Forecasting Irregularly Spaced UHF Financial Data: Realized Volatility vs UHF-GARCH Models

Transcription

1 Forecastng Irregularly Spaced UHF Fnancal Data: Realzed Volatlty vs UHF-GARCH Models Franços-Érc Raccot *, LRSP Département des scences admnstratves, UQO Raymond Théoret Département Stratége des affares, ESG-UQAM Alan Coën Département Stratége des affares, ESG-UQAM RePAd Workng Paper No * Adresses : Franços-Érc Raccot, Département des scences admnstratves, Unversté du Québec en Outaouas (UQO), Pavllon Lucen Brault, 0 rue Sant Jean Bosco, Gatneau, Québec, Canada, J8Y 3J5. LRSP : Laboratory for Research n Statstcs and Probablty afflated wth : Carlton Unversty, Unversty of Ottawa and UQO. Correspondance : francoserc.raccot@uqo.ca; Raymond Théoret, Département stratége des affares, Unversté du Québec à Montréal (UQAM), 35 est, Ste-Catherne, Montréal, HX-3X. Correspondance : theoret.raymond@uqam.ca; Alan Coën, Département stratége des affares, Unversté du Québec à Montréal (UQAM), 35 est, Ste-Catherne, Montréal, HX-3X. Correspondance : coen.alan@uqam.ca. Acknowledgments: We would lke to thank Pr. Chrstan Calmès hs helpful comments.

2 Forecastng Irregularly Spaced UHF Fnancal Data: Realzed Volatlty vs UHF-GARCH Models Abstract: A very promsng lterature has been recently devoted to the modelng of ultra-hgh-frequency (UHF) data. Our frst am s to develop an emprcal applcaton of Autoregressve Condtonal Duraton GARCH models and the realzed volatlty to forecast future volatltes on rregularly spaced data. We also compare the out sample performances of ACD GARCH models wth the realzed volatlty method. We propose a procedure to take nto account the tme deformaton and show how to use these models for computng daly VaR. Résumé : Dans cet artcle, nous comparons les prévsons de varance obtenues à partr de modèles à très haute fréquence. Nous analysons la performance des modèles ACD-GARCH, ACD-GARCH augmenté et celu de la varance réalsée. Pour ce fare, nous prenons en compte le phénomène de la déformaton temporelle, un problème souvent néglgé, et nous agrégeons les résultats de façon unforme d un modèle à l autre. Nos résultats montrent que la technque de la varance réalsée tend à surpasser les autres modèles d analyse à très haute fréquence. Cette étude peut se révéler utle pour le calcul de la VaR sur un horzon très rapproché. Keywords: Realzed volatlty, Ultra Hgh Frequency GARCH, tme deformaton, fnancal markets, Daly VaR. JEL classfcaton: C;C53;G4

3 . Introducton The very recent mplementaton of electronc order-matchng systems on fnancal markets has entaled ncreasng numbers and frequences of trades. Whle data on prces and volumes were regstered daly two decades ago, transactons (and especally those due to electronc systems) are now recorded nstantaneously wth an accuracy of a fracton of one second. The growng nterest devoted to ntra-daly models n the fnancal lterature s a drect consequence of the avalablty of hgher frequency measurements. Ths phenomenon stylzed by ncreasng frequences of observatons s at the orgn of the concept of ultra-hgh frequency. In ths context, the development of econometrc methods for the analyss of ultra-hgh frequency data seems to be promsng. But the other sde of the con s the problem nduced by the rregularty at whch the observatons arrve. For example, when we estmate a smple GARCH process on the S&P500, we usually use the returns observed every day or every week. In ths case, the nterval between each observaton s fxed: one day or one week. But when analyzng ntra-day observatons, the nformaton arrves sometmes n clusters and at dfferent tme ntervals. Ths problem s called tme deformaton because tme s not the same as calendar tme. The fact that the arrval of nformaton s rregularly spaced s a salent feature of ultra-hgh frequency data. Aggregates of these data up to fxed ntervals of tme ental an mportant loss of nformaton. To avod ths loss, Engle and Russell (998) and Engle (000) have recently developed methods that are drectly talored to rregular spacng of the data. The basc model s the autoregressve condtonal duraton (ACD) model whch s a type of Ths paper s based on prevous research done by Raccot (003). 3

