Long Memory versus Structural Breaks in Modeling and Forecasting Realized Volatility

Transcription

1 Long Memory versus Srucural Breaks in Modeling and Forecasing Realized Volailiy Kyongwook Choi 1, Wei-Choun Yu 2 and Eric Zivo 3 Absrac We explore he possibiliy of srucural breaks in he daily realized volailiy of he Deuschemark/Dollar, Yen/Dollar and Yen/Deuschemark spo exchange raes wih observed long-memory behavior. We find ha srucural breaks in he mean can parly explain he persisence of realized volailiy. We propose a VAR-RV-Break model ha provides superior predicive abiliy when he iming of fuure breaks is known. Wih unknown break daes and sizes, we find ha a VAR-RV-I(d) long memory model provides a robus forecasing mehod even when he rue financial volailiy series are generaed by srucural breaks. JEL classificaion: C32, C52, C53, G10 Keywords: Realize volailiy; Exchange rae; Long memory; Srucural break; Fracional inegraion; Volailiy forecasing 1 Deparmen of Economics, The Universiy of Seoul, 90 Cheonnong-dong, Dongdaemoon-gu, Seoul, , Souh Korea. kwchoi@uos.ac.kr Tel: Fax: Corresponding auhor: Economics and Finance Deparmen, Winona Sae Universiy, Somsen 319E, Winona, MN wyu@winona.edu. Tel: Fax: Deparmen of Economics, Universiy of Washingon, Box , Seale, WA ezivo@u.washingon.edu. Tel: Fax:

2 1. Inroducion Condiional volailiy and correlaion modeling has been one of he mos imporan areas of research in empirical finance and ime series economerics over he pas wo decades. Alhough daily financial asse reurns are approximaely unpredicable, reurn volailiy is imevarying bu highly predicable wih persisen dynamics and he dynamics of volailiy is well modeled as a long memory process 4. An inheren problem for measuring, modeling and forecasing condiional volailiy is ha he volailiy is unobservable or laen, which implies modeling mus be indirec. Typically, measuremens of volailiy are from parameric mehods, such as GARCH models or sochasic volailiy models for he underlying reurns. These parameric volailiy models, however, depend on specific disribuional assumpions and are subjec o misspecificaion problems. Given he availabiliy of inraday ulra-high-frequency price daa on asses, Andersen, Bollerslev, Diebold, and Labys (2003), henceforh ABDL, and Barndorff-Nielsen and Shephard (2001, 2002, 2004) inroduced a consisen nonparameric esimae of he price volailiy ha has ranspired over a given discree inerval, called realized volailiy. They compued daily Deuschemark/Dollar, Yen/Dollar, and Deuschemark/Yen spo exchange raes realized volailiies simply by summing high-frequency finely sampled inraday squared and crossproducs reurns. By sampling inraday reurns sufficienly frequenly, he model-free realized volailiy can be made arbirarily close o underlying inegraed volailiy, he inegral of insananeous volailiy over he inerval of ineres, which is a naural volailiy measure. ABDL found logarihmic realized volailiy could be modeled and accuraely forecas using simple parameric fracionally inegraed ARFIMA models. Their low-dimensional mulivariae realized volailiy model (VAR-RV-I(d) model, explained in Secion 4) provided 4 The findings sugges ha volailiy persisence is highly significan in daily daa bu will weaken as he daa frequency decreases. 2

3 superior ou-of-sample forecass for boh low-frequency and high-frequency movemens in he realized volailiies compared o GARCH and relaed approaches. Neverheless, many sudies have poined ou ha observed long memory may no only be generaed by linearly fracional inegraed process bu also by: (1) cross-secional aggregaion of shor memory saionary series (Granger and Ding, 1996); (2) mixure of numerous heerogeneous shor-run informaion arrivals (Andersen and Bollerslev, 1997); (3) non-linear models, such as srucural breaks (changes) or regime swiches (Granger and Hyung, 2004; Choi and Zivo, 2007; Diebold and Inoue, 2001). In paricular, i has been conjecured ha persisence of asse reurn volailiy may be oversaed due o he presence of srucural changes. In his paper, we focus on he possibiliy of srucural breaks and rends in he daily realized exchange rae volailiy series sudied by ABDL. Firs, we es for long memory and esimae long memory models for he realized volailiy series. We find srong evidence for long memory in exchange rae realized volailiy. Second, we es for and esimae a muliple mean break model based on Bai and Perron (1998, 2003) s mehod. We find several common srucural breaks wihin hese hree series. Using Beran and Ocker (2001) s semiparameric flexible rend long memory model, we do no find evidence of a smooh flexible rend in volailiies. Third, we examine he evidence for long memory in he break adjused daa. We find a parial reducion of persisence in realized volailiy afer he removal of breaks. The evidence suggess ha par of he observed long memory may be accouned for by he presence of srucural breaks in he exchange rae volailiy series. Sun and Phillips (2003) poined ou ha i is very difficul o separae low-frequency dynamics and high-frequency flucuaions, in paricular when shor-run flucuaions have high variance. Therefore, we propose an alernaive shor memory model, which adaps more quickly o he curren volailiy, wih deeced break informaion in realized volailiy. Our VAR-RV- Break model provides compeiive forecass compared wih mos of he forecasing models 3

