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1 Auhor's personal copy Journal of Economerics 147 (2008) Conens liss available a ScienceDirec Journal of Economerics journal homepage: Ou of sample forecass of quadraic variaion Yacine Aï-Sahalia a,b,, Loriano Mancini c a Deparmen of Economics, Princeon Universiy, Unied Saes b NBER, Unied Saes c Swiss Banking Insiue, Universiy of Zurich, Swizerland a r i c l e i n f o a b s r a c Aricle hisory: Available online 18 Sepember 2008 JEL classificaion: C14 C22 C53 We compare he forecass of Quadraic Variaion given by he Realized Volailiy (RV) and he wo Scales Realized Volailiy (SRV) compued from high frequency daa in he presence of marke microsrucure noise, under several differen dynamics for he volailiy process and assumpions on he noise. We show ha SRV largely ouperforms RV, wheher looking a bias, variance, RMSE or ou-of-sample forecasing abiliy. An empirical applicaion o all DJIA socks confirms he simulaion resuls Elsevier B.V. All righs reserved. Keywords: Marke microsrucure noise High frequency daa Measuremen error Realized volailiy wo scales realized volailiy Ou of sample forecass 1. Inroducion In financial economerics, one is ofen ineresed in esimaing he volailiy of he efficien log-price process dx = µ d + σ dw (1.1) using discreely sampled daa on he ransacion price process a imes 0,,..., n =. he volailiy of asse reurns plays an imporan role in derivaive pricing and hedging, asse allocaion, risk managemen prey much all of asse pricing. When σ is iself a sochasic process, a main objec of ineres is he quadraic variaion or inegraed variance (IV) Financial suppor from he NSF under grans SES and DMS (Aï-Sahalia) and from he Universiy Research Prioriy Program Finance and Financial Markes Universiy of Zurich and he NCCR-FinRisk Swiss Naional Science Foundaion (Mancini) is graefully acknowledged. For helpful commens, we hank wo anonymous referees, Parick Cheridio, Fulvio Corsi and seminar paricipans a he conferences Risk Measures & Risk Managemen for High- Frequency Daa, EURANDOM, 2006, Microsrucure of Financial and Money Markes, CRES, 2006, and he 33rd Annual Meeing of he European Finance Associaion, EFA, his research was underaken while Mancini visied he Deparmen of Operaions Research and Financial Engineering a Princeon Universiy. Corresponding auhor a: Deparmen of Economics, Princeon Universiy, Unied Saes. addresses: yacine@princeon.edu (Y. Aï-Sahalia), mancini@isb.uzh.ch (L. Mancini). X, X = 0 d (1.2) over a fixed ime period [0, ], say one day. he usual esimaor of X, X is he realized volailiy (RV), which is simply he sum of observed squared log-reurns n [X, X] = (X i+1 X i ) 2 (1.3) i=1 and his esimaor has been used exensively in he recen lieraure: see e.g., Andersen and Bollerslev (1998) and Barndorff- Nielsen and Shephard (2002). In heory, sampling a increasingly higher frequency should deliver, in he limi, a consisen esimaor of he quadraic variaion. he sum [X, X] consisenly esimaes he inegral X, X as has been well-known in he sochasic processes lieraure. Furher, he sum converges o he inegral wih a known disribuion, a resul daing back o Jacod (1994) and Jacod and Proer (1998). Selecing as small as possible (=n as large as possible) is opimal. Everyhing would be perfec if i were no for he fac ha financial asse reurns are subjec, especially a high frequency, o a vas array of fricions. As a resul, a perhaps more realisic model is one where he observed ransacion log-price is no X bu raher Y, he sum of an unobservable efficien price X and a noise componen due o he imperfecions of he rading process, ε: Y = X + ε. (1.4) /$ see fron maer 2008 Elsevier B.V. All righs reserved. doi: /j.jeconom

2 Auhor's personal copy 18 Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) ε summarizes marke microsrucure effecs, eiher informaional or no: bid-ask bounces, discreeness of price changes, differences in rade sizes or informaional conen of price changes, gradual response of prices o a block rade, he sraegic componen of he order flow, invenory conrol effecs, ec. In he presence of marke microsrucure noise as in model (1.4), afer suiable scaling, RV compued from he observed log-reurns, [Y, Y], is a consisen and asympoically normal esimaor of he quaniy 2nE[ε 2 ]. his quaniy is no he objec of ineres, X, X. In he high frequency limi where 0, marke microsrucure noise swamps he variance of he price signal. Differen soluions have been proposed for his problem. In he consan σ case, Zhou (1996) considers a bias correcing approach based on auocovariances ha is similar o he esimaor used in French e al. (1987). he behavior of his esimaor has been sudied by Zumbach e al. (2002). Efficien likelihood esimaion of σ is sudied by Aï-Sahalia e al. (2005), showing ha incorporaing ε explicily in he likelihood funcion of he observed log-reurns Y provides consisen, asympoically normal and efficien esimaors of he parameers ha are robus o various forms of misspecificaions of he noise erm. Hansen and Lunde (2006) sudy he Zhou esimaor and exensions in he case where volailiy is ime varying bu condiionally nonrandom. Relaed conribuions have been made by Oomen (2006) and Bandi and Russell (2006). he Zhou esimaor and is exensions, however, are inconsisen. his means in his paricular case ha, as he frequency of observaion increases, he esimaor diverges insead of converging o X, X : his was in fac recognized by Zhou, see p. 47 in Zhou (1996). In he sochasic volailiy case, Zhang e al. (2005b) propose a soluion o his problem which makes use of he full daa sample and delivers he firs consisen esimaors in he lieraure for X, X, as well as E[ε 2 ], in he presence of noise. he esimaor, wo Scales Realized Volailiy (SRV), is based on subsampling, averaging and bias-correcion. Our objecive in his paper is o examine and compare he ou-of-sample performance of RV and SRV as esimaors of he quadraic variaion, in a variey of conexs ha are ouside he conrolled seup ha has been used o derive he heoreical properies of hese esimaors. In oher words, does he beer performance of SRV ha is eviden when looking a is heoreical properies exend o he ou-of-sample forecass when he daa generaing mechanism incorporaes realisic feaures of he volailiy process such as ime series dependence in he noise, correlaion of he noise wih he price process, long memory, jumps and he like? his paper complemens oher papers which have sudied he forecasing abiliy of various volailiy esimaors. Andersen e al. (2003, 2004, 2005) sudy he sandard RV esimaor, looking a is poinwise forecass and he efficiency loss relaive o opimal, bu unfeasible, forecass based on he enire pah of volailiy. heir assumed model has an ARMA srucure for he log of he volailiy, and is one of he daa generaing processes we will consider. Under he eigenfuncion sochasic volailiy model of Meddahi (2001), hey obain analyical formulae for he auocovariance funcions of RV. Marke microsrucure noise is no incorporaed in hose papers, however. Analyical formulae for he auocovariance of RV wih noise are given independenly by Garcia and Meddahi (2006), and by Ghysels and Sinko (2006a) wihin he mixed daa sampling (MIDAS) framework. he laer auhors sudy he auocovariance correcions for RV proposed by Zhou (1996) and laer exended by Hansen and Lunde (2006), and show ha hey do no improve volailiy forecasing. Furher resuls are repored in Ghysels and Sinko (2006b) and Ghysels e al. (forhcoming), whose daa generaing process for he volailiy is of he same ype as in he papers jus cied, bu explicily include marke microsrucure noise. hey find ha a he high frequencies where marke microsrucure noise maers (alhough heir sudy does no consider observaion frequencies higher han one minue), SRV predics volailiy he bes; hey also find ha he noise componen consruced as a difference beween he sandard RV esimaor and SRV, is predicable on average. While wriing his paper, we became aware ha Andersen e al. (2006) are in he process of exending heir previous forecasing work o marke microsrucure robus noise measures. Corradi e al. (forhcoming, 2005) consruc condiional predicive densiies and confidence inervals for IV and esimae hese densiies using nonparameric kernel esimaors. hey use differen esimaors for IV, including RV and SRV and derive confidence bands around heir kernel esimaors. heir findings confirm he heoreical predicions of Zhang e