EFFICIENT ESTIMATION OF INTEGRATED VOLATILITY FUNCTIONALS VIA MULTISCALE JACKKNIFE

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1 EFFICIENT ESTIMATION OF INTEGRATED VOLATILITY FUNCTIONALS VIA MULTISCALE JACKKNIFE By Jia Li ad Yuxiao Liu ad Dacheg Xiu Duke Uiversity, Uiversity of North Carolia at Chapel Hill ad Uiversity of Chicago We propose semi-parametrically efficiet estimators for geeral itegrated volatility fuctioals of multivariate semimartigale processes. A plug-i method that uses oparametric estimates of spot volatilities is kow to iduce high-order biases that eed to be corrected to obey a cetral limit theorem. Such bias terms arise from boudary effects, the diffusive ad jump movemets of stochastic volatility, ad the samplig error from the oparametric spot volatility estimatio. We propose a ovel jackkife method for bias correctio. The jackkife estimator is simply formed as a liear combiatio of a few ucorrected estimators associated with differet local widow sizes used i the estimatio of spot volatility. We show theoretically that our estimator is asymptotically mixed Gaussia, semi-parametrically efficiet, ad more robust to the choice of local widows. To facilitate the practical use, we itroduce a simulatiobased estimator of the asymptotic variace, so that our iferece is derivative free ad hece is coveiet to implemet. 1. Itroductio. This paper cocers the efficiet estimatio of itegrated volatility fuctioals of the form T g (c s) ds, where c is the spot covariace matrix process of a d-dimesioal Itô semimartigale process X, g( ) is a smooth fuctio, ad T is the fixed time spa. A basic example of such fuctioals is the itegrated variace-covariace matrix ([3], [6]). Geeral itegrated volatility fuctioals have received much attetio i the recet literature of high-frequecy ecoometrics ad statistics, see, e.g., [15], [18], ad [25]. These fuctioals are broadly useful for measurig riskrelated quatities ([2], [17], [21]) for which the g( ) fuctio trasforms the spot covariace to quatities such as spot betas, correlatios, idiosycratic variaces, ad eigevalues. More geerally, itegrated volatility fuctioals ca be used as momet coditios for estimatig ecoomic models ([22]). I this case, g( ) ivolves geeral oliear fuctios implied by the uderlyig ecoomic theory (e.g., optio pricig theory or market microstructure theory), as well as weight fuctios (i.e., istrumets) i the estimatio AMS 2 subject classificatios: 6F5, 6G44, 62F12 Keywords ad phrases: high-frequecy data, jackkife, semimartigale, spot volatility. 1

2 2 JIA LI, YUNXIAO LIU AND DACHENG XIU procedure. A atural plug-i estimator of T g (c s) ds ca be formed by replacig the latet spot covariace matrix process with its oparametric estimate. Whe the volatility path is sufficietly smooth, [18] shows the plug-i estimator is cosistet ad asymptotically mixed Gaussia. However, the smoothess requiremet of [18] does ot hold for typical stochastic volatility models (e.g., jump-diffusios). I a recet work, [15] show that uder more geeral volatility dyamics, the raw plug-i estimator carries high-order asymptotic biases that are ot egligible i the secod-order asymptotics. [15] also propose a bias-correctio procedure ad show the bias-corrected estimator admits a (feasible) cetral limit theorem. Moreover, this estimator attais the semiparametric efficiecy boud established by [9] ad [25]. The theory of [15] is further exteded by [2] to allow for broader classes of test fuctios (i.e., g) via a spatial-localizatio techique. Although the aforemetioed bias-corrected estimator is kow i explicit form, implemetig i empirical applicatios ca be fairly cumbersome. The mai reaso is that the correctio term ivolves the secod partial derivatives of the test fuctio g( ) with respect to all elemets of the spot covariace matrix. Cosequetly, to implemet the bias correctio, the empirical researcher typically eeds to prepare a large umber of aalytical formulas of secod partial derivatives. Although such calculatios are feasible i priciple, they are quite costly i empirical research, especially whe the researcher may experimet with various choices of g( ) i search of a good specificatio. Such a cost may already be substatial eve for problems with moderate dimesios. For example, [21] study a volatility-spaig problem with d = 1, i which the bias-correctio term ivolves 99 distict secod partial derivatives for various highly oliear fuctios with 45 argumets (i.e., the umber of distict elemets i the spot covariace matrix). I geeral, the computatioal complexity grows at the rate of O(d 4 ). The task of bias correctio ca further be complicated whe the fuctio g( ) itself is ot kow i a aalytical form, but is calculated usig rather otrivial umerical procedures. For example, [22] demostrate itegrated volatility fuctioals ca be used as itegrated momet coditios for estimatig optio-pricig models with high-frequecy data. I such applicatios, g( ) ivolves a optio-pricig formula, which i tur eeds to be umerically evaluated by solvig ordiary differetial equatios ad Fourier trasforms (see, e.g., [1]). The computatioal complexity of evaluatig the secod partial derivatives of g( ) is two orders of magitude higher tha that of g( ) with lower umerical precisio, which reders a reliable implemetatio of the bias correctio very challegig.

