Forecasting Daily Volatility Using Range-based Data

Transcription

1 Forecasing Daily Volailiy Using Range-based Daa Yuanfang Wang and Mahew C. Robers* Seleced Paper prepared for presenaion a he American Agriculural Economics Associaion Annual Meeing, Denver, Colorado, Augus -4, 004 Correspondence Auhor: Yuanfang Wang The Ohio Sae Universiy 47 Agriculural Adminisraion Building 0 Fyffe Road Columbus, Ohio 430 Tel: (64) wang.586@osu.edu Copyrigh 004 by Yuanfang Wang and Mahew C. Robers. All righs reserved. Reader may make verbaim copies of his documen for non-commercial purposes by any means, provided ha his copyrigh noice appears on all such copies. *Graduae suden and Assisan Professor, Deparmen of Agriculural, Environmenal and Developmen Economics, The Ohio Sae Universiy, Columbus, OH 430. Fuure revisions of his paper will be available a hp://aede.osu.edu/people/robers.68/research

2 Forecasing Daily Volailiy Using Range-based Daa Absrac Users of agriculural markes frequenly need o esablish accurae represenaions of expeced fuure volailiy. The fac ha range-based volailiy esimaors are highly efficien has been acknowledged in he lieraure. However, i is no clear wheher using range-based daa leads o beer risk managemen decisions. This paper compares he performance of GARCH models, range-based GARCH models, and log-range based ARMA models in erms of heir forecasing abiliies. The realized volailiy will be used as he forecasing evaluaion crieria. The conclusion helps esablish an efficien forecasing framework for volailiy models. Keywords: range-based esimaor, log range, GARCH models, ARMA models, forecas

3 . Inroducion Users of agriculural markes always need o esablish accurae represenaions of expeced fuure volailiy. For example, fuure volailiy is he main ingredien is calculaing expeced daily opimal hedge raios. The applicaion of misspecified fuure volailiy has he poenial o induce inappropriae or even serious assessmen of asse risk and porfolio selecion. Thus, no surprisingly, seeking good volailiy forecass of agriculural marke volailiy has drawn increased aenion from financial academics and praciioners. On he one hand, he exisence of volailiy clusering a differen frequencies has been exensively documened in he finance lieraure. This high degree of volailiy persisence suggess ha financial marke volailiy is predicable. On he oher hand, forecasing he fuure level of volailiy is challenging for several reasons. For example, volailiy is no direcly observable; herefore he choice of evaluaion meric for forecasing performance is uncerain. Esablishing an appropriae framework for volailiy forecasing is an imporan heme for financial academics and is of grea relevance o praciioners. Numerous papers have employed ARCH (GARCH) models for forecasing. The ARCH family of models is specifically designed o model volailiy clusering effecs, and is use in forecasing is quie common. However, he forecasing performance of he ARCH family is raher conroversial. Conrary o sudies seeking new model specificaions, Andersen and Bollerslev (998) argue ha ARCH models provide good ou-of-sample forecass. However, ex-pos volailiy measures may no provide correc 3

4 appraisal of performances of volailiy models. Thus, esablishing more efficien volailiy measures is useful o evaluae he forecasing abiliy of exising ime series models. They reveal ha he coefficien of muliple deerminaion, R, is low when he daily squared reurns are used as a measure of ex-pos volailiy. They show ha realized volailiy, he sum of inraday squared reurns, is a much more efficien volailiy proxy. The fac ha range-based volailiy esimaors are highly efficien has been acknowledged by many auhors (For example, Parkinson (980), Garman and Klass (980), Beckers (983)). Their findings raise he quesion as o wheher he forecasing abiliy of ARCH models can be improved wih range-based daa. However, hese earlier works only focus on consrucing efficien volailiy esimaors and lile aenion is paid o he applicaion of hese esimaors. I was no unil Alizadeh, Brand and Diebold (00) ha he usefulness of a simple volailiy proxy, he log range, was formally esablished and applied o ime series models. They clarify ha he log range, defined as he log of he difference beween he high and low log prices during he day, is nearly Gaussian, robus o microsrucure noise and much less noisy han alernaive volailiy measures such as log absolue or squared reurns. Compared wih earlier sudies, heir work fully explois he disribuional properies of he log range esimaors and hus provides a heoreical underpinning for using he Gaussian ARMA class of models. Previous findings relaed o range-based volailiy esimaor are essenially saisical. I is no clear wheher using range-based daa leads o beer invesmen managemen decisions (e.g., more accurae opimal hedge raios). Furhermore, end-users of agriculural markes may no see addiional benefis from range-based models when compared o exan simple volailiy forecas framework, e.g., he Riskmerics TM model. 4

