Limit Theorems for the Empirical Distribution Function of Scaled Increments of Itô Semimartingales at high frequencies

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1 Limit Theorems for the Empirical Distributio Fuctio of Scaled Icremets of Itô Semimartigales at high frequecies Viktor Todorov ad George Tauche February 4, 03 Abstract We derive limit theorems for the empirical distributio fuctio of devolatilized icremets of a Itô semimartigale observed at high frequecies. These devolatilized icremets are formed by suitably rescalig ad trucatig the raw icremets to remove the effects of stochastic volatility ad large umps. We derive the limit of the empirical cdf of the adusted icremets for ay Itô semimartigale whose domiat compoet at high frequecies has activity idex of < β, where β = correspods to diffusio. We further derive a associated CLT i the ump-diffusio case. We use the developed limit theory to costruct a feasible ad pivotal test for the class of Itô semimartigales with o-vaishig diffusio coefficiet agaist Itô semimartigales with o diffusio compoet. Keywords: Itô semimartigale, Kolmogorov-Smirov test, high-frequecy data, stochastic volatility, umps, stable process. JEL classificatio: C5, C5, G. Research partially supported by NSF Grat SES Departmet of Fiace, Kellogg School of Maagemet, Northwester Uiversity, Evasto, IL 6008; v-todorov@orthwester.edu. Departmet of Ecoomics, Duke Uiversity, Durham, NC 7708; george.tauche@duke.edu.

2 Itroductio The stadard ump-diffusio model used for modelig may stochastic processes is a Itô semimartigale give by the followig differetial equatio dx t = α t dt + σ t dw t + dy t,. where α t ad σ t are processes with càdlàg paths, W t is a Browia motio ad Y t is a Itô semimartigale process of pure-ump type i.e., semimartigale with zero secod characteristic, Defiitio II..6 i Jacod ad Shiryaev 003. At high-frequecies, provided σ t does ot vaish, the domiat compoet of X t is its cotiuous martigale compoet ad at these frequecies the icremets of X t i. behave like scaled ad idepedet Gaussia radom variables. That is, for each fixed t, we have the followig covergece h X t+sh X t L σ t B t+s B s, as h 0 ad s [0, ],. where B t is a Browia motio ad the above covergece is for the Skorokhod topology. There are two distictive features of the covergece i.. The first is the scalig factor of the icremets o the left side of. is the squareroot of the legth of the high-frequecy iterval, a feature that has bee used i developig tests for presece of diffusio. The secod distictive feature is that the limitig distributio of the scaled icremets o the right side of. is mixed Gaussia the mixig give by σ t. Both these features of the local Gaussiaity result i. for models i. have bee key i the costructio of essetially all oparametric estimators of fuctioals of volatility. Examples iclude the ump-robust Bipower Variatio of Bardorff-Nielse ad Shephard 004, 006 ad the may other alterative measures of powers of volatility summarized i the recet book of Jacod ad rotter 0. Aother importat example is the geeral approach of Myklad ad Zhag 009 see also Myklad

3 et al. 0 where estimators of fuctios of volatility are formed by utilizig directly. ad workig as if volatility is costat over a block of decreasig legth. Despite the geerality of the ump-diffusio model i., however, there are several examples of stochastic processes cosidered i various applicatios that are ot ested i the model i.. Examples iclude pure-ump Itô semimartigales, semimartigales cotamiated with oise or more geerally o-semimartigales. I all these cases, both the scalig costat o the left side of. as well as the limitig process o the right side of. chage. Our goal i this paper, therefore, is to derive a limit theory for a feasible versio of the local Gaussiaity result i. based o high-frequecy record of X. A applicatio of the developed limit theory is a feasible ad pivotal test based o Kolmogorov-Smirov type distace for the class of Itô semimartigales with o-vaishig diffusio compoet. The result i. implies that the high-frequecy icremets are approximately Gaussia but the key obstacle of testig directly. is that the variace of the icremets, σt, is ukow ad further is approximately costat oly over a short iterval of time. Therefore, o a first step we split the highfrequecy icremets ito blocks with legth that shriks asymptotically to zero as we sample more frequetly ad form local estimators of volatility over the blocks. We the scale the high-frequecy icremets withi each of the blocks by our local estimates of the volatility. This makes the scaled highfrequecy icremets approximately i.i.d. cetered ormal radom variables with uit variace. To purge further the effect of big umps, we the discard the icremets that exceed a time-varyig threshold that shriks to zero asymptotically with time-variatio determied by our estimator of the local volatility. We derive a fuctioal Cetral Limit Theorem CLT for the covergece of the empirical cdf of the scaled high-frequecy icremets, ot exceedig the threshold, to the cdf of a stadard ormal radom variable. The rate of covergece ca be made arbitrary close to, by appropriately choosig the rate of 3

4 icrease of the block size, where is the umber of high-frequecy observatios withi the time iterval. This is achieved despite the use of the block estimators of volatility, each of which ca estimate the spot volatility σ t at a rate o faster tha /4. We further derive the limit behavior of the empirical cdf described above i two possible alteratives to the model.. The first is the case where X t does ot cotai a diffusive compoet, i.e., the secod term i. is abset. Models of these type have received a lot of attetio i various fields, see e.g., Bardorff-Nielse ad Shephard 00; Adrews et al. 009, Mikosch et al. 00, Klüppelberg et al. 00 ad Woerer 007. The secod alterative to. is the case i which the Itô semimartigale is distorted with measuremet error. I each of these two cases, the empirical cdf of the scaled high-frequecy icremets below the threshold coverges to a cdf of a distributio differet from the stadard ormal law. This is the stable distributio i the pure-ump case ad the distributio of the oise i the case of Itô semimartigale observed with error. The paper is orgaized as follows. I Sectio we itroduce the formal setup ad state the assumptios eeded for our theoretical results. I Sectio 3 we costruct our statistic ad i Sectios 4 ad 5 we derive its limit behavior. Sectio 6 costructs a feasible test for local Gaussiaity usig our limit theory ad i Sectios 7 ad 8 we apply the test o simulated ad real fiacial data respectively. The proofs are give i Sectio 9. Setup We start with the formal setup ad assumptios. We will geeralize the setup i. to accommodate also the alterative hypothesis i which X ca be of pure-ump type. Thus, the geeralized setup we cosider is the followig. The process X is defied o a filtered space Ω, F, F t t 0, ad has the followig 4

