Bootstrapping high-frequency jump tests

Transcription

1 Bootstrappig high-frequecy jump tests Prosper Dovoo Departmet of Ecoomics, Cocordia Uiversity Sílvia Goçalves Departmet of Ecoomics, Uiversity of Wester Otario Ulrich Houyo CREATES, Departmet of Ecoomics ad Busiess Ecoomics, Aarhus Uiversity Nour Meddahi Toulouse School of Ecoomics, Toulouse Uiversity December 19, 216 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 call {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. Our results show that the choice of {ˆv i } is crucial for the power of the test. I particular, we should choose {ˆv i } i a way that is robust to jumps. We the focus o a thresholdig-based estimator for {ˆv i } ad provide a set of primitive coditios uder which our bootstrap test is asymptotically valid. We also discuss the ability of the bootstrap to provide secod-order asymptotic refiemets uder the ull of o jumps. The cumulats expasios that we develop show that our proposed bootstrap test is uable to mimic the first-order cumulat of the test statistic. The mai reaso is that it does ot replicate the bias of the bipower variatio as a measure of itegrated volatility. We propose a modificatio of the origial bootstrap test which cotais a appropriate bias correctio term ad for which secod-order asymptotic refiemets obtai. 1 Itroductio A well accepted fact i fiacial ecoomics is the fact 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 the presece of jumps 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. For this reaso, may tests for jumps have bee 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. I additio, Ulrich 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 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, Shephard ad Sheppard 212; tests based o power variatio measures sampled at differet frequecies such as i Aït-Sahalia ad Jacod 29, Aït-Sahalia, Jacod ad Li 212, ad tests based o the maximum of a stadardized versio of itraday returs such as i Lee ad Myklad 28, 212. I additio, tests based o thresholdig or trucatio-based estimators of volatility have also bee proposed with the objective of disetaglig big from small jumps, as i Aït-Sahalia ad Jacod 29 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 Bardorff-Nielse ad Shephard 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. I particular, our aim is to improve fiite sample size while retaiig good power. I order to do so, 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 call {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 24, 26 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 rather tha trucatio-based methods to compute {ˆv i } 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. 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 statistic 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 a secod-order refiemet. We propose a modificatio of the bootstrap statistic that is able to do so. Specifically, the modified bootstrap test statistic cotais a correctio term that is based o a estimate of the cotributio to the first order cumulat of the test statistic due to the bias of bipower variatio. 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 2