al. (2005b) ha, when he ime inerval beween successive observaions becomes small, he signal o noise raio of he daa decreases, and realized volailiy and bipower variaion end o explode, insead of converging o he incremens of quadraic variaion. his resul is visible from heir predicive densiies, as he ranges of he densiies of hese wo uncorreced esimaors widen considerably as he observaion frequency increases. hey also find ha, SRV, as a microsrucurerobus measure of volailiy, is sable, and increasing he frequency a which he daa are sampled does no seem o induce any appreciable disorion in is densiy esimaor. Unlike hose papers, we perform a horse race beween differen esimaors by running Mincer Zarnowiz forecasing regressions, and use widely differen daa generaing processes. On he oher hand, unlike Corradi e al. (forhcoming, 2005), bu like he oher papers menioned, we look only a poin esimaes and no a he full forecas disribuion. Our resuls are unambiguous: no maer wha daa generaing process we consider, and wha assumpions we make on he noise (magniude wihin reason, iid or auocorrelaed, correlaed or no wih he price signal), we find ha SRV produces beer forecass of fuure IV han he sandard RV does, someimes by a wide margin. his paper is organized as follows. Secion 2 summarizes he RV and SRV esimaors we consider. Secion 3 presens our Mone Carlo resuls under differen assumpions: firs he sandard Heson sochasic volailiy model; second a jumpdiffusion model for he volailiy; hird a log-volailiy model; fourh a heerogeneous auoregressive realized volailiy (HAR- RV) model; fifh a long memory model for volailiy based on a fracional Ornsein Uhlenbeck process. In Secion 4, we sudy he forecasing abiliy of hese esimaors when applied o high frequency DJIA socks daa. Secion 5 concludes. 2. Esimaing he quadraic variaion: RV and SRV We sar wih a brief descripion of he wo esimaors, RV and SRV, and heir heoreical properies. If one uses all he logreurns daa available (say sampled every second), [Y, Y] (all) L X, X }{{ } + 2nE[ε 2 ] objec of ineres bias due o noise + 4nE[ε 4 ] + 2 σ 4 n d 0 due o noise due o discreizaion oal variance 1/2 Z oal, (2.1) condiionally on he X process, where L denoes sable convergence in law and Z denoes a sandard normal variable. So he bias

3 Auhor's personal copy Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) erm due o he noise, which is of order O(n), swamps he rue quadraic variaion X, X, which is of order O(1). Of course, sampling as prescribed by [Y, Y] (all) is no wha is done in pracice. Insead, he esimaor [Y, Y ] (sparse) consruced by summing squared log-reurns a some lower frequency: 5 min, or 10, 15, 30 min, is ypically used. Reducing he value of n, from say 23,400 (1 s sampling) o n sparse = 78 ( sparse = 5 min sampling over he same 6.5 h), has he advanage of reducing he magniude of he bias erm 2nE[ε 2 ]. Ye, one of he mos basic lessons of saisics is ha one should no do his. SRV is a simple mehod o ackle he problem: firs, pariion he original grid of observaion imes, G = { 0,..., n } ino subsamples, G (k), k = 1,..., K, where n/k as n. For example, for G (1) sar a he firs observaion and ake an observaion every 5 min; for G (2), sar a he second observaion and ake an observaion every 5 min, ec. hen we average he esimaors obained on he subsamples. o he exen ha here is a benefi o subsampling, his benefi can now be reained, while he variaion of he esimaor can be lessened by he averaging. his gives rise o he esimaor [Y, Y] (avg) = 1 K K k=1 [Y, Y](k) consruced by averaging he esimaors [Y, Y] (k) obained on K grids of average size n = n/k. he properies of his esimaor are given by [Y, Y] (avg) L X, X }{{ } + 2 ne[ε 2 ] objec of ineres bias due o noise + 4 n K E[ε4 ] + 4 σ 4 3 n d 0 due o noise due o discreizaion oal variance While a beer esimaor han [Y, Y] (all) 1/2 Z oal., [Y, Y] (avg) remains biased. he bias of [Y, Y] (avg) is 2 ne[ε 2 ]; of course, n < n, so progress is being made. Bu one can go one sep furher. Indeed, E[ε 2 ] can be consisenly approximaed using RV compued wih all he observaions: Ê[ε 2 ] = 1 [Y, Y](all). 2n (2.2) Hence he bias of [Y, Y] (avg) can be consisenly esimaed by n [Y, Y](all) n. SRV is he bias-adjused esimaor for X, X consruced as X, X (srv) = [Y, Y] (avg) n [Y, Y](all). (2.3) n slow ime scale fas ime scale If he number of subsamples is opimally seleced as K = cn 2/3, hen SRV has he following disribuion: X, X (srv) L X, X }{{ } objec of ineres + 1 n 1/6 8 c 2 E[ε2 ] 2 + c 4 σ 4 3 d 0 due o noise due o discreizaion oal variance 1/2 Z oal. (2.4) Unlike all he previously considered ones, his esimaor is now correcly cenered. he consan c can be se o minimize he oal asympoic variance above. In small samples, a small sample refinemen o X, X can be consruced as follows ( X, X (srv,adj) = 1 n ) 1 X, X (srv). (2.5) n he difference wih he esimaor (2.3) is of order O p (K 1 ), and hus he wo esimaors have he same asympoic behaviors o he order ha we consider. However, he esimaor (2.5) is unbiased o higher order. When he microsrucure noise ε is iid, he log-reurns follow he MA(1) model Y τi Y τi 1 = τi τ i 1 µ d + τi τ i 1 σ dw + ε τi ε τi 1. (2.6) Hence he corresponding negaive auocorrelaion can affec he esimae of quadraic variaions given by [Y, Y ] (sparse). An ad hoc remedy is o de-mean and filer he raw high frequency reurns using an MA(1) model before applying such an esimaor; see Andersen e al. (2001). In he simulaion and empirical work ha follows, RV will refer o he esimaor [Y, Y ] (sparse) (wih or wihou he pre- MA(1) filering of he raw daa), a differen values of he sparse sampling inerval sparse and SRV o he esimaor X, X (srv,adj) (wih no need for MA(1) filering), wih differen values of he parameer K conrolling he number of subgrids we use. Finally, while SRV provides he firs consisen and asympoic (mixed) normal esimaor of he quadraic variaion X, X, as can be seen from (2.4), i has he rae of convergence n 1/6. Zhang (2006) shows ha i is possible o generalize SRV o muliple ime scales, by averaging no jus on wo ime scales bu on muliple ime scales. For suiably seleced weighs, he resuling esimaor, MSRV, converges o X, X a he slighly faser rae n 1/4. Compuing his esimaor, however, requires addiional compuaions due o he exra averaging sep (over an asympoically increasing number of ime scales) and will in general produce close resuls o hose given by SRV. An alernaive derivaion of SRV and MSRV, based on kernels, has subsequenly been proposed by Barndorff-Nielsen e al. (2006). 3. Mone Carlo evidence using alernaive daa generaing processes In his secion we perform several experimens using widely differen volailiy models. In each experimen we compare he in- and ou-of-sample forecas performances of RV and SRV compued a differen frequencies Heson sochasic volailiy model In his Mone Carlo experimen, we use as he daa generaing process he sochasic volailiy model of Heson (1993) for he insananeous variance dx = (µ /2)d + σ dw 1, d = κ(α )d + γ σ dw 2,. (3.1) We se he parameers µ, κ, α, γ and ρ, he correlaion coefficien beween he wo Brownian moions W 1 and W 2, o parameer values which are reasonable for a sock price, as in Zhang e al. (2005b), namely, µ = 0.05, κ = 5, α = 0.04, γ = 0.5, ρ = 0.5. he volailiy parameers saisfy he Feller s condiion 2κα γ 2 which makes he zero boundary unaainable by he volailiy process. We simulae M = 10,000 sample pahs of he process using he Euler scheme a a ime inerval = 1 s. We