5 Informaion coss may make range-based GARCH models undesirable. If an ARMA model of he log range has compeiive forecasing abiliy, i will hold promise for pracical applicaions of range esimaors in agriculural asse pricing and risk managemen applicaions. Moivaed boh by he appeal of range-based models and by he pracical need o check heir forecasing abiliy in agriculural commodiy fuures marke, his sudy will invesigae wheher ARCH models exended wih he range daa provide beer ou of sample forecass of daily volailiy and wheher a simple ARMA model of he log range has compeiive forecasing abiliy. More imporanly, his sudy will check he forecasing performance under differen crieria using realized volailiy. The remainder of his paper is organized as follows. Secion presens he ime series models used in his sudy. Secion 3 summarizes he daa and he in sample fi of he models. Secion 4 presens he forecasing resuls and compare resuls under differen crieria. Secion 5 concludes.. Time Series Models. GARCH Models There are wo major caegories of ime-varying volailiy models, he ARCH family and he sochasic volailiy (SV) family. The auoregressive condiional heeroskedasiciy (ARCH) model was inroduced by Engle (98). Compared wih previous economeric models, ARCH processes are specifically designed o model and forecas condiional variances. This propery of he ARCH model makes i appealing for modeling he volailiy of economic ime series. Bollerslev (986) proposed an exension of he condiional variance funcion and inroduced he generalized ARCH (GARCH) 5

6 model. For many applicaions, he GARCH (,) model has been proved o be a parsimonious represenaion ha fis daa well. The represenaion of he GARCH (,) is y = c + ε () ε = σ η σ = ω + αε + βσ where η is a mean-zero, uni-variance, i.i.d. random variable. Engle and Kraf (983) were he firs o consider he effec of ARCH (GARCH) on forecasing. Akgiray (989) was he firs o apply he GARCH model o forecas volailiy. Afer ha, numerous papers have employed his mehod. However, he forecasing performance of he ARCH family is disappoining in many sudies. Differen conclusions have been drawn for differen sample periods and differen speculaive markes. Andersen and Bollerslev (998) provide a few insighs ino hese resuls. They find ha ARCH family performs beer if he ex-pos volailiy is esimaed by he sum of inraday squared reurns.. GARCH Models Exended wih Addiional Informaion Andersen and Bollerslev (998) demonsrae ha he daily squared reurn is a very noisy esimaor. If he previous rading day is quie volaile, bu he closing price happens o be he same as he opening price, he lagged daily squared reurn would be zero. Thus by exending he daily GARCH model wih informaion relaed o he real volailiy dynamics, he new model will provide a reasonable explanaion ha he previous day was volaile. The condiional variance equaion may be exended o allow for he inclusion of addiional regressors. For example, Bessembinder and Seguin (993) include daily 6

7 volume. Laux and Ng (993) include he number of price changes. Taylor and Xu (997) and Marens (00) use inraday reurns. Marens (00) use daily range. The specificaion of he variance equaion for he exended GARCH models is, σ () = ω + αε + βσ + ζi where I presens any rade relaed variables such as he raded volume, he sum of squared inraday reurns or he daily range..3 Simple Regression Model The Sochasic Volailiy (SV) models are more flexible han he ARCH family in he sense ha volailiy is driven by a noise erm which may or may no be relaed o he reurns process (Poon and Granger, 00). However, since SV models involve an unobservable, sochasic variance process, his precludes closed-form likelihood funcions; in urn esimaion of SV models is quie difficul. The idea of using he Simple Regression (SR) model insead of he SV model o forecas volailiy comes from Poon and Granger (00). They sugges ha One way o avoid his [SV] esimaion problem is o abandon he srucure of he mean and express he volailiy simply as a funcion of is pas values. This is known as he Simple Regression (SR) mehod. The SR mehod is principally auoregressive. If pas volailiy errors are included, one ges he ARMA model for volailiy. I would be ineresing o explore wheher alernaive volailiy proxies, such as he log range and squared inraday reurns, fi he class of SR models. The logic is ha if an esimaor is highly efficien, i is possible o exrac valuable informaion abou he fuure Page 8. 7