3 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. The rest of the paper is orgaized as follows. I Sectio 2, we provide the framework ad state our assumptios. I Sectio 3, we ivestigate the first-order asymptotic validity of the Gaussia wild bootstrap based o a give {ˆv i }. Specifically, 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. Sectio 3.2 provides a set of primitive assumptios uder which the bootstrap based o a thresholdig estimator {ˆv i } verifies these high level coditios ad is therefore asymptotically valid to first order. Sectio 4 ivestigates the ability of the bootstrap to provide asymptotic refiemets. I particular, 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. This result is crucial for establishig the properties of the bootstrap jump test based o the thresholdig approach. It is of idepedet iterest as it exteds some existig results i the literature, amely results by Jacod ad Protter 212, Jacod ad Rosebaum 213 ad Li, Todorov ad Tauche 216, who focused o smooth fuctios of a sigle local volatility estimate. 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 bootstrap cosistecy results preseted i Sectio 3 whereas Appedix S2 cotais the proofs of the results i Sectio 4 o the asymptotic refiemets of the bootstrap. Fially, Appedix S3 cotais formulas for the log versio of our tests. To ed this sectio, a word o otatio. As usual i the bootstrap literature, 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 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 semimartigale process. The jump process is defied as J t = t R t δ s, x I{ δs,x 1} µ ν ds, dx + 3 R δ s, x I{ 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 the Lebesgue measure o R, ad δ a real fuctio o Ω R + R. The first term i the defiitio of J t represets the small jumps of the process whereas the secod term represets the big jumps. 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 determiistic oegative fuctio 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, i.e. s t X s r < for all t >. 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 give 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 at frequecy 1/, r i X i/ X i 1/, i = 1,...,, where we omit the idex i r i to simplify the otatio. Our focus is o testig for o jumps usig the bootstrap. I particular, followig Aït-Sahalia ad Jacod 29, we would like to decide o the basis of the observed itraday returs {r i : i = 1,..., } i which of the two followig complemetary sets the path we actually observed falls: Ω = {ω : t X t ω is cotiuous o [, 1]} Ω 1 = {ω : t X t ω is discotiuous o [, 1]}, where Ω = Ω Ω 1 ad Ω Ω 1 =. Formally, our ull hypothesis ca be defied as H : ω Ω whereas the alterative hypothesis is H 1 : ω Ω 1. Let RV = i=1 r2 i deote the realized volatility ad let BV = 1 k1 2 r i 1 r i i=2 χ be the bipower variatio, where we let k 1 E 2 1/2 1 = E Z = 2/ π, where Z N, 1. χ This is a special case of k q = E 2 q/2 1 = E Z q 1+q = 2q/2 Γ 2 Γ 2, q >. 1 The class of statistics we cosider is based o the compariso betwee RV ad BV. It is ow well kow that uder certai regularity coditios icludig the assumptio that X is cotiuous see BN-S 26 ad Bardorff-Nielse, Graverse, Jacod, ad Shephard 26 the followig joit CLT holds: where RV IV BV IV st deotes stable covergece ad Σ = θ 4 st N, Σ, 4 IQ, 5

5 with IQ 1 σ4 udu ad θ = k k A implicatio of 4 is that uder o jumps, i.e. i restrictio to Ω, RV BV V st N, 1, where V τ IQ is the asymptotic variace of RV BV ad τ = θ 2. Hece, a liear versio of the test is give by RV BV T =, 6 ˆV where ˆV τ ÎQ with ÎQ = 3 r i 4/3 r i 1 4/3 r i 2 4/3. k4/3 Choosig the tripower realized quarticity esures that ÎQ IQ o both Ω ad Ω 1. Thus, T N, 1, i restrictio to Ω, ad the test that 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 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 >, i=3 lim P ω C S = α. Uder the alterative hypothesis, we ca show that T is alterative-cosistet, i.e. the probability that we make the icorrect decisio of acceptig the ull whe this is false goes to zero: P 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. 3 The bootstrap Whe bootstrappig hypothesis tests, imposig the ull hypothesis o the bootstrap data geeratig process is ot oly atural, but may be importat to miimize the probability of a type I error. I particular, Davidso ad MacKio 1999 see also MacKio 29 show that i order to miimize the error i rejectio probability uder the ull type I error of a bootstrap test, we should estimate the bootstrap DGP as efficietly as possible. This etails imposig the ull hypothesis o the bootstrap DGP. I this paper, we follow this rule ad impose the ull hypothesis of o jumps whe geeratig the bootstrap itraday returs. Specifically, we let st r i = ˆv i η i, i = 1,...,, 7 for some variace 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 agai write ri istead of ri,. Accordig to 7, 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 5

6 by the simplified model X t = t σ sdw s, where σ is idepedet of W ad there is o drift or jumps 1. Uder these assumptios, coditioally o the path of volatility, r i N, vi, idepedetly across i, where vi = i/ i 1/ σ2 udu. Thus, we ca thik of ˆv i as the sample aalogue of vi. The mai goal of Sectio 3.1 is to provide a set of geeral coditios o ˆv i uder which the bootstrap is asymptotically valid. I practice, we eed to choose ˆv i ad our recommedatio is to use a thresholdig estimator that we defie formally i Sectio 3.2. The bootstrap aalogues of RV ad BV are RV = i=1 r 2 i ad BV = 1 k 2 1 r i 1 r i. i=2 The first class of bootstrap statistics we cosider is described as T = RV BV E RV BV, 8 ˆV where ad E RV BV = ˆv i i=1 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 hypothesis of o jumps whe is large. Nevertheless, ad as we will study i Sectio 4, 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 i=2 i=3 T = RV BV E RV BV ˆV ˆv 1 + ˆv, 9 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. For istace, we could rely o local multipower realized volatility estimators of vi, followig the approach of Myklad, Shephard ad Sheppard 212 see also Myklad ad Zhag 29. Asymptotic refiemets of the bootstrap based o T will be discussed i Sectio 4. 1 Although 7 is motivated by this very simple model, as we will prove below, this does ot prevet the bootstrap method to be valid more geerally. I particular, its validity exteds to the case where there is a leverage effect ad the drift is o-zero. 6