3 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 3 Set agaist this backgroud, we propose a easy-to-implemet jackkife procedure for bias correctio, which avoids the estimatio of bias terms altogether. Jackkife methods have bee widely used i statistics (see, e.g., [11]). Here, we cosider two ucorrected estimators formed usig two differet sequeces of local widows i the estimatio of spot volatility. Because these estimators have similar bias terms but with differet loadigs, we ca elimiate their biases by formig a proper liear combiatio betwee them. We refer to the resultig estimator as the two-scale jackkife estimator. We show this estimator is asymptotically cetered mixed Gaussia. Importatly, its coditioal asymptotic variace is the same as that of the estimator i [15], ad hece the jackkife estimator is also semi-parametrically efficiet. To further facilitate iferece i practice, we propose a easy-to-implemet ad cosistet estimator for the asymptotic variace via a simulatio techique. Our motivatio is that existig estimators of the asymptotic variace ivolve the first partial derivatives of g( ) ad hece ca potetially be cumbersome to implemet i practice for reasos metioed above. I fact, our estimator is simply the sample variace of ucorrected estimators formed usig simulated data. This estimator does ot require aalytical or umerical evaluatio of the derivatives of g( ), ad the simulatio ca easily be parallelized. I a recet paper, [23] proposed a geeral approach for estimatig asymptotic variaces via the observed asymptotic variace that is formed as the quadratic variatio of miiature versios of the origial (full-sample) estimator. Our simulatio-based estimator for the asymptotic variace is coceptually differet from that of [23] because the former is costructed by re-computig the origial full-sample estimator for simulated data. I particular, we do ot eed to divide the sample ito blocks for estimatig miiature versios of the itegrated volatility fuctioal. That said, both estimators share the advatage that the user does ot have to kow the form of the asymptotic variace, which is clearly coveiet i practice. Hece, our method provides a useful complemet to that of [23] i empirical research. We ote that both the bias-corrected estimator of [15] ad the two-scale jackkife estimator are desiged to correct the oliearity bias that arises from the (squared) statistical error i the oparametric estimatio of spot volatility, for which the asymptotic justificatio relies o udersmoothig. That is, the local widow size for the spot volatility estimatio is relatively small, so that other biases due to the boudary effect, volatility of volatility, ad volatility jumps become asymptotically egligible. By cotrast, [16] cosider a specific choice of the local widow sequece i which all sources of biases are balaced at the same order, ad propose explicit correctio for

4 4 JIA LI, YUNXIAO LIU AND DACHENG XIU each of them. I the same vei, we show this complete bias correctio ca also be achieved via a multiscale jackkife estimator formed as a liear combiatio of three (or more) ucorrected estimators. The uderlyig idea is, agai, to cacel the biases usig ucorrected estimators without estimatig the biases explicitly. Theoretically, we show the multiscale jackkife estimator admits the same cetral limit theorem while allowig for a broad rage of growth behavior of the local widows. We show the multiscale bias correctio is sufficietly accurate (for obtaiig a cetral limit theorem) eve if all types of aforemetioed biases are explosive i each of the ucorrected estimators. I particular, the asymptotic behavior of the jackkife procedure is stable regardless of whether the oparametric volatility estimatio features udersmoothig or oversmoothig. This stability provides a theoretically guarateed robustess for the multiscale jackkife method. Our aalysis is limited to the settig i which the high-frequecy data are observed without oise. It is well kow that fiacial data at the ultra high frequecy are cotamiated by microstructure oise, which would lead to otrivial bias i the spot volatility estimates. I the multivariate settig, samplig asychroicity amog the uderlyig processes leads to aother type of bias i the estimatio of covariaces. Therefore, followig stadard practice, our method is maily applicable to data that are sparsely sampled, for which the effect of oise ad/or asychroicity is mild. A large ad growig literature exists o the estimatio of itegrated variace ad covariace matrix for oisy irregularly sampled data; see, e.g., [28], [4], [13], [12], [1], [27], [5], [7], ad [26]. The efficiecy problem i this cotext has bee addressed by [24] usig the equivalece-of-experimet approach. Extedig such results to the case of efficiet estimatio of geeral itegrated volatility fuctioals is a rather otrivial task that is beyod the scope of the curret paper. This paper is orgaized as follows. Sectio 2 presets the settig. Sectio 3 cotais our mai results. Sectio 4 reports simulatio results. Sectio 5 cocludes. The supplemetal appedix i [19] cotais all proofs. 2. The settig The uderlyig processes. We cosider a R d -valued process (X t ) t defied o a filtered probability space (Ω, F, (F t ) t, P). We observe X i for i =,..., T/ over a fixed time iterval [, T ], where asymptotically. Below, for ay variable Y t, we deote its jth compoet by Y (j) t ; the same covetio also applies to matrix- ad tesor-valued variables. We deote i X X i X (i 1), i 1.