4 dependent Posson process. The ACD model appled to IBM transactons arrval tmes by Engle (000) n GARCH framework has produced ultra-hgh frequency measures of volatlty. The results observed by Engle (000) are promsng and ndcate that ths theoretcal specfcaton seems to be accurate to estmate models for ultra-hgh frequency data or transacton data. The ACD GARCH and extended ACD GARCH volatlty models proposed by Engle (000) warrant ndeed very large gans n forecast error accuracy from a theoretcal perspectve. Consderng ths result, t would be very nterestng to apply these models to ultra-hgh frequency data. Therefore we propose two man contrbutons. Our frst am n ths paper s to develop an emprcal applcaton of ACD GARCH models n forecastng future volatltes. Then we propose another contrbuton n comparng the performance of ACD GARCH models to a new and smple way of modelng fnancal market volatlty usng hgh-frequency data recently developed by Bollerslev and Wrght (00): the ntegrated volatlty or recently called, the realzed volatlty (Dacarogna et al. (00), Barndorff- Nelsen et Shephard (00a), Andersen et al. (003)). Accordng to Bollerslev and Wrght (00), volatlty dynamcs may be modeled by fttng a long autoregressve (AR) representaton to ultra-hgh frequency data. The man nterrogaton n ther approach s that they gnore the fact that data arrve at rregular ntervals. Thus, we have to make an adjustment to take nto account these fundamental features of ultra-hgh frequency data. The plan of the rest of the paper s as follows. Secton s devoted to the presentaton of the models and ther necessary adjustments. We also show how to use 4

5 volatlty forecasts to compute daly VaR. Secton 3 contans a dscusson of the ultrahgh frequency data and the adjustment procedures employed. Secton 4 detals the volatlty calculatons and volatlty forecasts, followed by a comparson of the results and a short dscusson. Secton 5 concludes wth some suggestons for future research.. Ultra-hgh-frequency varance models. UHF GARCH(,) model Ultra-hgh frequency GARCH(,) model allows takng nto account the rregular character of market transactons even f the current duratons of these transactons are not explctly consdered as addtonal sources of nformaton. In ths sense, t s the smplest model of condtonal varance defned at ultra-hgh-frequency. Ths model may be wrtten as follows: σ () = + + ω αε βσ wth σ, the condtonal varance and ε, the nnovaton.. The ACD-GARCH model The autoregressve condtonal duraton (ACD) model was frstly developed by Engle and Russel (998). Later, Jasak (999), Gouréroux et al. (999), Gouréroux and Jasak (00), and Engle (000) refned and appled the model n a smlar context. 5

6 The basc formulaton of the ACD s specfed n terms of the condtonal densty of the duratons. The duraton s the nterval between two arrval tmes denoted by x = t t. The expectaton of the th duraton s gven by the followng functon: E ( x x,..., x ) = θ ( x,..., x; ψ ) θ () under the assumpton that x = θ e (3) where { e } ~..d., and ψ s a set of parameters to be estmated. By defnton, the condtonal expectaton of the duraton depends on past duratons. A more general formulaton of the model, called the ACD(p, q), may be gven by the followng specfcaton: p q α x j + j= 0 j= 0 θ = w + β θ (4) I j j where p and q are the orders of the lags. Ths specfcaton may be used to study the marks assocated wth the arrval tmes so that hypothess from the market mcrostructure theores may be tested. Engle (000) proposed a non-lnear generalzaton of ths model to defne a measure of prce volatlty usng transacton data and to analyse how the arrval tme nfluences ths volatlty. Assumng that r s the return from transacton - to, the condtonal varance per transacton s defned as ( r x ) h V = (5) where x s defned as prevously. The condtonal varance s dependent on current and past returns and duratons. As mentoned by Engle (000), volatlty s always measured 6

7 over a fxed nterval and s frequently reported n annualzed terms. Therefore, the condtonal volatlty per unt of tme s the most nterestng measure to be evaluated. It s gven by: r V x = σ (6) x These two varances mply: h = σ (7) x In ths case, the forecasted condtonal transacton varance may be defned as: ( h ) E ( x σ ) E = (8) The smple GARCH(, ) of Bollerslev (986) may be extended to compute σ wth the followng specfcaton: σ (9) = w + αe + βσ + γx - where x s the recprocal of duraton. Ths model s called ACD-GARCH. In ths specfcaton duratons are drectly ntroduced nto the condtonal varance. It should be noted that the standard GARCH(,) whtout ajustment for duraton, whle t s certanly not the best model for UHF data, s also used n the UHF lterature for computng volatlty forecasts. For example, daly VaR usng ths model are very smple to obtan. One smply have estmate the model usng standard econometrc package lke EVews and then to compute a one step-a-head forecast. The VaR number s then gven by : VaR σ. Secton 4. shows how t s smple to obtan forecast from =.65 amount + GARCH(,), ths mght explan the popularty of the model. For more detals on VaR computatons usng GARCH models, see Tsay (005). 7

