Activity Signature Functions for High-Frequency Data Analysis

Transcription

1 Activity Sigature Fuctios for High-Frequecy Data Aalysis Viktor Todorov ad George Tauche This Draft: July, 8 First Draft: November 7 Abstract We defie a ew cocept termed the activity sigature fuctio, which is costructed from discrete observatios of a process evolvig cotiuously i time. Uder quite geeral regularity coditios, we derive the asymptotic properties of the fuctio as the samplig frequecy icreases ad show that it is a useful device for makig ifereces about the activity level of a Itô semimartigale. Mote Carlo work cofirms the theoretical results. Oe empirical applicatio is from fiace. It idicates that the classical model comprised of a cotiuous compoet plus jumps is more plausible tha a pure-jump model for the spot $/DM exchage rate over A secod applicatio pertais to iteret traffic data at NASA servers. We fid that a pure-jump model with o cotiuous compoet ad paths of ifiite variatio is appropriate for modelig this data set. I both cases the evidece obtaied from the sigature fuctios is quite covicig, ad these two very disparate empirical outcomes illustrate the discrimiatory power of the methodology. Keywords: activity idex, Blumethal-Getoor idex, jumps, Lévy process, realized power variatio. We thak Cecilia Macii, Per Myklad, Neil Shephard, Jeaette Woerer as well as semiar participats at Duke Uiversity, Uiversity of Chicago workshop o fiace ad statistics, Imperial College 8 Fiacial Ecoometrics Coferece ad Purdue Uiversity Fiacial Mathematics semiar for may helpful commets. Departmet of Fiace, Kellogg School of Maagemet, Northwester Uiversity, Evasto, IL 68; v-todorov@kellogg.orthwester.edu Departmet of Ecoomics, Duke Uiversity, Durham, NC 778; george.tauche@duke.edu. 1

2 1 Itroductio We cosider measurig ad estimatig the activity level of a Itô semimartigale, which is a aalytically coveiet probability model for may stochastic processes evolvig i cotiuous time. The idex of activity of a Itô semimartigale is a extesio of the classical Blumethal- Getoor idex ad its geeralizatio proposed i Ait-Sahalia ad Jacod 7)) of a pure-jump process, where the leadig example is a stable process. The geeralized) Blumethal-Getoor idex for a jump process lies i the iterval [,], ad it idicates whether the process is relatively quiescet or highly vibrat. For example, whe the jump process is fiitely active, i.e. with oly a fiite umber of jumps i ay fiite time iterval, the Blumethal-Getoor idex is zero. Jump processes with Blumethal-Getoor idices above zero are ifiitely active, with paths of fiite variatio or ifiite variatio depedig upo where the idex lies relative to uity. The extesio of the geeralized) Blumethal-Getoor idex to Itô semimartigales cosidered i this paper also lies i the iterval [,], ad it is also a measure for the vibracy of the process. Furthermore, it applies also to cotiuous path processes. The activity idex of a cotiuous local) martigale is always, i.e. the highest possible value, ad thus it domiates that of all jump processes. The idex allows us to classify the differet processes used i cotiuous-time modelig i the followig order from low to high activity: fiite activity jumps, fiite variatio but ifiite activity) jumps, absolutely cotiuous processes, jumps of ifiite variatio ad cotiuous martigales. If the uderlyig process is the superpositio of several differet processes from the above list, the the measure will equal that of the most active compoet. Our activity idex is defied path-wise ad therefore is a radom variable; however, for may commo processes it is costat with probability oe. The focus of the paper is o the activity sigature fuctio, defied below, that proves useful for makig o-parametric iferece about the geeralized activity idex from a fiite sample. The estimatio strategy is simple. We first compute for a arbitrary iterval of time the realized power variatio over two differet time scales, a strategy derived from Ait-Sahalia ad Jacod 8a) ad Ait-Sahalia ad Jacod 7) see also Zhag et al. 5)) that removes a importat bias. The realized power variatio is just the sum of the absolute values of the icremets of the process withi the iterval raised to some power p. We the compute a oliear trasformatio of the ratio of realized power variatios ad plot it as a fuctio of the expoet over the rage p, 4]. This fuctio is the activity sigature fuctio. Uder mild regularity coditios, as the samplig frequecy icreases the asymptotically the activity sigature fuctio will have a kik at the value of the expoet that equals the geeralized idex of the activity of the process, which has to lie betwee ad. Mote Carlo evidece idicates that the kik is readily apparet from visual ispectio of the plots. I additio, we develop a ew test for the absece of a cotiuous martigale compoet, ad we propose a summary poit estimate of the activity based o the activity sigature fuctio. Mote Carlo results idicate very good performace of both the test ad the summary estimate. Although the activity idex asymptotically lies betwee ad, the characteristics of the activity sigature fuctios over the iterval, 4] are quite relevat as well, because that will idicate presece of jumps eve whe they are domiated by a cotiuous martigale. The ituitio behid our estimatio strategy is the followig. For a fixed time spa ad expoet, as we sample frequetly, the rate at which the realized power variatio coverges depeds solely o the activity of the process. Therefore, the ratio of the realized power variatio computed over differet scales asymptotically idetifies the activity of the observed process. There are several atecedets i the literature pertaiig to the results i this paper. I a paper that to our kowledge is upublished, Woerer 6) suggests examiatio of plots of a differet fuctio of the level of realized power variatio, ot the differece across time scales, i order to make iferece about the Blumethal-Getoor idex of a jump process; but this approach etails a sigificat bias as documeted i a Appedix. The two-scale approach