5 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 5 Our basic assumptio is that X is a Itô semimartigale (see, e.g., Sectio i [14]) with the followig form: t t t (2.1) X t = x + b s ds + σ s dw s + δ (s, z) µ (ds, dz), where W is a d -dimesioal Browia motio ad µ is a Poisso radom measure o R + E for some auxiliary Polish space E with compesator ν(dt, dz) = dt λ (dz) for some σ-fiite measure λ (dz). The stochastic volatility process σ t takes values i R d d. We deote the spot covariace matrix c t σ t σt, which takes value i the space M d of d-dimesioal positive semidefiite matrices. We suppose c t is also a Itô semimartigale with the form (2.2) c t = c t t t E E t bs ds + σ s dw s δ (s, z) 1 { δ(s,z) 1} (µ ν) (ds, dz) δ (s, z) 1 { δ(s,z) >1} µ (ds, dz), where we use a matrix otatio by defiig matrix-valued processes compoet by compoet. I particular, the (j, k) compoet of t σ sdw s is give by d t l=1 σ(jkl) s dw s (l). We ow collect some regularity coditios for the process X. Assumptio 1. The followig coditios hold for some costat r [, 1). There are a sequece (J m ) m 1 of oegative bouded λ-itegrable fuctios o E ad a sequece (τ m ) m 1 of stoppig times icreasig to, such that δ (t, ) r 1 J m ( ) ad δ (t, ) 2 1 J m ( ) o {t τ m }. The processes b t, b t ad σ t are càdlàg adapted. Assumptio 1 is fairly stadard for aalyzig the asymptotic properties of estimators formed usig high-frequecy data; see [14] for may examples. I particular, the costat r serves as a boud for the activity of jumps i X. The restrictio r < 1 is eeded for derivig cetral limit theorems for jump-robust estimators of volatility fuctioals. The primary iterest of the curret paper is o the efficiet estimatio of itegrated volatility fuctioals of the form (2.3) S (g) T g (c s ) ds, R

6 6 JIA LI, YUNXIAO LIU AND DACHENG XIU where g( ) : M d R k is a three-time cotiuously differetiable fuctio. For example, i a bivariate case, we take g(c) = c (12) /c (11), c (12) / c (11) c (22), ad c (22) (c (12) ) 2 /c (11) for estimatig the itegrated beta, correlatio, ad idiosycratic variace, respectively. More complicated trasformatios arise i oparametric specificatio tests for the covariace process ([21]) ad the estimatio of ecoomic models ([22]), i which the form of g( ) is determied by the scietific model uder ivestigatio The ucorrected estimator ad its high-order biases. We ow proceed to itroducig the ucorrected estimator, which is the buildig block of the jackkife estimator we propose below. The ucorrected estimator of S (g) is formed by replacig the latet covariace matrix process c with its oparametric estimate. To this ed, we choose a sequece k of local widows ad associate it with the followig spot covariace matrix estimator: for N T/ k + 1, (2.4) ĉ i (k ) 1 k 1 k i+jx i+jx 1 { i+j X u }, 1 i N, j= where u is a thresholdig sequece for elimiatig jumps i X that satisfies u ϖ, for some ϖ (, 1/2). Our otatio ĉ i (k ) emphasizes the depedece of this estimator o the local widow sequece k. The ucorrected estimator for S (g) is the costructed as its sample aalogue: N S (g; k ) g(ĉ i (k )). The asymptotic behavior of this ucorrected estimator is fairly complicated ad depeds crucially o the growth rate of k. The optimal rate for estimatig the spot volatility is kow to be attaied with k 1/2. Hece, a (seemigly) atural choice of k is such that i=1 k θ 1/2 for some θ (, ). Uder this rate coditio, [15] characterized the asymptotic behavior of the ucorrected estimator. To build ituitio for our later discussios, briefly recallig this result is istructive. We cosider the special case with d = d = 1 for simplicity. Below, we use L-s to deote stable covergece i law,

7 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 7 which meas the covergece i law is joit with ay bouded F-measurable radom variables. Theorem 3.1 i [15] shows 1/2 (S (g; k ) S (g)) L-s Z + B 1 (θ) + B 2 (θ), where Z is a F-coditioally cetered Gaussia variable with coditioal variace Σ (g) give by (2.5) Σ (g) 2 T g (c s ) 2 c 2 sds, ad B 1 (θ) ad B 2 (θ) are bias terms give by (2.6) T B 1 (θ) 1 2 g (c s ) c 2 θ sds, }{{} oliearity bias B 2 (θ) θ 2 (g (c ) + g (c T )) }{{} edge effect T θ 2 g (c s ) σ 12 sds 2 }{{} bias due to diffusive movemets i c 1 +θ <s T (g (c s + w c s ) (1 w)g (c s ) wg (c s )) dw. }{{} bias due to jumps i c We ote that, except for the edge effect, the bias terms show i (2.6) are preset oly whe the test fuctio g( ) is oliear. This result shows a importat qualitative departure of our aalysis o geeral volatility fuctioals from the baselie problem of estimatig itegrated variace-covariace matrices, as the latter correspods to g( ) beig the (liear) idetity fuctio. Explicitly de-biasig B 2 (θ) is clearly very difficult because it depeds o the volatility of volatility σ ad volatility jumps c, which ivolve a layer of latecy i additio to the latet volatility process. A simple ad elegat solutio proposed by [15] is to elimiate B 2 (θ) asymptotically via udersmoothig (i.e., k 1/2 ). As a result, the oliearity bias term B 1 (θ) becomes explosive ad eeds correctio. [15] proposed correctig B 1 (θ) via its sample aalogue: (2.7) B 1, (g) 1/2 k N 2 g (ĉ i (k )) ĉ i (k ) 2, i=1