8 To gve another fnancal llustraton to ths econometrc model we can use the results obtaned by Easley and O Hara (99). Accordng to the market mcrostructure model of Easley and O Hara (99), a fracton of the nvestors s nformed and knows f there are news concernng ther assets. When t s tme for them to do transactons, they wll buy f the news are favourable, sell on bad news and wll not trade f no news. In ths model, long ntervals (x ) wll be nterpreted as no news. Ths mples that we expect a postve value for γ n our extended ACD GARCH model. Long duratons ndcate ndeed that there are no news and consequently a lower volatlty 3..3 The extended ACD-GARCH Engle (000) suggests promsng extensons and proposes a rcher formulaton allowng both observed and expected duratons to enter the model. He also ntroduces a long run volatlty varable defned by the followng equaton: x σ (0) = α 0 + αe + βσ + γ x + γ + γ 3ξ + γ 4θ θ where ξ s the long run volatlty, θ s the condtonal duraton and mght be defned by the parsmonous ACD(,) model. Engle (000) proposes to compute the long run volatlty by a Exponental Weghted Movng Average (EWMA) model on r / x as r ξ = λξ + ( λ) () x In ths extended model for computng volatlty usng hgh frequency data, the nfluences of duratons on volatlty have been ncorporated n three parameters. They measure the 3 We can note that long duratons cannot nduce the condtonal varance to be negatve wth ths formulaton. 8

9 effect of surprse n duraton, the recprocal duraton and the expected recprocal duraton 4 respectvely. As n any other GARCH models, forecastng volatlty can be found smply by computng the condtonal expectaton and s gven by: E ( ) α + α e + βσ + γ E ( x ) + γ + γ ξ γ θ = σ () Ths calculaton led by Engle (000) reveals us that parameterγ s not persstent. However, parameters γ andγ 4 ndcate a long run nfluence on future volatltes because of the persstence 5 of duratons..4 A more parsmonous approach: realzed volatlty Whle Engle (000) approach for modelng and computng volatlty usng hghfrequency data seems promsng on the theoretcal sde of the con, ths approach s complcated by the fact that there s a lot of data manpulatons that must be done before havng an estmate of volatlty. The concept of realzed volatlty has been frstly developed by Andersen and Bollerslev (998) and later appled for computng daly volatlty forecasts of exchange rates and S&P 500 Index-Futures, respectvely, by Bollerslev and Wrght (00) and Martens (00) usng the appellaton ntegrated volatlty. The relaton between realzed volatlty and ntegrated volatlty s well explaned n Barndorff-Nelsen et Shephard (00a, 00b) and Meddah (00, 003). Smply put, the realzed volatlty s measured 4 The expected recprocal duraton s the expected rate of arrvals of transactons. 5 As mentoned by Engle (000) these models mght be estmated by QMLE (quas-maxmum lkelhood estmator) wthout specfyng the densty of the dsturbances and usng Bollerslev-Wooldrdge (99) robust standard errors. 9

10 by the squared value of ntra-daly returns. Ths measure s also consdered to be a more accurate measure of ex-post volatlty. As Anderson et al. (00) pont out, assumng that the returns follow a specal sem-martngale process, the quadratc varaton of ths process consttutes a natural measure of ex-post realzed volatlty. It also corresponds drectly to the theoretcal defnton of volatlty used n dffuson and stochastc volatlty models 6. Takng nto account tme deformaton, we can gve a mathematcal defnton of realzed volatlty as follows: N I ( m) = r m, n n n= σ (3) where r m,n s the nth squared return on day m. Because the returns are not observed at a constant nterval, the numbers of observatons N wll vary from day to day. Compared to the UHF-GARCH model, we can easly observe the smplcty of the calculatons requred for obtanng an estmate of the volatlty. As n the GARCH framework, t s possble to obtan a forecast of the ntegrated volatlty. The method mght be descrbed as follows. The forecasts are based on a long memory autoregressve model where the lag p of the autoregressve process must approach nfnty. The coeffcents obtaned from ths autoregresson are then used to construct a forecast functon that takes the followng form: ˆ σ ( m) = µ (4) N ( ˆ + ˆ ) ( m ) + n N ( m ) I v n N n= where = ˆ t+ t j j= vˆ α v (5) t j 6 See Barndorff-Nelson and Shephard (00), and Hull and Whte (987) for an example of such uses. 0