3 removes the bias as see below. Ulike this paper, the uderlyig results of Woerer 6) are derived uder the strog assumptio that the drivig jump measure is idepedet from the time-varyig itesity or jump size, ad this rules out some iterestig models, i particular may of the stochastic volatility models used i fiace. Secod, for the case of p = 4 our statistic is a oliear trasformatio of the test statistic for jumps proposed by Ait-Sahalia ad Jacod 8a). Fially, our focus o the activity level of the etire process differs from that of Ait-Sahalia ad Jacod 7), who cosider the challegig problem of estimatig the geeralized Blumethal-Getoor idex of the jumps i the presece of a cotiuous martigale. Ituitively, while our estimatio strategy idetifies the most active part of the discretely observed process, the estimator of Ait-Sahalia ad Jacod 7) recovers the least active compoet of the sum of two processes, give that the idex of the domiat compoet equals. Our estimatio strategy ca be useful for discrimiatig oparametrically across classes of models of stochastic processes. For istace, i fiacial ecoomics oe eeds to model both the asset price ad the stochastic volatility. A alterative to the classical model, which has a cotiuous martigale compoet, is a pure-jump model for which the price or volatility is comprised solely of jumps. The idea behid the pure-jump modelig is that small jumps ca elimiate the eed for a cotiuous martigale. The class of pure-jump models is extremely wide. It icludes the ormal iverse Gaussia ydberg, 1997; Bardorff-Nielse, 1997, 1998), the variace gamma Mada et al., 1998), the CGMY model of Carr et al. ), the timechaged Lévy models of Carr et al. 3), the COGACH model of Klüppelberg et al. 4) for the fiacial prices as well as the o-gaussia Orstei Uhlebeck-based models of Bardorff- Nielse ad Shephard 1) ad the Lévy-drive cotiuous-time movig average CAMA) models of Brockwell 1) for the stochastic volatility. Pure-jump models have bee extesively cosidered ad used for geeral optios pricig Huag ad Wu, 4; Broadie ad Detemple, 4; Levedorskii, 4; Schoutes, 6; Ivaov, 7), ad for foreig exchage optios pricig Huag ad Hug, 5; Daal ad Mada, 5; Carr ad Wu, 7). Other applicatios of pure-jump models iclude reliability theory Drose, 1986), isurace valuatio Ballotta, 5), ad fiacial equilibrium aalysis Mada, 5). The geeralized idex of activity of a pure-jump model is strictly less tha, while the idex of the classical model of fiace equals. The two classes of models are disjoit, ad oe does ot kow a priori which is empirically more plausible. The o-parametric evidece i Sectio 4 below comes dow o the side of the classical model for the spot $/DM exchage rate over A secod applicatio uses high-frequecy iteret traffic data o dowloads from NASA servers. The outcome is quite differet. We fid evidece that these data are best described by a model with jumps, o cotiuous compoet, ad paths of ifiite variatio. The sharp cotrasts betwee the outcomes obtaied by usig high-frequecy fiacial data or iteret traffic data illustrate the capability of the estimatio strategy to differetiate across the classes of models. This paper is orgaized as follows. I Sectio we itroduce the differet types of cotiuoustime models that our aalysis ecompasses. This sectio also cotais the assumptios eeded for all theoretical results i the paper. Sectio 3 defies a activity idex for a cotiuoustime process ad itroduces the activity sigature fuctio for estimatig this activity idex from discrete observatios. This sectio cotais also the asymptotic behavior of the activity sigature fuctio. I Sectio 4 we apply our theoretical results o simulated ad real data. Sectio 5 cocludes. Setup Our goal i this paper is to measure the activity defied formally i the ext sectio) of a cotiuous-time oe-dimesioal stochastic process from discrete observatios. We start our 3

4 aalysis with itroducig the differet models for the discretely-observed process, whose activity we ivestigate, ad statig the assumptios eeded for the asymptotic results of the ext sectio. Throughout, we always assume implicitly that each of the processes give below is defied o some probability space Ω, F, P). Further, we equip this probability space with some filtratio F, with respect to which all the processes used i the defiitios of the differet models below are adapted. I all models that we cosider the uderlyig process is a Itô semimartigale, i.e., a semimartigale whose characteristics, drift, diffusio ad jump compesator, are absolutely cotiuous with respect to time Jacod ad Shiryaev, 3). The differet models differ i whether the stochastic process cotais cotiuous martigale ad/or jumps. I what follows we will always assume that the processes i each of the models are well defied, for classical coditios Cot ad Takov, 3; Jacod ad Shiryaev, 3). Cotiuous Model: Y t = ad W t is a stadard Browia motio. Pure-Jump Model: X t = t b s ds + t t b 1s ds + t σ s κx) µds,dx) + σ 1s dw s,.1) t σ s κ x)µds,dx),.) where µ is a jump measure o + with compesator a s ds νx)dx, µds,dx) := µds,dx) a s dsνx)dx; κ x) = x κx) ad κx) is a cotiuous trucatio fuctio, i.e. a cotiuous fuctio with bouded support ad which coicides with the idetity aroud the origi; a t is some o-egative process. The two processes σ t ad a t i equatio.) geerate stochastic volatility i X t, but i two differet ways. σ t geerates stochastic volatility through timevaryig jump size, while a t geerates stochastic volatility through time-varyig the itesity of the jumps. Cotiuous plus Jumps Model: Z t = X t + Y t..3) We ext state the assumptios eeded for the asymptotic results i the paper. Assumptio A1. a) The processes σ 1s ad b 1s have càdlàg paths; σ 1s a.s. for every s >. b) The processes σ s, a s ad b s have càdlàg paths ad o fixed time of discotiuity. Assumptio A. The Lévy desity νx) ca be decomposed as ν 1 x) = νx) = ν 1 x) + ν x),.4) A x β+1 1 {x>} + B x β+1 1 {x<}, ν x) φx) x,.5) β +1 where A ad B are some o-egative costats; β < β < ; ad φx) is some oegative, slowly-varyig at zero fuctio which is bouded at zero. Assumptio A3. I additio to Assumptio A, assume that a) If β < 1, the t b s ds t a s σ s κx)dsνx)dx, for every t >..6) 4