8 8 JIA LI, YUNXIAO LIU AND DACHENG XIU ad showed that (2.8) 1/2 (S (g; k ) S (g)) B 1, (g) L-s Z. Furthermore, the asymptotic variace Σ (g) ca be cosistetly estimated via a plug-i estimator, which ca be used for coductig feasible iferece. This bias-corrected estimator has geerated much empirical iterest i the recet literature ([2], [17], [21], [22]). However, as metioed i the itroductio, estimatig the bias-correctio term ad the asymptotic variace ca be cumbersome i may empirically iterestig scearios, because of the large umber of derivatives to be calculated. The implemetatio becomes eve more complicated if oe also wats to estimate ad correct for various bias terms i B 2 (θ) as cosidered i [16]. We propose a simple-to-implemet jackkife method to address these issues, to which we ow tur. 3. Mai results Two-scale jackkife estimatio. We first itroduce the two-scale jackkife estimator. This estimator elimiates oly the oliearity bias ad hece ca be cosidered a couterpart of the estimator of [15]. The key idea uderlyig our costructio is to use ucorrected estimators with differet local widows. To this ed, we cosider two sequeces of local widows (k 1,, k 2, ) ad weights (ψ 1, ψ 2 ) such that (3.1) 2 ψ q = 1, q=1 2 q=1 ( ψ q kq, 1 = o 1/2 ). For example, with k 1, = k ad k 2, = 2k, the above coditio is satisfied with ψ 1 = 1 ad ψ 2 = 2. The two-scale jackkife estimator is the give by T S (g) = ψ 1 S (g; k 1, ) + ψ 2 S (g; k 2, ). The idea uderlyig this costructio is quite ituitive. Ideed, we observe from (2.6) that the oliearity bias B 1 (θ) is proportioal to 1/θ (hece 1/k ). The secod coditio i (3.1) thus esures the bias terms i the two ucorrected estimators cacel each other out up to desired precisio. Theorem 1, below, describes the asymptotic property of the two-scale estimator T S (g). Like [15], we impose the followig udersmoothig coditio (i.e., ς < 1/2) o the local widow sequeces. Assumptio 2. For q = 1, 2, k q, ς for some ς ( r 2 1 3, 1 2 ).

9 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 9 We also eed some smoothess coditio for the fuctio g( ) coupled with some mild pathwise regularity for the spot volatility process. Below, for a compact set K M d ad ε >, we deote the ε-elargemet about K by K ε {M M d : if M A < ε}. A K Assumptio 3. There exist a sequece of stoppig times (τ m ) m 1 icreasig to ifiity ad a sequece of covex compact subsets K m M d, m 1, such that c t K m for t τ m ad g is three-time cotiuously differetiable o Km ε for some ε >. Assumptio 3 requires the localized process c t τm to be compactly valued, ad g( ) is C 3 o a slight elargemet of this compact support. Importatly, we do ot require g( ) to satisfy the polyomial growth coditio as i [15], which is ofte violated i ecoomic ad fiacial applicatios. This geerality is achieved by usig a spatial localizatio argumet as i [2], [21], ad [22]. The ituitio uderlyig the spatial localizatio argumet is as follows. We first observe that the spot covariace estimates uiformly approximate the local averages of the true spot covariaces over the correspodig time itervals. Hece, with probability approachig 1, these estimates fall i a compact set that is slightly larger tha the (covex) set i which the spot covariace process takes values. As a result, for all our probabilistic calculatios, we ca restrict the test fuctio g( ) locally o a compact set, which avoids restrictios o the global growth rate of g( ). Theorem 1. The Suppose Assumptios 1, 2, ad 3 hold ad ϖ [ 1 ς 2 r, 1 2 ). 1/2 (T S (g) S (g)) L-s MN (, Σ (g)), where MN deotes the mixed ormal distributio ad d T Σ (g) jk g (c s ) lm g (c s ) (c (jl) s c (km) s + c (jm) s j,k,l,m=1 c (kl) s Theorem 1 shows the two-scale jackkife estimator T S (g) is a 1/2 - cosistet estimator of S(g). After ormalizatio, T S admits a cetral limit theorem with F-coditioal asymptotic covariace matrix Σ (g), which attais the efficiecy boud established i [9] ad [25]. To improve the fiite-sample performace, [15] suggest removig the boudary effect by adjustig the ucorrected estimator as S (g; k ) S (g; k ) + k 2 ( g (ĉ 1 (k )) + g ( ĉ N (k ) )). )ds.

