Bootstrapping high-frequency jump tests

Transcription

1 Bootstrappig high-frequecy jump tests Prosper Dovoo Departmet of Ecoomics, Cocordia Uiversity Sílvia Goçalves Departmet of Ecoomics, McGill Uiversity Ulrich Houyo Departmet of Ecoomics, Uiversity at Albay, SUNY ad CREATES Nour Meddahi Toulouse School of Ecoomics, Toulouse Uiversity November 15, 217 Abstract The mai cotributio of this paper is to propose a bootstrap test for jumps based o fuctios of realized volatility ad bipower variatio. Bootstrap itraday returs are radomly geerated from a mea zero Gaussia distributio with a variace give by a local measure of itegrated volatility which we deote by {ˆv i }. We first discuss a set of high level coditios o {ˆv i } such that ay bootstrap test of this form has the correct asymptotic size ad is alterative-cosistet. We the provide a set of primitive coditios that justify the choice of a thresholdig-based estimator for {ˆv i }. Our cumulats expasios show that the bootstrap is uable to mimic the higher-order bias of the test statistic. We propose a modificatio of the origial bootstrap test which cotais a appropriate bias correctio term ad for which secod-order asymptotic refiemets are obtaied. 1 Itroductio A well accepted fact i fiacial ecoomics is that asset prices do ot always evolve cotiuously over a give time iterval, beig istead subject to the possible occurrece of jumps or discotiuous movemets i prices. The detectio of such jumps is crucial for asset pricig ad risk maagemet because their presece has importat cosequeces for the performace of asset pricig models ad hedgig strategies, ofte itroducig parameters that are hard to estimate see e.g. Bakshi et al. 1997, Bates 1996, ad Johaes 24. I additio, jumps cotai useful market iformatio ad ca be used to improve asset pricig models oce detected. For istace, jumps are ofte associated with macro aoucemets as documeted by may studies, icludig Bardorff-Nielse ad Shephard 26, Aderse et al. 27, Lee ad Myklad 28 ad Lee 212. As show by Savor ad We are grateful for commets from participats at the SoFie Aual Coferece i Toroto, Jue 214, ad at the IAAE 214 Aual Coferece, Quee Mary, Uiversity of Lodo, Jue 214. We are also grateful to two aoymous referees ad a associate editor for may valuable suggestios. Dovoo, Goçalves ad Meddahi ackowledge fiacial support from a ANR-FQRSC grat ad Dovoo ad Goçalves from a Isight SSHRC grat. I additio, Houyo ackowledges support from CREATES - Ceter for Research i Ecoometric Aalysis of Time Series DNRF78, fuded by the Daish Natioal Research Foudatio, as well as support from the Oxford-Ma Istitute of Quatitative Fiace. Fially, Nour Meddahi has also beefited from the fiacial support of the chair Marché des risques et créatio de valeur Fodatio du risque/scor. 1

2 Wilso 214, it is easier to recocile the behavior of asset prices with the stadard CAPM model o those dates. Similarly, Li et al. 217a show that jumps i asset prices are ofte associated with aggregrate market jumps, suggestig that a stadard liear oe-factor model is appropriate to model jump regressios. Give the importace of jumps, may jump tests have bee proposed i the literature over the years, most of the recet oes exploitig the rich iformatio cotaied i high frequecy data. These iclude tests based o bipower variatio measures such as i Bardorff-Nielse ad Shephard 24, 26, heceforth BN-S 24, 26, Huag ad Tauche 25, Aderse et al. 27, Jiag ad Oome 28, ad more recetly Myklad et al. 212; tests based o power variatio measures sampled at differet frequecies such as i Aït-Sahalia ad Jacod 29, Aït-Sahalia et al. 212, ad tests based o the maximum of a stadardized versio of itraday returs such as i Lee ad Myklad 28, 212 ad Lee ad Haig 21. I additio, tests based o thresholdig or trucatio-based estimators of volatility have also bee proposed, as i Aït-Sahalia ad Jacod 29, Podolskij ad Ziggel 21 ad Cot ad Macii 211, based o Macii 21. See Aït-Sahalia ad Jacod 212, 214 for a review of the literature o the ecoometrics of high frequecy-based jump tests. I this paper, we focus o the class of tests based o bipower variatio origially proposed by BN- S 24, 26. Our mai cotributio is to propose a bootstrap implemetatio of these tests with better fiite sample properties tha the origial tests based o the asymptotic ormal distributio. Specifically, we geerate the bootstrap observatios uder the ull of o jumps, by drawig them radomly from a mea zero Gaussia distributio with a variace give by a local measure of itegrated volatility which we deote by {ˆv i }. Our first cotributio is to give a set of high level coditios o {ˆv i } such that ay bootstrap method of this form has the correct asymptotic size ad is alterative-cosistet. We the verify these coditios for a specific example of {ˆv i } based o a threshold-based volatility estimator costructed from blocks of itraday returs which are appropriately trucated to remove the effect of the jumps. I particular, we provide primitive assumptios o the cotiuous price process such that the bootstrap jump test based o the thresholdig local volatility estimator is able to replicate the ull distributio of the BN-S test uder both the ull ad the alterative of jumps. Our assumptios are very geeral, allowig for leverage effects ad geeral activity jumps both i prices ad volatility. We show that although trucatio is ot eeded for the bootstrap jump test to cotrol the asymptotic size uder the ull of o jumps, it is importat to esure that the bootstrap jump test is cosistet uder the alterative of jumps. Other choices of {ˆv i } could be cosidered provided they are robust to jumps. For istace, we could rely o multipower variatio volatility measures ad use our high level coditios to show the first-order validity of this bootstrap method. For brevity, we focus o the thresholdig-based volatility estimator, which is oe of the most popular methods of obtaiig jump robust test statistics. The secod cotributio of this paper is to prove that a appropriate versio of the bootstrap jump test based o thresholdig provides a secod-order asymptotic refiemet uder the ull of o jumps 1. To do so, we impose more restrictive assumptios o the data geeratig process that assume away the presece of drift ad leverage effects. For this simplified model, we develop secod-order asymptotic expasios of the first three cumulats of the BN-S test ad of its bootstrap versio. Our results show that the first-order cumulat of the BN-S test depeds o the bias of bipower variatio uder the ull of o jumps. Eve though this bias does ot impact the validity of the test to first-order because bipower variatio is a cosistet estimator of itegrated volatility uder the ull, it has a impact o the first-order cumulat of the statistic at the secod-order i.e. at the order O 1/2. Our bootstrap test statistic is uable to capture this higher-order bias ad therefore does ot provide 1 Secod-order refiemets are importat because bootstrap tests with this property are expected to have ull rejectio rates that coverge faster to the desired omial level tha those of the correspodig asymptotic theory-based test, hece esurig smaller fiite sample size distortios. 2