7 3.1 Bootstrap validity uder geeral coditios o ˆv i We first provide a set of coditios uder which a joit bootstrap CLT holds for RV, BV. I particular, we would like to establish that RV Σ 1/2 E RV d BV E BV N, I 2, i prob-p, where Σ V ar RV BV = V ar RV Cov RV, BV V ar BV, is such that Σ P Σ. The followig result gives the first ad secod order bootstrap momets of RV, BV. Note that sice ri = ˆv i η i, we ca write RV = ˆv i u i ad BV = 1 k1 2 i=1 1/2 ˆv i 1 ˆv i 1/2 w i i=2 where u i η 2 i ad w i η i 1 η i, with η i i.i.d. N, 1. The bootstrap momets of RV, BV deped o the momets ad depedece properties of u i, w i. The proof is trivial ad is omitted for brevity. Lemma 3.1 If r i = ˆv i η i, i = 1,...,, where η i i.i.d. N, 1, the a1 E RV = ˆv i. i=1 a2 E BV = ˆv 1/2 i 1 ˆv i 1/2. i=2 a3 V ar RV = 2 ˆv i 2. i=1 a4 V ar BV = k a5 Cov RV, BV = i=2 i=2 ˆv i ˆv i k ˆv ˆv i 1/2 i 1 ˆv 1/2 i 2. i=3 ˆv i ˆv 3/2 i 1 1/2 + ˆv ˆv i 1/2 3/2 i 1. i=2 Lemma 3.1 shows that the bootstrap momets of RV ad BV deped o multipower variatio measures of {ˆv i }. I particular, they deped o K 1+q/2 ˆv qk /2 i k+1, where K is a positive i=k k=1 iteger ad q K k=1 q k, with q k. The followig assumptio imposes a covergece coditio o these measures as well as other additioal high level coditios o ˆv i that are sufficiet for a bootstrap CLT to hold. 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: 7

8 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, 1+q/2 i=k k=1 K qk /2 P ˆv i k+1 1 σ q udu >, as. 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. 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, the bootstrap covariace matrix of RV, BV, is equal to Σ for this result, covergece of the multipower variatios of ˆv i with q = 4 suffices. 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 vector u i, w i is 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 The proof of Theorem 3.1 the follows by showig that P V = τ IQ uder Coditio Ai this follows from the covergece of the multi- small blocks is asymptotically egligible. ˆV = τ ÎQ power variatios of ˆv i of eighth order, explaiig why we require q 8. Uder this high level coditio, we ca prove the followig result. Theorem 3.1 Uder Coditio A, if, T d N, 1, i prob-p. st 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. More formally, let the bootstrap critical regio be defied as follows, where q,1 α ω is such that C = { ω : T ω > q,1 α ω }, P T, ω q,1 α ω = 1 α. The bootstrap test rejects H : ω Ω agaist H 1 : ω Ω 1 wheever ω C. 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 The results of the previous subsectio esure the cosistecy of the bootstrap distributio of T for ay choice of ˆv i that verifies Coditio A. I this sectio, we verify this coditio for the followig choice of ˆv i : ˆv j+i 1k = 1 k ri 1k 2 k +m 1 r i 1k +m u, m=1 8