11 vˆ k = ˆ + jv + t k t t+ k j t j= j= k α ˆ α v (6) j t+ k j and where the coeffcents α j mght be estmated n the tme doman by a long order autoregresson 7, log( v r ) µˆ and, where ˆ µ s the sample mean of log( ). It should t = t be noted that these coeffcents mght be estmated n a frequency doman usng a Wener-Kolmogorov flter. The results from usng these technques appear to be smlar (Bollerslev and Wrght, 00). So n the followng applcaton, we use the long order r t autoregresson on the log-squared returns 8 whch we assume to be a martngale dfference. More precsely, ( L )( r t µ ) = et log( 3 where α ( L) = α L α L α L... α ) 3, e t WN(0,σ ) and the lagged polynomal s assumed to converge. So to mplement the forecastng formula represented by equaton (4), we smply have to ft a long order autoregresson to the logsquared returns and use ths estmated equaton to compute our forecasts. Ths pont s made clearer n the followng secton. Because the log-squared returns may yeld large negatve numbers for returns close to zero, we have appled the followng transformaton: r * t τs = log( rt + τs ) (7) rt + τs where s s the sample varance of r t and τ s chosen to be equal to 0,0 as n Fuller (996) and Bredt and Carrqury (996). 7 More precsely, we make the assumpton that the tme seres of volatltes may be represented by an approprate proxy such as the log-squared returns whch has an autoregressve representaton. 8 As alternatve hypothess, we mght specfy that the squared or absolute returns has an autoregressve representaton (see Bollerslev and Wrght, 00).

12 3. Data and measurements We have to remnd that our frst am s to compare Engle s UHF-GARCH model (000) wth the realzed volatlty, whch s also called the ntegrated volatlty concept by Bollerslev and Wrght (00). Therefore, we use the sample of observatons used by Engle (000). The rregularly spaced ultra-hgh frequency data are the transacton quotes for IBM stocks. The data were abstracted from the Trades, Orders Reports, and Quotes (TORQ) data set constructed by Joel Hasbrouck and NYSE. Two types of random varables compose the transacton data: the tme of transactons and the marks at ths tme 9. In our applcaton, we defne a pont process as the tme at whch a transacton occurred. The marks are volumes of shares, prces, bd and ask prces of the traded contract at the transacton tme. Our data set ncludes around transactons made on the NYSE from November, 990 through January 3, 99. Only the trades occurrng between 9:30 AM and 4:00 PM are used for calculatons 0. Followng Engle (000), we delete transactons on Thanksgvng Frday and the day before Chrstmas and New Years, as well as all transactons wthout a reported set of quotes. Accordng to Engle (000), ths procedure leaves 546 unque transacton tmes. We consder only the unque tmes and remove all zero duratons. As justfed by Engle (000): Ths s consstent wth 9 The nformaton about bd and ask quote movements, the volume assocated wth the transactons, the transacton prces, and a tme stamp measured n seconds after, reflectng the tme at whch the transacton occurred, are consdered n the data set. 0 We have to menton that two days have been deleted from the 63 tradng days n the 3 month sample. As explaned by Engle (000) : A halt n IBM tradng of just over an hour and 5 mnutes occurred on Frday, November 3. On December 7 th there was a one and a half hour delay n the openng.

13 nterpretng a trade as a transfer of ownershp from one ore more sellers to one or more buyers at a pont n tme. We may add that the average volume correspondng to each tme stamp s 86 shares. Moreover, the mnmum tme between events s second and the maxmum duraton s 56 seconds (or 9 mnutes and seconds). A smple descrpton of the data used by Engle (000) shows that the average duraton between successve events for IBM s 8.38 seconds wth a standard devaton of As mentoned by Engle and Russell (998), we have to seasonally adjust the data to take out the tme of day effect. An mportant lterature has been devoted to ths effect. It nduces ndeed a hgher frequency of transactons near the openng and the closng of the market. The procedure we use s called by Engle and Russell a durnally adjustment. Therefore, we defne an adjusted duraton gven by the followng equaton: ~ x x = ϕ( t ; β ) where x = t t - s the duraton between trades and ϕ(.) s a pecewse lnear splne functon used to seasonally adjust the duratons. Exhbt gves an llustraton of a lnear splne. The knots are the ponts where the lnear peces of the splnes jon together. They appear at 9:30, 0:00, :00, :00, :00, :00, 3:00, 3:30, 4:00. [Please nsert exhbt ] 3