3 a secod-order refiemet. We propose a modificatio of the bootstrap statistic that is able to do so. Our simulatios show that although both bootstrap versios of the test outperform the asymptotic test, the modified bootstrap test statistic has lower size distortios tha the origial bootstrap statistic. I the empirical applicatio, where we apply the bootstrap jump tests to 5-miutes returs o the SPY idex over the period Jue 15, 24 through Jue 13, 214, this versio of the bootstrap test detects about half of the umber of jump days detected by the asymptotic theory-based tests. Although we focus o the BN-S test statistic, the local Gaussia bootstrap ca be applied more geerally to other jump tests. I particular, we provide a set of coditios uder which it ca be applied to the jump tests of Podolskij ad Ziggel 21 ad Lee ad Haig 21. The rest of the paper is orgaized as follows. I Sectio 2, we provide the framework ad state our assumptios. Sectio 3.1 cotais a set of high level coditios o {ˆv i } such that ay bootstrap method is asymptotically valid whe testig for jumps usig the BN-S test. Sectio 3.2 provides a set of primitive assumptios uder which the bootstrap based o a thresholdig estimator verifies these high level coditios. Sectio 4.1 cotais the secod-order expasios of the cumulats of the origial statistic whereas Sectio 4.2 cotais their bootstrap versios. Sectio 5 gives the Mote Carlo simulatios while Sectio 6 provides a empirical applicatio. Sectio 7 cocludes. Appedix A cotais a law of large umbers for smooth fuctios of cosecutive local trucated volatility estimates. I additio, a olie supplemetary appedix cotais the proofs of all the results i the mai text. Specifically, Appedix S1 cotais the proofs of the results preseted i Sectio 3 whereas Appedix S2 cotais the proofs of the results i Sectio 4. Appedix S3 cotais formulas for the log versio of our tests. Fially, Appedix S4 cotais the bootstrap theory for the jump tests of Podolskij ad Ziggel 21 ad Lee ad Haig 21. To ed this sectio, a word o otatio. We let P describe the probability of bootstrap radom variables, coditioal o the observed data. Similarly, we write E ad V ar to deote the expected value ad the variace with respect to P, respectively. For ay bootstrap statistic Z Z, ω ad ay measurable set A, we write P Z A = P Z, ω A = Pr Z, ω A X, where ω Ω, a sample space ad X deotes the observed sample. We say that Z P i prob-p or Z = o P 1 i prob-p if for ay ε, δ >, P P Z > ε > δ as. Similarly, we say that Z = O P 1 i prob-p if for ay δ >, there exists < M < such that P P Z M > δ as. For a sequece of radom variables or vectors Z, we also eed the defiitio of covergece i distributio i prob-p. I particular, we write Z d Z, i prob-p a.s.-p, if E f Z E f Z i prob-p for every bouded ad cotiuous fuctio f a.s. P. 2 Assumptios ad statistics of iterest We assume that the log-price process X t is a Itô semimartigale defied o a probability space Ω, F, P equipped with a filtratio F t t such that X t = Y t + J t, t, 1 where Y t is a cotiuous Browia semimartigale process ad J t is a jump process. Specifically, Y t is defied by the equatio Y t = Y + t a s ds + t σ s dw s, t, 2 where a ad σ are two real-valued radom processes ad W is a stadard Browia motio. The jump process is defied as J t = t R t δ s, x 1{ δs,x 1} µ ν ds, dx + 3 R δ s, x 1{ δs,x >1} µ ds, dx, 3

4 where µ is a Poisso radom measure o R + R with itesity measure ν ds, dx = ds λ dx, with λ a σ-fiite measure o R, ad δ a predictable fuctio o Ω R + R. We make the followig assumptios o a, σ ad J t, where r [, 2]. Assumptio H-r The process a is locally bouded, σ is càdlàg, ad there exists a sequece of stoppig times τ ad a sequece of determiistic oegative fuctios γ o R such that γ x r λ dx < ad δ ω, s, x 1 γ x for all ω, s, x satisfyig s τ ω. Assumptio H-r is rather stadard i this literature, implyig that the r th absolute power value of the jumps size is summable over ay fiite time iterval. Sice H-r for some r implies that H-r holds for all r > r, the weakest form of this assumptio occurs for r = 2 ad essetially correspods to the class of Itô semimartigales. As r decreases towards, fewer jumps of bigger size are allowed. I the limit, whe r =, we get the case of fiite activity jumps. The quadratic variatio process of X is give by [X] t = IV t + JV t, where IV t t σ2 sds is the quadratic variatio of Y t, also kow as the itegrated volatility, ad JV t s t J s 2 is the jump quadratic variatio, with J s = J s J s deotig the jumps i X. Without loss of geerality, we let t = 1 ad we omit the idex t. For istace, we write IV = IV 1 ad JV = JV 1. We assume that prices are observed withi the fixed time iterval [, 1] which we thik of as a day ad that the log-prices X t are recorded at regular time poits t i = i/, for i =,...,, from which we compute itraday returs r i X i/ X i 1/ at frequecy 1/; we omit the idex i r i to simplify the otatio. Our focus is o testig for o jumps usig the bootstrap. Specifically, followig Aït-Sahalia ad Jacod 29, let Ω = {ω : t X t ω is cotiuous o [, 1]} ad Ω 1 = {ω : t X t ω is discotiuous o [, 1]}, with Ω = Ω Ω 1 ad Ω Ω 1 =. Our ull hypothesis is defied as H : ω Ω ad the alterative hypothesis is H 1 : ω Ω 1. Let RV = i=1 r2 i deote the realized volatility ad let BV = 1 k 2 1 r i 1 r i i=2 be the bipower variatio, where we let k 1 = E Z = 2/ π be a special case of k q = E Z q, for q >, with Z N, 1. The test statistic whose distributio we bootstrap is defied as T = RV BV ˆV, 4 where ˆV τ ÎQ with ÎQ = 3 r i 4/3 r i 1 4/3 r i 2 4/3, k4/3 with τ = θ 2 ad θ The test rejects the ull of o jumps at sigificace level α wheever T > z 1 α, where z 1 α is the 1 1 α % percetile of the N, 1 distributio. This is justified by st the fact that T N, 1, i restrictio to Ω, where st deotes stable covergece see BN-S 26 ad Bardorff-Nielse et al. 26. I particular, we ca show that the test has asymptotically correct strog size, i.e. the critical regio C = {T > z 1 α } is such that for ay measurable set S Ω such that P S >, lim P ω C S = α. Uder the alterative hypothesis, we ca show that the test T is alterative-cosistet, i.e. lim P Ω 1 C =, where C is the complemet of C. Sice the above coditio implies that P C Ω 1, as, we have that P C Ω 1 1 as, which we ca iterpret as sayig that the test has asymptotic power equal to 1. 4 i=3