5 dyamics dx t = α t dt + σ t ds t + dy t,. where α t, σ t ad Y t are processes with càdlàg paths adapted to the filtratio ad Y t is of pure-ump type. S t is a stable process with a characteristic fuctio, see e.g., Sato 999, give by log [ Ee iust ] { = t cu β taβ/ if β, iγsiguφ, Φ = log u, if β =,. where β 0, ] ad γ [, ]. Whe β = ad c = / i., we recover our origial ump-diffusio specificatio i. i the itroductio. Whe β <, X is of pure-ump type. Y t i. will play the role of a residual ump compoet at high frequecies see assumptio A below. We ote that Y t ca have depedece with S t ad α t ad σ t, ad thus X t does ot iherit the tail properties of the stable process S t, e.g., X t ca be drive by a tempered stable process whose tail behavior is very differet from that of the stable process. Throughout the paper we will be iterested i the process X over a iterval of fixed legth ad hece without loss of geerality we will fix this iterval to be [0, ]. We collect our basic assumptio o the compoets i X ext. Assumptio A. X t satisfies.. A. σ t ad σ t are strictly positive o [0, ]. Further, there is a sequece of stoppig times T p icreasig to ifiity ad for each p a bouded process σ p t satisfyig t < T p = σ t = σ p t ad a positive costat K p such that E σ p t σ s p F s K p t s, for every 0 s t..3 A. There is a sequece of stoppig times T p icreasig to ifiity ad for each p a process Y p t satisfyig t < T p = Y t = Y p t ad a positive costat K p such that E Y p t Y p s q F s K p t s, for every 0 s t,.4 5

6 ad for every q β, where β < β. The assumptio i.3 ca be easily verified for Itô semimartigales which is the typical way of modelig σ t, but it is also satisfied for models outside of this class. The coditio i.4 ca be easily verified for pure-ump Itô semimartigales, see e.g., Corollary..9 of Jacod ad rotter 0. Uder assumptio A, we ca exted the local Gaussiaity result i. to see e.g., Lemma of Todorov ad Tauche 0a h /β X t+sh X t L σ t S t+s S t, as h 0 ad s [0, ],.5 for every t ad where S t is a Lévy process idetically distributed to S t ad the covergece i.5 beig for the Skorokhod topology. That is, the local behavior of the icremets of the process is like that of a stable process i the more geeral settig of.. For derivig the CLT for our statistic i the case of the ump-diffusio model i., we eed a stroger assumptio which we state ext. Assumptio B. X t satisfies. with β =, i.e., S t = W t. B. The process Y t is of the form Y t = t 0 E δ Y s, xµds, dx,.6 where µ is oisso measure o R + E with Lévy measure νdx ad δ Y t, x is some predictable fuctio o Ω R + E. B. σ t ad σ t are strictly positive o [0, ]. Further, σ t is a Itô semimartigale havig the followig represetatio t σ t = σ 0 + α u du + 0 t 0 σ u dw u + t 0 σ udw u + t 0 E δ σ s, xµds, dx,.7 where W t is a Browia motio idepedet from W t ; α t, σ t ad σ t are processes with càdlàg fuctios ad δ σ t, x is a predictable fuctio o Ω R + E. B3. σ t ad σ t are Itô semimartigales with coefficiets with càdlàg paths ad further umps beig itegrals of some predictable fuctios, δ σ ad δ σ, with respect to the ump measure µ. 6

7 B4. There is a sequece of stoppig times T p icreasig to ifiity ad for each p a determiistic oegative fuctio γ p x o E, satisfyig νx : γ p x 0 < ad such that δ Y t, x + δ σ t, x + δ σ t, x + δ σ t, x γ p x for t T p. The Itô semimartigale restrictio o σ t ad its coefficiets is satisfied i most applicatios ad from a practical poit of view is ot very restrictive. Similarly, we allow for geeral time-depedece i the umps i X which ecompasses most cases i the literature. B4 is the strogest assumptio ad it requires the umps to be of fiite activity. This assumptio ca be further relaxed to allow for ifiite activity umps. 3 Empirical CDF of the Devolatilized High Frequecy Icremets Throughout the paper we assume that X is observed o the equidistat grid 0,,..., with. I the derivatio of our statistic we will suppose that S t is a Browia motio ad the i the ext sectio we will derive its behavior uder the more geeral case whe S t is a stable process. The result i. suggests that the high-frequecy icremets i X = X i are approximately Xi Gaussia with variace give by the value of the process σt at the begiig of the icremet. Of course, the stochastic volatility σ t is ot kow ad varies over time. Hece to test for the local Gaussiaity of the high-frequecy icremets we first eed to estimate locally σ t ad the divide the high-frequecy icremets by this estimate. To this ed, we divide the iterval [0, ] ito blocks each of which cotais k icremets, for some determiistic sequece k with k / 0. O each of the blocks our local estimator of σ t is give by = k k i= k + i X i X, =,..., /k. 3. is the Bipower Variatio proposed by Bardorff-Nielse ad Shephard 004, 006 for measurig the quadratic variatio of the diffusio compoet of X. 7

8 We ote that a alterative measure of σ t ca be costructed usig the so-called trucated variatio. I turs out, however, that while the behavior of the two volatility measures i the case of the ump-diffusio model. is the same, it differs i the case whe S t is stable with β <. Usig a trucated variatio type estimator of σ t will lead to degeerate limit of our statistic, ulike the case of usig the Bipower Variatio estimator i 3.. For this reaso we prefer the latter i our aalysis. We use the first m icremets o each block, with m k, to test for local Gaussiaity. The case m = k amouts to usig all icremets i the block ad we will eed m < k for derivig a feasible CLT later o. Fially, we eed to remove the high-frequecy icremets that cotai big umps. The total umber of icremets used i our statistic is thus give by N α, ϖ = /k = k +m i= k + i X α ϖ, 3. where α > 0 a ϖ 0, /. We ote that we use a time-varyig threshold i our trucatio to accout for the time-varyig σ t. With this, we defie /k k F +m τ = N i X α, ϖ { τ = i= k + i X α ϖ}, 3.3 which is simply the empirical cdf of the devolatilized icremets that do ot cotai big umps. I the ump-diffusio case of., F τ should be approximately the cdf of a stadard ormal radom variable. 4 Covergece i probability of F τ We ext derive the limitig behavior of F τ both uder the ull of model. as well as uder a set of alteratives. We start with the case whe X t is give by.. Theorem Suppose assumptio A holds ad assume the block size grows at 8