14 Specfcally, the seasonal adjustment s done by regressng the duratons on the tme usng a lnear splne specfcaton that takes the followng form: x c + β t + β t + β t + β t + β t + β t + β t + t + e = β 8 8 where t - for =,,9 are vectors of tme varables constructed from the knots. From ths regresson we obtan x ˆ ϕ( t ; ˆ β ) =. The resultng varable x~, whch s free of the typcal tme of day effect, represents fractons of duratons below or above normal. 4. Analyss and results 4. Volatlty calculatons: a comparson We have to notce that the two methods dscussed to compute the daly volatlty present mportant dfferences. Thus an adjustment s necessary before we can make a comparson. The UHF-GARCH specfcaton gves volatlty calculatons per seconds whle the ntegrated volatlty gves a volatlty estmate for a day. We have to transform the models to obtan comparable unts. The UHF-GARCH calculaton s transformed on a daly bass 3. We ntroduce a new and obvous procedure whch has not been explored n the lterature. We proceed by analogy wth the ntegrated volatlty calculatons. More precsely, we suggest that we average out the ntra-day volatltes to defne a daly volatlty as: Note that we used a lnear splne. We mght have used a kth-order splne whch s a pecewse polynomal approxmaton, wth polynomals of degree k dfferentable k- tmes everywhere. The maxmum lkelhood estmator s used to estmate the parameters of all our UHF-GARCH models. 3 We can menton that ths transformaton would be accurate to evaluate one-day optons. 4

15 σ N d = N = where σ σ s obtaned by estmatng hgh-frequency GARCH models. In table, we summarze the results observed for the four dfferent methods used to compute the daly volatlty: ntegrated volatlty, GARCH(, ), ACD GARCH and Extended ACD GARCH. We compute volatltes for fve consecutve days usng our ntra-daly transactons on IBM stock for the frst week of our sample 4. A frst glance at the results confrms mmedately the accuracy of our proposton made to compare volatlty calculatons. [Please nsert table ] We observe ndeed that all the GARCH calculatons follow qute closely the realzed volatlty methodology. Ths result s reassurng and confrms our frst ntuton. As explaned n Bollerslev & Wrght (00) and as shown n table, the smple GARCH (.e. GARCH(, )) has the worst performance compared to the realzed volatlty for hghfrequency data. Ths observaton s not surprsng. [Please nsert table ] The Extended ACD GARCH seems to have the best performance among the GARCH models n our comparson. However, ths frst glance at volatlty calculatons needs to be completed by a more detaled analyss. To have a better dea on the performance of these volatlty models, we compute standard measures such as the R- 4 The frst week begns on a Thursday n November 990. There are no partcular reasons that explan why we have chosen ths specfc segment of tme to make our calculatons. It s smply for comparsons of calculatons. 5

16 squared of the Mncer-Zarnowtz (969) regresson. Bollerslev and Wrght (00) and Martens (00) underlne the accuracy of ths procedure to evaluate the performance of models of tme-varyng condtonal heteroskedastcty. They use ultra-hgh frequency data, but they make the assumpton that the data arrve at constant ntervals 5, whch s of course not the case n realty. As acknowledged by Engle (000) ultra-hgh frequency data arrve at rregular ntervals. Consderng these unsatsfactory approaches, we propose to use the concept of autoregressve condtonal duraton ntroduced by Engle and Russell (998) and mproved by Engle (000). In the next secton we mplement our approach. Then we present the forecastng performances of our four models. 4. Volatlty forecasts: a comparson Our am s to make volatlty forecasts based on our four models and then to compare ther performances. As suggested by Bollerslev and Wrght (00), we use the Mncer- Zarnowtz R-squared 6. For each model we proceed as follows to compute forecasts. The 5 Bollerslev and Wrght (00) consder a data set wth fve-mnute return seres. For 4-hour markets, there are 88 fve-mnute observatons n a day. An other applcaton of ths method may be found n Bollerslev and Zhang (003). 6 Other popular measures mght be used such as the mean absolute error (MAE), the root mean squared error (RMSE), the heteroskedastcy adjusted mean absolute error (HMAE) or the heteroskedastcy adjusted T Realzed root mean squared error (HRMSE). HMAE s defned as t T and HRMSE = = Forecast T t t Realzed t Forecast where the forecasted errors are adjusted for heteroskedastcty. See Andersen et T t= t al.(999) and Martens (00) for an applcaton of the last two measures. 6