5 3 The bootstrap We impose the ull hypothesis of o jumps whe geeratig the bootstrap itraday returs. 2 Specifically, we let r i = ˆv i η i, i = 1,...,, 5 for some volatility measure ˆv i based o {r i : i = 1,..., }, ad where η i is geerated idepedetly of the data as a i.i.d. N, 1 radom variable. For simplicity, we write ri istead of r i,. Accordig to 5, bootstrap itraday returs are coditioally o the origial sample Gaussia with mea zero ad volatility ˆv i, ad therefore do ot cotai jumps. This bootstrap DGP is motivated by the simplified model X t = t σ sdw s, where σ is idepedet of W ad there is o drift or jumps, but its cosistecy exteds to more geeral models with leverage ad drift, as our results i this sectio show. 3 The bootstrap aalogues of RV ad BV are RV = ri 2 ad BV = 1 r k1 2 i 1 r i. i=1 The first class of bootstrap statistics we cosider is described as RV T = BV E RV BV, 6 ˆV where ad E RV BV = ˆv i i=1 i=2 1/2 ˆv i 1 ˆv i 1/2, ˆV = τ ÎQ with ÎQ = 3 ri 4/3 r 4/3 i 1 r 4/3 i 2. k4/3 Thus, T is exactly as T except for the receterig of RV BV aroud the bootstrap expectatio E RV BV. This esures that the bootstrap distributio of T is cetered at zero, as is the case for T uder the ull of o jumps whe is large. As we will study i Sectio 4, it turs out that T has a higher-order bias uder the ull which is ot well mimicked by T, implyig that this test does ot yield asymptotic refiemets. For this reaso, we cosider a secod class of bootstrap statistics based o T = i=2 i=3 RV BV E RV BV ˆV ˆv 1 + ˆv, 7 ˆV where the secod term accouts for the higher-order bias i T. This correctio has a impact i fiite samples, as our simulatio results show. I particular, T has lower size distortios tha T uder the ull, especially for the smaller sample sizes. Next, we provide geeral coditios o ˆv i uder which T d N, 1, i prob-p idepedetly of whether ω Ω or ω Ω 1. The cosistecy of the bootstrap the follows by verifyig these high level coditios for a particular choice of ˆv i. We verify them for a thresholdig-based estimator, but other choices of ˆv i could be cosidered. Asymptotic refiemets of the bootstrap based o T will be discussed i Sectio 4. 2 This follows the recommedatios of Davidso ad Mackio 1999, who show that imposig the ull i the bootstrap samples is importat to miimize the probability of a type I error. 3 However, for the secod-order asymptotic refiemets of the bootstrap i Sectio 4 we do require the absece of leverage ad drift effects. 5

6 3.1 Bootstrap validity uder geeral coditios o ˆv i The first-order asymptotic validity of the local Gaussia bootstrap ca be established uder Coditio A below. Note that this is a high level coditio that does ot deped o specifyig whether we are o Ω or o Ω 1. Coditio A Suppose that {ˆv i } satisfies the followig coditios: i For ay K N ad ay sequece {q k R + : k = 1,..., K} of oegative umbers such that q K k=1 q k 8, as, 1+q/2 i=k k=1 K qk /2 P ˆv i k+1 1 σ q udu >. ii There exists α [, 3 7 such that [/L 2 +1] j=1 ˆv jl +1 = op 1, where L α ad [x] deotes the largest iteger smaller or equal to x. Theorem 3.1 Uder Coditio A, if, T d N, 1, i prob-p. Coditio Ai requires the multipower variatios of ˆv i to coverge to 1 σq udu for ay q 8. Uder this coditio, the probability limit of Σ V ar RV, BV is equal to Σ V ar RV, BV for this result, q = 4 suffices. See Lemma S1.1 i Appedix S1 ad the proof of Theorem 3.1. Together with Coditio Ai, Coditio Aii is used to show that a CLT holds for RV E RV, BV E BV i the bootstrap world. I particular, sice the summatio terms i BV are lag-oe depedet, we adopt a large-block-small-block argumet, where the large blocks are made of L cosecutive observatios ad the small block is made of a sigle elemet. Part ii esures that the cotributio of the small blocks is asymptotically egligible. The proof of Theorem 3.1 the follows by showig that ˆV = τ ÎQ V = τ IQ uder Coditio Ai this follows from the covergece of the multipower variatios of ˆv i of eighth order, explaiig why we require q 8. The bootstrap test rejects H : ω Ω agaist H 1 : ω Ω 1 wheever ω C, where the bootstrap critical regio is defied as C = { ω : T ω > q,1 α ω}, where q,1 α ω is such that P T, ω q,1 α ω st = 1 α. Sice T N, 1 o Ω, the fact that T d N, 1, i prob-p, esures that the test has correct size asymptotically. Uder the alterative i.e. o Ω 1 sice T diverges at rate, but we still have that T d N, 1, the test has power asymptotically. The followig theorem follows from Theorem 3.1 ad the asymptotic properties of T uder H ad uder H 1. st P Theorem 3.2 Suppose T N, 1, i restrictio to Ω, ad T + o Ω 1. If Coditio A holds, the the bootstrap test based o T cotrols the asymptotic strog size ad is alterativecosistet. 3.2 Bootstrap validity whe ˆv i is based o thresholdig I this sectio we verify Coditio A for the followig choice of ˆv i : ˆv j+i 1k = 1 k k m=1 P r 2 i 1k +m 1 { r i 1k+m u }, 6