9 the rate k q, for some q 0,, 4.4 ad m as. The if β, ], we have F τ F β τ, as, 4.5 where the above covergece is uiform i τ over compact subsets of, 0 0, + ; F β τ is the cdf of S E S S is the value of the β-stable process S t at time ad F τ equals the cdf of a stadard ormal variable Φτ. We ote that the covergece result i 4.5 is for values of τ excludig a eighborhood of zero. The reaso is that the cdf for those values of τ are affected by the presece of the drift term as well as Y t. Similarly, due to the big umps, we derive the behavior of the statistic oly o compact sets of τ as the estimate of the cdf for τ ± is affected by the trucatio. Remark. The limit result i 4.5 shows that whe S t is stable with β <, F τ estimates the cdf of a β-stable radom variable. We ote that whe β <, the correct scalig factor for the high-frequecy icremets is /β. However, i this case we eed also to scale by /β / i order for the latter to coverge to a o-degeerate limit that is proportioal to σ t. Hece the ratio i X = /β i X /β, 4.6 is appropriately scaled eve i the case whe β < ad importatly without kowig apriori the value of β. We further ote that the limitig cdf, F β τ, is of a radom variable that has the same scale regardless of the value of β. That is, i all cases of β, F β τ correspods to the cdf of a radom variable Z with E Z =. Therefore, the differece betwee β < ad the ull β = will be i the relative probability assiged to big versus small values of τ. We ote further that i Theorem we restrict β >. The reaso is that for β, the limit behavior of F τ is determied by the drift term i X whe preset ad ot S t. To allow for β ad still have a limit result of 9

10 the type i 4.5, we eed to use i X i X i the costructio of F τ which essetially elimiates the drift term. We ext derive the limitig behavior of F τ i the situatio whe the Itô semimartigale X is cotamiated by oise, which is of particular relevace i fiacial applicatios. Theorem Suppose assumptio A holds ad k q for some q 0, ad m as. Let F τ be give by 3.3 with i { } X replaced with i X for X i = X i + ɛ i ad where ɛ i are i.i.d. radom i=,..., variables defied o a product extesio of the origial probability space ad idepedet from F. Further, suppose E ɛ i +ι < for some ι > 0. Fially, assume that the cdf of µ ɛ i ɛ i, F ɛ τ, is cotiuous where we deote µ = E ɛ i ɛ i ɛ i ɛ i. The F τ F ɛ τ, as, 4.7 where the above covergece is uiform i τ over compact subsets of, 0 0, +. Remark. Whe X is observed with oise, the oise becomes the leadig compoet at high-frequecies. Hece, our statistic recovers the cdf of the appropriately scaled oise compoet. Similar to the pure-ump alterative of S t with β <, here is ot the right scalig for the icremets i X but this is offset i the ratio i F τ by a scalig factor for the local variace estimator that makes it o-degeerate. Ulike the pure-ump alterative, i the presece of oise the correct scalig of the umerator ad the deomiator i the ratio i F τ is give by i X that is, we eed to scale dow = i X, 4.8 to esure it coverges to o-degeerate limit. The limit result i 4.7 provides a importat isight ito the oise by studyig its distributio. We stress the fact that the presece of 0 i the

11 trucatio is very importat for the limit result i 4.7. This is because it esures that the threshold is sufficietly big so that it does ot matter i the asymptotic limit. If, o the other had, the threshold did ot cotai i.e., was replaced by i the threshold, the i this case the limit will be determied by the behavior of the desity of the oise aroud zero. We fially ote that whe ɛ i is ormally distributed, a case that has received a lot of attetio i the literature, the limitig cdf F ɛ τ is that of a cetered ormal but with variace that is below. Therefore, i this case F ɛ τ will be below the cdf of a stadard ormal variable, Φτ, whe τ < 0 ad the same relatioship will apply to F ɛ τ ad Φτ whe τ > 0. O a more geeral level, the above results show that the empirical cdf estimator F τ ca shed light o the potetial sources of violatio of the local Gaussiaity of high-frequecy data. It similarly ca provide isights o the performace of various estimators that deped o this hypothesis. 5 CLT for F τ uder Local Gaussiaity Theorem 3 Let X t satisfy. with S t beig a Browia motio ad assume that assumptio B holds. Further, let the block size grow at the rate m k 0, k q, for some q 0, /, whe. 5.9 We the have locally uiformly i subsets of, 0 0, + F τ Φτ = Ẑ τ + Ẑ τ + τ Φ τ τφ τ k 8 + o p, k /k m Ẑ τ /k k Ẑ τ L Z τ Z τ, 5. where Φτ is the cdf of a stadard ormal variable ad Z τ ad Z τ are

12 two idepedet Gaussia processes with covariace fuctios Cov Z τ, Z τ = Φτ τ Φτ Φτ, [ τ Φ τ τ Φ ] τ Cov Z τ, Z τ = + 3, τ, τ R \ We make several observatios regardig the limitig result i The first term of F τ Φτ i 5.0, Ẑ τ, coverges to Z τ which is the stadard Browia bridge appearig i the Dosker theorem for empirical processes, see e.g., va der Vaart 998. The secod ad third term o the right-had side of 5.0 are due to the estimatio error i recoverig the local variace, i.e., the presece of i F τ istead of the true uobserved σ t. Ẑ τ coverges to a cetered Gaussia process, idepedet from Z τ, while the third term o the right-had side of 5.0 is a asymptotic bias. Importatly, the asymptotic bias as well as the limitig variace of Z τ Z τ are all costats that deped oly o τ ad ot the stochastic volatility σ t. Therefore, feasible iferece based o 5.0 is straightforward. We ote that by pickig the rate of growth of m ad k arbitrary close to, we ca make the rate of covergece of F τ arbitrary close to. We should further poit out that this is ulike the rate of estimatig the spot σ t by with the same choice of k which is at most /4. The reaso for the better rate of covergece of our estimator is i the itegratio of the error due to the estimatio. The order of magitude of the three compoets o the right-had side of 5.0 are differet with the secod term always domiated by the other two. Its presece should provide a better fiite-sample performace of a test based o 5.0. The order of magitude of the other two compoets ca be made equal by settig m k3, which provides the optimal rate of covergece for F τ Φτ with respect to the choice of m. Fially, we poit out that a feasible CLT for F τ is available with oly arbitrarily close to rate of covergece ad ot exactly. This is due to the presece of the drift term i X. The latter leads to asymptotic bias which