5 b) If β = 1, the νx) ad κx) are symmetric ad b t for every t >. Assumptio A1 is a very mild regularity assumptio ad it is satisfied by most parametric applicatios. O the other had assumptio A is a o-trivial assumptio that we impose o the behavior of νx) aroud the origi. I the Lévy case it essetially amouts to restrictig the Lévy desity to be a power fuctio aroud zero like i the tempered) stable processes. I other words we cotrol the way i which νx) icreases to ifiity as x approaches zero. This is closely related with the Blumethal-Getoor idex which we explai i the ext sectio. A assumptio similar to A has bee made also i Woerer 6) ad Ait-Sahalia ad Jacod 7). Ulike the above cited papers however, we allow ν x) i.5) to be egative which is the case for example for the tempered stable process see 4.) below). Fially, assumptio A3 guaratees that whe the jumps are ot very active i a sese defied i the ext sectio), the drift term resultig from the compesatio of the small jumps ad/or the process b s is ot domiatig the activity i the pure-jump model. Uder assumptio A3, the process X t is a sum of its jumps. We will use this assumptio i Theorem 1 below i order to cocetrate o o-trivial cases. 3 Asymptotic esults For all our theoretical results we will fix the time iterval to be [,T] ad we will suppose that we observe the process Υ where Υ X, Y or Z) at the equidistat times,,,...,[t/ ] withi this iterval. We will thik of as beig small ad i this sectio we will derive statistics which allow us to determie the activity of the discretely-observed process as. We first eed a otio of activity of the uderlyig process. For this we itroduce the realized power variatio costructed from the discrete observatios of the process i the followig way V p,υ, ) t = [t/ ] i=1 i Υ p, p >, t,t], 3.1) where i Υ = Υ i Υ i 1) ad Υ is oe of the processes X, Y ad Z defied i the previous sectio. The we associate idex) the activity of the observed path of Υ with β Υ,T := if { r > : plim V r,υ, ) T < }. 3.) There are few thigs to ote about this idex of the activity. First, it is defied pathwise. Secod, it is always i the iterval [,]. Whe Υ Y, it is well kow that this idex is for every path equal to up to paths with measure zero). For a absolutely cotiuous process the idex is always 1. Whe Υ is the pure-jump process X, thigs are more complicated. As a simple example we ca cosider the case whe Υ is a pure-jump Lévy process. The the idex i 3.) will be the same o every observed path up to paths with measure ) ad it will coicide with the Blumethal-Getoor idex of the Lévy measure, see 3.4) below. Thus for a stable process with idex α, our activity idex i 3.) will coicide with α ad therefore ca take all the values i the rage [,]. Aother example is the compoud Poisso process ofte used i empirical applicatios; its activity idex is. Fially, the process Z defied i equatio.3) is a sum of cotiuous martigale drive by Browia motio) ad jumps, therefore its activity is determied by the most active part of it which is the cotiuous martigale part. Our goal will be to measure oparametrically the activity of the discretely-observed cotiuous time process. It is evidet from the discussio above that the activity idex will allow us to discrimiate betwee the differet classes of models defied i the previous sectio. For example, if the estimated activity idex is i the iterval,1), that meas that the observed process is from a pure-jump model with jumps havig fiite variatio. If the estimated idex is i the iterval 1,), the the appropriate model for the observed process is agai a pure-jump 5

6 model, but ow with jumps exhibitig ifiite variatio. Fially, if the estimated activity is, the to model appropriately the very small moves i the discretely-observed process, we eed a cotiuous martigale. Below we derive the asymptotic behavior of the realized power variatio uder quite geeral coditios ad the use it to propose activity sigature fuctios to estimate the activity idex from the discrete observatios of the process. 3.1 Asymptotics of ealized Power Variatio To determie the activity idex ad develop estimatio strategy for its estimatio from discrete observatios of the process Υ, we eed first to kow the asymptotic behavior of the realized power variatio for the differet models ad differet powers. I what follows u.c.p. deotes covergece i probability, locally uiformly i time Whe Υ Y Whe the partially) observed process is a cotiuous martigale plus a drift), the it follows from Bardorff-Nielse et al. 5) that uder assumptio A1a) ad for p > 1 p/ u.c.p. V p,y, ) t where A p is some costat depedig oly o p. t A p σ 1u p du, 3.3) 3.1. Whe Υ X I the pure-jump case, our activity idex ca be expressed directly as a fuctio of the jumps o the partially) observed path. I other words, whe Υ X, the activity idex i 3.) coicides with the followig geeralizatio of the Blumethal-Getoor idex due to Ait-Sahalia ad Jacod 7) BGX) T := if r > : X s r <, 3.4) s T where X s = X s X s. The Blumethal-Getoor idex was origially defied i Blumethal ad Getoor 1961) oly for pure-jump Lévy processes. The defiitio i 3.4) exteds it to a arbitrary jump semimartigale ad was proposed i Ait-Sahalia ad Jacod 7). It is implicit i the defiitio that BG T X) depeds o the time iterval [,T] ad also o the particular realizatio of the process, i.e. the idex is defied pathwise. The behavior of the idex is determied by the behavior of the small jumps jumps bigger tha ay fixed size are always fiite o a fiite iterval ad therefore are absolutely summable for ay power r). So, what is importat for our determiatio of the activity is essetially the very small icremets i X. This is ulike the problem of detectig jumps i discretely-sampled process, where we look to discrimiate the very small moves associated with the cotiuous part of the process) from the very big oes associated with the discotiuous part of the process). Fially, uder assumptio A, the geeralized Blumethal-Getoor idex BG T X) will coicide with β i equatio.5). Therefore, uder our assumptio A, the geeralized Blumethal-Getoor idex will be costat for each differet realizatio of the process. Turig to the behavior of the realized power variatio, whe p > BGX) T 1 it follows from Jacod 8) ad Bardorff-Nielse et al. 6) that uder assumptio A1b) we have u.c.p. V p,x, ) t X s p, 3.5) s t 6