7 where i = 1,..., k ad j = 1,..., k. Here, k is a arbitrary sequece of itegers such that k ad k / ad u is a sequece of threshold values defied as u = α ϖ for some costat α > ad < ϖ < 1/2. We will maitai these assumptios o k ad u throughout. The estimator ˆv i is equal to 1 times a spot volatility estimator that is popular i the high-frequecy ecoometrics literature uder jumps see e.g. Macii 21 ad Aït-Sahalia ad Jacod 29. By excludig all returs cotaiig jumps over a give threshold whe computig ˆv i, we guaratee that the bootstrap distributio of T coverges to a N, 1 radom variable, idepedetly of whether there are jumps or ot. This is crucial for the bootstrap test to cotrol asymptotic size ad at the same time have power. The followig lemma is auxiliary i verifyig Coditio A. Lemma 3.1 Assume that X satisfies 1, 2 ad 3 such that Assumptio H-2 holds. Let q = K k=1 q k with q k ad K N. If either of the followig coditios holds: a q > ad X is cotiuous; b q < 2; or c q 2, Assumptio H-r holds for some r [, 2, ad q 1 2q r ϖ < 1 2 ; the K 1+q/2 1 qk /2 P ˆv i k+1 σudu q >. i=k k=1 Lemma 3.1 follows from Theorem A.1 i Appedix A, a result that is of idepedet iterest ad ca be see as a extesio of Theorem of Jacod ad Protter 212 see also Jacod ad Rosebaum 213. I particular, Theorem A.1 provides a law of large umbers for smooth fuctios of cosecutive trucated local realized volatility estimators defied o o-overlappig time itervals. Istead, Theorem of Jacod ad Protter 212 oly allows for fuctios that deped o a sigle local realized volatility estimate eve though they are possibly based o overlappig itervals. Recetly, Li et al. 217b focus o a sigle local realized volatility estimate based o o-overlappig itervals ad exted the limit results of Theorem of Jacod ad Protter 212 to a more geeral class of volatility fuctioals that do ot have polyomial growth see also Li ad Xiu 216 for a extesio to overlappig itervals. Here, we restrict our attetio to fuctios that have at most polyomial growth, which is eough to accommodate the blocked multipower variatios measures of Lemma 3.1. Our coditios o ϖ deped o the polyomial growth rate of the test fuctio, requirig i particular a arrower rage of values for ϖ as q icreases as part c of Lemma 3.1 shows. It would be iterestig to exted the results of Li et al. 217b ad Li ad Xiu 216 to cosecutive trucated local realized volatility estimators so as to allow for more geeral test fuctios ad remove the depedece of ϖ o q eve if at the cost of itroducig more striget coditios o k. Give this result, we ca state the followig theorem. Theorem 3.3 Assume that X satisfies 1, 2, 3 such that Assumptio H-2 holds. If i additio, either of the two followig coditios holds: a X is cotiuous; or b Assumptio H-r holds for some 7 r [, 2 ad 16 r ϖ < 1 2 ; the the coclusio of Theorem 3.1 holds for the thresholdig-based bootstrap test T. Theorem 3.3 shows that the thresholdig-based statistic T is asymptotically distributed as N, 1 idepedetly of whether the ull of o jumps is true or ot. This guaratees that the bootstrap jump test has the correct asymptotic size ad is cosistet uder the alterative of jumps. Note that uder the ull, whe X is cotiuous, the result holds for ay level of trucatio, icludig the case where u =, which correspods to o trucatio. Nevertheless, to esure that T is also asymptotically ormal uder the alterative hypothesis of jumps some trucatio is required. Part b of Theorem 3.3 shows that we should choose u = α ϖ 7 with 16 r ϖ < 1 2, a coditio that is more striget tha the usual coditio o ϖ which is < ϖ < 1/2. The lower boud o ϖ is a icreasig fuctio 7

8 of r, a umber that is related to the degree of activity of jumps as specified by Assumptio H-r. For fiite activity jumps where r =, ϖ should be larger tha or equal to 7/16 but strictly smaller tha 1/2. As r icreases towards 2 allowig for a icreasig umber of small jumps, the rage of values of ϖ becomes arrower, implyig that we eed to choose a smaller level of trucatio i order to be able to separate the Browia motio from the jumps cotributios to returs. The followig result is a corollary to Theorem 3.3. Corollary 3.1 Assume that X satisfies 1, 2, 3 such that Assumptio H-r holds for some r [, 2 ad let u = α ϖ 7 with 16 r ϖ < 1 2. The, the coclusios of Theorem 3.2 are true for the thresholdig-based bootstrap test T. 4 Secod-order accuracy of the bootstrap I this sectio, we ivestigate the ability of the bootstrap test based o the thresholdig local realized volatility estimator to provide asymptotic higher-order refiemets uder the ull hypothesis of o jumps. Our aalysis is based o the followig simplified model for X t, X t = t σ s dw s, 8 where σ is càdlàg locally bouded away from ad t σ2 sds < for all t [, 1]. I additio, we assume that σ is idepedet of W. Thus, we ot oly impose the ull hypothesis of o jumps uder which J t =, but we also assume that there is o drift or leverage effects. Uder these assumptios, coditioally o the path of volatility, r i N, vi idepedetly across i, a result that we will use throughout this sectio. Allowig for the presece of drift ad leverage effects would complicate substatially our aalysis. I particular, we would ot be able to coditio o the volatility path σ whe derivig our expasios if we relaxed the assumptio of idepedece betwee σ ad W. Allowig for the presece of a drift would require a differet bootstrap method, the mai reaso beig that the effect of the drift o the test statistic is of order O 1/2 ad our bootstrap returs have mea zero by costructio see Goçalves ad Meddahi, 29. We leave these importat extesios for future research. To study the secod-order accuracy of the bootstrap, we rely o secod-order Edgeworth expasios of the distributio of our test statistics T ad T. As is well kow, the coefficiets of the polyomials eterig a secod-order Edgeworth expasio are a fuctio of the first three cumulats of the test statistics cf. Hall, I order to derive these higher-order cumulats, we make the followig additioal assumptio. We rely o it to obtai the limit of the first-order cumulat of T cf. κ 1,1 below. Assumptio V The volatility process σu 2 is pathwise cotiuous, bouded away from zero ad σ Hölder-cotiuous i L 2 P o [, 1] of order δ > 1/2, i.e., E 2 u σs 2 2 = O u s 2δ. This assumptio is useful to derive explicitly the probability limit of E RV BV F, which cotributes to the higher order bias of the BN-S statistic. It imposes that the volatility path is cotiuous ad, i additio, rules out stochastic volatility models drive by a Browia motio. Examples of processes that satisfy Assumptio V iclude the fractioal Browia motio with Hurst parameter larger tha 1/2 as well as the fractioal stochastic volatility model itroduced by Comte ad Reault