4 Auhor's personal copy 20 Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 1 In-sample esimaes of daily IV 10 4 based on 10,000 simulaed pahs under he Heson model, d = 5(0.04 )d σ dw 2 Bias Var RMSE Rel. bias Rel. var Rel. RMSE RV 5 min RV 5 min MA(1) SRV 5 min RV 10 min SRV 10 min RV 15 min SRV 15 min RV 30 min SRV 30 min SRV minimum variance he efficien log-price dx = (0.05 /2)d + σ dw 1, and correlaion ρ = 0.5 beween Brownian moions. he observed log-price Y = X + ε, where ε NID(0, ). Euler discreizaion scheme wih ime sep = 1 s. assume ha he marke microsrucure noise, ε, has a Gaussian disribuion, and (Eε 2 ) 1/2 = 0.001, i.e. he sandard deviaion of he noise is 0.1% of he value of he asse price. We simulae coninuous sample pahs of m + 1 = 101 days, [0, 1 ],..., [ 99, 100 ], [ 100, 101 ], for he observed log-price process Y, ha is each sample pah consiss of ,400 logreurns, assuming ha each rading day consiss of 6.5 h, as is he case on he NYSE and NASDAQ. When consrucing he RV esimaor, he firs m = 100 days are used o esimae an MA(1) filer in order o remove as much of he noise as possible, and consisenly wih he common pracice in he empirical RV lieraure. his sep is no needed for he SRV esimaor. On each simulaed sample pah, he MA(1) model is esimaed using = 7800 log-reurns a he 5 min frequency and he de-meaned MA(1) filered reurns on he 100h day are used o esimae m m 1 d; a similar procedure is applied, for insance, by Andersen e al. (2001) o invesigae he IV s of he DJIA socks. he 100h day is used for he in-sample esimae of he IV and he las 101s day is saved for he ou-of-sample forecas of he IV One day ahead forecass of IV Denoing by 1 0 = 2 1 he one day ime inerval, i follows ha E[ 1 F 0 ] = a 0 + b, (3.2) where a = e κ( 1 0 ), b = α(1 e κ( 1 0 ) ), and follows he model (3.1): see e.g., Cox e al. (1985). F = σ { ; } is he informaion se or σ -field generaed by he insananeous variance process up o ime. Inerchanging he inegraion operaors, we have E [ 1 0 d F 0 ] = a in 0 + b in, (3.3) where a in = 1/κ(1 e κ( 1 0 ) ) and b in = α( 1 0 ) α/κ(1 e κ( 1 0 ) ). hen using (3.3) and (3.2) gives [ [ 2 ] ] E E d F 1 F 0 1 E = E[a in 1 + b in F 0 ] = a in E[ 1 F 0 ] + b in = a in (a 0 + b) + b in = a(a in 0 ) + a in b + b in ( [ 1 ] ) = a E d F 0 b in + a in b + b in 0 [ 2 1 d F 0 ] = ae [ 1 0 d F 0 ] + b( 1 0 ), (3.4) which will give he ou-of-sample forecas Eq. (3.6). Using (3.4), Bollerslev and Zhou (2002) propose a GMM-ype esimaor for he Heson model based on inra-day reurns for he foreign exchange markes. Noice ha he exac condiional forecas of 2 E [ 2 1 d is 1 d F 1 ] = a in 1 + b in, (3.5) which is no feasible as 1 is no observed In-sample esimaions of IV We presen he in-sample esimaion of IV given by SRV and RV under he Heson volailiy model. he same procedure is adoped in he oher experimens. We esimae he IV on he 100h day, m m 1 d, using he RV esimaor [Y, Y](sparse) and he SRV esimaor X, X (srv,adj) based on unfilered reurns and arbirarilydeermined frequencies of 5, 10, 15 and 30 min. We also repor he resuls for [Y, Y] (sparse) based on de-meaned MA(1) filered 5 min reurns and X, X (srv,adj) under he opimal sampling frequency. able 1 shows he in-sample resuls for he differen esimaion sraegies and SRV esimaors always largely ouperform all RV esimaors in erms of bias, variance and RMSE a each frequency. In-sample simulaions for his model are also repored in Zhang e al. (2005b), bu wih half variance for he microsrucure noise, ha is (Eε 2 ) 1/2 = Hence he ouperformance of SRV over RV is robus o an increase in he magniude of he noise, as expeced. Of course, going in he reverse direcion by decreasing he amoun of noise diminishes he advanage of SRV over RV, bu empirical sudies such as hose cied in Aï-Sahalia e al. (2005) sugges level of noise ha are significanly higher han able 1 also shows ha he MA(1) filer ends o improve he esimaion performance of RV esimaors, which remains largely inferior o he SRV performance. Similar in-sample resuls are obained for all he subsequen Mone Carlo experimens and o save space hey are no repored here, bu colleced in Aï-Sahalia and Mancini (2007). he minimum variance SRV is compued using K 100 subsamples which correspond o a slow ime scale of less han wo minues. 1 Fig. 1 shows he finie sample disribuion of he minimum variance SRV esimaor (2.5) and he corresponding asympoic Gaussian disribuion. Incidenally, he wo disribuions appear o be very close, confirming he finding of Zhang e al. (2005a), ha are robus o an increase in he magniude of microsrucure noise. In all he subsequen Mone Carlo experimens he finie sample disribuions of he minimum variance SRV are very close o he asympoic disribuion and he corresponding plos will be omied. 1 Roughly he same number of subsamples is also used in he subsequen Mone Carlo experimens.

5 Auhor's personal copy Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) Fig. 1. Asympoic and small sample disribuions of he minimum variance SRV esimaor based on 10,000 simulaed pahs of he insananeous variance under he Heson model, d = 5 (0.04 )d σ dw 2. he efficien log-price dx = (0.05 /2)d + σ dw 1, and correlaion ρ = 0.5 beween Brownian moions. he observed log-price Y = X + ε, where ε NID(0, ). Euler discreizaion scheme wih ime sep = 1 s Ou-of-sample forecass of IV In his secion we compare he one day ahead ou-of-sample forecass of he IV given by SRV and RV. As discussed in Secion 3.1.1, he relaion beween he condiional means of IV is [ m+1 ] [ ] m E d F m 1 = e κd E d F m 1 m m 1 + α(1 e κd )D, (3.6) where D = m+1 m = m m 1 is one day on an annualized base. Replacing he condiional mean in he righ-hand-side by he esimae of he IV on day m provides a simple mehod o forecas he IV on day m + 1. Precisely we proceed as follows. We spli each simulaed sample pah in wo pars. he firs par of ,400 high frequency log-reurns is used o esimae he ime series of 100 daily IV and an AR(1) model for he IV iself or equivalenly o esimae inercep and slope in he forecas Eq. (3.6) where he condiional expecaion in he righ-hand-side is replaced by he esimaed IV. he second par of he simulaed sample pah consiss of 23,400 high frequency log-reurns, i represens he las 101s rading day of high frequency daa and i is saved for ouof-sample purposes. he ou-of-sample forecas of IV on he day 101s is provided by he AR(1) model for IV and compared wih rue IV realized on ha day. his procedure is repeaed for each sample pah and for boh SRV and RV compued a he differen frequencies. In empirical applicaions he rue underlying model parameers are unknown and finie sample properies of sochasic volailiy models, such as mean reversion, can be quie differen from he corresponding asympoic properies. Hence esimaion of Eq. (3.6) is required o be realisic. In he radiion of Mincer and Zarnowiz (1969) and Chong and Hendry (1986), we compare he alernaive volailiy forecass by projecing he rue realized IV on day m + 1, m+1 m d, on a consan and he various model forecass. he forecas evaluaion regressions ake he form { m+1 = b 0 + b 1 {SRV m+1 m } j + b 2 {RV m+1 m } j m d } j + error j, (3.7) for j = 1,..., 10,000. SRV m+1 m is he ou-of-sample forecas of IV from day m o day m + 1 given by he AR(1) model for IV esimaed using he ime series of SRV daily IV. RV m+1 m is similarly compued. Fig. 2. Ou-of-sample forecas error, (i.e. forecas minus rue value of IV 10 4 ), ploed versus rue value of IV 10 4 for he 10,000 simulaed pahs of he insananeous variance under he Heson model, d = (0.04 )d +0.5 σ dw 2. he efficien log-price dx = (0.05 /2)d + σ dw 1, and correlaion ρ = 0.5 beween Brownian moions. he observed log-price Y = X + ε, where ε NID(0, ). Euler discreizaion scheme wih ime sep = 1 s. he OLS esimaes of he forecas evaluaion regressions (3.7) are repored in able 2. In all cases SRV forecass largely ouperform RV forecass; Fig. 2 visually confirms his resul. Moreover, in almos all cases, adding any RV forecass o SRV forecass gives no addiional explanaory power, ha is he R 2 of he exended regressions do no increase. he only excepion is he exreme siuaion of SRV compued a he (excessively long) 30 min frequency. In all cases, he null hypohesis H 0 : b 0 = 0 and H 0 : b 1 = 1 are rejeced. However, also using he rue in-sample IV on day m = 100 o forecas he IV on day m + 1, he previous hypohesis are rejeced. his is mainly due o a Jensen s inequaliy effec which links in a nonlinear way he parameers of he Heson model and he IV process. We performed a number of robusness checks o suppor he previous resuls. o save space, and because he resuls are consisen wih hose above, hese addiional resuls are no repored in he presen paper, bu colleced in he supplemenary appendix (Aï-Sahalia and Mancini, 2007). For insance, we invesigae he ou-of-sample forecass of he inegraed volailiy IV 1/2 insead of he inegraed variance, IV. hen in he forecas evaluaion regressions (3.7), IV and he corresponding esimaes are replaced by IV 1/2. he overall conclusions are he same as in able 2, meaning ha he previous findings are robus o Jensen s inequaliy effecs. We also repea he previous simulaion sudy using a nearly inegraed Heson model o approximae long memory feaures exhibied in he real world by he volailiy of some asses; see Secion 3.5 below for a more heoreically-based approach o long memory. In he Heson model, we se he mean reversion coefficien κ = 1 and he local volailiy parameer γ = in order o saisfy Feller s condiion. he findings of he robusness checks confirm he resuls repored above. he main difference is ha RV has an even larger relaive variance when esimaing IV in-sample. his is due o a larger over-esimaion of small inegraed variances Jump-diffusion model Empirically i has been observed ha he Heson model has some difficulies in fiing asse and opion prices (see for insance Jones (2003)) and hence i has been exended in several direcions, in paricular by including jump componens o he volailiy process. o invesigae he impac of jumps on RV and SRV, we now consider a model where he log-price dynamic