10 1 JIA LI, YUNXIAO LIU AND DACHENG XIU Likewise, we ca adjust the jackkife estimator as T S (g) ψ 1 S (g; k 1, ) + ψ 2 S (g; k 2, ). Because 1/2 (S (g; k ) S (g; k )) = O p (k 1/2 ), which vaishes asymptotically i the curret case with udersmoothig, these fiite-sample adjustmets do ot result i ay chage for Theorem Cosistet estimatio of Σ (g) via simulatio. We eed a cosistet estimator for Σ (g) so as to coduct feasible iferece based o Theorem 1. A atural choice is the plug-i estimator Σ (g) give by d j,k,l,m=1 N i=1 jk g (ĉ i ) lm g (ĉ i ) (ĉ,(jl) i ĉ,(km) i + ĉ,(jm) i ĉ,(kl) i ), where ĉ i = ĉ i (k ) ad k satisfies k ad k ; see, for example, Theorem i [14]. As metioed i the Itroductio, this plug-i estimator ca be cumbersome to implemet because it requires calculatios of the partial derivatives of g. For this reaso, we propose a simulatio-based estimator for the asymptotic covariace matrix that avoids the calculatio of g altogether. Algorithm 1, below, describes this estimator, which we deote by Σ (g). Algorithm 1. The procedure comprises of four steps: 1. Estimate ĉ i (k ) for some k. 2. For each block b {1,..., T/k }, simulate ( (b 1)k +i X ) 1 i k as i.i.d. draws from N (, ĉ (b 1)k +1 ). 3. Compute S (g) as T/k k b=1 g 1 k k 1 j= ( ( (b 1)k+1+j X ) (b 1)k+1+j X ). 4. Repeat Steps 2 ad 3 for a large umber of times. Compute Σ (g) as the sample covariace matrix of the simulated 1/2 S (g). Implemetig Algorithm 1 is easy because it oly requires recomputig the ucorrected estimator usig simulated returs. We ote that to implemet this algorithm, oe oly eeds to re-estimate the spot covariace matrix i each simulatio for T/k ooverlappig blocks (istead of for N overlappig widows). Theorem 2, below, shows Σ (g) is a cosistet estimator of Σ (g).

11 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 11 Theorem 2. Suppose Assumptios 1 ad 3 hold, ϖ [ 1 ς 2 r, 1 2 ), k ad k. The Σ P (g) Σ (g) Multiscale jackkife estimatio ad its robustess property. The twoscale jackkife estimator ad the bias-corrected estimator of [15], implicitly or explicitly, correct the oliearity bias (i.e., the B 1 (θ) term i (2.6)). However, other sources of biases (i.e., the B 2 (θ) term i (2.6)), icludig those drive by volatility-of-volatility ad volatility jumps, are elimiated essetially by makig the udersmoothig assumptio. Although this type of theoretical argumet is quite commo i oparametric statistics, these bias terms may still have a otrivial effect i fiite samples. This effect should be relatively large durig sample periods with large fluctuatios i volatility, such as crisis ad macro ews-aoucemet times. Such periods are of importat empirical iterest i ecoomic applicatios because the sizable variatio i volatility helps i idetifyig ad testig ecoomic models. Motivated by this cocer, we cosider ext a complete bias correctio that elimiates all bias terms idetified i (2.6). Alog this lie, [16] made a iterestig cotributio by costructig a estimator for each bias term ad the correctig the biases explicitly. However, implemetig the estimator of [16] is difficult i practice because the (direct) estimatio of volatility-ofvolatility ad volatility jumps are otoriously difficult i fiite samples. As a result, the udersmoothig approach has bee preferred i ecoomic ad fiacial applicatios; see, for example, [2], [17], [2], [21], ad [22]. We propose a easy-to-implemet alterative by usig a multiscale jackkife estimator that is formed as a liear combiatio of three (or more) ucorrected estimators associated with differet local widows. To this ed, we cosider local widows k q, ad weights ψ q, 1 q Q, such that (3.2) Q ψ q = 1, q=1 Q q=1 ( ψ q kq, 1 = o 1/2 ), Q q=1 ( ψ q k q, = o For example, whe Q = 3, k j, = a j k, the weights are solved as ψ ψ 2 = a 1 a 2 a 3. ψ 3 1/a 1 1/a 2 1/a 3 The multiscale jackkife estimator is the give by MS (g) Q ψ q S (g; k q, ). q=1 1/2 ).

12 12 JIA LI, YUNXIAO LIU AND DACHENG XIU Compared with the weights i the two-scale jackkife (recall (3.1)), the weights i the multiscale costructio (3.2) satisfy a additioal coditio Q q=1 ψ qk q, = o( 1/2 ). This coditio is used to esure bias terms such as B 2 (θ) (which is proportioal to θ ad k ) cacel each other out i the liear combiatio of ucorrected estimators. Theorem 3, below, describes the asymptotic behavior of the multiscale jackkife estimator MS (g), for which we eed the followig coditio o the local widows. Assumptio 4. For all 1 q Q, k q, ς for some ς ( r 2 1 3, 2 3 ). Theorem 3. Suppose Assumptios 1, 3, 4, ad ϖ [ 1 ς 2 r, 1 2 ). The 1/2 (MS (g) S (g)) L-s MN (, Σ (g)), where Σ (g) is defied as i Theorem 1. Theorem 3 shows the multiscale jackkife estimator has the same asymptotic distributio as its two-scale couterpart show i Theorem 1. Hece, the multiscale estimator is also semi-parametrically efficiet ad its asymptotic variace ca be estimated usig the method described i Sectio 3.2. Note that Theorem 3 holds uder Assumptio 4, which is much weaker tha Assumptio 2 that is used i Theorem 1. The key differece is that Theorem 3 ot oly holds with udersmoothig (i.e., k 1/2 ), but also holds with oversmoothig (i.e., k 1/2 ). I the latter case, the two-scale jackkife estimator ad the bias-corrected estimator of [15] do ot admit a cetral limit theorem, because the biases due to the boudary effect, volatility-of-volatility, ad volatility jumps are o loger asymptotically egligible (these bias terms are actually explosive). But such biases are implicitly corrected via the multiscale jackkife, which makes the resultig estimator MS (g) much less sesitive to the growth rate of local widows. This result thus provides a well-defied sese of robustess for the multiscale jackkife estimator. We ca further compare Theorem 3 with the result of [16]. [16] also cosider correctig all bias terms idetified i (2.6) by explicitly estimatig each of them. These authors proved a cetral limit theorem for their estimator uder a specific choice of the local widow, amely, k 1/2. With this choice, the bias terms are balaced ad have the same order as the term that drives the cetral limit theorem. Theorem 3, however, does ot require the local widows to grow at this particular rate, but allows them to exhibit a broad rage of asymptotic behavior. Importatly, it shows the multiscale