8 value of volailiy by jus using simple echnique. Given he findings described by Alizadeh, Brand and Diebold (00), i is naural o assume ha he log-range process falls wihin he Gaussian ARMA models. If rue, his will grealy reduce he compuaional coss. Sandard forecasing echniques may be applied o generae predicions of fuure log range. Through simple ransformaions, he forecass of volailiy can be obained. Specifically, he SR model for he range daa is, R = α R + α R v (3) where v ~ iid (0,) ; R denoes he log range. 3. Daa and In-Sample Fi 3. Daa Descripion The daa se consiss of Chicago Board of Trade (CBOT) soybean fuures inraday ransacion prices and daily prices. The daily daa were obained from he CRB/Bridge Fuures Daabase. The sample consiss of daily soybean fuures high/low/closing prices from January, 985 o July 3, 00. The inraday daa are ime and sales ransacion prices, which were obained from he Fuures Indusry Insiue. The full sample covers he period January, 990 o July 3, 00. The firs,64 rading days (January, 985 December 9, 989) are used o esimae he parameers of he various models. The nex,909 rading days (for which inraday daa are available) are used o es he ou-of-sample forecasing performance. In calculaing he reurns series, he nearby conracs are used o consruc he coninuous reurns series. However, reurns are calculaed from he second nearby conrac when he nearby conrac is in he delivery monh. This swich guaranees ha 8

9 reurns are nearly always calculaed from he prices of he conrac ha has he highes rading volume. Figure plos he prices of he fuures daa. Figure plos he reurns series acually used in his sudy, which is 00 ln( P / P ) of he fuures daa. Table repors summary saisics for daily reurns. Soybean fuures reurns conform o several sylized facs which have been exensively documened for financial variables. The disribuion of he reurns is almos symmeric and has fa ails and a subsanial peak a zero. Excess kurosis of he series indicaes ha he disribuion of daily reurns is far from Gaussian. The auocorrelaions of reurns are close o zero. The Q-saisics are smaller han he criical values a 5% level. In conras, he squared reurns are significanly auocorrelaed. Figure reflecs anoher sylized fac, he clusering effec. Variances of reurns change over ime and large (small) changes end o be followed by large (small) changes. In his sudy, he daily range is defined as, Range Max h, c ) Min( l, c ) (4) = ( where h and l denoe highes and lowes prices on day respecively and c represens he closing price on day. Since he curren soybean daily daa only cover he full floor rading from 9:30 a.m. o :5 p.m., equaion (4) capures informaion abou he overnigh marke aciviy. While some of he markes previously sudied in he lieraure do have rading limis in place, such as he S&P 500, hey are much less frequenly invoked han in physical commodiy markes. When he equilibrium price moves beyond he rading limis, rading ceases. Since no rades are recorded during hese moves, equaion (4) provides a reasonable range proxy for limi move days. Following Alizadeh, Brand and Diebold (00), he daily log range is defined as, 9

10 R = log[log( Max( h, c )) log( Min( l, c ))] (5) The volailiy lieraure primarily uses absolue or squared reurns as volailiy proxies. To jusify he superior efficiency of he log range, Table presens descripive saisics for log absolue reurns and he log range. Firsly, he log range is preferable in erms of is smaller sandard deviaion. Secondly, he skewness and kurosis of he log range are and.93, respecively. These values are closer o he corresponding values of 0 and 3 for a normal random variable compared wih hose for log absolue reurns. This conclusion is confirmed by checking he Jarque-Bera saisic. I is more obvious by looking a Figure 3, which shows he quanile-quanile (Q-Q) plo. The Q-Q plo for he log range falls nearly on a sraigh line and indicaes ha he log range has a disribuion close o Normal. In conras, he Q-Q plo of he log absolue reurns curves downward a he lef end and upward a he righ. Finally, he log range proxy is superior in erms of is ime series dynamics. The large and slowly decaying auocorrelaions of he log range clearly manifes srong volailiy persisence. The erraic flucuaion of log absolue reurns masks he volailiy persisence. 3. In-Sample Fi 3.. Esimaion of GARCH Models Maximum likelihood esimaion of he GARCH model is easy o implemen once he densiy funcion of ε is specified. If he residuals are no condiionally normally disribued, quasi-maximum likelihood (QML) esimaor will sill be consisen provided ha he mean and variance funcions are correcly specified. 0