6 Auhor's personal copy 22 Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 2 Ou-of-sample, one day ahead forecass of daily IV in percenage based on 10,000 simulaed pahs under he Heson model, d = 5 (0.04 )d σ dw 2 b 0 b 1 b 2 R 2 RV 5 min (0.016) (0.005) RV 5 min MA(1) (0.013) (0.004) RV 10 min (0.013) (0.005) RV 15 min (0.013) (0.005) RV 30 min (0.014) (0.006) SRV 5 min (0.006) (0.003) SRV 5 min + RV 5 min (0.016) (0.008) (0.009) SRV 5 min + RV 5 min MA(1) (0.014) (0.010) (0.010) SRV 5 min + RV 10 min (0.010) (0.008) (0.008) SRV 5 min + RV 15 min (0.008) (0.007) (0.007) SRV 5 min + RV 30 min (0.007) (0.006) (0.006) SRV 10 min (0.007) (0.004) SRV 10 min + RV 5 min (0.019) (0.010) (0.011) SRV 10 min + RV 5 min MA(1) (0.018) (0.013) (0.013) SRV 10 min + RV 10 min (0.013) (0.011) (0.011) SRV 10 min + RV 15 min (0.010) (0.010) (0.011) SRV 10 min + RV 30 min (0.008) (0.008) (0.008) SRV 15 min (0.008) (0.004) SRV 15 min + RV 5 min (0.020) (0.011) (0.011) SRV 15 min + RV 5 min MA(1) (0.019) (0.014) (0.013) SRV 15 min + RV 10 min (0.014) (0.013) (0.013) SRV 15 min + RV 15 min (0.012) (0.013) (0.013) SRV 15 min + RV 30 min (0.010) (0.010) (0.011) SRV 30 min (0.011) (0.006) SRV 30 min + RV 5 min MA(1) (0.017) (0.013) (0.011) SRV 30 min + RV 30 min (0.013) (0.017) (0.017) SRV minimum variance (0.004) (0.002) he efficien log-price dx = (0.05 /2)d + σ dw 1, and correlaion ρ = 0.5 beween Brownian moions. he observed log-price Y = X + ε, where ε NID(0, ). Euler discreizaion scheme wih ime sep = 1 s. OLS sandard errors in parenhesis. Eq. (3.6) is esimaed using he previous 100 simulaed days and regressing he corresponding esimaes of i+1 i σ 2 d on a consan and i i 1 d, for i = 1,..., 99. is given by Eq. (3.1), bu he volailiy follows a jump-diffusion Heson ype model d = κ(α )d + γ σ dw 2, + J dq, (3.8) where = lim s s, q is a Poisson process wih inensiy λ and J is he jump size, assumed o be exponenially disribued wih parameer ξ. he processes q and J are independen of he wo Brownian moions. Wihou jumps, λ = 0, Eq. (3.8) reduces o he Heson s model (3.1). he long run mean of he volailiy process is E[] = α + ξλ/κ.2 We se he jump coefficiens λ = 23,400/2 which implies (on average) wo volailiy jumps per day, and ξ = , ha is he jump size is abou 2% of he uncondiional variance. he diffusion coefficiens are as in Secion 3.1, bu α = In-sample esimaion resuls are raher close o he findings under he Heson model and repored in Aï-Sahalia and Mancini (2007). Hence, as in he previous experimen, SRV ouperforms RV in erms of bias, variance and RMSE a each frequency. able 3 shows he ou-of-sample simulaion resuls. Again as in he previous cases, SRV provides more efficien and unbiased forecass han RV and a each frequency; SRV resuls in larger R 2 han RV. I is known ha empirically asse reurns end o display a jump componen. Hence we simulae high frequency daa using he same Heson volailiy model as in Secion 3.1, bu adding a jump componen o he log-price process X. All simulaion resuls are colleced in Aï-Sahalia and Mancini (2007) and show ha SRV ouperforms RV also in ha seing. 2 his can be seen inegraing Eq. (3.8) and aking he uncondiional expecaion [ ] [ ] [ ] E[ ] = E[σ q 0] + E κ(α s )ds + E γ σ s dw 2,s + E J j. 0 0 j=1 he Iô inegral has zero mean, and E[ q j=1 J j] = ξλ. Denoing E[] = and using Fubini s heorem gives = + κ(α ) + ξλ, which implies = α + ξλ/κ Log-volailiy model In empirical work, volailiy is someimes modeled and prediced wih ARMA models ha are consruced using logs of realized volailiy; see Andersen e al. (2001, 2003) for he use of log-rv for sock reurn and exchange rae volailiy. he raionale for doing his is ha he disribuion of log-rv can be closer o Gaussian han ha of RV. Goncalves and Meddahi (2005b) generalize his o he use of Box Cox ransforms. Wihou log or oher ransformaions, Goncalves and Meddahi (2005a) and Zhang e al. (2005a) provide Edgeworh expansions for RV and SRV wih and wihou noise. Here, i urns ou ha lile of he difference in predicabiliy can be aribued o non-normaliy. We run simulaions using a log-volailiy model calibraed o he parameer values repored by hese papers and add he same iny amoun of microsrucure noise as above. Le IV (d) denoe he daily inegraed variance IV (d) d s = ds, where d = 1 day. Following Andersen e al. (2003), he daily logarihmic sandard deviaion l is assumed o evolve according o an AR(5) model 3 l = 1 2 log(iv (d) ) = φ φ i l id + u, (3.9) i=1 where u is a srong whie noise, u NID(0, Eu 2 ). In conras o he previous volailiy models, he log-volailiy model (3.9) does no specify he insananeous variance dynamic and is defined in discree ime. Inraday efficien log-reurns r are given by he daily inegraed volailiy imes a noise erm r = IV (d) z, (3.10) 3 Precisely, o capure long memory feaures in IV, Andersen e al. (2003) use a fracionally inegraed AR process. We invesigae he long memory issue in Secions 3.4 and 3.5.

7 Auhor's personal copy Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 3 Ou-of-sample forecass, one day ahead of daily IV in percenage based on 10,000 simulaed pahs under he Heson jump-diffusion model, d = 5 (0.035 )d σ dw 2 + J dq, q is a Poisson process wih inensiy λ = 23,400/2 and J Exp(0.0007) b 0 b 1 b 2 R 2 RV 5 min (0.020) (0.005) RV 5 min MA(1) (0.018) (0.004) RV 10 min (0.019) (0.005) RV 15 min (0.019) (0.006) RV 30 min (0.021) (0.007) SRV 5 min (0.009) (0.003) SRV 5 min + RV 5 min (0.019) (0.009) (0.010) SRV 5 min + RV 5 min MA(1) (0.018) (0.011) (0.011) SRV 5 min + RV 10 min (0.013) (0.008) (0.009) SRV 5 min + RV 15 min (0.011) (0.007) (0.008) SRV 5 min + RV 30 min (0.010) (0.006) (0.006) SRV 10 min (0.012) (0.004) SRV 10 min + RV 5 min (0.023) (0.011) (0.011) SRV 10 min + RV 5 min MA(1) (0.022) (0.014) (0.014) SRV 10 min + RV 10 min (0.017) (0.012) (0.012) SRV 10 min + RV 15 min (0.014) (0.011) (0.011) SRV 10 min + RV 30 min (0.013) (0.008) (0.009) SRV 15 min (0.013) (0.005) SRV 15 min + RV 5 min (0.023) (0.011) (0.011) SRV 15 min + RV 5 min MA(1) (0.022) (0.014) (0.013) SRV 15 min + RV 10 min (0.018) (0.014) (0.014) SRV 15 min + RV 15 min (0.017) (0.014) (0.014) SRV 15 min + RV 30 min (0.015) (0.011) (0.011) SRV 30 min (0.017) (0.006) SRV 30 min + RV 5 min MA(1) (0.020) (0.012) (0.010) SRV 30 min + RV 30 min (0.019) (0.017) (0.017) SRV minimum variance (0.006) (0.002) he efficien log-price dx = (0.05 /2)d + σ dw 1, and correlaion ρ = 0.5 beween Brownian moions. he observed log-price Y = X + ε, where ε NID(0, ). Euler discreizaion scheme wih ime sep = 1 s. OLS sandard errors in parenhesis. he corresponding Eq. (3.6) is esimaed using he previous 100 days of esimaed IV, regressing he corresponding esimaes of i+1 i σ 2 d on a consan and i i 1 d, for i = 1,..., 99. where z NID(0, 1). In our simulaion experimen we se he parameer φ 0 = , φ 1 = 0.35, φ 2 = 0.25, φ 3 = 0.20, φ 4 = 0.10, φ 5 = 0.09, and (E u 2 ) 1/2 = 0.02 o induce approximaely he same level of daily logarihmic sandard deviaions documened by Andersen e al. (2001) for he DJIA socks. We simulae 10,000 sample pahs of ,400 log-reurns, i.e. each sample pah represens 201 days of log-reurns sampled a one second inervals. he firs 200 days are used o esimae he log-volailiy model (3.9) and he las day is saved for he ou-of-sample forecass. In he curren experimen we simulae longer sample pahs (201 days insead of 101 days) han in he previous experimens o achieve a reasonable sample size for he esimaion of he logvolailiy model. he ou-of-sample forecass of IV are obained by Eq. (3.9). As he logarihmic sandard deviaion l is updaed on a daily base, AR-ype forecas equaions are correcly specified and direcly implied by he assumed log-volailiy model. In-sample