13 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 13 jackkife bias correctio is effective eve if the biases are explosive. From a techical poit of view, to establish such results, we eed to use refied techiques to aalyze the biases ad their correctio, which have ot bee see i prior work Discussio o issues related to microstructure oise ad samplig asychroicity. The estimators proposed above rely o regularly sampled data i a oise-free settig. However, fiacial trasactio data are ofte cotamiated with microstructure oise ad are sampled irregularly i time. These complicatios will result i biases i the spot covariace estimates ad, subsequetly, biases i the estimate of itegrated volatility fuctioals. I order to mitigate these microstructural effects, a stadard practice is to apply the estimator to sparsely sampled data. By doig so, oe reduces the relative effect of microstructure oise ad asychroous samplig by icreasig the sigal level cotaied i the efficiet price icremets. As a result, the spot covariace estimates based o sparsely sampled oisy data are closer to those formed usig the efficiet price. I this subsectio, we discuss sufficiet coditios that justify the use of the sparse samplig method for makig iferece about itegrated volatility fuctioals (although these coditios ca likely be further improved upo). To fix ideas, let ˆv i (k ) deote the spot covariace estimate formed usig sparsely sampled oisy data. As i previous sectios, we use ĉ i (k ) to deote the spot covariace estimate costructed from the efficiet price. For simplicity, we refer to ˆv i (k ) ad ĉ i (k ) as the oisy ad the oise-free estimates, respectively. I view of the above heuristics cocerig sparse samplig, we assume that the oisy estimate ˆv i (k ) cosistetly approximates its oise-free couterpart uiformly: sup ˆv i (k ) ĉ i (k ) = o p (1). 1 i N This coditio is relatively mild because it does ot require ay specific rate of covergece. Like i the proof of our mai theorems above, this uiform approximatio coditio allows us to ivoke the spatial localizatio argumet so that we ca suppose without loss of geerality that g( ) is compactly supported ad, hece, Lipschitz cotiuous. I order to show that microstructural complicatios have egligible effect for iferece, we eed the differece betwee ˆv i (k ) ad ĉ i (k ) to be sufficietly small o average. More precisely, we cosider the followig high-level

14 14 JIA LI, YUNXIAO LIU AND DACHENG XIU coditio: for some a, N (3.3) ˆv i (k ) ĉ i (k ) = O p (a ), i=1 where a depicts the rate of covergece. The microstructural effect o the ucorrected estimator for the itegrated volatility fuctioal ca the be bouded as N N g(ˆv i (k )) g(ĉ i (k )) i=1 i=1 N K ˆv i (k ) ĉ i (k ) = O p (a ). i=1 Sice the jackkife estimators are formed as liear combiatios of ucorrected estimators, the microstructural effect is asymptotically egligible for derivig their cetral limit theorems if (3.4) a = o( 1/2 ). This coditio ca be further iterpreted i terms of the extet to which the observatio oise i the price ad/or the degree of asychroicity should be small for sparsely sampled data. We start with the case i which the price is observed with oise: istead observig the efficiet price X i, we ow observe Y i = X i + ε,i where the oise terms ε,i are mutually idepedet ad idepedet of X. Suppose that the oise terms are small i the sese that their stadard deviatio is uiformly bouded by some sequece ã. We the observe that k ˆv i (k ) ĉ 2 i (k ) k i+jx(ε,i+j ε,i+j 1 ) + 1 k j=1 k j=1 ε,i+j ε,i+j 1 2. It is easy to see that the two terms o the majorat side of the above estimate are of orders O p (ã / k ) ad O p (ã 2 / ), respectively, which yields a = max {ã / } k, ã 2 /. For the multiscale jackkife estimator, we ca take k 1/2. As a result, a sufficiet coditio for (3.4) is ã = o( 3/4 ).