7 provided there is at least oe jump o the observed path. Whe BGX) T < 1 ad b s a sσ s κx)νx)dx for all s [,T]) o a give path, the drift is domiatig the jumps ad therefore for p 1 we easily have 1 p u.c.p. V p,x, ) t t b s p a s σ s κx)νx)dx ds. 3.6) So we are left with the case whe BGX) T > 1 or assumptio A3 holds i.e. jumps always domiate the drift) ad p < BGX) T. For this case thigs are more complicated ad we eed to assume the stroger assumptio A uder which BG T X) = β ad is therefore oradom). The behavior of the realized power variatio i this case is give by the followig theorem. Theorem 1 For the process X t defied i.) assume that A1b), A ad A3 hold. Deote with L s a pure-jump Lévy process defied o some probability space) which has Lévy measure ds ν 1 x)dx ad drift { b = κ x)ν 1 x)dx if β 1 κx)ν 1x)dx if β < 1, with respect to the trucatio fuctio κ used i the defiitio of X t. If g p s) = E L s p ), the for p < β we have 1 p/β u.c.p. V p,x, ) t t g p a s ) σ s p ds. 3.7) A precursor special case of this theorem has bee proved i Woerer 3), but oly for the case whe a s 1 ad σ s is idepedet from the jump measure µ ad more restrictive specificatio for νx)). Also related is a result i Ait-Sahalia ad Jacod 7) o the asymptotic behavior of the cout of the icremets bigger tha a decreasig threshold Whe Υ Z We are fially left with the case whe the observed process is a superpositio of a cotiuous martigale, jumps ad drift. We will assume that Z cotais at least oe jump o the observed path, otherwise the behavior of the realized power variatio is as i the pure-cotiuous model. Whe p <, the limitig behavior of the realized power variatio is determied by the most active compoet i Z, which is the cotiuous martigale. Whe p =, the both the jumps ad the cotiuous martigale determie the behavior of the realized power variatio. Fially, whe p >, the limitig behavior of the realized power variatio is govered by the jumps. The precise results are as follows. Uder assumptio A1 we have 1 p/ u.c.p. t V p,z, ) t A p σ 1u p du if < p < u.c.p. V,Z, ) t t σ 1udu + s t Z s if p = u.c.p. V p,z, ) t s t Z s p if p >, where A p is the same costat that appears i equatio 3.3). These results are trivial cosequeces of or follow directly from) the results i Bardorff-Nielse et al. 5), Jacod 8) ad Bardorff-Nielse et al. 6). 3. Measurig Activity via the Activity Sigature Fuctio Havig characterized the asymptotic behavior of the realized power variatio for the differet models, we are ow ready to develop strategy for iferrig the activity idex from the discrete observatios of the process. The idea is to compute the realized power variatio at two differet frequecies ad use the fact that the scalig factors at the two differet frequecies will differ, provided the activity idex is above the power that is used. This two-scale approach was first 7 3.8)

8 proposed i Ait-Sahalia ad Jacod 8a) for desigig tests for presece of jumps ad further used i Ait-Sahalia ad Jacod 7) for estimatio of the geeralized Blumethal-Getoor idex of the jumps. To this ed set < p < β Υ,T, the for arbitrary iteger k 1 ad uder A1-A3 we have k 1 p/βυ,t 1 p/βυ,t P V p,υ,k ) T C T p), where C t p) is some stochastic process, depedig o Υ. This suggests that if we fix p sufficietly low ad compute the p-th realized power variatio over differet samplig frequecies we ca recover β from the slope coefficiet i a regressio of {l V p,υ, ) T )} k o a costat ad {l k )} k. For example, i the case whe we use oly the two samplig frequecies ad k we ca determie β usig β,t] Υ,p) = l k) p l k) + l V p,υ,k ) T ) l V p,υ, ) T ). 3.9) We ote that our estimator β,t] Υ,p) makes use of all icremets of the process Υ. I fact the small icremets carry the most importat iformatio about the activity idex. Our measure is thus differet from the statistic proposed i Ait-Sahalia ad Jacod 7), which is essetially based o the cout of the icremets of the discretely-observed process that are bigger tha a threshold level the threshold level decreases to zero at a give rate). While 3.9) is estimatig the activity idex i 3.), their estimator is desiged to ifer the geeralized Blumethal-Getoor idex of the jumps i the possible) presece of a cotiuous compoet i.e. they estimate 3.4) whe Υ Z). Thus, sice the cotiuous martigale domiates the activity of the process Z = X +Y, Ait-Sahalia ad Jacod 7) eed to discard the very small icremets which are domiated by the cotiuous martigale. I cotrast, whe Υ Z 3.9) uses the very small icremets ad estimates the activity of the domiatig compoet i the observed process, i.e. that of the cotiuous martigale. Of course, our estimator ad the oe i Ait-Sahalia ad Jacod 7) will have the same asymptotic limit oly i the pure-jump case, i.e. whe Υ X. We ca look at β,t],p) as a fuctio of p ad aalyze the behavior of this fuctio uder the three differet models itroduced i Sectio. I the followig theorem by local uiform covergece o a give ope) set we mea covergece that is uiform o each compact subset of that set. Theorem a) Cotiuous Semimartigale: Suppose Υ Y ad assumptio A1a) holds. The for fixed T > we have { P if p β,t] Υ,p) 3.1) if p >, where the covergece is locally uiform i p o, ). b) Pure-Jump Semimartigale: Suppose Υ X ad assumptios A1b), A ad A3 hold. The for fixed T > we have { P β if p < β β,t] Υ,p) 3.11) p if p > β, where the covergece is locally uiform i p o,β) β, ). c) Cotiuous plus Jumps Semimartigale: Suppose Υ Z, assumptio A1 holds ad there is at least oe jump o the observed path. The for fixed T > we have { P if p β,t] Υ,p) 3.1) p if p >, where the covergece is locally uiform i p o, ). 8

9 From this theorem it follows that the activity of the discretely-observed process ca be iferred from the kik i β,t],p) or its absece i the cotiuous semimartigale case). We refer to β,t],p) as the activity sigature fuctio, as it allows us to detect the activity of the discretely-observed process. We ote that for that we eed to look at values of p,) oly ad check util what power the fuctio stays flat. However, higher values of p ca allow us to determie whether there is a kik at i the activity sigature fuctio β,t],p) or its stays flat, which i tur determies whether the process cotais jumps i additio to the cotiuous martigale the test for jumps of Ait-Sahalia ad Jacod 8a) is a mootoe trasformatio of β,t],p) for p = 4). This meas that usig the activity sigature fuctio we ca discrimiate betwee the three types of models defied i Sectio for the discretely-observed process. Apart from visual check for a kik i the activity sigature fuctio, we ca form differet estimators of the activity level associated with the observed path usig our activity sigature fuctio. Key for that is that the covergece of the activity sigature fuctio i Theorem is locally) uiform i p). I the umerical sectio we will illustrate with a particular activity estimator. 3.3 Testig Jump Diffusios Versus Pure-Jump Models usig the Activity Sigature Fuctio A activity level of has a special meaig sice it separates pure-jump models part b) of Theorem ) from models cotaiig cotiuous martigale parts a) ad c) of Theorem ). As oted i the Itroductio, pure-jump models have bee proposed as a alterative to stadard jump-diffusios i fields such as fiace ad isurace. Thus, it is useful to have a formal test for the presece of a cotiuous compoet agaist the alterative of a pure-jump process. We costruct such a test usig the activity sigature fuctio evaluated at a fixed power. The ext theorem gives the relevat ull distributio for this test. I the theorem, L s deotes stable covergece i law. Theorem 3 Set Υ Y or Υ Z. Assume that A1 holds ad that BG T X) < 1 ad p BGT X) BG T X),1 ), where BG T X) is the Blumethal-Getoor idex of the jumps i Z defied i 3.4) which ca be radom). The, if the process σ 1 is a Itô semimartigale with locally bouded coefficiets, we have a) ) 1/ L s log β,t] Υ,p)) log) K p ǫ, 3.13) K p = T pµ p lk σ 1u p du T σ 1u p du k + 1)µ p k 1 p/ µ p k) + k 1)µ p, 3.14) where ǫ is stadard ormal defied o a extesio of the origial probability space; for the defiitio of µ p ad µ p k) itroduce u 1 ad u two idepedet stadard ormal variables, the we set µ p = E u 1 p ad µ p k) = E u 1 p u 1 + k 1u p). b) A cosistet estimator for K p i 3.14) uder the coditios of the Theorem is give by K p = 1/ pµ p lk [T/ ] i=4 i Υ p/ i 1 Υ p/ i Υ p/ i 3 Υ p/ [T/ ] i= i Υ p/ i 1 Υ p/ k + 1)µ p k 1 p/ µ p k) + k 1)µ p. 3.15) 9