4 considered by ABDL if he iming of fuure break daes and heir sizes are known. If we have lile knowledge abou break daes and sizes, he VAR-RV-I(d) model is he bes forecasing model even when he rue volailiy series are generaed by srucural breaks. We suppor our findings wih an exensive Mone Carlo experimen. The res of he paper is organized as follows. Secion 2 presens resuls from he esimaion of various long memory models for realized volailiy. Secion 3 presens resuls from he esimaion of a srucural breaks model and re-examines he evidence for long memory in he break-adjused series. Secion 4 repors he evaluaion of ou-of-sample forecasing performance of various models. Secion 5 concludes. 2. Daa and Long Memory Model 2.1. Daa We use he same daa as ABDL, which are spo exchange raes for he U.S. dollar, he Deuschemark, and he Japanese yen from December 1, 1986 hrough June 30, Their daa se consiss of 3,045-days of inra-day 30-minue log reurns on he DM/$ and Yen/$ exchange raes. The inraday reurn is denoed r, where = h, 2h, 3h,.. 47h, 1, 49h,.., 3045, and h = ( h ) 1/48 = Realized daily volailiies of DM/$ and Yen/$ are compued as he diagonal elemens of 1/ h ( h) ( h)' r + 1 ih r + 1 ih i= 1. By absence of riangular arbirage, he Yen/DM reurns can be calculaed direcly from he difference beween he DM/$ and Yen/$ reurns. Realized volailiies, 5 The raw daa include all inerbank DM/$ and Yen/$ bid/ask quoes shown on he Reuers FX screen provided by Olsen & Associaes. These hree currencies were he mos acively raded in he foreign exchange marke during he sample period. Thiry-minue prices are consruced from he linearly inerpolaed logarihmic average of he bid and ask quoes for he wo icks immediaely before and afer he hiry-minue ime samps over he global 24-hour rading day. Thiry-minue reurns are obained from he firs difference of he logarihmic prices. Reurns from Friday 21:00 Greenwich Mean Time (GMT) o Sunday 21:00 GMT and cerain holiday periods o avoid weekend and holiday effecs are excluded. 4

5 also called realized sandard deviaions, are calculaed from he square roo of he realized variances. Following ABDL, we consider models for he naural logarihm of realized volailiy Long Memory Models Before conducing furher modeling and forecasing, i is crucial o deermine wheher he ime series of log realized volailiies is saionary or no. The dichoomy beween I(0) and I(1), however, may be far oo narrow. A long memory model ha allows fracional orders of inegraion, I(d), provides more flexibiliy. For an I(0) process, shocks decay a an exponenial rae; for an I(1) process, shocks have permanen effec; for an I(d) process, shocks dissipae a a slow hyperbolic rae. Long memory behavior in volailiy has been well esablished, see for example, Ding, Granger, and Engle (1993), Baillie, Bollerslev and Mikkelsen (1996), and Andersen and Bollerslev (1997). Granger and Joyeux (1980) and Hosking (1981) showed ha a long memory process for y can be modeled as a fracionally inegraed, I(d), process d (1 L) ( y μ ) = ε, (1) where L denoes he lag operaor, d is fracional difference parameer, μ is he uncondiional mean of y, and ε is saionary wih zero mean and finie variance. The process y is saionary provided -½ < d < ½, and exhibis non-saionary long memory if ½ d < 1. Two commonly used semi-parameric esimaors for d are he log-periodogram esimaor of Geweke and Porer-Hudak (1983) and he local While esimaor of Künsh (1987) and Robinson (1995b). A flexible parameric process called he ARFIMA (p, d, q) model incorporaes boh longerm and shor-erm memory, φ ( L)(1 L) d ( y μ) = θ( L) ε, (2) 5

6 where φ ( L) and θ ( L) are auoregressive and moving average polynominals, respecively, wih roos ha lie ouside he uni circle and ε is Gaussian whie noise. Sowell (1992) and Beran (1995) discussed maximum likelihood esimaion of (2). To allow for a daa-driven disincion of long memory, shor memory, sochasic rends, and smooh deerminisic rends wihou any prior knowledge, Beran and Ocker (2001) considered he semiparameric fracional auoregressive (SEMIFAR) model δ φ ( L)(1 L) ((1 L) m y g( i )) = ε, (3) where δ is he long memory parameer, and gi ( ) is a smooh rend funcion on [0,1] wih i = / T. In (3), y mus be differenced o achieve saionariy by he parameer d = δ + m. The value of m deermines wheher he rend should be esimaed from he original daa (when m = 0) or he firs difference (when m = 1). When δ > 0, y is long memory. When δ < 0, y is anipersisen. When δ = 0, y has shor memory. For more deailed discussions of long memory esing and esimaion mehods, we refer he reader o Baillie (1996) and Robinson (1995a) Long Memory Esimaion As in ABDL, we consider modeling log realized volailiy. According o he slow decay of auocorrelaions in Figure 1, i is eviden ha log realized volailiy of he exchange rae series has long memory dynamics. We esimae he long memory parameer using he local While, ARFIMA(p, d, q) and SEMIFAR models menioned in Secion 2.2. The esimaes of d for logarihmic realized volailiy are repored in Table 1. Wheher we use nonparameric, semiparameric, or parameric mehods, all of he esimaes of d are in he range beween 0.37 and 0.56, which confirms he exisence of long memory in he logarihmic realized volailiy. 6

7 [Inser Figure 1 here] [Inser Table 1 here] 3. Srucural Break Model 3.1. Muliple Srucural Break Model I is well known ha srucural change and uni roos are easily confused (see Perron 1989; Zivo and Andrews 1992). Recenly he confusion beween long memory and srucural change has been geing more and more aenion. Granger and Ding (1996), Granger and Hyung (2004), and Choi and Zivo (2007) suggesed ha he observed long memory propery in volailiy can be explained by he presence of srucural breaks. To invesigae his conjecure for realized volailiy, we use he pure muliple mean break mehod proposed by Bai and Perron (1998, 2003), henceforh BP. The m-break model (m + 1 regimes) is defined as y = c + u, = T + 1, T + 2,..., T, (4) j j 1 j 1 j where j = 1, 2,, m + 1, y is he logarihmic realized volailiy, and c j is he mean of he logarihmic realized volailiy. The break poins ( T 1, T 2,..., T) m are reaed as unknown. The error erm u may be serial correlaed and heeroskedasic. Esimaion is based on he leas-squares principle as described in BP. We use he ess for srucural change proposed in BP. Le sup F ( l ) denoe he F saisic for he null of no srucural breaks versus an alernaive hypohesis conaining an arbirary number of breaks, and le M denoe he maximum number of breaks allowed. We se M = 5 wih rimming value = Define he double maximum saisic UD = max sup F ( l), and he weighed double max saisic max 1 l M T T 7