9 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 3.2 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 ; c q 2, Assumptio H-r holds for some r [, 2, ad q 1 2q r ϖ < 1 2 ; the 1+q/2 i=k k=1 K qk /2 P ˆv i k+1 1 σ q udu >. Lemma 3.2 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, Todorov ad Tauche 216 focus o 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. 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 3.2. 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 r [, 2 ad 7 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 a stadard ormal radom variable 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 9

10 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 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 or equal tha 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. This result shows that the thresholdig-based bootstrap jump test has the correct asymptotic size ad is cosistet uder the alterative of jumps provided we choose a trucatio level u = α ϖ with 7 16 r ϖ < 1 2, where r is [, 2. I particular, if we choose ϖ i the viciity of 1/2, as commoly doe i applicatios, the bootstrap test is cosistet uder the alterative of jumps for a wide spectrum of jump activities icludig fiite activity. For example, if ϖ =.45, the set of r such that ϖ 7/ 16 r is [,.444] ad whe ϖ =.48, this set becomes [, 1.417]. 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, 1 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. 1

11 Assumptio V The volatility process σu 2 is pathwise cotiuous, bouded away from zero ad σ Holder-cotiuous i L 2 P o [, 1] of order δ > 1/2, i.e., E 2 u σs 2 2 = O u s 2δ. Thus, we ot oly impose that the volatility path is cotiuous, but we also rule out stochastic volatility models drive by a Browia motio. Examples of processes that satisfy Assumptio V iclude fractioal Browia motio with Hurst parameter H > 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 1 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 = θ 2 = 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 ormal approximatio assumes that the values of these cumulats are zero, this iduces a error of order O 1/2 for the asymptotic ormal approximatio whe approximatig the ull distributio of T. 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, 11 ˆV 11

12 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 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 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=1 i=2 i=2 v i 1 1/2 vi 1/2, where IV = i=1 v i, ad otig that by Lemma S2.3 i Appedix S2, uder Assumptio V, vi v i 1 1/2 v i 1/2 P 1 σ σ1 2, 12 P ad V τσ 4. 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 Let κ 1 ad κ 3 deote the leadig terms of κ 1 T ad κ 3 T, the first ad third order cumulats of T, respectively. I particular, 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 deped o sice they are a fuctio of the origial sample. 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 1 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 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

13 Theorem 4.2 shows that the bootstrap test based o T oly captures the first order cumulat 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, where R ca be writte as R = V A ˆV with A = 1 ˆv 1 + ˆv. 2 V Sice ˆv i is equal to a spot volatility estimator, it follows that A = 1 ˆv 1 + ˆv P 1 σ 2 + σ1 2 κ 2 V 1,1 2 τσ 4 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 where κ 1 ad κ 3 are defied as i Theorem Mote Carlo simulatios κ 1 = κ 1 ad κ 3 = κ 3 I this sectio, we assess by Mote Carlo simulatios the performace of our bootstrap tests. Alog with the asymptotic test of BN-S 26, we report bootstrap results usig ˆv i based o the thresholdig estimator cf. Sectio 3.2. We follow Jacod ad Rosebaum 213 ad set k = [ ], the iteger part of. As their results show see also Jacod ad Protter 212, this yields the optimal rate of covergece for the spot volatility estimator ˆv i. 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 2 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. 2 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 13