esimaes of IV are repored in Aï-Sahalia and Mancini (2007) and show ha SRV ouperforms RV in erms of bias, variance and RMSE as in he previous simulaion experimens. able 4 shows he ou-of-sample forecass of IV and largely confirms he previous findings. For insance, RV forecass have no addiional explanaory power in erms of R 2 when added o SRV forecass in he Mincer Zarnowiz regressions Heerogeneous auoregressive realized volailiy model In his secion, we simulae he inegraed volailiy process using he heerogeneous auoregressive realized volailiy (HAR- RV) model proposed by Corsi (2004). he model is inspired by he heerogeneiy of agens operaing in financial markes, and in paricular o he differen ime horizons (such as daily or monhly) relevan for differen invesors (such as day raders or fund managers); see also Müller e al. (1997) and Dacorogna e al. (1998). Such heerogeneiy could explain he srong posiive correlaion beween volailiy and marke paricipaion observed empirically. Indeed, if all he invesors were equal, he more agens aced in he marke, he faser he price converged o is marke value on which all agens agreed. hus, he volailiy had o be negaively correlaed wih marke paricipaion. Anoher aspec capured by he HAR-RV model is ha shor erm raders are concerned abou long erm volailiy as i poenially affecs expeced fuure rends and riskiness. Hence on he one hand hey reac o changes in long erm volailiy by revising heir rading sraegies and causing volailiy. On he oher hand, he level of shor erm volailiy does no affec he rading sraegies of long erm invesors such as pension fund mangers or insiuional invesors. Such an asymmeric dependence among volailiies a differen ime scales can be formalized using a cascade volailiy model from low frequencies o high frequencies. o simplify he model we will consider only hree frequencies, i.e. daily, weekly, and monhly as in Corsi (2004). In he following, irrespecive of he frequency, all he quaniies are expressed on an annual basis. As in he previous secion, IV (d) denoes he daily inegraed variance and RV (d) he daily realized variance RV (d) = d/ j=0 r j 2, where r j = x j x (j+1), x is he rue underlying log-price and = 1 s. Weekly and monhly realized variances, RV (w) and RV (m), are similarly defined by summing up squared log-reurns over he las week and monh, respecively. he HAR-RV model for he daily inegraed variance is IV (d) + = β 0 + β (d) RV (d) + β (w) RV (w) + β (m) RV (m) + ω (d) +, (3.11) where ω (d) NID(0, σω 2 ). Similarly o he log-volailiy model (3.9), he HAR-RV model does no specify he insananeous variance dynamic. Hence inraday efficien log-reurns, r, are given by Eq. (3.10). he Eq. (3.11) has a simple AR srucure and formally IV (d) does no belong o he class of long memory processes. However, he inegraed variance can exhibi an apparen long memory behavior wih an auocorrelogram decaying (approximaely) hyperbolically

8 Auhor's personal copy 24 Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 4 Ou-of-sample, one day ahead forecass of daily IV in percenage based on 10,000 simulaed pahs under he log-volailiy model (3.9), l = l d l 2d l 3d l 4d l 5d + u, where u NID(0, ) and d = 1 day b 0 b 1 b 2 R 2 RV 5 min (0.025) (0.008) RV 5 min MA(1) (0.021) (0.007) RV 10 min (0.020) (0.009) RV 15 min (0.018) (0.009) RV 30 min (0.017) (0.010) SRV 5 min (0.008) (0.005) SRV 5 min + RV 5 min (0.019) (0.009) (0.009) SRV 5 min + RV 5 min MA(1) (0.018) (0.010) (0.010) SRV 5 min + RV 10 min (0.013) (0.009) (0.009) SRV 5 min + RV 15 min (0.011) (0.008) (0.009) SRV 5 min + RV 30 min (0.009) (0.007) (0.008) SRV 10 min (0.010) (0.006) SRV 10 min + RV 5 min (0.022) (0.011) (0.011) SRV 10 min + RV 5 min MA(1) (0.021) (0.013) (0.012) SRV 10 min + RV 10 min (0.016) (0.012) (0.012) SRV 10 min + RV 15 min (0.013) (0.011) (0.012) SRV 10 min + RV 30 min (0.011) (0.009) (0.010) SRV 15 min (0.011) (0.007) SRV 15 min + RV 5 min (0.024) (0.012) (0.012) SRV 15 min + RV 5 min MA(1) (0.022) (0.014) (0.013) SRV 15 min + RV 10 min (0.017) (0.013) (0.013) SRV 15 min + RV 15 min (0.015) (0.013) (0.013) SRV 15 min + RV 30 min (0.013) (0.011) (0.011) SRV 30 min (0.013) (0.009) SRV 30 min + RV 5 min MA(1) (0.021) (0.014) (0.012) SRV 30 min + RV 30 min (0.015) (0.016) (0.014) SRV minimum variance (0.006) (0.003) he efficien log-reurn r = X X = parenhesis. Eq. (3.9) is esimaed using he previous 200 days of esimaed daily IV. IV (d) z, where z NID(0, 1) and = 1 s. he observed log-price Y = X + ε, where ε NID(0, ). OLS sandard errors in as shown for insance in Fig. 3 4 ; see Corsi (2004) for furher deails. In general, a process obained aggregaing shor memory processes can exhibi long memory feaures when he aggregaion level is no large enough compared o he lowes frequency componen of he aggregaed process. Indeed, Granger (1980) shows ha aggregaing an infinie number of shor memory processes can induce a long memory process; see also Ding and Granger (1996). Recenly, LeBaron (2001) shows ha he sum of only hree AR(1) processes (each process operaing on a differen ime scale) can exhibi long memory when he half life of he longes componen is large enough. 5 In our experimens, we se he parameers as follows: β 0 = 0.002, β (d) = 0.45, β (w) = 0.30, β (m) = 0.20, and σ ω = he low variance of ω (d) prevens negaive values of IV (d). As in he previous secion, we simulae 10,000 sample pahs of ,400 log-reurns, i.e. each sample pah represens 201 days of log-reurns sampled every second. he firs 200 days are used o esimae he HAR-RV model (3.11) and he las day is saved for ou-of-sample forecass. he ou-of-sample forecass of IV are correcly specified using Eq. (3.11), which implies ARype forecas equaions. In-sample esimaions are colleced in Aï-Sahalia and Mancini (2007) and ou-of-sample forecass are repored in able 5. hese resuls confirm all he previous findings abou he ouperformance of SRV over RV. Fig. 3. Firs plo: simulaed pah of daily, weekly and monhly IV 1/2 on an annual base under he HAR-RV model, IV (d) = RV (d) RV (w) RV (m) + ω (d), where ω (d) NID(0, ). he efficien log-reurn r = X X = IV (d) z, where z NID(0, 1) and = 3 h. Second plo: esimaed auocorrelaion funcion of daily IV 1/2 wih lags measured in days. 4 he long sample pah of 4000 days displayed in Fig. 3 is obained by simulaing IV (d) wih ime sep of hree hours and persisence of he process β (d) + β (w) + β (m) = 0.9. Empirically, i has been observed ha he higher he sampling frequency of reurns, he more persisen is he volailiy process. In our Mone Carlo experimen, we will simulae reurns a frequency of one second and hen we will increase he persisence of IV (d) o In boh cases, he uncondiional inegraed variance β 0 /(1 β (d) β (w) β (m) ) = A closely relaed approach o modeling long-range dependence is given by he superposiion of Ornsein Uhlenbeck or CEV models; see Barndorff-Nielsen (2001) and Barndorff-Nielsen and Shephard (2001) Fracional Ornsein Uhlenbeck model A subsanial amoun of empirical evidence suggess ha he volailiy process may exhibi long memory. Above, we have considered models which approximae his behavior. Bu a powerful class of models designed o capure his feaure are processes driven by fracional Brownian moions. We will use a fracional Ornsein Uhlenbeck (OU) process, as in Come and Renaul (1998), wih a wis, for he purpose of represening

