, the SV model and other models</p><p>using a group of six FX rates. They find that</p><p>GARCH is more accurate (using Root Mean</p><p>Squared Error (RMSE)) than SV for five out of the</p><p>six rates over a 1-month horizon and for four out of</p><p>the six rates for a 3-month horizon. Sadorsky (2005)</p><p>uses a range-based SV model to forecast volatility</p><p>1 day ahead. For a variety of assets, an equity index,</p><p>crude oil, bonds and FX, he found that SV forecasts</p><p>Forecasting accuracy under different volatility scenarios 773</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f N</p><p>orth</p><p> Tex</p><p>as] </p><p>at 1</p><p>7:25</p><p> 26 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>were less accurate than EWMA and other naivemodels.</p><p>We refer to our implementation of MCL estima-tion of the SV model as SV(MCL). The full details ofthis c...</p></li></ul>