17 forecasts nduced by the realzed volatlty 7 are gven by our equaton (6). We know that an approprate proxy for the tme seres of volatltes has an autoregressve representaton. One can easly demonstrate that ths specfcaton may be approxmated by a smple ARMA(, ) 8. Our forecasts are based on the estmaton of that process. The computaton of forecasts from a smple GARCH(, ), commonly used n fnance, can be done by usng the followng formula 9 : n ( α + β ) E tσ t+ n = α 0 + σ t ( α + β ) α β n = α α ( α + β ) α (8) n 0 0 n + σ t ( α + β ) β α β α 0 = α β + ( α + β ) n σ t α 0. α β We have to menton that equaton (8) mght be used for computng forecasts at any horzon smply by replacng n by a value of nterest 0. The computaton of forecasts based on the ACD and extended ACD GARCH models seems much more complcated. We need ndeed expected values of duratons. Ths 7 For a recent applcaton of ths method to forecastng and a dscusson of the problem of tme aggregaton, see Goser, Madhavan, Serbn and Yang (005). 8 Bollerslev and Wrght (00) have used an AR (050) to ft ther ultra-hgh frequency data observed at fxed ntervals. It s well known n the econometrc lterature that hgh order autoregressve models may be approxmated by parsmonous ARMA models. For example see Mlls (990), Hamlton (994) or Raccot and Théoret (00). 9 The ndex n represents the number of steps. 0 When α + β, the condtonal expectaton of volatlty n perods ahead s nstead E = t ( σ t+ n ) = σ t + nα0. 7

18 problem may be solved easly. We know that x s related to θ. So, we assume that the expectaton of the duratons may be represented by a smple ARMA(, ) process. Then, we use the forecasted values of the duratons nduced to nclude them n the ACD GARCH and n the extended ACD GARCH models. The man dsadvantage of ths procedure s the ncreasng number of computatons we have to perform to get a forecast. For our purpose we use a smple ARMA(, ) specfcaton. Thus the process s defned by the followng equaton: x = v + α x + βε + ε (9) whereε x θ s a martngale dfference (.e. ε = x E - ( x )). Forecasted values of x can be obtaned from (9) and then ncluded n equaton (). Table 3, presented below, shows the results of the performance of the GARCH models compared wth the realzed volatlty. [Please nsert table 3] As we can see f we compare the RMSE, MAE or the R of the Mncer-Zarnowtz 3 (969) regresson, the realzed volatlty method outperforms all the GARCH models. However, we have to moderate ths result. It must be noted that none of the numbers presented n ths We observe that the expectaton of the duraton has the same type of representaton as the condtonal duraton. As suggested by Engle and Russell (998), we use an ARMA process. Another approach s to assume that the duratons may be represented by log lnear regresson models. 3 To obtan the Mncer-Zarnowtz regressons we regress the ex post realzed values of the varable under scrutny on the forecasted values of ths varable plus a constant term. For example, n our case, we forecast the IBM prces for dfferent sample szes: 700, 400 and 00, and then we run the regressons (.e. * * y = c + β + ε ). The resultng R are shown n table 3. t y ft t 8

19 table s sgnfcant. Nevertheless, for the realzed volatlty, the t statstcs of Mncer- Zarnowtz are near sgnfcance level 4. The poor performances of the four models are not so surprsng and the results are consstent wth prevous studes devoted to forecastng models 5. As mentoned by Andersen, Bollerslev and Lange (999), the standard GARCH volatlty models tend to perform very poorly when they are appled drectly to UHF data. But t s decevng that the ACD GARCH models do not perform better n comparson wth the smple realzed volatlty method. In fact, t would be possble to conclude that the gans n forecast error accuracy from ACD GARCH and extended ACD GARCH models reman large from a pure theoretcal perspectve. The smple way of modelng fnancal markets volatlty usng ultra-hgh frequency data ntroduced by Bollerslev and Wrght (00), the realzed volatlty, tends to perform better than the standard and autoregressve condtonal duraton GARCH models. However, mprovements have to be made to take nto account the fact that data arrve at rregular ntervals. 5. Conclusons In ths paper we have proposed an applcaton of the autoregressve condtonal duratons GARCH models (ACD GARCH) recently developed by Engle and Russel (998) and 4 More precsely, the t statstcs are.75 (0.08),.8 (0.06), and.80 (0.07) for sample szes of 700, 400, and 00 respectvely. Ther correspondng p-values are n brackets. 5 See Mlls (999). 9