9 Auhor's personal copy Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 5 Ou-of-sample, one day ahead forecass of daily IV in percenage based on 10,000 simulaed pahs under he HAR-RV model, IV (d) = RV (d) RV (w) RV (m) + ω (d), where ω (d) NID(0, ) b 0 b 1 b 2 R 2 RV 5 min (0.017) (0.005) RV 5 min MA(1) (0.014) (0.005) RV 10 min (0.013) (0.006) RV 15 min (0.012) (0.006) RV 30 min (0.012) (0.006) SRV 5 min (0.006) (0.004) SRV 5 min + RV 5 min (0.017) (0.009) (0.009) SRV 5 min + RV 5 min MA(1) (0.016) (0.011) (0.011) SRV 5 min + RV 10 min (0.011) (0.009) (0.009) SRV 5 min + RV 15 min (0.009) (0.008) (0.009) SRV 5 min + RV 30 min (0.007) (0.007) (0.007) SRV 10 min (0.007) (0.004) SRV 10 min + RV 5 min (0.019) (0.010) (0.010) SRV 10 min + RV 5 min MA(1) (0.017) (0.013) (0.012) SRV 10 min + RV 10 min (0.012) (0.011) (0.011) SRV 10 min + RV 15 min (0.010) (0.010) (0.011) SRV 10 min + RV 30 min (0.008) (0.009) (0.009) SRV 15 min (0.007) (0.005) SRV 15 min + RV 5 min (0.019) (0.010) (0.010) SRV 15 min + RV 5 min MA(1) (0.017) (0.013) (0.012) SRV 15 min + RV 10 min (0.013) (0.012) (0.011) SRV 15 min + RV 15 min (0.011) (0.012) (0.012) SRV 15 min + RV 30 min (0.009) (0.010) (0.010) SRV 30 min (0.009) (0.006) SRV 30 min + RV 5 min MA(1) (0.016) (0.012) (0.010) SRV 30 min + RV 30 min (0.010) (0.013) (0.012) SRV minimum variance (0.005) (0.003) he efficien log-reurn r = X X = IV (d) z, where z NID(0, 1) and = 1 s. he observed log-price Y = X + ε, where ε NID(0, ). OLS sandard errors in parenhesis. Eq. (3.11) is esimaed using he previous 200 days of esimaed daily IV, and hen aggregaing such esimaes o obain weekly, and monhly IV. a process wih long memory in volailiy. he price dynamics coninue o be given by Eq. (3.1). Le us sar wih a brief descripion of fracional OU processes; for deails, we refer he reader o Cheridio e al. (2003). A fracional Brownian moion (FBM) wih Hurs parameer H (0, 1] is an almos surely coninuous, mean zero ( Gaussian process wih covariance kernel Cov(W H,, W H,s ) = 1 2 2H + s 2H s 2H) for all, s R. Furher, W H,0 = 0 almos surely and W H is a process wih saionary incremens; in general, however, he incremens are no independen, excep in he special case H = 1/2 which corresponds o a (wo-sided) sandard Brownian moion. When H 1/2, W H is no a semimaringale and for 0 < < s Cov(W H,τ+ W H,τ, W H,τ+s+ W H,τ+s ) = 1 ( s 2H + s + 2H 2 s 2H) 2 ( = s2h 2 1 2H ) s H s 2 ( ) N 2n 2n 1 = (2H k) s 2H 2n + O(s 2H 2N 2 ) (2n)! n=1 k=0 as s (3.12) for every ineger N 1. hus, for H (1/2, 1], he incremens process (W H,(n+1) W H,n ) n=0 exhibis long memory, namely n=0 Cov(W H,, W H,(n+1) W H,n ) =. An alernaive characerizaion of FBM is hrough is sochasic represenaion in erms of he sandard BM, W : ( 1 0 ( W H, = s H 1/2 s H 1/2) dw s Γ (H + 1/2) ) + s H 1/2 dw s, (3.13) 0 where Γ is he Gamma funcion. he represenaion (3.13) is in general no unique. A fracional OU process is he soluion of dz H, = κ(α Z H, )d + γ dw H,, (3.14) where we assume mean reversion (κ > 0) o a posiive mean (α > 0). Firs, here exiss a saionary, almos surely coninuous, soluion Z H, o (3.14). Second, he soluion is given by Z H, = α + γ e κ e κu dw H,u (3.15) and is Gaussian and ergodic. 6 hird, he auocovariance funcion of Z H decays a he same rae as ha of W H (see heorem 2.3 in Cheridio e al. (2003)): ( ) Cov(Z H,, Z H,+s ) = γ 2 N 2n 1 κ 2n (2H k) s 2H 2n 2 n=1 k=0 + O(s 2H 2N 2 ) as s so Z H exhibis long memory when H (1/2, 1]. Come and Renaul (1998) invesigae he long memory properies of a volailiy process driven by he fracional Ornsein Uhlenbeck process Z H,, where Z H, = log(σ ). Bu in order o be able o forecas IV exacly under he assumed model, we will modify ha specificaion o one where Z H, = σ direcly. his has he drawback of allowing for negaive values of σ, which is no prey bu oherwise creaes no mahemaical difficulies. As in he case of a sandard OU process Z 1/2,, a high enough long erm mean α can make he probabiliy of his occurring very small. 7 6 In Eq. (3.15) he inegral eκu dw H,u appears, where W H is he FBM. Since he FBM is no a semimaringale, sochasic inegrals wih respec o FBM migh no be coninuous funcion wih respec o he inegrand. However, he special case of deerminisic inegrand urns ou o be sufficien for he presen purpose; see, for insance, Gripenberg and Norros (1996) for a shor and elemenary FBM-specific inroducion o he subjec. 7 Indeed, all simulaed insananeous variances were sricly posiive.

10 Auhor's personal copy 26 Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) I urns ou ha we now have all we need o be able o compue a forecasing equaion for IV. Le Iσ m = m+1 m σ d. I is clear ha (Iσ m, Iσ m 1 ) is joinly Gaussian. As a resul, E[Iσ m Iσ m 1 ] = E[Iσ m ] + Cov(Iσ m, Iσ m 1 ) Var(Iσ m 1 ) (Iσ m 1 E[Iσ m 1 ]). (3.16) he key poin is ha his is an affine expression in Iσ m 1, hence one ha is amenable o an exacly-specified forecasing regression. he coefficiens of ha regression expression can be compued explicily as E[Iσ m ] = α( m+1 m ) and [ ( m+1 ) Cov(Iσ m, Iσ m 1 ) = E Z H, d E[Iσ m ] m ( )] m Z H,r dr E[Iσ m 1 ] m 1 = E [ ( γ ( γ m+1 m m = γ 2 m+1 r m 1 m ) e κ( u) dw H,u d )] e κ(r v) dw H,v dr [( ) e κu dw H,u e κ(+r) E m m 1 ( r )] e κv dw H,v drd (3.17) and we conclude wih he fac ha [( ) ( r )] E f (u)dw H,u g(v)dw H,v = r f (u)g(v) u v 2H 2 dvdu (3.18) (see Pipiras and aqqu (2000)). he calculaion for Var(Iσ m 1 ) is idenical. In our experimens, we need o simulae relaively long sample pahs of he fracional OU process. Several approaches are available o simulae fracional Gaussian processes. For insance, using a runcaed version of he FBM, Come and Renaul (1998) propose a racable represenaion of he fracional OU process for simulaion purposes; see also Come and Renaul (1996). Unforunaely, his approach relies on a numerical evaluaion of sochasic inegrals and i is quie compuaionally demanding for simulaing he long sample pahs needed in he presen seing. 8 For he simulaion of he fracional Gaussian noise (FGN), ha is he incremen of FBM, we use he mehod proposed by Davies and Hare (1987). he aim is o compue a marix G such ha Σ = GG, where G is he ranspose of G and Σ is he covariance marix of he FGN wih auocovariance funcion ρ(k) = Var(W H,+ W H, ) 2 ( k + 1 2H + k 1 2H 2 k 2H). (3.19) Suppose ha a sample pah of size N has o be simulaed, he main idea is o embed Σ in a so-called circulan marix C of size 2N. he firs row of such a marix is [ρ(0) ρ(1) ρ(2) ρ(n 1) 0 ρ(n 1) ρ(n 2) ρ(1)] 8 Similar compuaional difficulies arise for insance in he Hosking mehod, also known as he Durbin or Levinson mehod. and he i-h row is obained by shifing he firs row i 1 places o he righ and padding he removed elemens o he lef. he upper lef corner of size N is he covariance marix Σ. he circulan marix C can be decomposed as C = Q ΛQ, where Λ is he diagonal marix of eigenvalues of C, Q is a uniary marix and Q is he complex conjugae of Q. Hence QQ is he ideniy marix. A sample pah of he FGN is obained by Q Λ 1/2 Q ζ, where ζ is a vecor wih iid Gaussian random variables. Davies and Hare (1987) provide an algorihm in hree seps o efficienly compue he las quaniy; see also Craigmile (2003) and Dieker (2004) and references herein. 1. Compue he Λ eigenvalues. he eigenvalues λ k, k = 0,..., 2N 1, can be efficienly compued using he fas Fourier ransform of he elemens in he firs row of C. 2. Compue U = Q ζ. Using he covariance srucure of U leads o he following simulaion scheme for U = [U 0 U 1 U N 1 U N U N+1 U 2N 1 ]. Generae wo independen Gaussian random variables U 0 and U N. For 1 s N 1, generae wo independen Gaussian random variables ζ (1) s and ζ (2) s and U s = 1 (ζ (1) s + ıζ (2) s ), U 2N s = 1 (ζ (1) s ıζ (2) s ), 2 2 (3.20) where ı = 1. he resuling vecor U has he same disribuion as Q ζ. 3. Compue Q Λ 1/2 U. his quaniy can be efficienly compued using he inverse fas Fourier ransform of he elemens λ s U s, s = 0,..., 2N 1. A sample pah of he FGN is obained by aking he firs N elemens of Q Λ 1/2 U. he main advanage of he Davies Hare mehod is he low compuaional cos which is of order N log(n) for N sample poins (when log 2 (N) is an ineger), as opposed for insance o he Hosking mehod which is of order N 2. Once a sample pah of he FGN is available, he fracional OU can be simulaed using he Euler discreizaion of Eq. (3.14). We se he parameer α = 0.2, κ = 20, γ = 0.012, H = 0.7 and we simulae 10,000 sample pahs of 87 days for he fracional OU process using a ime inerval = 1 s. Hence each sample pah consiss of 87 23,401 log-prices. 9 Eq. (3.16) is esimaed using he previous 86 days of IV and he las day is saved for he ou-of-sample forecas of IV. In-sample simulaion resuls are repored in Aï-Sahalia and Mancini (2007) and confirm ha SRV largely ouperforms RV in he esimaion of IV. able 6 shows he ou-of-sample simulaion resuls. Forecass of IV given by RV are quie biased and inefficien, while SRV forecass are much more accurae for insance in erms of R 2. Oher Mone Carlo simulaions (no repored here) confirm ha he levels of R 2 depend on he amoun of predicabiliy (or memory) in he daa generaing process. Changing model parameers will change R 2, bu he ouperformance of SRV over RV and he large differences in R 2 remain. 9 he odd number of days allows o simulae N = 2 21 sample poins using he sandard power of wo fas Fourier ransform algorihm and o speed up he simulaion. Since 2 21 > 87 23,401, he firs unnecessary sample poins are discarded.