11 The seasonal effecs of price volailiy are widely documened in many surveys. In ime series modeling, one can ake care of seasonaliy firs and fi a model wih he deseasonalized daa. Or a model can be esimaed for seasonally unadjused daa by adding a seasonal componen in he model. This sudy follows he second approach. Robers (00) models he seasonal effecs in volailiy by including a Fourier expansion for he inercep of he GARCH volailiy equaion. The specificaion of he GARCH model is hus of he form, 00 ln( P / ) = µ + ε (6) P ε = σ η σ = ω + αε + βσ M ω = κ + φm sin(mπτ ) + ψ m cos(mπτ ) m= 0 τ where τ denoes he ime of year of he observaion. Esimaion resuls (based on daily daa) for GARCH (,) models are given in Table 3. All of he specificaions capure well he auocorrelaion in he volailiy of reurns. For he GARCH (,) model wihou seasonaliy, he esimaes of parameers α and β are highly significan. The persisence in volailiy is quie large, wih α + β larger han The Ljung-Box pormaneau es saisic for up o enh order serial correlaion in he sandardized residuals η akes he value Q (0) = 8.094, which is no significan for he χ 0 disribuion. However, he Q-ess sugges ha here exiss serial dependence in he residuals squares a lag 5 and lag 0. Furhermore, he GARCH (,) model is considered o be a parsimonious represenaion, since resuls no repored here show ha higher orders have nohing exra o offer.

12 uncondiional sample kurosis for he residuals is 4.04, which exceeds he normal value of hree. And he residuals coninue o display asymmery. The second se of resuls in Table 3 include a firs order seasonal expansion. Only one of he wo seasonal parameers is significan a 5% level. However, he LR es saisic equals 9.04, which is significan a.5% level in he corresponding asympoic χ disribuion. The addiion of wo parameers is also preferred from an AIC (Akaike informaion crieria) perspecive. Moreover, he inclusion of a Fourier series reduces he sample kurosis in residuals. For he second order seasonaliy, only wo seasonal parameers are significan a 5% level alhough he LR es and AIC prefer he inclusion of wo addiional parameers. Including a hird order seasonaliy is rejeced no only from a LR es perspecive bu also from a -es perspecive. Noe, he esimaed value for α decreases as more parameers are added ino he variance equaion. This indicaes ha he reliance of he condiional volailiy on he previous dae is reduced. From he above resuls, a firs order seasonaliy model represens a reasonable radeoff beween he need of model fi and he need of parsimony. Also he GARCH (,) model wih no seasonaliy is used as he reference basis for forecas analysis. 3.. GARCH Models Exended wih Daily Range In his secion, he daily soybean daa are fied o he GARCH models exended wih daily range. The inercep of he variance equaion is defined as,