8 WD = max w sup F ( l), where he weighs w l are such ha he marginal p-values are max 1 l M l T equal across values of l. The null hypohesis of boh ess is no srucural breaks agains he alernaive of an unknown number of breaks given some specific upper bound M. The sequenial sup F ( l+ 1 l) ess he null of l breaks versus he alernaive l + 1 breaks. To deermine he T number of breaks, we firs use UD max and WD max o deermine if a leas one break occurred. If here is evidence for srucural change, we selec he number of srucural breaks using sup F ( l+ 1 l). To allow for a penaly facor for he increased dimension of a model, he above T procedure may be complemened by selecing he number of breaks by minimizing a Bayesian Informaion Crierion (BIC) or a modified Schwarz Crierion (LWZ) Muliple Srucural Break Esimaion Table 2 displays values of all he ess used o deermine he number of breaks for he logarihmic realized volailiy series. The UDmax, WDmax and sup F ( l ) ess poin o he presence of muliple breaks for all series. For DM/$, he sup F ( l+ 1 l) is significan a 1% level when l = 4, which suggess 5 breaks. BIC suggess 5 breaks as well while LWZ suggess 2 breaks. Therefore, we choose 5 breaks for DM/$. For Yen/$, sup F ( l+ 1 l) is significan when l = 3 bu no significan when l = 4, which sugges 4 breaks. We follow BIC and choose 5 breaks for Yen/$. For Yen/DM, sup F ( l+ 1 l) sugges 4 breaks as well as BIC. Hence 4 breaks are chosen for Yen/DM. T T T T [Inser Table 2 here] 8

9 In Table 2 we also repor he esimaes of he break daes wih heir respecive 90% confidence inervals. The break daes esimaed for DM/$ and Yen/$ are very similar, which suggess common break daes for he process: May 1989, March May 1991, March 1993, June Augus 1995, and May July The esimaes of he mean parameers ( c ˆ j ) for regimes (m + 1) are also provided on he boom of Table 2. Figure 2 shows he logarihmic realized volailiy wih he esimaed ĉ values superimposed. We noe ha he mean shifs coincide wih hisorical financial or currency crisis. For example, he highes volailiy regime of DM/$ exchange rae occurred beween wo breaks: March 1991 and March 1993 and he second highes volailiy regime of Yen/DM exchange rae occurred beween wo breaks: June 1992 and May Boh regimes can be aribued o he Exchange Rae Mechanism (ERM) crisis of in Europe. The Asian financial crisis occurred in 1997 and appears o cause he breaks in July 1997 for DM/$, May 1997 for Yen/$, and May 1997 for Yen/DM. Since he breaks in May 1997, Yen/$ and Yen/DM have urned o heir highes volailiy regimes among he whole sample. [Inser Figure 2 here] 3.3. Long Memory Esimaion Afer Adjusing for Srucural Breaks Table 3 shows he long memory parameer esimaes of he hree series afer adjusmen for he esimaed srucural breaks 6. The parameer d is esimaed using he residual series y cˆ. j All esimaes of d are lower afer using break-adjused series, especially in Yen/DM series. The long memory parameer of Yen/DM decreases from o 0.283, from o , and from o for While, ARFIMA, and SEMIFAR mehods, respecively. Figure 3 6 In conras o our wo-sep approach, Hsu (2005) proposed a one-sep approach which esimaes he long memory parameer direcly from he daa wih unknown mean changes. Using such a procedure, he found a smaller long memory parameer for G7 inflaion raes. 9

10 displays he auocorrelaion funcion for he adjused volailiy series. Compared o Figure 1 for he auocorrelaion before adjusmen for breaks, i is eviden ha he persisence of volailiy has been reduced afer removing he esimaed breaks. The presence of srucure breaks, however, can no oally explain he persisence of exchange rae realized volailiy. Oher reasons for he observed long memory migh be (1) aggregaion of inraday squared reurn series; (2) mixure of numerous heerogeneous shor-run informaion arrivals (Andersen and Bollerslev, 1997). [Inser Table 3 here] [Inser Figure 3 here] The paern of mean shifs in Figure 2, suggess ha here migh be a smooh upward rend in he volailiy series, especially in he Yen/DM series which has he mos persisence, which is an alernaive o abrup mean breaks. To invesigae his alernaive, we esimae he SEMIFA model wih flexible rend in (3). The resuls for he esimaed rend are shown in Figure 4. We see ha he rend is no saisically significan. Our resuls show ha he realized volailiy series is beer characerized by abrup mean shifs han by a smooh flexible rend. [Inser Figure 4 here] Baillie and Kapeanios (2007) proposed a es for nonlineariy of unknown form in addiion o he long memory componen in a ime series process. Using he same ABDL daa, hey rejeced he null of lineariy for Yen/$ and Yen/DM. Their nonlineariy forms are essenially approximaions o smooh ransiion auoregressive (STAR) models. Such STAR models could generae series similar o srucural breaks, especially wih he larger speed of ransiion parameer in some exponenial STAR (ESTAR) or logisic STAR (LSTAR) models. I is difficul 10