9 4.1 Secod-order expasios of the cumulats of T Next we provide asymptotic expasios for the cumulats of T. For ay positive iteger i, let κ i T deote the i th cumulat of T. I particular, recall that κ 1 T = ET, κ 2 T = V art ad κ 3 T = ET ET 3. I additio, for ay q >, we let σ q = 1 σq udu. Theorem 4.1 Assume that X satisfies 8 ad Assumptio V holds, where σ is idepedet of W. The, coditioally o σ, we have that κ 1 T = σ 2 + σ2 1 a 1 σ 6 1 τ σ 4 2 3/2 + O ; σ 4 }{{} κ 2 T = 1 + O 1 ; ad κ 1 =κ 1,1 +κ 1,2 κ 3 T = 1 a a 1 a 3 σ 6 σ 4 3/2 } {{ } κ 3 + O 1, where τ = k k ad the costats a 1, a 2 ad a 3 also deped o k q = E Z q, Z N, 1, for certai values of q > ; their specific values are give i Lemma S2.5 i the Appedix. Theorem 4.1 shows that the first ad third order cumulats of T are subject to a higher-order bias of order O 1/2, give by the costats κ 1 ad κ 3. Sice the asymptotic N, 1 approximatio assumes that the values of these cumulats are zero, the error of this approximatio is of the order O 1/2. The bootstrap is asymptotically secod-order accurate if the bootstrap first ad third order cumulats mimic κ 1 ad κ 3. As it turs out, this is ot true for the bootstrap test based o T. The mai reaso is that it fails to capture κ 1,1, a bias term that is due to the fact that bipower variatio is a biased but cosistet estimator of IV. To uderstad how this bias impacts the first-order cumulat of T, ote that we ca write T = RV BV = S + A /2 U + B, 9 ˆV where S = A = B = RV BV E RV BV ; V E RV BV ˆV E ˆV ; U = ; V V E ˆV V V, ad V = V ar RV BV. By costructio, coditioally o σ, E S = ad V ar S = 1; the variable S drives the usual asymptotic ormal approximatio. The term A is determiistic coditioally o σ ad reflects the fact that E RV BV uder the ull of o jumps. I particular, we ca easily see that 9

10 E RV BV = IV E BV. Thus, A reflects the bias of BV as a estimator of IV. We ca show that A = O 1/2, implyig that to order O 1, the first-order cumulat of T is κ 1 T = 1 A 1 2 E S U + O }{{} κ 1,1 +κ 1,2 κ 1 The limit of A is κ 1,1. This follows by writig A = E IV BV = vi V V i=1 i=2 v i /2 vi 1/2, where IV = i=1 v i, ad otig that by Lemma S2.3 i Appedix S2, vi v i 1 1/2 vi 1/2 P 1 σ σ1 2, 1 i=1 i=2 P ad V τσ 4, uder Assumptio V. Next we show that the bootstrap test based o T does ot replicate κ 1,1 ad therefore is ot secod-order correct. We the propose a correctio of this test ad show that it matches κ 1 ad κ Secod-order expasios of the bootstrap cumulats Write κ 1 T = 1 1 κ 1 + o P ad κ 3 T = 1 1 κ 3 + o P, where κ 1 ad κ 3 are the leadig terms of the first ad third order cumulats of T ; they are a fuctio of the origial sample ad hece deped o. Their probability limits are deoted by κ 1 ad κ 3 ad the followig theorem derives their values. Theorem 4.2 Assume that X satisfies 8 ad Assumptio V holds, where σ is idepedet of W. Suppose that k such that k /, /k is bouded ad u is a sequece of threshold values defied as u = α ϖ for some costat α > ad < ϖ < 1/2. The, coditioally o σ, we have that κ 1 = κ 1,2 κ 1 ad κ 3 = κ 3, where κ 1,2, κ 1 ad κ 3 are defied as i Theorem 4.1. Theorem 4.2 shows that the bootstrap test based o T oly captures κ 1 partially ad therefore fails to provide a secod-order asymptotic refiemet. The mai reaso is that by costructio the bootstrap aalogue of A which we deote by A is zero for T. Because the origial test has A, the bootstrap fails to capture this source of ucertaity. Note that the coditios o u used by Theorem 4.2 specify that ϖ, 1/2, but the result actually follows uder o restrictios o u sice we assume that X is cotiuous this explais also why we do ot require stregtheig the restrictios o ϖ as we did whe provig Theorem 3.3. Our solutio is to add a bias correctio term to T that relies o the explicit form of the limit of A. I particular, our adjusted bootstrap statistic is give by T = RV BV E RV BV ˆV ˆv 1 + ˆv ˆV = T + R, 1

11 where R ca be writte as R = V A ˆV. Sice ˆv i is equal to a spot volatility estimator, it follows that σ 2 + σ1 2 A = 1 2 ˆv 1 + ˆv V P 1 2 τσ 4 κ 1,1 uder our assumptios. Hece, T is able to replicate the first ad third order cumulats through order O 1/2 ad therefore provides a secod-order refiemet. The followig theorem provides the formal derivatio of the cumulats of T. We let κ 1 ad κ 3 deote the probability limits of κ 1 ad κ 3, the leadig terms of the first-order ad third-order bootstrap cumulats of T. Theorem 4.3 Uder the same assumptios as Theorem 4.2, coditioally o σ, we have that κ 1 = κ 1 ad κ 3 = κ 3, where κ 1 ad κ 3 are defied as i Theorem Mote Carlo simulatios I this sectio, we assess by Mote Carlo simulatios the performace of our bootstrap tests. Alog with the asymptotic test 4 of BN-S 26, we report bootstrap results usig ˆv i based o the thresholdig estimator. We follow Jacod ad Rosebaum 213 ad set k = [ ]. We also follow Podolskij ad Ziggel 21 ad choose ϖ =.4 ad α = 2.3 BV for the trucatio parameters. We preset results for the SV2F model give by 5 dx t = adt + σ u,t σ sv,t dw t + dj t, σ u,t = C + A exp a 1 t + B exp a 2 1 t, σ sv,t = s-exp β + β 1 τ 1,t + β 2 τ 2,t, dτ 1,t = α 1 τ 1,t dt + db 1,t, dτ 2,t = α 2 τ 2,t dt φτ 2,t db 2,t, corr dw t, db 1,t = ρ 1, corr dw t, db 2,t = ρ 2. The processes σ u,t ad σ sv,t represet the compoets of the time-varyig volatility i prices. We follow Huag ad Tauche 25 ad set a =.3, β = 1.2, β 1 =.4, β 2 = 1.5, α 1 =.137, α 2 = 1.386, φ =.25, ρ 1 = ρ 2 =.3. At the start of each iterval, we iitialize the persistet factor τ 1 by τ 1, N, 1 2α 1, its ucoditioal distributio. The strogly mea-revertig factor τ 2 is started at τ 2, =. The process σ u,t models the diural U-shaped patter i itraday volatility. I particular, we follow Hasbrouck 1999 ad Aderse et al. 212 ad set the costats A =.75, B =.25, C = , ad a 1 = a 2 = 1. These parameters are calibrated so as to produce a strog asymmetric U-shaped patter, with variace at the ope close more tha times that at the middle of the day. Settig C = 1 ad A = B = yields σ u,t = 1 for t [, 1] ad rules out diural effects from the observed process X. Fially, for our power aalysis, we cosider two alterative data geeratig processes. Specifically, we first geerate J t as a fiite activity jump process modeled as a compoud Poisso process with costat jump itesity λ ad radom jump size distributed as N, σjmp 2. We let σ2 jmp = uder the ull hypothesis of o jumps. Uder the alterative, we let λ =.58, ad σjmp 2 = Followig commo practice, we implemet the BN-S test statistic usig a versio of BV ad ÎQ that cotais a fiite-sample correctio of the bias itroduced by boudary effects. I particular, we multiply BV with the factor / 1 ad ÎQ with / 2. These same correctios are used whe costructig the bootstrap statistics. 5 The fuctio s-exp is the usual expoetial fuctio with a liear growth fuctio splied i at high values of its argumet: s-expx = expx if x x ad s-expx = expx if x > x, with x x 2 x = log1.5. +x2 11