14 The processes σ u,t ad σ sv,t represet the compoets of the time-varyig volatility i prices. I particular, σ sv,t deotes the two factors stochastic volatility model commoly used i this literature. 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. I our experimet, we first cosider a setup without diural effects followed by oe with diural effects i X. Fially, J t is 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 i the retur process. Uder the alterative, we let λ =.58, ad σjmp 2 = 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. 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, 96, 288, ad 576, correspodig approximately to 8-miute, 4-miute, 1,35-miute, ad 4-secod frequecies. Table 1 gives the rejectio rates. We report results without jumps ad with fiite activity jumps. Test results from both the liear test statistic ad its log versio 3 are reported usig asymptotic-theory based critical value as well as bootstrap critical values. All tests are carried out at 5% omial level. The rejectio rates 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 three times larger tha the omial level of the test at 15.69% for = 48. Although this rate drops as icreases, it remais sigificatly larger tha the omial level eve whe = 576, with a value equal to 8.27%. As expected, the log versio of the test statistic has smaller size distortios: the rejectio rates are ow 13.4% ad 7.68% 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 8 ad 9 deoted Boot1 ad Boot2, respectively ad for both the liear ad the log versios of the test. 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. 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, Croux ad Lauret See the olie Appedix S3 for details o the log-trasform of the test statistic T ad the bootstrap-related formulas. 14

15 for diural patters correctio. I the process, stadardized returs are obtaied usig a estimate ˆσ u,i of itraday volatility patter from 2, simulated days. The results for the tests based o the raw returs without diurality correctio appear i the middle pael of Table 1 whereas the bottom pael cotais results for tests based o the trasformed returs. We ca see that the test based o the asymptotic theory of BN-S 26 has large distortios drive by the differece i volatility across blocks, eve if the sample size is large. For = 576, the ull rejectio rate is 13.29% for the liear versio of the test ad 11.74% for the log versio. These are more tha twice as large as the desired omial level of 5%. The overrejectio is magified for smaller sample sizes. For istace, for = 48 they are equal to 32.61% ad 28.91%, respectively. As expected, i the bottom part of Table 1 correctios for diural effects help reduce the distortios. For = 48, the rates are ow equal to 14.82% ad 12.1%, whereas for = 576 they are 8.47% ad 7.91%. The bootstrap ull rejectio rates are always smaller tha those of the asymptotic theory-based tests, implyig that the bootstrap outperforms the latter. 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, which shows that the results for bootstrap tests especially Boot2 with diural effects correctio are systematically closer to 5% tha those with o correctio of diural effects. These results also reveal that Boot2 outperforms Boot1, i particular for small sample sizes. This shows that takig ito accout the asymptotically egligible bias i T, oly relevat at the secodorder, is very useful for smaller values of. Overall, the left pael of Table 1 shows that the bootstrap reduces dramatically the size distortios that we ca see from asymptotic tests ad this across sample sizes whether the liear or log versio of the test is used. Turig ow to the power aalysis, results i Table 1 right pael show that the mai feature of otice is that the bootstrap tests have lower power tha their asymptotic couterparts, especially i presece of diural effects. This is expected give that the asymptotic tests have much larger rejectios uder the ull tha the bootstrap tests. I particular, this explais the large discrepacy betwee the bootstrap ad the asymptotic test whe both are applied to the o-trasformed data. As icreases, we see that this differece decreases. The results also show that power is largest for tests both asymptotic ad bootstrap-based 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. Overall, Table 1 shows that Boot2 is the best choice. This is especially true whe usig smaller values of. Therefore, our recommedatio is to choose Boot2. 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. Data o SPY have bee used by Myklad, Shephard ad Sheppard 212 see also Bollerslev, Law ad Tauche 28. 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. This tick-by-tick dataset has bee cleaed accordig to the procedure outlied by Bardorff-Nielse, Hase, Lude, ad Shephard 29. We 15