20 Engle (000). We have used them to make volatlty forecasts before comparng ther performances wth the recent concept of realzed volatlty ntroduced by Bollerslev and Wrght (00). To take nto account the tme deformaton nduced by the fact that ultrahgh frequency data arrve at rregular ntervals, we have made some assumptons and proposed adjustments. Our results show that the realzed volatlty seems to be better, n terms of RMSE, MAE, and Mncer-Zarnowtz (969) crterons, than any of UHF-GARCH models. Although none of the tested models has well performed on the IBM stock data used for the emprcal analyss, t s qute decevng that the ACD GARCH has not performed better than the realzed volatlty. It s well known n the lterature that when usng GARCH models to forecast hgher frequency data, they perform very poorly 6. Nevertheless, the theoretcal mprovement developed by Engle to take nto account tme deformaton seems to show poor performances when usng to forecast volatltes. As suggested n an other framework by Donaldson and Kamstra (997), we thnk t would be very nterestng to mprove the forecastng power by addng an Artfcal Neural Network component n the ACD GARCH model. Moreover, as n Engle (000) we have used lnear splnes to adjust the data, usng non-lnear splnes may mprove the results. We leave all these ssues for future research. 6 See Andersen et al. (998) and (00). 0

21 References Andersen, T.G. and T. Bollerslev, 998, Answerng the skeptcs: yes, standard volatlty models do provde accurate forecasts, Internatonal Economc Revew 39 (4), Andersen, T.G., Bollerslev, T. and S., Lange, 999, Forecastng fnancal market volatlty: samplng frequency vs-à-vs forecast horzon, Journal of Emprcal Fnance 6, Andersen, T.G., Bollerslev, T., Debold, F.X. and P. Labys, 00, The dstrbuton of realzed exchange rate volatlty, Journal of the Amercan Statstcal Assocaton 96, Barndorff-Nelsen, O.E. and N. Shephard, 00, Non-gausan Ornsten-Uhlenbeck models and ther uses n fnancal economcs, Journal of the Amercan Statstcal Assocaton 96. Barndorff-Nelsen, O. et N. Shephard., 00a, Econometrc analyss of realzed volatlty and ts use n estmatng stochastc volatlty models, Journal of the Royal Statstcal Socety, 64, Barndorff-Nelsen, O. et N. Shephard, 00b, Estmatng quadratc varaton usng realzed varance, Journal of Appled econometrcs, 7, Bollerslev, T. and J. Wooldrge, 99, Quas-maxmum lkelhood estmaton and nference n dynamc models wth tme varyng covarances, Econometrc Revews, Bollerslev, T. and J.H. Wrght, 00, Hgh-frequency data, frequency doman nference, and volatlty forecastng, Revew of Economcs and Statstcs 83(4), Bollerslev, T. and B. Zhang, 003, Measurng and modelng systematc rsk n factor prcng models usng hgh frequency data, Journal of Emprcal Fnance 0,

22 Bredt, F.J. and A.L., Carrqury, 996, Improved quas-maxmum lkelhood estmators for stochastc volatlty models, n A. Zellner, Lee, J. S. (Eds.), Modelng and Predcton: Honorng Seymour Gesel. Sprnger-Verlag, New York. Dacorogna, M.M., Gençay, R., Müller, U., Olsen, R.B. and O.V. Pctet, 00, An Introducton to hgh-frequency Fnance, Academc Press, San Dego. Donaldson, R.G. and M. Kamstra, 997, An artfcal neural network-garch model for nternatonal stock return volatlty, Journal of Emprcal Fnance 4, Easley, D. and M. O Hara, 99, Tme and the process of securty prce adjustment, Journal of Fnance 47, Engle, R.F. and J.R. Russel, 998, Autoregressve condtonal duraton: a new model for rregularly spaced transacton data, Econometrca 66 (5), 7-6. Engle, R.F. 000, The econometrcs of ultra-hgh-frequency data, Econometrca 68(), -. Fuller, W.A., 996, Introducton to Statstcal Tme Seres, Wley, New York. Goser, K., Madhavan, A., Serbn V. and J. Yang, 005, Toward Better Rsk Forecats : Usng data wth hgher frequency than the forecastng horzon, The Journal of Portfolo Management, Sprng 005. Gouréroux, C., Jasak, J. and G. Le Fol, 999, Intra-day market actvty, Journal of Fnancal Markets (3), Gouréroux, C. and J. Jasak, 00, Duratons n a Companon to Theoretcal Econometrcs, chap., Blackwell, Massachusetts.