11 Auhor's personal copy Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 6 Ou-of-sample, one day ahead forecass of daily IV in percenage based on 10,000 simulaed pahs under he fracional OU model, dσ = 20 (0.2 σ )d dW H, where dw H is a FBM wih Hurs parameer 0.7 b 0 b 1 b 2 R 2 RV 5 min (0.032) (0.010) RV 5 min MA(1) (0.026) (0.009) RV 10 min (0.024) (0.010) RV 15 min (0.022) (0.010) RV 30 min (0.021) (0.011) SRV 5 min (0.009) (0.006) SRV 5 min + RV 5 min (0.024) (0.010) (0.011) SRV 5 min + RV 5 min MA(1) (0.021) (0.011) (0.011) SRV 5 min + RV 10 min (0.017) (0.009) (0.011) SRV 5 min + RV 15 min (0.014) (0.009) (0.010) SRV 5 min + RV 30 min (0.012) (0.008) (0.009) SRV 10 min (0.012) (0.007) SRV 10 min + RV 5 min (0.029) (0.013) (0.014) SRV 10 min + RV 5 min MA(1) (0.026) (0.016) (0.015) SRV 10 min + RV 10 min (0.021) (0.014) (0.015) SRV 10 min + RV 15 min (0.018) (0.014) (0.014) SRV 10 min + RV 30 min (0.015) (0.012) (0.013) SRV 15 min (0.014) (0.009) SRV 15 min + RV 5 min (0.032) (0.015) (0.015) SRV 15 min + RV 5 min MA(1) (0.028) (0.018) (0.017) SRV 15 min + RV 10 min (0.023) (0.017) (0.017) SRV 15 min + RV 15 min (0.020) (0.017) (0.018) SRV 15 min + RV 30 min (0.016) (0.016) (0.016) SRV 30 min (0.016) (0.011) SRV 30 min + RV 5 min MA(1) (0.028) (0.019) (0.016) SRV 30 min + RV 30 min (0.019) (0.024) (0.022) SRV minimum variance (0.006) (0.004) he efficien log-price dx = (0.05 /2) d + σ dw 1. he observed log-price Y = X + ε, where ε NID(0, ). Euler discreizaion scheme wih ime sep = 1 s. OLS sandard errors in parenhesis. Eq. (3.16) is esimaed using he previous 86 days of esimaed IV, regressing he corresponding esimaes of i+1 and i i 1 d, for i = 1,..., 85. i d on a consan 3.6. Alernaive correlaion srucures for he marke microsrucure noise In he Mone Carlo experimens above, he microsrucure noise erm ε was assumed o be iid. In he parameric volailiy conex, Aï-Sahalia e al. (2005) discuss how he likelihood funcion is o be modified in he case of serially correlaed noise and noise ha is correlaed wih he price process. Aï-Sahalia e al. (2006) propose a version of he SRV esimaor which can deal wih such serial dependence in marke microsrucure noise. Hansen and Lunde (2006) considered such deparures from he iid noise assumpion in he case of he Zhou esimaor for σ ime varying. he lieraure repors some evidence agains he iid noise assumpion. For insance, Griffin and Oomen (forhcoming) provide an ineresing analysis of he ick vs. ransacion ime sampling schemes showing ha he naure of he sampling mechanism can generae disinc auocorrelogram paerns for he resuling logreurns: in ransacion ime he microsrucure noise can be close o iid, bu in ick ime i is srongly auocorrelaed. Hansen and Lunde (2006) argue ha he microsrucure noise is auocorrelaed and negaively correlaed wih he laen reurn process, in paricular when prices are sampled from quoaions. In his secion, we remove he independen noise assumpion and ε will be neiher ime independen nor independen of he laen price X. he underlying daa generaing mechanism for X is given by he Heson model (3.1) and we examine he wo esimaors, RV and SRV in ha conex Auocorrelaed microsrucure noise Following Aï-Sahalia e al. (2006), a simple model o capure he ime series dependence in he microsrucure noise ε is ε = U + V, (3.21) where U is iid, V is an AR(1) process wih firs order coefficien ϱ, ϱ < 1, and U V are normally disribued. We se E[U 2 ] = E[V 2 ] = and ϱ = 0.2, which imply ha (E ε 2 ) 1/2 = as in he previous experimens. In-sample esimaions of IV are raher close o he ones in he iid microsrucure noise case (documened in able 1) and he corresponding resuls are colleced in Aï-Sahalia and Mancini (2007). Hence SRV ouperforms RV in esimaing he IV also under auocorrelaed noise. able 7 shows he ou-of-sample forecass of IV and largely confirms all he previous findings. For insance RV forecass have no addiional explanaory power in erms of R 2 when added o SRV forecass in he Mincer Zarnowiz regressions Microsrucure noise correlaed wih he laen price process We simulae a microsrucure noise ε no auocorrelaed, bu correlaed wih he reurn laen process X Corr(ε, (X X )) = ϑ (3.22) and we consider boh posiive, ϑ = 0.2, and negaive, ϑ = 0.2, correlaions. Boh in- and ou-of-sample resuls are very close o he findings in he iid noise case documened in ables 1 and 2 and are no repored here, bu colleced in Aï-Sahalia and Mancini (2007) Auocorrelaed microsrucure noise correlaed wih he laen price process In his secion he microsrucure noise ε is auocorrelaed and correlaed wih he laen reurn process, ha is ε = U + V, where V is he same AR(1) process as in Secion 3.6.1, and U is negaively correlaed wih he laen reurns, Corr(ε, (X X )) = 0.2. In- and ou-of-sample simulaion resuls are very close o he resuls in he auocorrelaed noise case discussed in Secion and are repored in Aï-Sahalia and Mancini (2007). he correlaion beween microsrucure noise and laen reurn process has a minor impac on he measuremen and he

12 Auhor's personal copy 28 Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 7 Ou-of-sample, one day ahead forecass of daily IV in percenage based on 10,000 simulaed pahs under he Heson model, d = 5 (0.04 )d σ dw 2 b 0 b 1 b 2 R 2 RV 5 min (0.016) (0.005) RV 5 min MA(1) (0.013) (0.004) RV 10 min (0.013) (0.005) RV 15 min (0.013) (0.005) RV 30 min (0.014) (0.006) SRV 5 min (0.005) (0.003) SRV 5 min + RV 5 min (0.017) (0.008) (0.009) SRV 5 min + RV 5 min MA(1) (0.016) (0.010) (0.010) SRV 5 min + RV 10 min (0.010) (0.008) (0.008) SRV 5 min + RV 15 min (0.008) (0.007) (0.007) SRV 5 min + RV 30 min (0.007) (0.006) (0.006) SRV 10 min (0.007) (0.004) SRV 10 min + RV 5 min (0.020) (0.010) (0.011) SRV 10 min + RV 5 min MA(1) (0.019) (0.013) (0.013) SRV 10 min + RV 10 min (0.013) (0.011) (0.011) SRV 10 min + RV 15 min (0.011) (0.010) (0.011) SRV 10 min + RV 30 min (0.008) (0.008) (0.008) SRV 15 min (0.008) (0.004) SRV 15 min + RV 5 min (0.021) (0.011) (0.011) SRV 15 min + RV 5 min MA(1) (0.020) (0.014) (0.013) SRV 15 min + RV 10 min (0.015) (0.013) (0.013) SRV 15 min + RV 15 min (0.012) (0.013) (0.013) SRV 15 min + RV 30 min (0.010) (0.010) (0.011) SRV 30 min (0.010) (0.006) SRV 30 min + RV 5 min MA(1) (0.017) (0.012) (0.011) SRV 30 min + RV 30 min (0.013) (0.017) (0.017) SRV minimum variance (0.004) (0.003) he efficien log-price dx = (0.05 /2)d + σ dw 1, and correlaion ρ = 0.5 beween Brownian moions. he observed log-price Y = X + ε, where ε = U + V, U is iid, V is AR(1) wih firs order coefficien ϱ = 0.2, U V, and E ε 2 = Euler discreizaion scheme wih ime sep = 1 s. OLS sandard errors in parenhesis. Eq. (3.6) is esimaed using he previous 100 simulaed days and regressing he corresponding esimaes of i+1 i d on a consan and i i 1 d, for i = 1,..., 99. forecas of he IV, while he auocorrelaion in he microsrucure noise plays a major role. Overall, SRV ouperforms RV in esimaing and forecasing IV also when he microsrucure noise is auocorrelaed and correlaed wih he laen price process Comparison beween SRV and MSRV We have seen ha SRV provides he firs consisen and asympoic (mixed) normal esimaor of IV and has he rae of convergence n 1/6. A he cos of higher complexiy SRV can be generalized o muliple ime scales, by averaging no on wo ime scales bu on muliple ime scales. he resuling esimaor, muliple scale realized volailiy (MSRV), is X, X (msrv) = M i=1 a i [Y, Y] (K i) weighed sum of M slow ime scales + 2 Ê[ε 2 ], (3.23) fas ime scale where Ê[ε2 ] is given in (2.2). SRV corresponds o he special case M = 1 and uses only one slow ime scale (and he fas ime scale o bias correc i). For suiably seleced weighs a i s, X, X (msrv) converges o he rue X, X a he rae n 1/4. his generalizaion is due o Zhang (2006). We repea all he previous Mone Carlo experimens o compare in-sample esimaions and ou-of-sample forecass of SRV and MSRV. o save space we repor all he deails and simulaion resuls in Aï-Sahalia and Mancini (2007). Overall, under all scenarios SRV and MSRV provide raher close (and accurae) esimaions and forecass of IV. he efficiency loss in using SRV insead of MSRV is essenially negligible. hese resuls are in agreemen wih he empirical findings of Aï-Sahalia e al. (2006) as hey show ha SRV and MSRV give raher close esimaes of IV for he Inel and Microsof socks. 