12 These parameters are motivated by empirical studies by Huag ad Tauche 25 ad Bardorff- Nielse, Shephard, ad Wikel 26, which suggest that the jump compoet accouts for 1% of the variatio of the price process. Secod, we cosider J t as a symmetric tempered stable process with e Lévy measure νdx = c d 1 x 1 dx, where c x 1+r 1 >, d 1 >, ad r [, 2] measures the degree of jump activity. We let d 1 = 3 ad r =.5. We ote that this choice of r produces a ifiite-activity, fiitevariatio jump process. c 1 is calibrated so that J t accouts for 1% of the quadratic variatio. For a similar parameterizatio, see Ait-Sahalia ad Xiu 216 ad Houyo 217. We follow Todorov et al. 214 ad geerate J t as the differece betwee two spectrally positive tempered stable processes, which are simulated usig the acceptace-rejectio algorithm of Baeumer ad Meerschaert 21. We simulate data for the uit iterval [, 1] ad ormalize oe secod to be 1/23, 4, so that [, 1] is meat to spa 6.5 hours. The observed process X is geerated usig a Euler scheme. We the costruct the 1/-horizo returs r i = X i/ X i 1/ based o samples of size. Results are preseted for four differet samples sizes: = 48, 78, 288, ad 576, correspodig approximately to 8-miute, 5-miute, 1,35-miute, ad 4-secod frequecies. Table 1 gives the 5% omial level rejectio rates. Those reported i the left part of Table 1 uder o jumps are obtaied from 1, Mote Carlo replicatios with 999 bootstrap samples for each simulated sample for the bootstrap tests. For fiite activity jumps, sice J t is a compoud Poisso process, eve uder the alterative, it is possible that o jump occurs i some sample over the iterval [,1] cosidered. Thus, to compute the rejectio rates uder the alterative of jumps we rely o the umber of replicatios, out of 1,, for which at least oe jump has occurred. For our parameter cofiguratio, = 57. Startig with size, the results show that the liear versio of the test based o the asymptotic theory of BN-S 26 labeled AT i Table 1 is substatially distorted for the smaller sample sizes. I particular, for the SV2F model without diural effects, the rejectio rate is 15.44% for = 48, decreasig to 8.45% for = 576. As expected, the log versio of the test has smaller size distortios: the rejectio rates are ow 12.66% ad 7.67% for = 48 ad = 576, respectively. The rejectio rates of the bootstrap tests are always smaller tha those of the asymptotic tests ad therefore the bootstrap outperforms the latter uder the ull. This is true for both bootstrap jump tests based o 6 ad 7 deoted Boot1 ad Boot2, respectively ad for both the liear ad the log versios 6 of the test. Whe X has diurality patters i volatility, we apply the tests to both raw returs ad to trasformed returs with volatility corrected for diural patters. We use the oparametric jump robust estimatio of itraday periodicity i volatility suggested by Boudt et al The results based o the raw returs appear i the middle pael of Table 1 whereas the bottom pael cotais results based o the trasformed returs. We ca see that the test based o the asymptotic theory of BN-S has large distortios drive by the differece i volatility across blocks, eve if the sample size is large. As expected, correctios for diural effects help reduce the distortios. The bootstrap ull rejectio rates are always smaller tha those of the asymptotic theory-based tests. This is true eve for the bootstrap test applied to the o-trasformed itraday returs, which yields rejectio rates that are closer to the omial level tha those obtaied with the asymptotic tests based o the correctio of the diural effects compare Boot2 i the middle pael with AT i the bottom pael. This is a very iterestig fidig sice it implies that our bootstrap method is more robust to the presece of diural effects tha the asymptotic theory-based tests. Of course, eve better results ca be obtaied for the bootstrap tests by resamplig the trasformed itraday returs ad this is cofirmed by Table 1. These results also reveal that Boot2 outperforms Boot1, i particular for smaller sample sizes. Turig ow to the power aalysis, for fiite activity jumps, results i Table 1 ceter pael show 6 Note that our bias correctio adjustmet of the bootstrap test is specific to the liear versio of the statistic as it depeds o its cumulats. Sice we have ot developed cumulat expasios for the log versio of the statistic, we do ot report the aalogue of Boot2 for this test. 12

13 that the bootstrap tests have lower power tha their asymptotic couterparts, especially i presece of diural effects. This is expected give that the latter have much larger rejectios uder the ull tha the bootstrap tests. The results also show that power is largest for tests applied to the trasformed returs. For these tests, the differece i power betwee the bootstrap ad the asymptotic tests is very small. Give that the bootstrap essetially elimiates the size distortios of the asymptotic test, these two fidigs strogly favor the bootstrap over the asymptotic tests. For ifiite activity jumps, the right pael of Table 1 shows that power drops sigificatly for all tests whe there are o diural effects, cofirmig that the BN-S tests are ot always the most powerful oes uder ifiite activity jumps. I ureported simulatios, we foud that the small jumps test by Lee ad Haig 21 has more power i these situatios, which is i lie with previous results i the literature. Iterestigly, the combiatio of stochastic volatility with diurality ad ifiite activity jumps seems to restore power across all tests. 6 Empirical results This empirical applicatio uses trade data o the SPDR S&P 5 ETF SPY, which is a exchage traded fud ETF that tracks the S&P 5 idex. Our primary sample comprises 1 years of trade data o SPY startig from Jue 15, 24 through Jue 13, 214 as available i the New York Stock Exchage Trade ad Quote TAQ database. After cleaig this data set usig the procedure suggested by Bardorff-Nielse et al. 29 ad removig short tradig days, we are left with 2497 observatios for the whole period. I additio, we cosider three subperiods: Before crisis, through August tradig days; Crisis, from September 2, 28 through May 29, tradig days, ad After crisis, from Jue 1, 29 through Jue 13, tradig days. Table 2 shows the percetage of days idetified with a jump jump days by the asymptotic ad bootstrap tests. We cosider the asymptotic versio of the liear ad the log test statistics as well as their bootstrap versios. For the liear bootstrap test, we rely o Boot2, which does best i fiite samples accordig to our simulatios. For the log versio of the bootstrap test, we rely o Boot1. These tests are applied to data with ad without correctio for diural effects ad are based o 5-mi returs throughout. This yields 78 daily observatios over the 6.5 hours of the tradig sessio. I lie with the simulatio fidigs, the asymptotic tests ted to substatially over detect jumps compared to the bootstrap tests, which throughout detect about half of the umber of jump days detected by the asymptotic tests. More precisely, with o accout for diural effects, the asymptotic liear ad log tests detect 26.31% ad 23.27% jump days, respectively, while the bootstrap tests detect 13.7% ad 16.9% jump days. These percetages are about the same as what is obtaied before ad after crisis. Durig the crisis though, less jump days i proportio are detected. We also report test results applied to returs corrected for diural effects. This is particularly relevat because Figure 1 suggests that these are importat i our applicatio. The U-shape of these graphs highlights the fact that the market seems to be more volatile early ad late i daily tradig sessios compared to the mid-day volatility. We ca also see that the gap betwee early/late ad mid-day volatilities is magified i the crisis period. After correctio for diural effects, less jumps days are detected by all the tests before ad after crisis. However, i the crisis period, while the bootstrap still detects about the same umber of jump days, the asymptotic tests detect substatially more jumps after diural effects correctio. It is also worthwhile to poit out that the gap betwee the bootstrap liear ad log tests arrows as diural effects are accouted for. Overall, the bootstrap tests seem more robust to diurality tha the asymptotic tests. 13