16 Table 1: Rejectio rates of of asymptotic ad bootstrap tests, omial level α =.5. Size Power Liear test Log test Liear test Log test AT Boot1 Boot2 AT Boot1 AT Boot1 Boot2 AT Boot1 SV2F model without diural effects, o jumps SV2F model without diural effects, jumps SV2F model with diural effects, o correctio, o jumps SV2F model with diural effects, o correctio, jumps SV2F model with diural effects, correctio, o jumps SV2F model with diural effects, correctio, jumps Notes: AT is based o 6, i.e., the asymptotic theory of BN-S 26; Boot1 ad Boot2 are based o bootstrap test statistics T cf. 8 ad T cf. 9, respectively. Boot2 takes ito accout the asymptotically egligible bias i T which may be relevat at the secodorder, ad uder certai coditios provides the refiemet for the bootstrap method. We use 1, Mote Carlo trials with 999 bootstrap replicatios each. also removed short tradig days leavig us with 2497 days of trade data. Figure 1 shows the series of daily returs o SPY over the 2497 tradig days cosidered. The 28 fiacial crisis is oticeable with large returs appearig i the third quarter of 28 ad the first quarter of 29. We ca actually distiguish three subperiods for SPY: Before crisis, from the begiig of the sample Jue 15, 24 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 left pael gives some summary statistics o daily returs ad 5-mi-retur-based realized volatility RV ad realized bipower variatio BV over the metioed periods. The average daily returs before ad after the crisis are positive 1.53 ad 6.42 basis poits, respectively whereas the average retur over the crisis is basis poits. Daily averages of RV ad BV are also quite high durig the crisis period with both culmiatig to 6 times their respective levels across the whole sample. The average cotributio of jumps to realized volatility as measured by RJ = RV BV /RV also deepes durig the crisis period to 5%, whereas the 7% foud for the full sample ad i pre- ad post-crisis periods seems i lie with the fidigs of Huag ad Tauche 25 for S&P 5 future idex. 16

17 .15.1 Returs Ju 4 Ju 5 Ju 6 Ju 7 Ju 8 Ju 9 Ju 1 Ju 11 Ju 12 Ju 13 Ju 14 Days Figure 1: Daily returs o SPY from Jue 15, 24 through Jue 13, 214. Table 2: This table gives the average daily retur, realized volatility RV, realized bipower variatio BV ad the cotributio of jumps to realized volatility RJ of SPY over each period alog with their stadard deviatios SD. RV ad BV are based o 5-mi itra-day returs. These statistics are also reported over days idetified with ad without jumps by the asymptotic approximatio of the log-test-statistic ad the bootstrap approximatio of the liear test statistic. α =.5. I the table, Ret. stads for retur. Ret. 1 4 RV 1 4 BV 1 4 RJ Ret. 1 4 RV 1 4 BV 1 4 RJ Full sample: Jue 15, 24 through Jue 13, days Days idetified with jumps by the asymptotic log test 581 days Mea SD Before crisis: Jue 15, 24 through August 29, days Days idetified without jumps by the asymptotic log test 1916 days Mea SD Durig crisis: September 2, 28 Days idetified with jumps by through May 29, days the bootstrap liear test 342 days Mea SD After crisis: Jue 1, 29 Days idetified without jumps by through Jue 13, days the bootstrap liear test 2155 days Mea SD

18 Table 3 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, the adjusted bootstrap statistic that promises secod-order refiemets ad 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 out of the 2497 days i our sample, 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, with the asymptotic tests detectig aroud 2%, while the bootstrap liear ad log tests detect about 1.8% ad 13.3% jump days, respectively. Give the results of the jump tests both asymptotic ad bootstrap-based, we ca compute the summary statistics for days with ad without jumps. The results are cotaied i the right pael of Table 2. Besides the fact that the bootstrap fids less jump days, average returs are higher by aroud 4 basis poits o bootstrap-jump-days with higher stadard deviatio. The average cotributio of jumps to realized volatility is substatially higher o jump days tha o o-jump-days by a ratio of about 1-to-1 for the asymptotic log test ad 5-to-1 for the bootstrap test. Table 3: Percetage of days idetified as jumps day by daily statistics omial level α =.5 usig itra-day 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 test uses the secod-order corrected bootstrap test statistic for asymptotic refiemet. We also report test results applied to returs corrected for diural effects. This is of particular relevace i the curret applicatio sice, as show by Figure 2, diural patters seem to be i display i our samples. Figure 2 displays graphs of average absolute 5-mi returs over the days i the specified sample. See Aderse ad Bollerslev The U-shape of these graphs highlights the fact that 18