11 o compare he in-sample fi of he BP model and STAR model for he realized volailiy in he presence of long memory. Therefore he choice beween he models is likely o be seled in erms of economic plausibiliy and research ineress. Inuiively, if he exchange rae volailiy has more recurren changes, hen he STAR model is more appropriae. If changes of exchange rae volailiy are more likely o be nonrecuren, hen he mean-shif model is favored Mone Carlo Simulaion of Long Memory Process We discussed previously ha srucural change is easily confused wih long memory. Granger and Hyung (2004) poined ou ha here exiss anoher perplexiy: a long memory model wihou breaks may cause breaks o be deeced spuriously by sandard esimaion mehods. To illusrae his phenomenon, we generaed six long memory series wih d = 0.1, 0.2, 0.3, 0.35, 0.4, 0.45, respecively, wih mean: -0.5, sandard deviaion: 0.4, and sample size: 3,045. These series are similar o our sample logarihmic realized volailiy. Table 4 shows resuls for he srucural break ess of BP for he differen daa-generaing processes (DGPs). The resuls sugges a posiive relaionship beween he number of breaks and he value of d as found in Granger and Hyung (2004). This reveals a possibiliy ha a long memory/fracionally inegraed process iself conains some porion of a permanen shock, which ofen appears as a break in some siuaions 7. The imporan implicaion from his Mone Carlo evidence is ha he long memory DGP provides a good parsimonious alernaive of in-sample fi for he rue srucural-break DGP when we have lile knowledge for he pas break daes and size 8. [Inser Table 5 here] 7 Currenly here is no formal es available for muliple srucural changes in he I(d) process wih unknown number of breaks. The nonlinear es developed by Baillie and Kapeanios (2007) conribue he lieraure a his poin. 8 This propery, which is rivial here, will become much more imporan when we discuss he long memory and srucural breaks for ou-of-sample forecasing in Secion 5. 11

12 4. Forecas Evaluaion and Simulaion 4.1. I(d) versus Breaks Model Many models have been provided for forecasing asse reurn volailiy and he success of a volailiy model lies in is ou-of-sample forecasing power. For example, ABDL proposed he following rivariae VAR-RV-I(d) (fracionally inegraed Gaussian vecor auoregressiverealized volailiy) model, d Φ( L)(1 L) ( Y μ) = ε, (5) where Y is ( 3 1) vecor of logarihmic realized exchange rae volailiies, Φ ( L) is a lag polynomial, μ is he uncondiional mean, and ε is a vecor whie noise process. They fixed he value of d for each series a 0.401, which is close o our long memory esimaes in Table 1. They se he order of Φ ( L) o five o capure lagged effecs up o a week. They compared volailiy forecass from several popular models, and hey found ha heir VAR-RV-I(d) model produced superior ou-of-sample one-sep-ahead and en-sep-ahead forecass. Granger and Joyeux (1980), however, poined ou ha a long memory model would no necessarily produce clearly superior shor-run forecass, which is of ineres in financial forecasing. As an alernaive o (5), we consider a shor memory model for break-adjused series o produce shor-run forecass. Our alernaive model, denoed VAR-RV-Break, is * Φ( L)( Y μ ) = ε, (6) where * Y is he vecor of break-adjused logarihmic realized exchange rae volailiies and he order of Φ ( L) is equal o five following ABDL. Forecass are obained by esimaing rolling models. We esimae iniially over he firs 2449 observaions, December 2, 1986 o December 1, 1996, and using he in-sample parameer esimaes one-day-ahead forecass are made for he nex 12

13 day, day The process is hen rolled forward 1 day, deleing he firs observaion and adding on he 2450h observaion, he model is re-esimaed and he second forecas is made for The rolling mehod is repeaed unil observaion 3045, he end of he ou-of-sample forecas period. We ge 596 one-sep-ahead predicions in he ou-of-sample period, which is from December 2, 1996 o June 30, Known versus Unknown Breaks We consider wo siuaions. Firs, we assume ha he fuure (ou-of-sample) break daes and sizes are known. In real-ime forecass by financial praciioners, i is someimes possible o idenify breaks and make adjusmens using human judgmen. For example, when he Bank of Japan announced ha i would abandon is zero-ineres rae policy, hey knew o some exen ha srucural breaks in Japanese bonds marke s volailiy would happen. In his case, when he breaks happened, hey quickly adjused he forecass based on he given ou-of-sample breaks daes and means. Second, we assume ha we have lile knowledge abou ou-of-sample breaks and sizes. For example, i was no known ha a srucural break occurred in 1984 for he US real oupu volailiy unil recenly. As a resul, many macroeconomiss did no do any real-ime adjusmens even hough hey made sysemaic forecass errors. They could have used he BP mehod o deec srucural breaks by rolling over he ou-of-sample period, bu hey would sill need o wai a sufficien amoun of ime o esimae i Forecas Evaluaion and Comparison 9 We choose his in-sample period o compare our resul o hose in ABDL. 10 Andrews (1993) suggess a resriced sample inerval [0.15, 0.85] insead of he full inerval for rimming o avoid he much reduced power of es saisics. 13

14 Figure 5 displays he DM/$, Yen/$, and DM/Yen realized volailiy along wih he corresponding one-day-ahead VAR-RV-Break forecass under he assumpion ha he fuure breaks are known. I appears ha our forecass capure movemen of he realized volailiies well. To deermine which model provides more informaion abou he fuure value, we use he Mincer and Zarnowiz (1969) encompassing regression 11 VAR RV Break Model + 1, i β0 β1 + 1, i β2 + 1, i ε vol = + vol + vol +, (7) VAR RV Break where vol+ 1 denoes predicions by our benchmark VAR-RV-Break model, and vol Model 1 + denoes predicions from some oher candidae model. The alernaive models are he ones used in ABDL 12. [Inser Figure 5 here] In addiion o (7), we also evaluae ou-of-sample forecass using he relaive mean squared error (MSE), Model vol+ 1 Break vol+ 1 ( vol ) ( vol ) (8) 11 This is a regression-based mehod where he predicion is unbiased only if β 0 =0 and β 1 =1. When here are more han one forecasing models, addiional forecass are added o he righ-hand-side o check for incremenal explanaory power. The firs forecas is said o subsume informaion in oher forecass if hese 2 addiional forecass do no significanly increase he R. 12 The VAR-RV-I(d) model is he main model. The VAR-ABS model is a fracionally inegraed vecor auoregressive using daily absolue reurns insead of realized volailiy. The daily AR(1) GARCH (1,1) models for DM/$, Yen/$, and DM/Yen use AR coefficiens 0.986, 0.968, and 0.99, respecively. The RiskMerics model is an exponenially weighed moving average of he cross producs of daily reurns wih smoohing facor λ =0.94. The fracionally inegraed exponenial GARCH (FIEGARCH) (1,d,0) by Bollerslev and Mikkelsen (1996) is a varian of FIGARCH model by Baillie, Bollerslev, and Mikkelsen (1996). The las one is he high-frequency FIEGARCH model using he deseasonalized and filered 30- minues reurns. 14