10 The above theorem ca be used to coduct a oe-sided test of the statistical sigificace of the discrepacy betwee log[ β,t] Υ,p)] ad the ull value of log); egative values of the studetized left-had side of 3.13) discredit the ull hypothesis of the presece of a cotiuous compoet. For practical purposes a value of p very close to 1 is desirable, so that the requiremet for p i the theorem is satisfied, whe Z cotais jumps of fiite variatio. Whe Z cotais jumps of ifiite variatio 3.13) does ot hold, sice the jumps slow dow the rate of covergece. Therefore i this case the oe-sided test based o Theorem 3 will be rather coservative. Extesive Mote Carlo work, documeted i a compaio appedix, idicates that the test is oly slightly coservative, i.e., udersized, ad overall it has good size ad power properties whe usig the large, dese, high-frequecy data sets for which it is iteded. 4 Numerical Study 4.1 Computig Activity Sigature Fuctios We show the geeral methodology by computig sigature fuctios o simulated data from the followig model for the observed process Υ: Υ t = Υ + ρ t Υ s ds + σ 1 W t + σ s t Υ s, 4.1) where W t is a stadard Browia motio, ad ρ, σ 1 ad σ are costats. The compesator for the jumps is either of these two: A e λx x β+1 dxds, β [,) or, λ Jδ {x=±τ} dxds, 4.) where δ ǫ deotes the Dirac poit mass. The left compesator i 4.) correspods to a symmetric tempered stable process osiński, 7; Carr et al.,, 3), also called the CGMY process, ad β is the Blumethal-Getoor idex. The right-had compesator i 4.) correspods to a compoud Poisso process with itesity λ J ad jumps τ or τ with equal probability, frequetly termed rare-jumps. If jumps ad a cotiuous compoet are preset, we fix the proportio of the jumps i the total variace of Υ to be percet, a upper limit foud empirically i fiace. If ρ <, the process Υ is a Orstei-Uhlebeck OU) process drive by Browia motio or jumps. We use ρ =.693 correspodig to half-life of 1 uits of time; the results are isesitive to the value of ρ. Table 1 cotais additioal details o the simulatios. I each sceario we simulate a realizatio {Υ t } N of N = 3 uits of time ad 88 icremets withi each uit of time, which correspod to 5-mi samplig over a 4-hour day. For each time iterval t 1,t] we compute the realizatio of the sigature fuctio β t 1,t] Υ,p). For short we set β t p) := β t 1,t] Υ,p) for t = 1,...,N. For each p >, let β 1) p) β ) p)... β N) p) deote the order statistics of { β t p)} t=1,...,n, ad deote the α-th quatile by BΥ,p,α) := β [αn] p). 4.3) BΥ,p,α) is the quatile activity sigature fuctio of p >, α fixed. We fid usig the quartiles α =.5, α =.5, ad α =.75 iformative. I Figure 1, the first three simulatio settigs i the left-side paels of correspod to Υ as a symmetric tempered stable process, a ifiitely active pure-jump process. From Theorem β t p) P max{β,p} as the samplig frequecy icreases, uiformly i p, for t. Therefore, there should be o importat differeces betwee the three BΥ, p, α) if the asymptotic approximatios 1

11 Table 1: Parameter Settig for the Mote Carlo Case ρ σ 1 σ Jump Specificatio TS tempered stable with A = 1, β =.5 ad λ =.5 TS tempered stable with A = 1, β = 1. ad λ =.5 TS tempered stable with A = 1, β = 1.5 ad λ =.5 C..8. oe C-JL rare-jump with λ J =.4, τ =.361 C-JM rare-jump with λ J =.3333, τ =.7746 C-JH rare-jump with λ J =., τ =.316 OU-TS tempered stable with A = 1, β =.5 ad λ =.5 OU-TS tempered stable with A = 1, β = 1. ad λ =.5 OU-TS tempered stable with A = 1, β = 1.5 ad λ =.5 OU-C oe are accurate, as appears to be the case. Also, the reader ca very easily visually determie the kik i BΥ, p, α) ad thereby make iferece regardig the activity of the process. For the bottom left-side pael of Figure 1, the process is cotiuous with flat activity fuctios very close to.. For the four cases show i the right-side paels of Figure 1, we use the tempered stable or Browia motio as the backgroud drivig Lévy processes for a OU process, so the icremets exhibit temporal depedece. The asymptotic patter of the quatile activity plots chages oly for the tempered stable case with β =.5, where the drift is more active tha the jumps. Sice the estimator β t p) idetifies the activity of the most active compoet i Υ, which i this case is the drift, β t p) P 1 for p 1, although β =.5. Figure 1 idicate that the ability to idetify o-parametrically from the activity plots the most active compoet i Υ is ot affected by the itroductio of temporal depedece i Υ. For the three cases show i Figure we add to Browia motio rare jumps as for the cases C-JL, C-JM, ad C-JH of Table 1. The itesities correspod to a jump every 5, 3, ad.5 uits of time respectively, o average. Now the Browia motio is the most active compoet of Υ. For a uit of time t o which there is a jump β t p) P max{,p}, while if o jump β t p) P., a flat lie at.. For the low itesity jump case i the top pael of Figure, the three quatile fuctios are flat sice the jumps are very ifrequet, so this case looks like the pure-cotiuous case eve whe usig 5 75-th quatiles. For the medium itesity case i the middle pael of Figure, we see that for α =.75 the fuctio BΥ,p,α) icreases after p =, but for the other two quatiles it stays flat. I the high itesity case i the bottom pael of Figure, the three quatiles icrease after p = idicatig presece of very high itesity jumps. Aother gauge of the samplig variability of the realized sigature fuctio is see i Figure 3, which shows replicates of the media sigature fuctio. The up-dow spread i the simulated fuctios is ot large, although the bias, the locatio of the bed i the curve relative to the populatio value, is larger. We retur to this i Subsectio 4.3 below, which summarizes a extesive Mote Carlo study of poit estimates based o the realized sigature fuctio. 11