13 is of order / ad removig it via de-biasig is i geeral impossible as we caot estimate the latter from high-frequecy record of X. 6 Test for Local Gaussiaity of High-Frequecy Data We proceed with a feasible test for a ump-diffusio model of the type give i. usig the developed limit theory above. The critical regio of our proposed test is give by { C = sup τ A } N α, ϖ F τ Φτ > q α, A 6.3 where recall Φτ deotes the cdf of a stadard ormal radom variable, α 0,, A R\0 is a fiite uio of compact sets with positive Lebesgue measure, ad q α, A is the α-quatile of sup τ A Z m m τ Φ τ τφ τ τ + Z τ + + 3, k k k with Z τ ad Z τ beig the Gaussia processes defied i Theorem 3. We ca easily evaluate q α, A via simulatio. The test i 6.3 resembles a Kolmogorov-Smirov type test for equality of cotiuous oe-dimesioal distributios. There are two differeces betwee our test ad the origial Kolmogorov-Smirov test. First, i our test we scale the high-frequecy icremets by a oparametric local estimator of the volatility ad this has a asymptotic effect o the test statistic, as evidet from Theorem 3. The secod differece is i the regio A over which the differece F τ Φτ is evaluated. For reasos we already discussed, that are particular to our problem here, we eed to exclude arbitrary high i magitude values of τ as well as a arbitrary small iterval excludig the zero. Now, i terms of the size ad power of the test, uder assumptios A ad B, usig Theorem ad Theorem 3, we have lim C = α, if β = ad lim if C =, if β,, 6.5 3

14 where we make also use of the fact that the stable ad stadard ormal variables have differet cdf-s o compact subsets of R\ 0 with positive Lebesgue measure. By Theorem, the above power result applies also to the case whe we observe X i + ɛ i, provided of course the limitig cdf of the oise i 4.7 differs from that of the stadard ormal o the set A. We ote that existig tests for presece of diffusive compoet i X are based oly o the scalig factor of the high-frequecy icremets o the left side of.5. However, the limitig result i.5 implies much more. Maily, the distributio of the devolatilized icremets should be stable ad i particular ormal i the ump-diffusio case. icorporates this distributio implicatio of.5 as well. 7 Mote Carlo Our test i 6.3, ulike earlier work, We ow evaluate the performace of our test o simulated data. We cosider the followig two models. The first is dx t = V t dw t + xµds, dx, dv t = V t dt + 0. V t db t, 7. R where W t, B t is a vector of Browia motios with CorrW t, B t = 0.5 ad µ is a homogeous oisso measure with compesator νdt, dx = dt 0.5e x / dx which correspods to double expoetial ump process with itesity of 0.5 i.e., a ump every secod day o average. This model is calibrated to fiacial data by settig the meas of cotiuous ad ump variatio similar to those foud i earlier empirical work. Similarly, we allow for depedece betwee X t ad V t, i.e., leverage effect. The secod model is give by X t = S Tt, with T t = t 0 V s ds, 7. where S t is a symmetric tempered stable martigale with Lévy measure 0.089e x x +.8 ad V t is the square-root diffusio give i 7.. The process i 7. is a timechaged tempered stable process. The parameters of S t are chose such that it behaves locally like.8-stable process ad it has variace at time equal to 4

15 as the model i 7.. For this process the local Gaussiaity does ot hold ad hece the behavior of the test o data from the model i 7. will allow us to ivestigate the power of the test. We tur ext to the implemetatio of the test. We apply the test o oe year worth of simulated data which cosists of 5 days our uit of time is oe tradig day. We cosider two samplig frequecies: = 00 ad = 00 which correspod to samplig every 5 ad miutes respectively i a typical tradig day. We experimet with -3 blocks per day. I each block we use 75% or 70% of the icremets i the formatio of the test, i.e., we set m /k = 0.75 for = 00 ad m /k = 0.70 for = 00. We foud very little sesitivity of the test with respect to the choice of the ratio m /k. For the trucatio of the icremets, as typical i the literature, we set α = 3.0 ad ϖ = Fially, the set A over which the differece F τ F τ i our test is evaluated is set to A = [Q0.0 : Q0.40] [Q0.60 : Q0.99], 7.3 where Qα is the α-quatile of stadard ormal. The results of the Mote Carlo are reported i Tables ad. For the smaller sample size, = 00, ad with o blockig at all k = to accout for volatility movemets over the day, there are size distortios most oticeable at the covetioal 5 percet level. With two blocks /k =, size is appropriate ad, if aythig, the test slightly favors the ull, while it is see to have excellet power i Table. But with three blocks o = 00, there are size distortios because the oisy estimates of local volatility distort the test. Cosiderig the larger sample size = 00, ow with three blocks the test s size is correct ad agai the test slightly favors the ull while power is excellet. For larger values of k relative to /k = or /k = the time variatio i volatility over the day coupled with the relatively high precisio of estimatig a biased versio of local volatility, leads to departures from Gaussiaity of the small scaled icremets ad hece the over-reectios. 5

16 Table : Mote Carlo Results for Jump-Diffusio Model 7. Nomial Size Reectio Rate Samplig Frequecy = 00 k = 33 k = 50 k = 00 α = % α = 5% Samplig Frequecy = 00 k = 67 k = 00 k = 00 α = % α = 5% Note: For the cases with = 00 we set m /k = 0.75 ad for the cases with = 00 we set m /k = Empirical Illustratio We ow apply our test to two differet fiacial assets, the IBM stock price ad the VIX volatility idex. The aalyzed period is ad like i the Mote Carlo we cosider two ad five miute samplig frequecies. The test is performed for each of the years i the sample. We set A as i 7.3 ad /k = for the five-miute samplig frequecy ad /k = 3 for the two-miute frequecy. As i the Mote Carlo, the ratio m /k is set to 0.75 ad 0.70 for the five-miute ad two-miute respectively samplig frequecies. Fially, to accout for the well-kow diural patter i volatility we stadardize the raw high-frequecy returs by a time-of-day scale factor exactly as i Todorov ad Tauche 0b. The results from the test are show o Figure. We ca see from the figure that the local Gaussiaity hypothesis works very well for the 5-miute IBM returs. At -miute samplig frequecy for the IBM stock price, however, our test reects the local Gaussiaity hypothesis at covetioal sigificace levels. Nevertheless, the values of the test are still fairly close to the critical oes. The explaatio of the differet outcomes of the test o the two samplig frequecies is to be foud i the presece of microstructure oise. The latter becomes more 6