15 where he denominaor is he benchmark model mean squared forecas error and he numeraor is he candidae mehod s mean squared forecas error. If he relaive MSE is less han one, he candidae model forecas is deermined o have performed beer han he benchmark. The forecas evaluaion saisics are presened in Table 5. In general, our VAR-RV-Break model ouof-sample forecass perform as well as ABDL s VAR-RV-I(d) model, and ouperform mos of he oher models. [Inser Table 5 here] 4.4. Forecas Evaluaion Resuls Firs, he regression 2 R from he VAR-RV-Break model is similar o ha from VAR- RV-I(d) model and is higher han mos of he oher models. Second, we canno rejec he hypohesis ha β 0 = 0 and β 1 = 1 in he VAR-RV-Break model using ess while we rejec he hypohesis ha β 0 = 0 and/or β 2 = 1 for all he oher models excep he VAR-RV-I(d) model. Third, in he encompassing regression ha includes boh he break model and an alernaive forecas, he esimaes for β 1 are closer o uniy and he esimaes for β 2 are closer o zero. Fourh, including an alernaive forecas mehod has lile conribuion o increasing mos of he relaive MSEs are bigger han one 2 R. Finally, The resuls in Table 5 show he superior forecasing abiliy of he VAR-RV-Break model when he fuure break daes and sizes are known in he ou-of-sample period. This resul is consisen wih Hyung, Poon and Granger (2006), who invesigaed ou-of-sample forecasing of S&P 500 reurn volailiy. Pesaran, Peenuzzo, and Timmermann (2006) provided a Bayesian esimaion and predicion procedure allowing breaks o occur over he forecas horizon. They 15

16 found ha heir mehod, which has similar inference wih known break informaion (prior), leads o beer ou-of-sample forecass han alernaive models. As shown in Table 6, wihou knowledge of ou-of-sample breaks, unsurprisingly, he predicion abiliy of he VAR-RV-Break deerioraes. The degree of deerioraion depends on he numbers and sizes of he ou-of-sample breaks. For he DM/$ series, he VAR-RV-Break model sill ouperforms all models excep he VAR-RV-I(d) model because he ou-of-sample break is no large as shown in Figure 2. Bu for he Yen/$ and Yen/DM, which show larger breaks, he VAR-RV-Break model s predicion abiliy becomes inferior o he oher models (excep VAR- ABS). In his case, he VAR-RV-I(d) is he bes forecasing model. [Inser Table 6 here] 4.5. Forecas Simulaion for Break and Long Memory Models To check he robusness of our comparison of he VAR-RV-Break and he VAR-RV-(d) ou-of-sample forecass, we perform he following simulaion experimen. To mimic he daily logarihmic exchange rae realized volailiy daa, we generae 3000 observaions from an AR(1) process wih auoregressive coefficien equal o 0.41 and uncondiional variance equal o We pariion he series ino six periods by four ad hoc breaks as shown in Figure 6.A. Each period s range and mean are as follows: Period 1[1:700; 0.5], Period 2[701:1500; -1.3], Period 3[1501:2000; -0.5], Period 4[2001:2300; -0.5], Period 5[2301:2700; -1.2], and Period 6[2701:3000, 0.7]. The 2000 observaions in Period 1 o Period 3 are reaed as he in-sample period and he 1000 observaions in Period 4 o Period 6 are reaed as he ou-of-sample period.. [Inser Figure 6 here] 16

17 For he AR-Break model, we perform one-sep-ahead forecass based on he rue DGP. Again, we consider wo siuaions. Firs, when he ou-of-sample breaks are known, we adjus he mean for he forecas evaluaion. If he break informaion is no known, we don do any adjusmen. For he AR-I(d) model, we use in-sample daa (Figure 6.A P1 o P3) o iniially esimae he long memory parameer and he AR(1) coefficien based on he SEMIFAR model (3) 13 We ge d = and AR(1) = We hen compue rolling one-sep-ahead forecass using he updaed SEMIFAR model. Figure 6.B shows he resuls for Period 4 in which an ou-ofsample break has no occurred. Wheher breaks are known or no, he break model performs slighly beer han he I(d) model. Surprisingly, in Period 5 and 6 afer breaks occurred, he I(d) model sill accuraely predics while he break model deerioraes subsanially when he breaks are unknown. Table 7 conains he roo mean squared error (MSE), 2 1/2 y Model + 1 y + 1 N, (9) [ ( ) / ] and relaive mean squared error, I( d) y+ 1 Break y+ 1 ( y ) ( y ), (10) o evaluae N ou-of-sample forecass from he known break model, unknown break model, and I(d) model. In Period 4, he relaive MSEs for he known break and unknown break models are equal o 1.02 implying ha break models perform slighly beer han he I(d) model. In period 5, 6, and he whole sample, he relaive MSEs for he known break model are 1.117, 1.589, and 1.222, respecively. Unsurprisingly, he known break model forecass beer for he rue break process han he I(d) model. Compared wih he resuls in Table 6 for he exchange rae realized 13 The number of lagged values o use for predicion is 50. k= is chosen o deermine he Fourier frequencies o use in evaluaing he heoreical specrum of an ARFIMA model. 17