12 4 Tempered Stable, β=.5 4 OU Drive by Tempered Stable,.5 β β Tempered Stable, β=1. 4 OU Drive by Tempered Stable, 1. β β Tempered Stable, β=1.5 4 OU Drive by Tempered Stable, 1.5 β β Browia Motio 4 OU Drive by Browia Motio β β Figure 1: Quatile activity sigature plots for processes drive by the symmetric tempered stable process. The figures show BΥ, p, α) defied i 4.3) as a fuctio of p for the quatiles α =.5, α =.5, ad α =.75, readig bottom up. The left-side paels correspod to values of β ad other parameters give by Cases TS.5, TS1., TS1.5, ad Case C of Table 1; right-side paels pertai to Cases OU-TS.5, OU-TS1., OU-TS1.5, ad Case OU-CC of Table Applicatio to Observed Data 4..1 Exchage ates I a first applicatio we use high-frequecy data o the log of the spot $/DM exchage rate for the period 1986:1 1999:6, 345 tradig days. Each tradig day is 4 hours ad samplig every five-miutes gives 88 log returs icremets). For each day we calculate β t 1,t] p) where t = 1,..., 345 ad the evaluate the quatile activity fuctio BΥ, p, α) defied i 4.3). The top pael of Figure 4 shows the 5-th, 5-th ad 75-th quatile activity fuctios computed o the observed data. The sharp drop ear the origi is due to the effects of roudig, sice the exchage rate is quoted to five digits, whereas simulatios are computed to machie precisio. From Figure 4, the evidece idicates rather sharply that the exchage rate process cotais a cotiuous martigale compoet, which has activity level of.. The bottom two paels of Figure 4 idicate that the empirical fidig is robust to usig either the first or secod half of the sample. Overall, the o-parametric aalysis of the activity level of the exchage rate 1

13 4 Quatile Activity Sigature Plots, Browia Motio + Compoud Poisso Low Itesity) β Browia Motio + Compoud Poisso Medium Itesity) 4 β Browia Motio + Compoud Poisso High Itesity) 4 β Figure : Activity sigature Plots for Browia Motio plus rare jumps. The rare jumps are a compoud Poisso of three differet itesity levels ad parameters give by Cases C-JL, C-JM, ad C-JH of Table 1; the figures show BΥ, p, α) defied i 4.3) as a fuctio of p for the quatiles α =.5, α =.5, ad α =.75, readig bottom up. data shows that pure-jump models for asset prices as proposed i Carr et al., 3) ad elsewhere are probably ot good descriptios of the high-frequecy exchage rate data. The data strogly suggest that a appropriate model for the exchage rate is of the form of a Cotiuous plus Jumps Model as i.3), which has may small moves. The sigature plots provide idirect evidece o the activity level of the jumps. The approach is differet from Ait-Sahalia ad Jacod 7), who address the statistically challegig problem of estimatig directly the activity level of the jumps give that cotiuous compoet is preset i Υ. As see i Figure 4, the media ad the 75-th quatile icrease for p >, so may of the days i the sample cotai jumps. A model defied by Browia motio plus a compoud Poisso with rare jumps appears ulikely to accout for the activity sigature plots obtaied from the exchage rate data. The cases C-JL, C-JM, ad C-JH i Table 1 are realistically calibrated so that the pure-jump part accouts for percet of the total variace of Z. However, oly with very high jump itesities of oe jump every three days, or two jumps every day, ca activity sigature plots computed from simulatios as i Figure be cosistet with the sigature plots computed from observed data i Figures 4. However, all empirical evidece we kow of suggests a itesity level betwee five to twety jumps per year if the data are presumed to follow a Browia motio plus a compoud Poisso process. As see i the top pael of Figure, at such low itesity levels the process would geerate quatile activity sigature plots very much ulike those of Figure 4 computed o the data. Cosistet with Ait-Sahalia ad Jacod 7, 8b), this evidece idicates that the jump compoet i Υ is ifiitely active with a rather high idex. Fially, we implemet the formal test for a cotiuous semimartigale compoet i the spot $/DM exchage rate discussed i Subsectio 3.3. Figure 5 is a plot of the day-by-day poit estimate log β,1] Υ,p)) for p =.9 alog with a 95 percet lower cofidece boud 13

14 1 Case TS.5, β= Case TS1., β= Case TS1.5, β= Case C, β= Figure 3: Over-plots of 1 Mote Carlo eplicates of the Media Activity Sigature Fuctio for Cases TS.5, TS1., TS1.5, ad C of Table 1 with 5-mi samplig ad 15 periods days) of data. 14