17 Table : Mote Carlo Results for ure-jump Model 7. Nomial Size Reectio Rate Samplig Frequecy = 00 k = 33 k = 50 k = 00 α = % α = 5% Samplig Frequecy = 00 k = 67 k = 00 k = 00 α = % α = 5% Note: For the cases with = 00 we set m /k = 0.75 ad for the cases with = 00 we set m /k = promiet at the higher frequecy. Turig to the VIX idex data, we see a markedly differet outcome. For this data set, the local Gaussiaity hypothesis is strogly reected at both frequecies. The explaatio for this is that the uderlyig model is of pure-ump type, i.e., the model. with β <. 9 roofs We start with itroducig some otatio that we will make use of i the proofs. A t = t α s ds, B t = t 0 0 σ s ds s, σ t = σ t s t σ s. Ṽ = k = k k i= k + k i= k + V i = V σ k i X i X, V = k i B i B, V = σ k k i= k + k i A+ i B i A+ i B, k i= k + i S i S, k i S i S + i S i+s, i =,...,, where we set 0 S = +S = 0. We also deote F τ = N α, ϖ /k m F τ. 9. 7

18 IBM, 5 mi VIX, 5 mi IBM, mi 8 VIX, mi Year Year Figure : Kolmogorov-Smirov tests for local Gaussiaity. The correspods to the value of the test sup τ A N α, ϖ F τ F τ ad the solid lies are the critical values q α, A for α = 5% ad α = %. Fially, i the proofs we will deote with K a positive costat that might chage from lie to lie but importatly does ot deped o ad τ. We will also use the shorthad otatio E i F = E i. 9. Localizatio We will prove Theorems -3 uder the followig stroger versios of assumptio A ad B: SA. We have assumptio A with α t, σ t ad σ t beig all uiformly bouded o [0, ]. Further,.3 ad.4 hold for σ t ad Y t respectively. SB. We have assumptio B with all processes α t, α t, σ t, σt, σ t, σ t, Y t ad 8

19 the coefficiets of the Itô semimartigale represetatios of σ t ad σ t beig uiformly bouded o [0, ]. Further δ Y t, x + δ σ t, x + δ σ t, x + δ σ t, x γx for some o-egative valued fuctio γx o E satisfyig νx : γx = E 0dx < ad γx K for some costat K. Extedig the proofs to the weaker assumptios A ad B follows by stadard localizatio techiques exactly as Lemma of Jacod ad rotter roof of Theorem Without loss of geerality, we will assume that τ < 0, the case τ > 0 beig dealt with aalogously by workig with F τ istead. We first aalyze the behavior of. We deote with η a determiistic sequece that depeds oly o ad vaishes as. Usig the triagular iequality, the Chebyshev iequality, successive coditioig, as well as Hölder iequality ad assumptio SA, we get for =,..., /k /β V +ι η K /β /β, ι > 0. η Similarly, usig triagular iequality, Chebyshev iequality as well as Hölder iequality, we get for =,..., /k /β V Ṽ η K /β +ι, ι > 0. η Next, usig, triagular iequality, Chebyshev iequality, Hölder iequality, Burkholder- Davis-Gudy iequality as well as assumptio SA, we get for =,..., /k /β Ṽ V η K k/ ι, ι > 0. / ι η Fially, usig the self-similarity of the stable process ad Burkholder-Davis- Gudy iequality for discrete martigales, we get for =,..., /k /β V σ k E S η K k β ι η β ι, ι 0, β. 9

20 Combiig these results, we get altogether for ι 0, β /β σ k E S η /β /β +ι / ι k K η / ι η k β ι η β ι. 9. Next, for i = k +..., k + m ad =,..., /k, we deote ξi, = /β i A + i Y + i i σ u σ k ds u ξ i, = /β σ k i S{ i X >α ϖ }. With this otatio, usig similar iequalities as before, we get ξi, /β /β +ι β/+ι/ k η K η β/+ι/ η β+ι. 9.3 Next, usig the result i 9. above as well as Hölder iequality, we get ξi, / ϖβ+ι /β /β +ι k / / ι k +ι β η K, ι > 0. η ι 9.4 We ext deote the set ote that by assumptio SA, σ t is strictly above zero o the time iterval [0, ] A i, = ω : ξ i, + ξ i, E S > η /β / σ k for i = k +,..., k + m ad =,..., /k. We ow ca set recall 4.4 η = x, 0 < x < [ β q β E S > η, 9.5 ] qβ, 9.6 β ad this choice is possible because of the restrictio o the rate of icrease of the block size k relative to give i 4.4. With this choice of η, the results i 9., 9.3 ad 9.4 imply /k m /k = k +m i= k + A i, = o

21 Therefore, for ay compact subset A of, 0, where we deote Ĝ τ = /k m /k = sup F τ Ĝτ = o p, 9.8 τ A k +m i= k + i X { i X α ϖ} τ {A i, }. c Takig ito accout the defiitio of the set A i,, we get { } /k k+m Ĝ τ /k m = i= k /β i S τ η + η, { E S } /k k+m Ĝ τ /k m = i= k /β i S τ + η + + η. E S Usig Gliveko-Catelli theorem, see e.g., Theorem 9. of va der Vaart 998, we have sup τ sup τ /k m /k m /k = /k = k +m i= k + k +m i= k + { /β i S E S τ η η } F β τ η η { /β i S E S τ + η + η } F β τ + η + η 0, 0, ad further usig the smoothess of cdf of the stable distributio we have sup τ F β τ η η F β τ 0, sup F β τ + η + η F β τ 0. τ These two results altogether imply sup Ĝτ F β τ τ 0, ad from here, usig 9.8, we have sup τ A F τ F β τ = o p for ay compact subset A of, 0. Hece, to prove 4.5, we eed oly to show N α, ϖ /k m, as. 9.9