18 volailiy daa, he relaive MSE values are 0.98, 1.02, and 1.04, respecively, which means he known break model s forecasing is no as good as ha for he simulaed daa. This is reasonable because he rue series in Table 5 is no a pure AR-Break process. For he unknown break model, he relaive MSEs in Period 5, 6 and he whole ample are 0.301, 0.155, and 0.257, respecively. The I(d) model performs far beer han he unknown break model. [Inser Table 7 here] The lower panel of Table 7 shows he forecass evaluaion saisics for en-sep ahead forecass. We see a similar paern in hese forecas evaluaion saisics. The known break model is beer han he I(d) model, and he I(d) model is beer han he unknown break model. I is worh noing ha when we compare he roo MSE resuls in Table 7, we see ha he I(d) model s 10-sep-ahead forecass worsen more han break model s forecass. For he I(d) model, is whole ou-of-sample roo MSE increases from o when i forecass from one-sep-ahead o en-sep-ahead. In conras o he known break model, is whole ou-of-sample roo MSE only increases from o when i forecass from one-sep-ahead o en-sep-ahead. Alhough he long memory is expeced o have beer muli-sep-ahead forecass compared o oher models, he findings here do no suppor his expecaion. In summary, when he DGP is a pure mean break series wihou any long memory, we can ge very good ou-of-sample forecas performance using a simple AR-I(d) model. This resul shows ha a long memory/fracional inegraed model may sill be he bes forecasing model when he rue financial volailiy series are generaed by srucural breaks and we have lile knowledge abou he iming and sizes of he breaks. 18

19 5. Conclusions Modeling realized volailiy consruced from inraday high-frequency daa provides improved ou-of-sample forecass compared wih radiional volailiy models. ABDL s VAR- RV-I(d) model beas popular GARCH-ype models for forecasing daily realized volailiies for he DM/$, Yen/$ and Yen/ DM spo exchange raes. The main reason is ha he former model, which explois he inraday volailiy informaion, provides an accurae and fas-adaping esimae of curren volailiy while he laer model, depending on slowly decaying pas squared reurns, adaps only gradually o curren volailiy shocks. In ligh of his propery, we show ha superior volailiy forecass could also be produced using a shor memory model for realized volailiy which incorporaes mean shif informaion on realized volailiy insead of long memory. Our shor-memory-break model generally performs he bes provided he iming of fuure break daes and heir sizes are known. Wih lile knowledge abou ou-of-sample break daes and heir sizes, we show ha a parsimonious long memory model is a robus forecasing model even when he rue financial volailiy series are generaed by srucural breaks. Acknowledgemens We hank Torben Andersen and Tim Bollerslev for sharing he realized variance daa for he exchange rae and relaed daa in Andersen, Bollerslev, Diebold, and Labys (2003). We are also graeful o he helpful commens from anonymous referees. 19

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22 Granger, C.W.J., and N. Hyung. (2004). Occasional Srucural Breaks and Long Memory wih an Applicaion o he S&P 500 Absolue Sock Reurns, Journal of Empirical Finance, 11, pp Henry, M. (2001). Robus Auomaic Bandwidh for Long Memory, Journal of Time Series Analysis, 22:3, pp Hosking, J. R. M. (1981). Fracional Differencing, Biomerika, 68, pp Hyung, N., S. H. Poon, and C. W. J. Granger. (2006). A Source of Long Memory in Volailiy, Working paper. Hsu, C.C. (2005). Long Memory or Srucural Changes: An Empirical Examinaion on Inflaion Raes, Economics Leers, 88, pp Künsch, H. (1987). Saisical Aspecs of Self-Similar Processes, Proceedings of he Firs World Congress of he Bernoulli Sociey, Vol, I, VNU Science Press, Urech, pp Mincer, J., and V. Zarnowiz. (1969). The Evaluaion of Economic Forecass, in Economic Forecass and Expecaions, edied by W. F. Sharpe, and C. M. Cooner. Englewood Cliffs, New Jersey: Prenice-Hall. Perron, P. (1989). The Grea Crash, he Oil Price Shock and he Uni Roo Hypohesis, Economerica, 57, pp Pesaran, M. H., D. Peenuzzo, and A. Timmermann. (2006). Forecasing Time Series Subjec o Muliple Srucural Breaks, Review of Economic Sudies, 73:4, pp Robinson, P. M. (1995a). Gaussian Semiparameric Esimaion of Long Range Dependence, Annals of Saisics, 23, pp Robinson, P. M. (1995b). Log-Periodogram Regression of Time Series wih Long-Range Dependence, Annals of Saisics, 23, pp Sowell, F. (1992). Maximum Likelihood Esimaion of Saionary Univariae Fracionally Inegraed Time Series Models, Journal of Economerics, 53, pp

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24 Table 1. Esimaion for Long and Shor Memory of Log Realized Volailiy d AR(1) MA(1) Local While DM/$ Yen/$ Yen/DM (0.0269) (0.0269) (0.0269) N/A N/A N/A N/A N/A N/A ARFIMA DM/$ Yen/$ Yen/DM (0.0142) (0.0142) (0.0316) (0.05) 0.49 (0.02) DM/$ 0 N/A (0.0142) SEMIFAR Yen/$ 0 N/A (0.0142) Yen/DM N/A (0.0212) (0.03) 1. The numbers in he parenheses indicae sandard errors. 2. Local While s mehod is based on Gaussian maximum likelihood esimaion wih he choice of bandwidh equal o 0.5, suggesed by Henry (2001). δ 3. ARFIMA model is based on Beran (1995). φ ( L)(1 L) [(1 L) m y μ] = θ( L) ε where -0.5 < d < 0.5. The ineger m is he number of imes ha y mus be differenced o achieve saionariy, and he long memory parameer is given by d = δ + m. The mehod uses BIC o choose he shor memory parameers p and q. When m = 0, μ is he expecaion of y ; when m = 1, μ is he slope of linear rend componen in y. 4. SEMIFAR (Semiparameric Fracional Auoregressive) model is based on Beran and Ocker (2001). δ φ( L)(1 L) [(1 L) m y g( i)] = ε. By using a nonparameric kernel esimae of g( i ) insead of consan erm μ. The mehod uses BIC o choose he shor memory parameer p. 24