14 Table 1: Rejectio rates of asymptotic ad bootstrap tests, omial level α =.5. Uder H: o jumps Uder H1: presece of jumps Size Power: with fiite activity jumps Power: with ifiite activity jumps Liear test Log test Liear test Log test Liear test Log test AT Boot1 Boot2 AT Boot1 AT Boot1 Boot2 AT Boot1 AT Boot1 Boot2 AT Boot1 SV2F model without diural effects, SV2F model without diural effects, SV2F model without diural effects, o jumps fiite activity jumps ifiite activity jumps SV2F model with diural effects, SV2F model with diural effects, SV2F model with diural effects, o correctio, o jumps o correctio, fiite activity jumps o correctio, ifiite activity jumps SV2F model with diural effects, SV2F model with diural effects, SV2F model with diural effects, correctio, o jumps correctio, fiite activity jumps correctio, ifiite activity jumps Notes: AT is based o 4, i.e., the asymptotic theory of BN-S 26; Boot1 ad Boot2 are based o bootstrap test statistics T cf. 6 ad T cf. 7, respectively. Boot2 takes ito accout the asymptotically egligible bias i T which may be relevat at the secod-order, ad uder certai coditios provides the refiemet for the bootstrap method. We use 1, Mote Carlo trials with 999 bootstrap replicatios each. 14

15 Table 2: Percetage of days idetified as jumps day by daily statistics omial level α =.5 usig 5-mi returs. No correctio for diural effects With correctio for diural effects AT-li AT-log Boot2-li Boot1-log AT-li AT-log Boot2-li Boot1-log Full sample: Jue 15, 24 through Jue 13, days Before crisis: Jue 15, 24 through August 29, days Durig crisis: September 2, 28 through May 29, days After crisis: Jue 1, 29 through Jue 13, days Notes: AT-li ad Boot2-li AT-log ad Boot1-log stad for asymptotic ad bootstrap tests usig the liear log versio of the test statistic. Boot2-li uses the secod-order corrected bootstrap test statistic for asymptotic refiemet. Average absolute retur 3.5 x Before crisis Durig crisis After crisis Full sample.5 1: 11: 12: 13: 14: 15: 16: Five miute iterval Figure 1: Diural patter of SPY. The graph displays the average over the specified samples of absolute 5-mi returs of each tradig day. Before crisis refers to the sample from Jue 15, 24 through August 29, 28; Durig crisis refers to the period from September 2, 28 through May 29, 29, ad After crisis refers to the period from Jue 1, 29 through Jue 13, Coclusio The mai cotributio of this paper is to propose bootstrap methods for testig the ull hypothesis of o jumps. The methods geerate bootstrap returs from a Gaussia distributio with variace give by a local realized measure of itegrated volatility {ˆv i }. We first provide a set of high level 15

16 coditios o {ˆv i } such that ay bootstrap method of this form is asymptotically valid whe testig for jumps usig the BN-S test statistic. We the provide a detailed aalysis of the bootstrap test based o a thresholdig estimator for {ˆv i }. A secod cotributio of this paper is to discuss the ability of the bootstrap to provide secodorder asymptotic refiemets over the usual asymptotic mixed Gaussia distributio uder the ull of o jumps. Our results show that our bootstrap test is ot secod-order accurate because it is ot able to match the first-order cumulat of the test statistic at higher order. We therefore propose a modificatio of the origial bootstrap test for which a asymptotic refiemet exists. The modificatio cosists of addig a bias correctio term that estimates the cotributio of the bipower variatio bias to the first-order cumulat of the origial test. Our simulatios show that this adjustmet is importat i fiite samples, especially for the smaller sample sizes whe samplig is more sparse. We illustrate the usefuless of our bootstrap jumps test by applyig it to 5-mi returs o the SPY idex over the period from Jue 15, 24 through Jue 13, 214. Overall, the mai fidig is that the bootstrap detects about half of the umber of jump days detected by the asymptotic-theory based tests. Appedix A: A law of large umbers for fuctios of o-overlappig local volatility estimates I this sectio, we state ad prove Theorem A.1, a result that is auxiliary i provig Lemma 3.1. As oted i the mai text, Theorem A.1 has merit o its ow right as it exteds Theorem of Jacod ad Protter 212 to the case of smooth fuctios of cosecutive local realized volatility estimates rather tha a sigle local estimate. Let ĉ j, = k k m=1 r 2 j 1k +m 1 { r j 1k+m u }, j = 1,..., k, with r i X i X i 1, i = 1,..., ; u = α ϖ, ϖ, 1 2 ad k is a sequece of itegers satisfyig k ad k as. Theorem A.1 Assume that X satisfies Assumptio H-2, ad let g be a cotiuous fuctio o R l + such that gx 1,..., x l K 1 + x 1 p + + x l p for some p. If either: a X is cotiuous; b p < 1; or c Assumptio H-r holds for some r [, 2 ad p 1, ϖ 2p 1 4p r ; the G k /k j=l g ĉ j,, ĉ j 1,,..., ĉ j l+1, P 1 gσ 2 s,..., σ 2 sds. I the proof, we will follow the stadard localizatio argumet of Jacod ad Protter 212 ad assume without loss of geerality that the followig stroger versio of Assumptio H-r holds: Assumptio SH-r Assumptio H-r holds, ad i additio the processes a ad σ are bouded, ad δω, t, x 1 γx with γx r dx <. Proof. We follow the proof of Theorem of Jacod ad Protter 212. By localizatio, we assume without loss of geerality that SH-r holds with r = 2 for p < 1 ad that g. Step 1 : We first assume that g is bouded. For all s [, 1] ad l = 1,..., l, let ĉ l s = ĉ j+l, whe j 1 k s < j k, where ĉ j, = if j > /k. We have G = k 1 l k g ĉ l,, ĉ l 1,,..., ĉ 1, + 16 g ĉ l s,..., ĉ 1 s ds.