15 4 Quatile Activity Sigature Plots, Dollar/DM Exchage ate, Dollar/DM Exchage ate, First Half of Sample Dollar/DM Exchage ate, Secod Half of Sample Figure 4: Activity sigature plots, the Dollar/DM exchage rate, ad subperiods; the figures show BΥ, p, α) defied i 4.3) as a fuctio of p for the quatiles α =.5, α =.5, ad α =.75, readig bottom up. The poits to.; the value of β is a ukow parameter. based o the asymptotic approximatio give i Theorem 3. Oly 1.5 percet of the poit estimates lie below their correspodig 95 percet cofidece boud ad thus are statistically sigificatly differet from the ull value of log). Cosistet with cotemporaeous evidece obtaied usig a differet test Cot ad Macii, 7), Figure 5 idicates very little statistical evidece agaist the classical jump-diffusio model i favor of a pure-jump model. 4.. Iteret Traffic Uder a so-called slow-growth coditio, processes related to the classical α-stable process are importat for modelig iteret traffic Mikosch et al.,, Theorem 1, p. 33). Parameter estimates of the idex obtaied o the maitaied hypothesis of a classical α-stable model ca rage from.7 to 1.67 depedig upo the data set Xiaohu et al., 4, Table 1, p. 45). The top pael of Figure 6 shows the umber of megabytes dowloaded from NASA servers over te secod itervals for the period August 4 31, aw data are from the file NASA_access_log_Aug95 i the public domai. NASA servers experiece heavy demad, sometimes betwee two to three large requests per secod at peak periods. The time stamps show the hour, miute, ad secod of the data-request, with multiple records per secod i the data file. We aggregated to te secod itervals as is commo practice. We exclude August 1 3, 1995, because of various aomalies ad we use segmets of legth oe hour, reflectig the slowly varyig overall level of traffic revealed i iitial aalysis. There are 36 observatios per segmet, ad 4 hours) 8 days) = 67 segmets to compute BΥ,p,α) i 4.3). The data are cetered usig the full sample mea. From the middle pael of Figure 6, the oparametric evidece suggests that the activity idex of this NASA iteret traffic series is i the rage of , which is slightly more active tha the Cauchy process ad of ifiite variatio. I the bottom pael of Figure 6 15

16 .5 Daily Estimated log β ad 95 Percet Lower Cofidece Boud, p=.9 log β Figure 5: Daily poit estimates log β,1] Υ, p)) marked by for p =.9 ad 95 percet lower cofidece bouds. A poit estimate below its correspodig cofidece boud would be cosidered statistically sigificatly differet from the ull value of log) =.69. early all poit estimates lie well below their correspodig cofidece boud ad would thus be cosidered statistically sigificatly differet from the ull value of log) =.69, which is very strog evidece for the alterative hypothesis of a pure jump model. Figure 6 provides a useful startig poit for buildig a fully parametric pure jump model of the time series. The close resemblace betwee the sigature plots from the NASA series i Figure 6 ad those obtaied from the tempered stable process i Figure 1 is very strikig. The tempered stable appears a excellet cadidate for a iitial parametric model that could be refied uder the guidace of various specificatio tests, a task beyod the scope of this paper. 4.3 Poit Estimatio The precedig iferece strategy i Subsectios regardig the level of the activity idex is heavily graphical, which is justified by the theory of Sectio 3. Plots of the activity sigature fuctio reveal immediately the flat regio, ad the sharpess of the kik gives a visual meas to assess the precisio to which the idex ca be iferred from a particular data set. Noetheless, some readers, especially i ecoometrics, have a preferece for poit estimates, ad ideed it is coveiet to come away from a study with a sigle umber to report regardig the idex. For this purpose, we preset a straightforward two-step strategy to compute such a estimate from a realizatio of the activity sigature fuctio. Our period t estimate of the activity idex is give by β t 1,t] Υ) = βtτ) τ β t p)dp, 4.4) β t τ) τ where τ is a small umber ad ˆβ t τ) is our iitial estimate of the idex. I the first step, a good value of τ is determied by ispectig plots of the sigature fuctio, ad we fid values of τ aroud.1 ofte works very well. The secod step is usig iema sums over a fie grid to compute the itegral i 4.4). The value β t 1,t] Υ) i 4.4) is a simple equi-weighted average of the activity sigature fuctio over the data-determied iterval [τ, β t ˆτ)]. Fially, we recommed ) β = media {β t 1,t] Υ)} T t=1 4.5) as the overall estimate from the data set. 16

17 4 x 15 NASA Iteret Traffic August 4 31, 1995, MBytes Every 1 Secods Quatile Activity Sigature Plots, NASA Iteret Traffic August 4 31, 1995, 1 Secods x Estimated log β ad 95% lower cofidece boud uder ull cotiuous+jumps, p= Figure 6: aw data, quatile plots, ad tests for o cotiuous compoet. Although poit estimatio per se is ot the focus of the paper, we did coduct a very extesive Mote Carlo study of the estimate 4.5), sice doig so idirectly coveys iformatio about the activity sigature fuctio upo which it is based. The Mote Carlo study covered a much wider rage of scearios tha Table 1 above alog with a broad rage of samplig frequecies ad spas of the data set. The volumious Mote Carlo output is relegated to a supplemetary appedix, though the fidigs ca be readily summarized. Overall, the situatio is oe of small variace with a small to modest bias, although the bias ca be large i some extreme cases. Not surprisigly, the key cotrol parameter is the samplig frequecy, which strogly affects the bias, while the data spa oly mildly affects the variace. For samplig itervals correspodig to 1-mi ad 5-mi, the poit estimator is geerally quite accurate across the estimatio cotexts cosidered. O the other had, for samplig at 1-mi or 3-mi itervals the estimate 4.5) ca sometimes be quite off the mark, especially at 3-mi samplig. The activity sigature fuctio is ot advisable or should be used with extreme cautio o coarse data sets, but it appears to be a useful tool for iferece about the activity of a process observed o a large dese data set like those ow available i various disciplies. The estimator β i 4.5) is cosistet as samplig frequecy icreases, ad the Mote Carlo assessmet of error rates suggested that with a appropriate data set the activity idex, which perforce lies i the iterval [,], ca be determied to withi ±.1 ad eve ±.5, which seems reasoably accurate for may purposes. 17