22 We have i X > α /β / ϖ σ k E S > /β i X > 0.5α σ k E S / ϖ. From here we ca use the bouds i 9. ad 9.3 to coclude i X > α ϖ K, for some sufficietly small ι > 0, 9.0 ι hece the covergece i 9.9 holds which implies the result i roof of Theorem The proof follows the same steps as that of Theorem. We deote with η a determiistic sequece depedig oly o ad vaishig as. The, usig triagular iequality ad successive coditioig, we have µ η K /, 9. η ɛ i ɛ i { η i X >α ϖ} K ϖ / η ι. 9. We deote Bi, = ω : i X { i X α ϖ} ɛ i ɛ i > η µ > η, for i = k +,..., k + m ad =,..., /k. We set η = x for 0 < x < ι / ϖ /. With this choice /k m /k = k +m i= k + B i, = o. Therefore, for ay compact subset A of, 0, we have where we deote Ĝ τ = /k m /k = sup F τ Ĝτ = o p, τ A k +m i= k + i X { i X α ϖ} τ {B i, }. c

23 Takig ito accout the defiitio of the set Bi,, we get /k { k+m Ĝ τ /k m = i= k + µ ɛ i /k { k+m Ĝ τ /k m = i= k + µ ɛ i ɛ i ɛ i } τ η η µ, } τ + η + η µ. From here we ca proceed exactly i the same way as i the proof of Theorem to show that Ĝτ show N α,ϖ /k m F ɛ τ locally uiformly i τ. Hece we eed oly as. This follows from i X > α ϖ µ > i X > 0.5αµ / ϖ K, for some sufficietly small ι > 0, ι which ca be show usig 9., the fact that the oise term has a fiite first momet ad the Burkholder-Davis-Gudy iequality. 9.4 roof of Theorem 3 As i the proof of Theorem, without loss of geerality we will assume τ < 0. First, give the fact that m /k 0, it is o limitatio to assume k m > ad we will do so heceforth. Here we eed to make some additioal decompositio of the differece Ṽ V. It is give by the followig Ṽ V = R + R + R 3 + R 4, =,..., /k, 9.3 R R = k k i= k + = k σ k R 3 = k σ k [ i B i B σ i i W i W k i= k + k i= k + ] + σ i σ k i W i W, [ σ i σ k [ i k i k i k σ k dw u σ k dw u + i W i W i, σ k dw u ] i W i W, k σ k dw u ] 3

24 R 4 = σ k k k i= k + [ i k For i = k +,..., k we set R 4 i = R 4 σ k k [ i k i i i σ k dw u + k i σ k dw u + i σ k dw u ]. σ k dw u We further deote for i = k +,..., k +m ad =,..., /k, ξ = ξ i, = ξ = ξ = ξ i,3 = ξ i,4 = + σ k σ k, ξ = σ k, 8σ 4 k V i + R 4 i σ k, σ ξ i, = k V + R 4 σ k V σ k σ k i W [ σ k σ k σ k σ k, ξ = σ k [ σ k, ξ = W i W i V i + R 4 i σ k 8σ 4 k V + R 4 σ k V σ k 8σ 4 k W k W k i 8σ 4 k, + σ k W i + σ k W i, ] W k, W k With this otatio we set for i = k +,..., k + m ad =,..., /k χ i, = i i A + i Y + σ u σ i dw u { i X α ϖ} + i W { i X >α ϖ} χ i, = σ k i W σ k, ]. ]. σ i σ k ξi,3 { i X α ϖ}, ξ + ξ + ξ ξ ξ i, + ξ i,. 4

25 Fially, we deote Ĝ τ = /k m = /k m /k = /k = k +m i= k + k +m i= k + The proof cosists of three parts: i X σ { k i X α ϖ} τ σ k χ i, τχ i, i W τ + τ ξ i, τ ξ i, ξi,3. the first is showig the egligibility of k F τ Ĝτ, the secod is derivig the limitig behavior of Ĝτ Φτ ad third part is showig egligibility of k F τ F τ The differece F τ Ĝτ We first collect some prelimiary results that we the make use of i aalyzig F τ Ĝτ. We start with max i=,..., i B. We have i max i=,..., i B max σ i=,..., u σ i dw u i + sup s [0,] σ s max i=,..., i W. Usig the the results for the maxima of a Browia motio, see e.g., Embrechts et al. 00, as well as the fact that σ t is uiformly bouded o [0, ], we have Next, we have max i=,..., sup σ s max s [0,] i=,..., i W = o p /+ι, ι > 0. i σ u σ i dw u i sup s,t [0,] t 0 σ udw u, where σ u = σ u σ i for s [i, i ]. From here, usig Burkholder- Davis-Gudy iequality ad the fact that σ t is a Itô semimartigale we have E t s σ udw u +ι K t s +ι/, 0 s t, ι > 0. Therefore, the process / ι t 0 σ udw u is C-tight see e.g., Theorem.3 i Billigsley 968. The applyig Theorem VI.3.6 i Jacod ad Shiryaev 5

26 003, we get i / ι max i=,..., σ u σ i dw u i > ζ Combiig these results, we get altogether 0, ζ > 0, ι > 0. max i=,..., i B = o p /+ι, ι > Next, usig assumptio SB i particular that umps are of fiite activity, we have k k E δ φ z, x 0 µdz, dx K k, φ = Y, σ, σ ad σ. 9.5 We ow provide bouds for the elemets of χ i, ad χ i,. I what follows we deote with η some determiistic sequece of positive umbers that depeds oly o. We first have recall the defiitio of σ t i i σ u σ i dw u η k k i + E i δ Y s, x 0 µds, dx σ u σ i dw u η. For the secod term o the right had side of the above iequality, we ca use Chebyshev iequality as well as Burkholder-Davis-Gudy iequality, to get for p : i i σ u σ i dw u η p/ E i i σ u σ i du Therefore, applyig agai Burkholder-Davis-Gudy iequality, we have altogether i i σ u σ i dw u η K η p p/ [ k ] p/ η p, p > Similar calculatios usig the fact that σ t ad σ t are Itô semimartigales, yields for p > 0 i W σ k σ i σ k ξi,3 η K [ k p ] k. η