25 Table 2. Muliple Srucural Changes Tes Resuls \ Series Saisics DM/$ Yen/$ Yen/DM Tess sup F T (1) sup F T (2) sup F T (3) sup F T (4) sup F T (5) UDmax 55.64** ** ** WDmax 84.95** ** ** sup F T (2 1) ** sup F T (3 2) sup F T (4 3) sup F T (5 4) Numbers of Changes Seleced BIC LWZ Sequenial Muliple Srucural Changes Daes Esimaion T ˆ 1 [ ] [ ] [ ] T ˆ 2 [ ] [ ] [ ] T ˆ 3 [ ] [ ] [ ] T ˆ 4 [ ] [ ] [ ] T ˆ 5 [ ] [ ] Esimaions of Mean for Each Regime ĉ (0.015) (0.017) (0.012) ĉ (0.017) (0.018) -0.5 (0.013) ĉ (0.016) (0.017) (0.015) ĉ (0.015) (0.017) (0.012) ĉ (0.017) (0.018) (0.014) ĉ (0.017) (0.018) 1. * indicaes 5% significance level 2. ** indicaes 1% significance level 3. In bracke are he 90% confidence inervals 4. In parenheses are sandard errors 5. Number of Changes Seleced From Sequenial Mehod is based on 1% level 25

26 Table 3. Esimaion for Long and Shor Memory of Log Realized Volailiy Afer Adjusmen for Breaks d AR(1) MA(1) Local While DM/$ Yen/$ Yen/DM (0.0675) (0.0675) (0.0675) N/A N/A N/A N/A N/A N/A ARFIMA DM/$ Yen/$ Yen/DM (0.0142) (0.0142) (0.0142) DM/$ (0.0142) 0 N/A SEMIFAR Yen/$ (0.0142) 0 N/A Yen/DM (0.0142) 0 N/A 1. The numbers in he parenheses indicae sandard errors. 2. The While, AFIMA, and SEMIFAR models are explained in he deails below Table 1. 26

27 Table 4. Esimaed Spurious Breaks for Long Memory Simulaion Breaks exis or no Number of Breaks Seleced d Udmax Wdmax sup FT ( l+ 1 l) BIC LWZ Sequenial 0.1 No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Six differen long memory parameers DGP based on Mone Carlo Simulaion for 3045 observaions. 2. Srucural breaks ess are based on Bai and Perron (1998, 2003). 3. The ess are based on 1% significance level. 27

28 Table 5. Ou-of-Sample Forecas Evaluaion When Fuure Breaks Are Known β 0 β 1 β 2 2 R Rel MSE DM/$ VAR-RV-Break (0.048) (0.091) VAR-RV-I(d) (0.049) (0.092) VAR-ABS (0.028) (0.089) Daily GARCH (0.063) (0.105) Daily RiskMerics (0.042) (0.075) Daily FIEGARCH (0.052) (0.083) Inraday FIEGARCH deseason/filer (0.060) (0.099) VAR-RV-Break + VAR-RV-I(d) (0.049) (0.332) (0.327) VAR-RV-Break + VAR-ABS (0.046) (0.102) (0.096) VAR-RV-Break + Daily GARCH (0.060) (0.120) (0.137) VAR-RV-Break + Daily RiskMerics (0.047) (0.119) (0.098) VAR-RV-Break + Daily FIEGARCH (0.052) (0.109) (0.100) VAR-RV-Break + Inraday FIEGARCH deseason/filer (0.059) (0.207) (0.217) Yen/$ VAR-RV-Break (0.106) (0.144) VAR-RV-I(d) (0.110) (0.151) VAR-ABS (0.086) (0.241) Daily GARCH (0.147) (0.187) Daily RiskMerics (0.108) (0.131) Daily FIEGARCH (0.193) (0.236) Inraday FIEGARCH deseason/filer (0189) (0.263) VAR-RV-Break + VAR-RV-I(d) (0.101) (0.564) (0.662) VAR-RV-Break + VAR-ABS (0.109) (0.148) (0.136) VAR-RV-Break + Daily GARCH (0.141) (0.131) (0.263) VAR-RV-Break + Daily RiskMerics (0.112) (0.102) (0.134) VAR-RV-Break + Daily FIEGARCH (0.209) (0.260) (0.484) VAR-RV-Break + Inraday FIEGARCH deseason/filer (0.252) (0.375) (0.734) DM/Yen VAR-RV-Break (0.096) (0.132) VAR-RV-I(d) (0.101) (0.143) VAR-ABS (0.062) (0.175) Daily GARCH (0.092) (0.119) Daily RiskMerics (0.084) (0.107) Daily FIEGARCH (0.105) (0.144) Inraday FIEGARCH deseason/filer (0.150) (0.217) VAR-RV-Break + VAR-RV-I(d) (0.099) (0.452) (0.548) VAR-RV-Break + VAR-ABS (0.094) (0.148) (0.140) VAR-RV-Break + Daily GARCH (0.082) (0.135) (0.167) VAR-RV-Break + Daily RiskMerics (0.089) (0.117) (0.121) VAR-RV-Break + Daily FIEGARCH (0.106) (0.118) (0.143) VAR-RV-Break + Inraday FIEGARCH deseason/filer (0.156) (0.294) (0.530) In parenheses are sandard errors. 28