17 G Thus, E 1 g σs, 2..., σs 2 1 l k ds K k + a sds, with a s = E g ĉ l s,..., ĉ 1 s g σs, 2..., σs 2 l. Sice g is bouded, we ca claim that ĉ s P σs 2 for all s [, 1 by usig the same argumet as i the proof of Theorem 9.3.2a of Jacod ad Protter 212, implyig that a s teds to as for each s ad is bouded uiformly i, s. The result the follows by the domiated covergece theorem. Step 2 : Let ψ be a C fuctio: R + [, 1] with 1 [, x ψx 1 [ 1, x, ad ψ εx = 2 ψ x /ε ad ψ ε = 1 ψ ε. For m 2, let g mx 1,..., x l = gx 1,..., x l l l=1 ψ mx l ad g m = g g m. Sice g m is cotiuous ad bouded, for ay fixed m, by Step 1, k /k j=l g m ĉ j,, ĉ j 1,,..., ĉ j l+1, P 1 g mσ 2 s,..., σ 2 sds. Note that 1 g mσs, 2..., σsds 2 = 1 gσ2 s,..., σsds 2 for m large eough sice σs 2 is bouded ad ψ mx = 1 for x m/2. Thus, the result follows by showig that k /k j=l g m ĉ j,, ĉ j 1,,..., ĉ j l+1, is egligible for large ad m. By assumptio, l l g m x 1,..., x l K 1 + x l 1 p ψ mx l l=1 where 1 l l=1 ψ mx l l l=1 1 { x l m 2 }, sice ψ mx l = 1 if x l m/2 ad both sides of the l iequality are il if x l m/2 for all l. If x l > m/2 for some l, the 1 { xl m 2 } 1 1 l ψ mx l. Also, if x l > m/2 for some l, we have that 1+ l x l p 2 l x l p. Thus, g m x 1,..., x l l=1 2K l l,l =1 x l p 1 { xl m 2 }. Therefore, to complete the proof, it suffices to show that lim m lim sup E k /k j=l l=1 l=1 l=1 ĉ p j l+1, 1 {ĉ j l +1, >m} =, l=1 A.1 for all l, l = 1,..., l. Cosider first p < 1. Lettig κ = whe X is cotiuous ad κ = 1 otherwise, for q 2, we ca show that E r i q K 1 1 q + κ ad, from the c q/2 q/2 1 r -iequality, E ĉ p j, K q 1 + κ 1. By successive applicatios of the Hölder ad Markov iequalities, for ay q > p: q 1 q E ĉ p i, 1 {ĉ j, >m} E ĉ q p q i, P ĉ j, m 1 p q E ĉ q p q 1 i, ĉ m q E q 1 p q K q j, m q p 1 + κ 1 q 1 q Take q = 2p if X is cotiuous ad q = 1 > p otherwise ad coclude A.1. Next, cosider p 1. With the same alterative decompositio of X as that i Jacod ad Protter 212, Eq , we write r i = r 1i + r 2i, with r 1i ad r 2i the icremets of the process X ad X, respectively. We have that r ri 2 1 { ri u } 2 r1i 2 + u 2 2 2i u 2 1 K r1i 2 + u 2 ϖ r 2i 1 2, 17.

18 where we use for the last iequality the fact that a/b 1 max1, 1/b[a 1], with a, b >. Thus, ĉ j, ζ j, + ζ j, with ζ j, = K 1 k k m=1 with v = u. Notig that r 1,j 1k+m 2, ζ j, = K v2 ĉ i, 1 {ĉj, m} ĉi,ĉ j, m 1 2m k k m=1 ϖ r 2,j 1k+m 1 2, ĉ2 i, + ĉ 2 1 j, ζ i, 2 + ζ i, 2 + ζ j, 2 + ζ j, 2, m by the c r -iequality, E ĉ p i, 1 {ĉ j, m} 4p 1 m Eζ p i, 2p i, + Eζ 2p j, + Eζ 2p j,. Moreover, Eqs ad of Jacod ad Protter 212 esure that E r 1i q F i 1 K q ad E ϖ r 2i 2 1 F i 1 K 1+rϖ φ,with φ as. Thus, by a further applicatio of the c r -iequality, Eζ 2p j, Eζ 2p j, K v4p < K, whereas k k m=1 E ϖ r 2,j 1k+m 1 4p K v4p k k m=1 E ϖ r 2,j 1k+m 1 2 K 4p ϖ rϖ φ = K w φ, with w = 1 2p + ϖ4p r. Thus, E ĉ p i, 1 {ĉ j, m} K m 1 + w φ p. Sice w uder the maitaied assumptios, A.1 follows. Refereces [1] Aït-Sahalia, Y., ad J. Jacod, 29. Testig for jumps i a discretely observed process, Aals of Statistics 37 1: [2] Aït-Sahalia, Y., ad J. Jacod, 212. Aalyzig the spectrum of asset returs: jump ad volatility compoets i high frequecy data, Joural of Ecoomic Literature, 54, [3] Aït-Sahalia, Y. ad J. Jacod, 214. High Frequecy Fiacial Ecoometrics, Priceto Uiversity Press. [4] Aït-Sahalia, Y., J. Jacod ad J. Li, 212. Testig for jumps i oisy high frequecy data, Joural of Ecoometrics, 168, [5] Aït-Sahalia, Y., ad D. Xiu, 216. Icreased correlatio amog asset classes: Are volatility or jumps to blame, or both?, Joural of Ecoometrics, 1942, [6] Aderse, T.G., T. Bollerslev, ad F. X. Diebold, 27. Roughig it up: icludig jump compoets i the measuremet, modelig, ad forecastig of retur volatility, Review of Ecoomics ad Statistics 89 4, [7] Aderse, T.G., D. Dobrev ad E. Schaumburg, 212. Jump-robust volatility estimatio usig earest eighbor trucatio, Joural of Ecoometrics, 169, [8] Baeumer, B., ad M. M. Meerschaert 21. Tempered stable Levy motio ad trasiet superdiffusio, Joural of Computatioal ad Applied Mathematics 2331,