27 Next, applyig Chebyshev iequality ad the elemetary i a i p i a i p for p 0, ], we get ι/ i ι E i E δy s, x µds, dx i Y η η ι ι/ i E i E δy s, x ι µds, dx η ι K +ι/ η ι, ι 0,. 9.8 Further, Chebyshev iequality ad the boudedess of a t easily implies i A η p/ i A p η p K p/ η p. 9.9 We tur ext to the differece V η k max i=,..., i A + i B V. Usig triagular iequality, we have k k E δ Y s, x µds, dx η Takig ito accout that k / 0 from 5.9, as well as 9.4, we have k k max i=,..., i A + i B δ Y s, x µds, dx η k E /+ι k [/ ι max i=,..., i A + i B ] k k E δ Y s, x µds, dx η /+ι ι K E[ / ι max η k i=,..., i A + i B ] ι k E δ Y s, x µds, dx k E /+ι ι k K, ι > 0. η k Thus altogether we get V η K /+ι η k ι ι k

28 We cotiue ext with the differece V iequality gives i A + i B i A + i B i B i B Ṽ. Applicatio of triagular i A + i B i A + i A i B. Usig this iequality ad applyig Chebyshev iequality, we get V Ṽ η K η p, p, 9. ad this iequality ca be further stregtheed but suffices for our aalysis. Turig ext to R, usig triagular iequality, Burkholder-Davis-Gudy iequality as well as 9.5, we ca easily get k R η + R η, R η, K k + K k k k η p + K E E k δ σ s, x 0µds, dx η δ σ s, x 0µds, dx = 0 p, p. Similar calculatios, ad utilizig the fact that σ t σ t are themselves Itô semimartigales, yield R η K k + K Next, by splittig i W i W = i W we ca decompose R 3 Davis-Gudy iequality, we get p p k + K, p. 9. η η i W + i W, ito two discrete martigales. The applyig Burkholder- R 3 η K η p, p. 9.3 Fially, we trivially have p R 4 R 4 i η K η, p 9.4 E R 4 R 4 i p K, p > 0, 8

29 V V p i η K k η, E V V p 9.5 i p K k, p > 0, V σ k Ṽ V 0.5σ k p η K k η, 9.6 K k, p. Further, applicatio of Burkholder-Davis-Gudy iequality gives { E V σ p K, k k p/ E R 4 p K k p/ 9.7, p. Now we ca use the above results for the compoets of σ, to aalyze k the first term i χ i, ivolvig algebraic iequality x y x y y + x y 8y y σ k. We make use of the followig x y4 x y 3 + 8y 7/ y, 5/ for every x 0 ad y > 0. Usig this iequality with x ad y replaced with ad σ respectively, as well the bouds i 9.0, 9. ad 9.6, we k get ξi, + ξi, σ k η K [ /+ι η /3 k ι k k η p/3 [ p/ /k p/ ] for p ad ι > 0. Similarly, usig the followig iequality η p/3 k p ], 9.8 x y ɛ x y 0.5ɛ + y K + x y 0.5ɛ/K, for ay radom variables x ad y ad costats ɛ > 0 ad K > 0, together with the bouds i 9.0 ad 9., we have ξ i, ξi, ξ i, + ξ i, η [ /+ι ι ] k k K η k η[ p, p/ /k p ] 9.9 for every p ad arbitrary small ι > 0. 9

30 We fially provide a boud for the last term i χ i,. We ca use Chebyshev iequality as well as Hölder iequality to get σ k i W i X > α ϖ K η [ ] /+ι i X > α ϖ. η ι 9.30 We ca further write i X > α ϖ + σ k i X > 0.5ασ k 0.5σ k From here we ca use the bouds i 9.0, 9. ad 9.6 as well as 9.8 ϖ. ad coclude σ k i W i X > α ϖ η K /+ι k η ι, ι > Combiig the results i 9.6, 9.7, 9.8, 9.9, 9.8, 9.9 ad 9.3, we get χ i, + χ i, > η [ /+ι ι k k K η k i X σ { k i X α χ i, + χ i, > η τ + τ + E Φ ξ i, τ ξ i, + η + τ ξi, 4 ] /+ι k η[ p p/ /k p ] η ι. From here, usig the fact that the probability desity of a stadard ormal variable is uiformly bouded, we get E i X { i X α ϖ} τ χ i, + χ i, > η + Kητ. ϖ} τ σ k χ i, τχ i, τ + τ ξ i, τ Φ ξ i, η + τ ξi, 4 30

31 Therefore, upo pickig η q ι for ι 0, / q, we get fially for ay compact subset A of, 0 sup F τ Ĝτ = o p. 9.3 τ A k 9.4. The asymptotic behavior of Ĝτ Φτ We have Ĝ τ Φτ = A = a i = /k 5 A i, A = /k m i= /k = /k = k +m i= k + Φ τ + τξ τξ Φτ, A 3 = i W τ + τ ξ i, τ ξ i, ξ i, 4 A 4 = /k m A 5 = /k m /k = /k = k +m i= k + k +m i= k + [ i W τ Φτ ], /k 9.33 k +m a i, /k m = i= k + i W τ τ + τ ξ i, τ +Φτ Φ ξ i, ξi, 4, [ ] Φ τ+τ ξ i, τ ξ i, Φ τ+τξ τξ, [ Φ τ + τ ξ i, τ ξ i, ξ i, 4 ] Φ τ+τ ξ i, τ ξ i,. We first derive a boud for the order of magitude of A 3, A 4 ad A 5 ad the aalyze the limitig behavior of A ad A. Usig the idepedece of i W, h W, i W, h W from each other ad F k, the fact that ξi, 4 is adapted to Fi as well as successive coditioig, we have for i h ad i, h = k +,..., k + m E a i a h = 0, E a i Kτ k, i h, i, h =,...,. Therefore, we have A 3 = k τ O p /k m For A 4, usig a secod-order Taylor expasio, the bouds i 9.4, 9.5 ad 9.7, as well as the uiform boudedess of the probability desity of the 3