29 Table 6. Ou-of-Sample Forecas Evaluaion When Fuure Breaks Are Unknown β 0 β 1 β 2 2 R Rel MSE DM/$ VAR-RV-Break (0.050) (0.088) VAR-RV-I(d) (0.049) (0.092) VAR-ABS (0.028) (0.089) Daily GARCH (0.063) (0.105) Daily RiskMerics (0.042) (0.075) Daily FIEGARCH (0.052) (0.083) Inraday FIEGARCH deseason/filer (0.060) (0.099) VAR-RV-Break + VAR-RV-I(d) (0.050) (0.179) (0.196) VAR-RV-Break + VAR-ABS (0.047) (0.103) (0.104) VAR-RV-Break + Daily GARCH (0.060) (0.131) (0.161) VAR-RV-Break + Daily RiskMerics (0.046) (0.131) (0.116) VAR-RV-Break + Daily FIEGARCH (0.051) (0.111) (0.108) VAR-RV-Break + Inraday FIEGARCH deseason/filer (0.059) (0.189) (0.210) Yen/$ VAR-RV-Break (0.127) (0.218) VAR-RV-I(d) (0.110) (0.151) VAR-ABS (0.086) (0.241) Daily GARCH (0.147) (0.187) Daily RiskMerics (0.108) (0.131) Daily FIEGARCH (0.193) (0.236) Inraday FIEGARCH deseason/filer (0189) (0.263) VAR-RV-Break + VAR-RV-I(d) (0.120) (0.151) (0.146) VAR-RV-Break + VAR-ABS (0.137) (0.212) (0.155) VAR-RV-Break + Daily GARCH (0.147) (0.129) (0.203) VAR-RV-Break + Daily RiskMerics (0.132) (0.137) (0.108) VAR-RV-Break + Daily FIEGARCH (0.185) (0.146) (0.295) VAR-RV-Break + Inraday FIEGARCH deseason/filer (0.190) (0.260) (0.450) DM/Yen VAR-RV-Break (0.138) (0.248) VAR-RV-I(d) (0.101) (0.143) VAR-ABS (0.062) (0.175) Daily GARCH (0.092) (0.119) Daily RiskMerics (0.084) (0.107) Daily FIEGARCH (0.105) (0.144) Inraday FIEGARCH deseason/filer (0.150) (0.217) VAR-RV-Break + VAR-RV-I(d) (0.131) (0.175) (0.096) VAR-RV-Break + VAR-ABS (0.142) (0.269) (0.143) VAR-RV-Break + Daily GARCH (0.123) (0.162) (0.110) VAR-RV-Break + Daily RiskMerics (0.132) (0.177) (0.085) VAR-RV-Break + Daily FIEGARCH (0.148) (0.194) (0.107) VAR-RV-Break + Inraday FIEGARCH deseason/filer (0.133) (0.192) (0.295) In parenheses are sandard errors. 29

30 Table 7. Ou-of-Sample Forecas Evaluaion for Simulaions Period 4 Period 5 Period 6 Whole Ou-of-Sample One-Sep Ahead AR-Break Model Known Break Unknown Break AR-I(d) Model Roo MSE Relaive MSE Roo MSE Relaive MSE Roo MSE Relaive MSE Roo MSE Relaive MSE Period 4 Period 5 Period 6 Ten-Sep-Ahead AR-Break Model Known Break Unknown Break AR-I(d) Model Roo MSE Relaive MSE Roo MSE Relaive MSE Roo MSE Relaive MSE Whole Roo MSE Ou-of-Sample Relaive MSE /2 1. Roo Mean Squared Error (MSE) is calculaed as [ ( y Model 10 y 10 ) / N]. 2. Relaive MSE is calculaed as I( d) y+ 1 ( y ) Break + 1 y+ 1 ( y ) 2 30

31 A u o co rre la io n s fo r D M /$ L o g R e a liz e d V o la iliy ACF Lag Auocorrelaions for Yen/$ Log Realized Volailiy ACF Lag A u o co rre la io n s fo r Y e n /D M L o g R e a lize d V o la iliy ACF Lag Figure 1. Auocorrelaions for Log Realized Volailiy 31

32 LVDM_D BS_DM_D LVYEN_D BS_YEN_D LVYEN_DM BS_YEN_DM Figure 2. Esimaed Srucural Breaks Means and Daes for Daily Exchange Rae Log Realized Volailiy ( ) 32

33 o correlaions for DM /$ Log Realized Volailiy Afer Adjusing B ACF Lag o correlaions for Yen/$ Log Realized Volailiy Afer Adjusing B ACF Lag c o rre la io n s fo r Y e n /D M L o g R e a lize d V o la iliy A fe r A d ju sin g ACF Lag Figure 3. Auocorrelaions for Log Realized Volailiy Afer Adjusing for Srucural Breaks 33

34 DM/$ Log Realized Volailiy Original Series Smoohed Trend series x$rend Fied Values Residuals fie d x$residuals Yen/$ Log Realized Volailiy Original Series Smoohed Trend series x $ r e n d Fied Values Residuals fie d x $ r e s id u a ls Yen/DM Log Realized Volailiy Original Series Smoohed Trend series x$rend Fied Values Residuals fie d x$residuals Figure 4. Semiparameric Fracional Auoregressive Model Decomposiion Noes: Based on Beran and Ocker s mehod (2001) 34

35 D M /$ re a liz e d v o la iliy P r e d ic e d V a lu e R e a liz e d V a lu e Q 1 Q 2 Q 3 Q 4 Q 1 Q 2 Q 3 Q 4 Q 1 Q 2 Q Yen/$ realized volailiy Prediced Value R ealized Value Q 1 Q 2 Q 3 Q 4 Q 1 Q 2 Q 3 Q 4 Q 1 Q 2 Q D M /Y e n r e a liz e d v o la iliy P r e d ic e d V a lu e R e a liz e d V a lu e Q 1 Q 2 Q 3 Q 4 Q 1 Q 2 Q 3 Q 4 Q 1 Q 2 Q Figure 5. Realized Volailiy and Ou-of-Sample VAR-RV-Break Forecass 35