18 5 Coclusio Our o-parametric strategy for iferece about the activity level of a discretely-observed semimartigale is ituitive ad graphical, ad it gives a o-parametric test for presece of cotiuous martigale. I a fiace applicatio to the $/DM exchage rate, the fidigs idicate that pure-jump models are empirically less plausible tha the classical model with rare jumps. I the secod applicatio, the fidigs suggest that NASA iteret traffic follows a pure-jump model without a cotiuous compoet ad of ifiite variatio. The two disparate fidigs idicate the rage of applicability of the methods. 6 Proofs 6.1 Prelimiary esults First, it is easy to see that the law of the pure-jump model X t, give with equatio.), ca be geerated with the followig process defied o some differet probability space we will still call it X t ) X t = t t + b s ds + t + σ s κx)1 {y<as} µds,dx,dy) + σ s κ x)1 {y<as}µds,dx,dy), 6.1) where ow µ is a Poisso measure but o + + ad with compesator ds νx)dx dy; µds,dx,dy) is the compesated versio of µ. The process X t i 6.1) is writte as a stochastic) itegral with respect to a time-homogeous Poisso radom measure, but we ote that µ is defied o a three dimesioal space. Ituitively, the first dimesio is the time, the secod oe is the size of the jumps but ote that the jumps are multiplied by σ t ). The role of the third dimesio is to geerate thiig of the jumps accordig to the process a t ad thus create time-varyig itesity of the jumps i the process X t. I all of the proofs i this sectio we will assume the represetatio i 6.1) for X t, ad we will work with the correspodig probability space ad filtratio which support it. This is coveiet as whe chages we do ot eed to make a chage of the probability space i our proofs. We start with showig some prelimiary results. Deote with L covergece i law o the space D) of oe-dimesioal càdlàg fuctios: +, which is equipped with the Skorokhod topology. We have the followig two Lemmas. Lemma 1 Let Ψ t deote a Lévy process with characteristic triplet b,,f) with respect to some trucatio fuctio κx), where F is a Lévy measure with Fdx) = νx)1 { x ǫ} dx ad νx) is the desity defied i.4), while ǫ is a arbitrary positive umber. Further, let b = κx)1 { x ǫ}νx)dx if β < 1 ad b = if β 1. Fially, assume that ν is symmetric whe β = 1. Deote with L t aother Lévy process with characteristic triplet b 1,,F 1 ) with respect to the same trucatio fuctio). b 1 = κ x)ν 1 x)dx if β 1 ad b 1 = κx)ν 1x)dx for β < 1. F 1 is a Lévy measure with F 1 dx) = ν 1 x)dx where ν 1 x) is defied i.5). Fially, assume κx) is F 1 -a.s. cotiuous ad i additio that it is symmetric whe β = 1. The as we have: a) b) 1/β Ψ t L L t, 6.) p/β E Ψ t p ) E L t p ), locally uiformly i t for some p < β. 6.3) 18

19 Proof: Part a). Sice 1/β Ψ t is a Lévy process to prove the covergece of the sequece we eed to show the covergece of its characteristics, see e.g. Jacod ad Shiryaev 3), Corollary III.3.6. The case β < 1 is straightforward, we show the case β 1. We eed to establish the followig for κ 1/β ) x) 1/β κx) νx)1 { x ǫ} dx κ 1/β x)νx)1 { x ǫ} dx g 1/β x)νx)1 { x ǫ} dx κ x)ν 1 x)dx, 6.4) κ x)ν 1 x)dx, 6.5) gx)ν 1 x)dx, 6.6) where g is a arbitrary cotiuous ad bouded fuctio o, which is aroud. First, the covergece i 6.5) ad 6.6) trivially follow from a chage of variable i the itegratio ad the fact that the fuctio φ i.5) is bouded aroud the origi. Therefore, we are left with showig 6.4). For β = 1 this follows from the symmetry of the trucatio fuctio ad ν. We show 6.4) therefore oly i the case β > 1. First, we have = κ 1/β κ 1/β x) 1/β κx) ) x) 1/β x ) ν 1 x)1 { x ǫ} dx ν 1 x)1 { x ǫ} dx + 1 1/β x κx))ν 1 x)1 { x ǫ} dx κ x)ν 1 x)dx. 6.7) Thus we will prove 6.4) if we ca show κ 1/β x) 1/β κx) ) ν x)dx. 6.8) It suffices to show 6.4) for κx) = x1{ x 1}. For small eough we have ) 1/β κx) κ 1/β x) ν x)dx = 1 1/β 1 1/β 1 1/β 1 1/β xν x)dx + 1 1/β x β φx)dx + 1 1/β 1/β 1 1/β 1 xν x)dx x β φx)dx. 6.9) For sufficietly small, usig the fact that φx) slowly varies aroud zero, has a fiite limit at ad is itegrable aroud, we have 1 1/β x β φx)dx K + Kφ 1/β ) 1/β β /β, 6.1) for some costat K. The same result obviously holds for 1/β x β φx)dx, ad from here 1 recall we are lookig at the case β > 1) we have the result i 6.8). Thus, combiig 6.7) ad 6.8), we have the result i 6.4) ad therefore 6.) is established. Part b). First we establish the poitwise covergece for arbitrary t >. Give the result i Part a), we eed oly show that for some p such that p < p < β we have sup E p /β Ψ t p ) <. 6.11) 19

20 I order to cosider the cases β 1 ad β < 1 together we make a slight abuse of otatio. I what follows we assume that κx) ad tur off the drift b whe β < 1, while whe β 1 we make o chage ad κx) cotiues to be a true trucatio fuctio i.e. it has bouded support ad coicides with the idetity aroud the origi). With this assumptio o κx), deotig with µ the jump measure associated with the process Ψ t ad with µ its compesated versio, we ca write 1/β 1/β Ψ t = 1/β Ψ t i all cases of β) as κx) µds,dx) + 1/β t t κ x)µds,dx). 6.1) The, we ca proceed with the followig decompositio t p 1/β Ψ t p K 1/β κx) µds,dx) x >s 1/β t p +K 1/β κx) µds,dx) x s 1/β t p +K κ x)µds,dx), 6.13) 1/β where K is some costat. Obviously, give the fact that κx) if β < 1, for the first two terms o the right had side of 6.13) we look oly at the case β 1. For the secod itegral o the right-had side of 6.11) for some 1 q such that q > β we have t p t q) p /q E 1/β κx) µds,dx) E x s 1/β κx) µds,dx) 1/β x s 1/β t )) p /q K q/β E κx) q dsνx)dx K, 6.14) x s 1/β where the first iequality follows from the fact that p < q; the secod oe is a applicatio of Burkholder-Davis-Gudy iequality see e.g. Protter 4)); ad the third oe uses the fact that q, the properties of the fuctio φx) ad a similar argumet as i 6.1). For the first itegral o the right-had side of 6.13) if β > 1 we ca pick some q such that max{p,1} q < β ad make exactly the same steps as i 6.14) to get t p E 1/β κx) µds,dx) K. 6.15) x >s 1/β Whe β = 1, ν is symmetric so the itegratio with respect to µ is the same as the itegratio with respect to the compesated measure µ aroud the origi. Thus, whe β = 1, for the first itegral o the right-had side of 6.13) we ca write t p E 1/β κx) µds,dx) x >s 1/β t t p = p /β E κx)µds,dx) κx)dsνdx) x >s 1/β x >s 1/β t ) p /β E κx) p νds, dx) + K K, 6.16) x >s 1/β