4. Empirical analysis 4.1. he daa Our empirical analysis is based on he ransacion prices from he NYSE s rade and Quoe (AQ) daabase for he hiry DJIA socks. he sample period exends from January 3, 2000 unil December 31, 2004, for a oal of 1256 rading days and he daily ransacion records exend from 9:30 unil 16:00. he DJIA socks are among he mos acively raded US equiies and he sample period covers high and low volailiy periods. Hence he comparisons beween SRV and RV and in paricular he ou-ofsample forecass of he inegraed volailiies will be ineresing exercises. he original daa are pre-processed eliminaing obvious daa errors (such as ransacion prices repored a zero, ransacion imes ha are ou of order, ec.) and bounce back ouliers larger han he cuoff Admiedly, oher cuoff hresholds could be conceived, bu hey would be all equally arbirary. Aï- Sahalia e al. (2006) invesigae he dependence of SRV and RV on cuoff levels, and hey show empirically ha SRV is much less affeced han RV by he differen level choices. Such ouliers can be viewed as a form of microsrucure noise and he robusness of SRV o differen ways of pre-processing he daa is a desirable propery. he pre-processed daa will be used o esimae he daily inegraed variances applying he SRV and RV esimaors Uncondiional reurn and inegraed volailiy disribuions In his secion we presen he saisical properies of he uncondiional reurn and he inegraed variance disribuions. For each sock, daily inegraed variances are esimaed using he SRV esimaor (2.5) wih a slow ime scale of five minues and wo RV esimaors based on de-meaned MA(1)-filered and unfilered 5 min log-reurns. In he consrucion of he RV esimaors we

13 Auhor's personal copy Y. Aï-Sahalia, L. Mancini / Journal of Economerics 147 (2008) able 8 Summary saisics for he daily log-reurn disribuions in percenage of he DJIA socks, r i, Sock r i, Mean Sd. Skew. Kur. Min Max Mean Sd he sample period exends from January 3, 2000 o December 31, 2004, for a oal of 1256 observaions. Fig. 4. Inegraed volailiy, IV 1/2, for Inel sock (icker INC) on an annual base (252 days) given by SRV and RV from January 3, 2000 o December 31, follow Andersen e al. (2001). he five minues log-reurn series are obained by aking he difference beween log-prices recorded a or immediaely before he corresponding 5 min marks. hen o compue he firs RV esimaor, he raw log-reurns are demeaned and for each sock an MA(1) model is esimaed using he 5 min log-reurn series of he full five years sample. he second RV esimaor is compued using he unfilered 5 min logreurns. In our sample, he IV esimaes given by he wo RV esimaors are very similar, confirming he findings in Andersen e al. (2001). For all socks and all rading days he differences beween he corresponding annualized inegraed volailiies are always below 1%. herefore, o save space he subsequen analysis will only involve he RV esimaes based on 5 min de-meaned MA(1) filered reurns. his choice is consisen wih he previous Mone Carlo analysis, where he RV forecass based on MA(1) filered reurns ends o ouperform he corresponding forecass based on unfilered reurns. For compleeness, we repor all he summary saisics for boh RV esimaors for each DJIA sock in Aï-Sahalia and Mancini (2007). able 8 presens summary saisics for he daily log-reurns of he DJIA socks, r i,, i = 1,..., 30, over he sample period, = 1,..., 1256, i.e. January 3, 2000 o December 31, he average reurns across all socks are very close o zero, he majoriy of he sock reurns have posiive skewness, and all sock reurns display excess kurosis. hese findings are broadly consisen wih he large empirical lieraure on speculaive asse reurns, daing back a leas o Mandelbro (1963) and Fama (1965). As observed for insance in Andersen e al. (2001), when he log-reurns are sandardized by he corresponding inegraed volailiy, r i, /IV 1/2 i,, he uncondiional disribuions are very close o he Gaussian disribuion. able 9 shows he previous summary saisics for he log-reurns sandardized using SRV and RV and confirms his finding. For insance he average kurosis is reduced from for he raw log-reurns r i, o and for he log-reurns sandardized using SRV and RV, respecively. We remark ha he inegraed volailiies for sandardizing he reurns are only observable ex-pos, and no ex-ane as for insance in GARCHype models. Noneheless, he approximae Gaussian disribuion of sandardized reurns, r i, /IV 1/2 i,, is in conras o he ypical finding ha, when sandardizing daily reurns by he one day ahead GARCH-ype volailiy forecass, he resuling disribuions are invariably lepokuric, albei less so han he raw reurns, r i,. able 10 summarizes he uncondiional disribuion of he inegraed volailiies, IV 1/2, on an annual base (252 days), given by SRV and RV for he DJIA socks. On average, he inegraed volailiies esimaed using SRV are lower, and less volaile, posiively skewed and lepokuric han he inegraed volailiies given by RV. hese findings are consisen wih he fac ha he RV esimaes can be adversely affeced by microsrucure noise as demonsraed in he Mone Carlo analysis. In paricular, he IV 1/2 esimaed using RV are 2% higher wih a kurosis 23% larger han he IV 1/2 esimaed using SRV. hese differences are economically significan, resuling in differen asse prices and opimal asse allocaions. For example, wih a ypical hree-monh opion wih a vega sensiiviy o volailiy of 10, such a 2% difference in volailiy resuls in an opion price ha is 20% higher. o illusrae ypical differences recorded on a single sock, Fig. 4 shows he inegraed volailiies of he Inel sock (icker INC) esimaed using SRV and RV: as can be seen, in several occasions he wo esimaes are quie differen. As discussed earlier, given he high skewness and kurosis of he inegraed volailiies, a logarihmic ransformaion of IV 1/2 is someimes adoped o induce an approximae Gaussian disribuion. able 11 shows he summary saisics for he ) disribuions esimaed using SRV and RV. Boh log(iv 1/2 i, disribuions are closer o he Gaussian one when compared o he uncondiional disribuions of IV 1/2. For insance, he average kurosis across all he hiry socks are now reduced o and for SRV and RV esimaes, compared o and for he IV 1/2 in able 10. As observed in Andersen e al. (2003), he approximae normaliy of he log-inegraed volailiy, log(iv 1/2 ), suggess he use of sandard linear Gaussian approaches for modeling and forecasing logarihmic volailiy. In our in- and ou-of-sample forecass of he DJIA inegraed volailiies, we will use AR models for log(iv 1/2 ) o obain IV 1/2 forecass. We will also experimen wih AR models for he inegraed variance processes, IV, o obain inegraed volailiy forecass emporal dependence and long memory of inegraed volailiy emporal dependence and long memory are well documened empirical feaures of financial asse reurns. able 12 repors summary saisics o invesigae he emporal dependence of he inegraed volailiy, IV 1/2 i,, in Panel A, and he logarihmic inegraed volailiy, log(iv 1/2 i, ), in Panel B, esimaed using SRV and RV. Consisenly wih he empirical lieraure, all he Ljung Box pormaneau es srongly rejecs he join null hypohesis of zero auocorrelaions up o lag 22, ha is abou one monh of rading days. All he es saisics are well above he criical value a 1% confidence level, documening he well-known volailiy clusering phenomenon. Similar slow decay raes in he auocorrelaion funcions have been documened in he empirical lieraure wih daily ime series of absolue and squared reurns spanning several decades (see, for example, Ding e al. (1993)), bu he resuls in able 12 are noeworhy in ha he sample only spans five years. In spie of he slow decay in he auocorrelaions, abou 90% of he augmened Dickey Fuller ess