23 Gouréroux, C. and J. Jasak, 00, Fnancal Econometrcs, Prnceton Seres n Fnance, Prnceton Unversty Press, Prnceton. Martens, M., 00, Measurng and forecastng S&P 500 ndex-futures volatlty usng hgh-frequency data, Journal of Futures Markets (6), Meddah, N., 003, ARMA representaton of ntegrated and realzed varances, Econometrcs Journal, 6, Meddah, N., 00, A theoretcal comparson between ntegrated and realzed volatlty, Journal of Appled Econometrcs, 7, Mlls, T.C., 990, Tme Seres Technques for Economst, Cambrdge Unversty Press, Cambrdge. Mlls, T.C., 999, The Econometrc Modelng of Fnancal Tme Seres, Second Edton, Cambrdge Unversty Press, Cambrdge. Mncer, J. and V. Zarnowtz, 969, The evaluaton of economc forecasts, n J. Mncer (Ed.), Economc Forecasts and Expectatons, 3-46, NBER, Cambrdge. F.E. Raccot and A. Coën, 004, Integrated volatlty and UHF-GACH models : A comparson usng hgh frequency fnancal data, Proceedngs of the Global Fnance Conference (GFC), Las Vegas, Nevada, USA. Raccot, F.E., 003, Classcal and Advanced Nonlnear Models n Emprcal Fnance n Tros essas sur l analyse des données économques et fnancère, Ph.D. thess, ESG- UQAM. 3

24 Raccot, F.E. and R. Théoret, 00, Traté d économétre fnancère : modélsaton fnancère, Presses de l Unversté du Québec (PUQ), Ste-Foy, Québec. Tsay, R.S., 005, Analyss of Fnancal Tme Seres, Second Edton, Wley Seres n Probablty and Statstcs. 4

25 Exhbt. Representaton of the lnear splne Duraton 9:30 4:00 tme Day Realzed volatlty Table Volatlty computatons (n sample) * Smple GARCH ACD GARCH Extended ACD GARCH Thurs Frday Monday Tuesday Wednes Number of observatons * The smple GARCH(,) model s estmated by usng equaton (), the ACD GARCH, by usng equaton (9) and the extended ACD GARCH, by usng equaton (). Realzed volatlty s gven by equaton (3). 5

26 Table Average absolute percentage changes (n sample) * Day Smple GARCH ACD GARCH Extended ACD GARCH Thursday 5.7% 6.04%.89% Frday 3.87% 33.30% 7.96% Monday 4.7% 7.37%.6% Tuesday 3.5%.99% 6.49% Wednesday.%.% 8.66% Average 7.3% 6.6%.3% * Table s smply a recast of table. As realzed varance s the benchmark, we express the data of table as percentage devatons from ths benchmark. Ths table s used to compare the three GARCH models to the model of realzed varance, whch s the smplest to compute. Table 3 Forecasts evaluaton of GARCH models and realzed volatlty * Number of observatons Smple GARCH ACD GARCH Extended ACD GARCH 700 RMSE : 3. RMSE : 3. RMSE : 0.94 MAE : 3.84 MAE : 3.84 MAE : 3.9 R : R : R : RMSE : 4.83 RMSE : 4.83 RMSE :.47 MAE : 4.35 MAE : 4.35 MAE : 4.34 R : R : R : RMSE : 5.55 RMSE : 5.54 RMSE : 3.0 MAE : 4.3 MAE : 4.3 MAE : 4.3 R : R : R : Realzed Volatlty RMSE :.6 MAE :.03 R : RMSE :.3 MAE :.99 R : RMSE :.7 MAE :.97 R : * The evaluaton of forecasts s performed for forecastng samples of 700, 400 and 00 observatons, whch are supposed to represent the number of possble transactons n one day. These numbers are based on table. To compare these samples, we aggregate the observatons of everyone by usng the formula: σ N d = σ N = the ACD GARCH models., wth N the number of transactons n one day and where σ s obtaned by estmatng 6

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