32 stadard ormal distributio ad its derivative, we get E A 4 Kτ τ k 3/ Next, for A 5, we ca use the boudedess of the probability desity of the stadard ormal as well as a secod-order Taylor expasio, to get for ι > 0 ad sufficietly high Φ τ + τ ξ i, τ ξ i, ξ i, 4 Φ τ + τ ξ i, τ ξ i, = b i + b i + b i 3, { τ + τ b ξ i, τ i = Φ ξ i, ξi, 4 Φ τ + τ ξ i, τ ξ i, } { ξ i, 4 k / ι}, b i = Φ τ + τ ξ i, τ ξ i, τ+τ ξ i, τ ξ i,ξ i,4 { ξ i, 4 < k / ι}, For b i ad b i 3, we have b i 3 K τ + τ ξ i, τ ξ i, k / / ι 3 ξ i,4. E b i + b i 3 Kτ k. For b i, by a applicatio of Hölder iequality, we first have E b i Φ τ τξi,4 { ξi, 4 < k / ι} Kτ. The, E /k m /k = k +m i= k + Therefore, altogether we get Φ τ τξi,4 { ξ i, 4 < k / ι} K k. E A 5 Kτ τ k We tur ow to A ad A. Usig seco-order Taylor expasio, we ca extract the leadig terms i A. I particular, we deote { A = /k /k A = /k = Φ ττξ, /k = 0.5Φ ττ ξ Φ ττξ. 3

33 With this otatio, usig the bouds i 9.7, as well as the boudedess of Φ, we have [ k 3/ ] 3/ E A A A Kτ Further, upo deotig with  ad  the couterparts of A ad A with ξ ad ξ replaced with ξ ad ξ respectively, we have usig the bouds i 9.7 as well as the restrictio o the rate of growth of k i 5.9 E A + A   Kτ τ k k Thus we are left with the terms A,  ad Â. For Â, usig E k ξ = E ξ k = + 3, 4 we have k  τ Φ τ τφ τ 8 locally uiformly i τ. We fially will show that /k m A /k k  k + 3, 9.39 L Z τ Z τ, 9.40 locally uiformly i τ. We have /k m A /k k  = /k k i= ζi Φ ττ ζi + ζ i Φ ττ ζ, with ζi = [ i W τ Φτ] /k m /k k i W i W i W /k k, i I, where I = {i = k +,..., k + m, =,..., /k }, ad for i =,..., \ I, ζi is exactly as above with oly the first elemet beig 33

34 replaced with zero, ad fially ζ = / /k [ k /k k W k W + = + k W where we set 0 W = 0. With this otatio, we have Further, E i ζ i = 0, E ζ K. k /k k i= ], E i ζ i +ι 0, ι > 0, /k k E [ i ζ i ζi ] Φτ Φτ i= 0 Combiig the last two results, we have the covergece i 9.40, poitwise i τ, by a applicatio of Theorem IX.7.8 i Jacod ad Shiryaev 003. Applicatio of Theorem.3 i Billigsley 968, exteds the covergece to local uiform i τ. Altogether, the limit behavior of Ĝτ Φτ is completely characterized by the limits i ad sup Ĝτ Φτ A Â Â = o p, 9.4 τ A k where A is a compact subset of, 0, with the result i 9.4 followig from the bouds o the order of magitude derived above The differece F τ F τ To aalyze the differece F τ F τ, we use the followig iequality i X > α ϖ > i X > 0.5ασ k ϖ. σ k 34

35 For the first probability o the right-had side of the above iequality we ca use the bouds i 9.6, 9.7 ad 9.8, while for the secod oe we ca use the expoetial iequality for cotiuous martigales with bouded variatio, see e.g., Revuz ad Yor 999, as well as the algebraic iequality i a i p i a i p for p 0, ], to coclude i X > α ϖ K [ ] k +ιϖ, ι > Sice k / 0 ad from the result of the previous two subsectios F τ Φτ = O p k, we get from here sup F τ F τ = o p, 9.43 τ A k for ay compact subset A of, 0. Refereces Adrews, B., M. Calder, ad R. Davis 009. Maximum Likelihood Estimatio of α-stable Autoregressive rocesses. Aals of Statistics 37, Bardorff-Nielse, O. ad N. Shephard 00. No-Gaussia Orstei Uhlebeck- Based Models ad some of Their Uses i Fiacial Fcoomics. Joural of the Royal Statistical Society Series B, 63, Bardorff-Nielse, O. ad N. Shephard 004. ower ad Bipower Variatio with Stochastic Volatility ad Jumps. Joural of Fiacial Ecoometrics, 37. Bardorff-Nielse, O. ad N. Shephard 006. Ecoometrics of Testig for Jumps i Fiacial Ecoomics usig Bipower Variatio. Joural of Fiacial Ecoometrics 4, 30. Billigsley, Covergece of robability Measures. New York: Wiley. Embrechts,., C. Kluppelberg, ad T. Mikosch 00. Modellig Extremal Evets 3rd ed.. Berli: Spriger-Verlag. Jacod, J. ad. rotter 0. Discretizatio of rocesses. Berli: Spriger-Verlag. Jacod, J. ad A. Shiryaev 003. Limit Theorems For Stochastic rocesses d ed.. Berli: Spriger-Verlag. Klüppelberg, C., T. Meyer-Bradis, ad A. Schmidt 00. Electricity Spot rice Modellig with a View Towards Extreme Spike Risk. Quatitative Fiace 0,

36 Mikosch, T., S. Resick, H. Rootze, ad A. Stegema 00. Is Network Traffic Approximated by Stable Levy Motio or Fractioal Browia Motio? Aals of Applied robability, Myklad,., N. Shephard, ad K. Sheppard 0. Efficiet ad Feasible Iferece for the Compoets of Fiacial Variatio usig Blocked Multipower Variatio. Techical report. Myklad,. ad L. Zhag 009. Iferece for Cotiuous Semimartigales Observed at High Frequecy. Ecoometrica 77, Revuz, D. ad M. Yor 999. Spriger. Cotiuous Martigales ad Browia Motio. Sato, K Lévy rocesses ad Ifiitely Divisible Distributios. Cambridge, UK: Cambridge Uiversity ress. Todorov, V. ad G. Tauche 0a. Realized Laplace Trasforms for ure-jump Semimartigales. Aals of Statistics 40, Todorov, V. ad G. Tauche 0b. The Realized Laplace Trasform of Volatility. Ecoometrica 80, va der Vaart, A Asymptotic Statistics. Cambridge: Cambridge Uiversity ress. Woerer, J Iferece i Lévy-type Stochastic Volatility Models. Advaces i Applied robability 39,