15 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 15 Turig to the case with asychroous samplig, we measure the degree of asychroicity as the proportio of mismatch betwee the irregular samplig scheme ad the regular samplig scheme, which we deote by b. For example, if the miimal distace (e.g., measured by Hausdorff distace) betwee the set of (irregularly spaced) trasactio times ad the regular 5- miute samplig grid is less tha 1 secod o average, the b is bouded by 1/3. Measurig the degree of asychroicity as such, it is easy to see that a = O( b ). Hece, b = o( 1/2 ) is a sufficiet coditio for the asymptotic egligibility of asychroous samplig i the cotext of the curret paper. 4. Mote Carlo. I this sectio, we examie the fiite-sample performace of our estimators i Mote Carlo simulatios. We also compare them with the bias-corrected estimators of [15], [16], ad [18]. To this ed, we focus o the estimatio of itegrated quarticity i a uivariate settig (i.e., g(c) = c 2 ) for which the bias-correctio terms of the completely bias-corrected estimator of [16] were give i closed-form. Below, we suppress the depedece o g i our otatio for simplicity. Throughout, we fix T = 21 days ad cosider two samplig frequecies: = 1 or 5 miutes. We coduct 1, Mote Carlo trials i total. We simulate X ad its volatility process σ accordig to (4.1) { dxt = σ t dw t + djt X, σ t = exp ( F t ), df t = 5F t dt + 2d W t + djt F, with E[dW t d W t ] =.75dt. We simulate Jt X as a compesated Poisso jump process with jump-size distributio N (.1,.2 2 ) ad itesity λ = 36, ad simulate Jt F as a compesated tempered-stable process (or CGMY process i [8]) with the Lévy jump measure give by (4.2) ν(x) = α x 1+β e γ x 1 {x<} + α x 1+β e γ +x 1 {x>}, where γ + = 3, γ = 5, β =.8, ad α = 4.5. The percetage of expected quadratic variatio of F t due to its jumps is about 3%. I this settig, the volatility of volatility i (2.2) is give by σ t = 4c t, ad the bias terms B 1 (θ) ad B 2 (θ) i (2.6) ca be calculated explicitly as B 1 (θ) = 2 θ T c 2 sds, B 2 (θ) = θ 2 (c2 + c 2 T ) θ 6 T σ sds 2 θ ( c s ) 2. 6 s T

16 16 JIA LI, YUNXIAO LIU AND DACHENG XIU We cosider several estimators for the itegrated quarticity, icludig the two-scale jackkife estimator T S ad its boudary-adjusted versio T S, the multiscale jackkife estimator MS, the udersmoothig estimator JR (1) S 1/2 B 1, proposed by [15] ad its boudary-adjusted versio JR (2) S 1/2 B 1,, the estimator with complete bias correctio proposed by [16]: that is, JR (3) S 1/2 (B 1, + B 2, ), where B 2, k 1/2 2 1/2 4 (ĉ 1 (k ) 2 + ĉ N (k ) 2 ) + 1/2 k N k i=1 (ĉ i+k (k ) ĉ i (k )) 2. N ĉ i (k ) 2 I additio, we cosider the kerel-based estimator of [18] give by [T/ ] K i=h +1 g(ˆk i (h )), where, writig K h ( ) K( /h)/h for the kerel fuctio K(z) = 6(1 + 3z + 2z 2 )1 { 1 z } ad badwidth h, ˆk i (h ) i=1 i K h ((j i) )( j X) 2 1 { j X u }. j=1 Ulike the other estimators above, K does ot ivolve bias-correctio. As is typical i covetioal oparametric statistics, this estimator was desiged for settigs where the volatility path is smooth, but this is ot the case i our stochastic volatility model. Nevertheless, we iclude this estimator i our compariso for completeess. The asymptotic variace for all estimators is Σ = 8 T c4 sds, which we estimate usig Algorithm 1 with 1, simulated samples. I Table 1, we fid the multiscale estimator MS achieves the smallest relative bias for each samplig frequecy: its bias is moderately smaller tha that of JR (3) (which also corrects all sources of biases), ad is substatially smaller tha those of the other estimators. This fidig suggests both MS ad JR (3) are effective for bias correctio, with the former performig better tha the latter. That said, the multiscale estimator has a slightly larger relative root-mea-square error tha JR (3), which reflects a bias-variace tradeoff i fiite samples. Geerally speakig, the jackkife estimators perform

17 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 17 = 5 miutes = 1 miute BIAS(%) RMSE(%) BIAS(%) RMSE(%) Jackkife Estimators T S T S MS Estimators with Explicit Bias Correctio JR (1) JR (2) JR (3) Kerel-based Estimator K Table 1 I this table, we report the relative biases ad relative root-mea-square-errors (i percetage uit) for seve estimators. For the multiscale estimator, we choose (ψ 1, ψ 2, ψ 3) = ( 2.5, 8, 4.5), ad correspodigly (k 1,, k 2,, k 3,) = (15, 3, 45) for data sampled every 5 miutes, or (4, 8, 12) for data sampled every miute. For the two-scale estimator, we choose (ψ 1, ψ 2) = ( 1, 2) alog with the same (k 1,, k 2,) above. For JR estimators, we choose k = 3 for 5-miute data, ad 8 for 1-miute data, respectively. For the kerel-based estimator, we choose h = 3k for both 5-miute ad 1-miute data, which miimizes approximately the RMSE amog various choices ex post. We fix T = 1 moth.