13 M ω = κ + ζi + φm sin(mπτ ) + ψ m cos(mπτ ) (7) m= where I = Max ( h, c ) Min( l, c ). Table 4 presens he esimaion resuls. Firs, he log-likelihood values of hree GARCH-I models are greaer han hose of he corresponding GARCH models in secion 4... This resul suggess ha range daa improve in sample model fiing and reflecs he greaer precision of he range as a volailiy proxy. Second, GARCH-I models are also desirable from he AIC and SC (Schwarz crieria) perspecives since boh values of AIC and SC fall as he daily range is added in. Third, he esimaes of α become no significan a % level or even a 5% level for second order and hird order seasonaliy. In conras, he esimaes of range parameerζ are highly significan. This resul is foreseeable sinceε and I are compeing facors o presen las period s variance and he inclusion of range daa reduces he proporions ofε in accouning for las period s volailiy. This resul also confirms he fac ha daily range is a relaively less noisy volailiy proxy han daily squared reurns. Omiing range, he esimaion resuls for he GARCH-I models are quie similar o he resuls for he GARCH models in erms of seasonaliy. AIC, SC and LR es all sugges ha GARCH-I (,) wih hird order seasonaliy is readily rejeced. The addiion of one order of seasonaliy performs beer han he second order seasonaliy model in erms of LR es. This conclusion is confirmed by he esimaes of second order seasonaliy model. φ is significan a 5% level and ψ is no significan judged by he sandard errors. Addiionally, he values of AIC are close for boh models. Finally, he skewness and kurosis coefficiens of he sandardized residuals for hree models do no 3

14 provide much informaion abou model selecion. The p-values for Q-es are also close and ell a similar sory for four models. All in all, he firs order seasonaliy model works bes for GARCH-I models. However, in order o avoid he possible over-fiing problem in forecasing, GARCH-I model wihou seasonaliy is also included as one of he forecasing frameworks. 4. Ou-of-Sample Daily Volailiy Forecass 4. Forecas Evaluaion Crieria I is difficul o compare forecasing performance of compeing models since here is a variey of evaluaion crieria used in he lieraure. Saisical analysis is one of he evaluaion measures frequenly used. Poon and Granger (00) sugges ha uiliy-based economic crieria are cosly o apply and saisical analysis provides a pracical way for forecas evaluaion. Wes and Cho (995) consider alernaive saisical measures. Basically, saisical measures evaluae he difference beween forecass a ime and realized values a ime + k. However, asse price volailiy is no direcly observable and measuring he realized values of volailiy is challenging. Much effor has been devoed o exracing volailiy from oher observable marke aciviies. The daily squared reurn has been widely used in he lieraure as ex-pos volailiy. However, Andersen and Bollerslev (998) show ha i is a very noisy volailiy esimaor and does no provide reliable inferences regarding he underlying laen volailiy in daily samples. They inroduce a new volailiy measure, ermed realized volailiy. Realized volailiy esimaes volailiy by summing squared inraday reurns. Volailiy esimaes so consruced are close o he underlying inegraed volailiy. Thus, he volailiy of a price 4

15 process can be reaed as an observable process. In his sudy, realized volailiy is calculaed based on 5-minue reurn series. For he performance evaluaion, wo forecas evaluaion crieria, Roo Mean Square Error (RMSE) and Mean Absolue Error (MAE), are defined by, RMSE = T T = ( σ σ ) (8) rv, where T denoes he forecas horizon. σ denoes one sep ahead daily forecas and σ rv, denoes realized volailiy. MAE = T T σ σ rv, = (9) In order o accoun for he heeroskedasiciy, wo alernaive measures, he heeroskedasiciy adjused roo mean squared error (HRMSE) and mean absolue error (HMAE) are included. These wo measures are compued as follows, HRMSE = T rv, = σ ( T σ ) (0) HMAE = T σ T rv, = σ () The second meric used o evaluae daily volailiy forecass is he regression-based mehod. The coefficien of deerminaion ( R ) of he regression of realized volailiy on forecased volailiy resuls from, rv, = ϕ0 + ϕ σ σ + v () 5

16 4. Resuls Table 5 repors he ou of sample forecass based on evaluaion crieria RMSE, MAE, HRMSE, HMAE and R. The forecass are based on parameer esimaes from rolling samples wih fixed sample size of 64 days. A number of conclusions may be drawn. Firs, he GARCH (,) model is inferior o oher hree models: he firs order seasonaliy GARCH (,) model, he GARCH (,) model exended wih range daa, and he firs order seasonaliy GARCH model exended wih daily range. The daily GARCH (,) model has he smalles regression R and highes values for RMSE, MAE, HRMSE and HMAE. Second, he regression based mehod and summary saisics boh sugges ha GARCH (,) models exended wih he difference beween daily high and low are beer han he use of GARCH models ignoring daily range. Third, including seasonaliy improves he ou of sample forecass of he daily GARCH (,) model. The coefficien of deerminaion R, increases from for he GARCH (,) model o 0.05 for he firs order seasonaliy GARCH (,) model. Resuls are qualiaively consisen across four differen saisical measures. Fourh, ineresingly, he firs order seasonaliy GARCH model exended wih daily range is no he bes model based on R, MAE and HMAE. The firs order seasonaliy GARCH model exended wih daily range has an R 0.33, whereas he GARCH (,) model exended wih daily range has an R The MAE is 0.65 for he GARCH (,) model exended wih daily range, whereas i is for he firs order seasonaliy GARCH model exended wih daily range. Similarly, he HMAE drops from o 0.47 when ignoring seasonaliy. The use of Fourier series does no lead o a superior forecasing performance for he exended GARCH (,) models. 6

17 5. Conclusion Previous sudies reveal ha range-based volailiy esimaor is highly efficien. However, lile aenion is paid o he applicaion of hese esimaors. This paper compares he performance of GARCH models, range based GARCH models, and logrange based ARMA models in erms of heir forecasing abiliies. The empirical analysis so far makes he following poins: For forecasing soybean fuures marke volailiy i is imporan o include he daily range, defined as he difference beween daily high and low. For he exended GARCH models, he adding of seasonaliy become less imporan, bu i sill improves forecass resuls in erms of RMSE and HRMSE. 7

18 Mean Sandard Deviaion Skewness Kurosis Minimum Maximum Lags Lag Lag Lag 5 Lag 0 Lag 5 Q-Tes Resuls Reurns Q-Saisics P-Value 0.33 (0.565) (0.844) (0.33).306 (0.65) 9.86 (0.77) Squared Reurns Q-Saisics P-Value 77.7 (0.000) 9.53 (0.000) (0.000) 03.0 (0.000) (0.000) Table : Summary Saisics for Soybean Fuures Reurns ( 00 ln( P / P ) ) Mean Sandard Deviaion Skewness Kurosis Log Absolue Reurns Log Range Jarque-Bera Saisics & P-Value Auocorrelaions Lag Lag Lag 5 Lag 0 Lag (0.000) (0.00) Table : Summary Saisics for Soybean Fuures Log Absolue Reurns and Log Range 8

19 µ α β κ φ ψ φ ψ φ 3 ψ 3 No Seasonaliy Esimae Sd. Error *** *** ** Firs Order Seasonaliy Esimae Sd. Error *** *** ** ** Second Order Seasonaliy Esimae Sd. Error *** *** *** ** ** Third Order Seasonaliy Esimae Sd. Error *** *** *** * ** *** * AIC SC Log-likelihood Skewness Kurosis Q-es P-values# Lag Lag Lag 3 Lag 5 Lag *, **, *** Significan a he 0%, 5%, and % level, respecively. # P-values for η and η. Table 3: GARCH Esimaion Resuls 9

20 µ α β κ Range φ ψ φ ψ φ 3 ψ 3 No Seasonaliy Esimae Sd. Error ** *** *** Firs Order Seasonaliy Esimae Sd. Error ** *** *** ** Second Order Seasonaliy Esimae Sd. Error * *** *** ** * Third Order Seasonaliy Esimae Sd. Error * *** *** ** ** AIC SC Log-likelihood Skewness Kurosis Q-es P-values# Lag Lag Lag 3 Lag 5 Lag *, **, *** Significan a he 0%, 5%, and % level, respecively. # P-values for η and η. Table 4: Esimaion Resuls of GARCH Exended wih Daily Range 0

21 RMSE MAE GARCH (,) Firs Order Seasonaliy GARCH (,) GARCH (,) Exended wih Daily Range Firs Order Seasonaliy GARCH (,) Exended wih Daily Range HRMSE HMAE R Table 5: Daily Volailiy Forecas Performance

22 Figure : Soybean Fuures Prices (985/0-989/) Figure : Soybean Fuures Reurns (985/0-989/)

23 4 3 Normal Quanile Log Absolue Reurns 4 3 Normal Quanile Log Range Figure 3: Q-Q Plos of Log Range and Log Absolue Reurns 3

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