18 18 JIA LI, YUNXIAO LIU AND DACHENG XIU TS JR (1) TS JR (2) MS JR (3) Fig 1. I this figure, we compare the histograms of the studetized statistics with the stadard ormal desity. The estimators are implemeted as described i Table 1. The asymptotic variace is estimated usig Algorithm 1 with 1, simulated samples, for which we set k = 39 for 5-miute data, ad k = 78 for 1-miute data. similarly to the bias-corrected estimators of [15] ad [16], ad outperform the kerel-based estimator. Note the jackkife method does ot ivolve ay explicit estimatio of the biases, which would typically require substatial effort i practical implemetatio. Figure 1 compares the histograms of these estimators studetized by the estimated asymptotic stadard error. We observe that the estimators that rely o udersmoothig, amely, T S ad JR (1), are ot well cetered. This issue is evidetly mitigated after adjustig for the boudary effect, as show by the histograms of T S ad JR (2), ad further improvemet is attaied by MS ad JR (3). I particular, the histograms of studetized versios of MS ad JR (3) match the stadard ormal desity very well, suggestig the feasible cetral limit theorem works well i fiite samples for these estimators. To evaluate the impact of microstructure oise o these estimators, we add a i.i.d. Gaussia oise to each observed price i our sample. The stadard deviatio of the oise is 1 4, which is realistically calibrated. The results are reported i Table 2. Compared with the results i Table 1, the impact of the oise is somewhat oticeable at a 1-miute frequecy but is less clear for

19 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 19 = 5 miutes = 1 miute BIAS(%) RMSE(%) BIAS(%) RMSE(%) Jackkife Estimators T S T S MS Estimators with Explicit Bias Correctio JR (1) JR (2) JR (3) Kerel-Based Estimator K Table 2 I this table, we report the relative biases ad relative root-mea-square errors (i percetage uit) for seve estimators. The settig is idetical to that of Table 1, except that we add i.i.d. Gaussia oises to the observed log prices. a 5-miute frequecy. This fidig is ot surprisig, give that subsamplig is the stadard ad most widely used procedure i practice to deal with the microstructure oise, ad that the sigal-to-oise ratio icreases as samplig becomes more sparse. All estimators cosidered above ivolve a choice of local widows. I practice, the widow size should ot be chose too small so as to have eough aggregatio withi each local widow, ad should ot be take too big so that the time-variatio of various uderlyig processes remai moderate. Data-drive choices of local widows, whe available, ofte deped o user-specified loss fuctios ad ukow latet quatities that are hard to estimate. Give this difficulty, it is advisable to experimet with differet local widow sizes as a robustess check, to which we ow tur. I Table 3, we demostrate the robustess of our multiscale estimator agaist differet choices of local widows (i.e., k q, ). Recall that i Table 1, we choose k 2, = 3 ad 8 for 5-miute ad 1-miute data, respectively, while fixig k q, = a q k 1, for (a 1, a 2, a 3 ) = (1, 2, 3). Here, we use the same values of the k 2, s as before, but choose wider rages of (a 1, a 2, a 3 ): (1, 3, 9) for 5-miute data ad (1, 4, 12) for 1-miute data. I the case with 1-miute samplig, for example, the jackkife estimator ow ivolves widow sizes k 1, = 2 ad k 3, = 24, which clearly spa a wide rage i practical terms.

20 2 JIA LI, YUNXIAO LIU AND DACHENG XIU = 5 miutes = 1 miute BIAS(%) RMSE(%) BIAS(%) RMSE(%) Jackkife Estimators T S T S MS Estimators with Explicit Bias Correctio k = 1 k = k = 9 k = JR (1) JR (2) JR (3) JR (1) JR (2) JR (3) Table 3 I this table, we report the relative biases ad relative root-mea-square errors (i percetage uit) for six estimators. The settig is idetical to that of Table 1, except that we choose differet choices of badwidths. For the multiscale estimator, we choose (k 1,, k 2,, k 3,) = (1, 3, 9) for data sampled every 5 miutes, or (2, 8, 24) for data sampled every miute. For JR estimators, we choose k = 1 ad 9 for 5-miute data, ad 2 ad 24 for 1-miute data, respectively.

21 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 21 This robustess check thus poses a otrivial challege for the proposed estimators. For compariso, we also report results for the JR estimates correspodig to k = k 1, ad k 3,. From Table 3 we see that the performace of the multiscale estimator is isesitive to these alterative choices of k q,, but the performace of the JR estimators deteriorates. I particular, the bechmark estimator JR (1), which relies o udersmoothig, shows large bias whe the local widow size is large. I cotrast, eve though the multiscale estimator depeds o the same choice of local widows, the bias terms from its buildig blocks are effectively caceled out so that its performace is more robust to the choice of widow size. Overall, the Mote Carlo evidece cofirms the effectiveess for biascorrectio of the multiscale jackkife ad the two-scale jackkife with a boudary adjustmet. I additio, we fid the stadard error geerated by Algorithm 1 captures the samplig variabilities of these bias-corrected estimators well. Take together, the proposed iferece procedure shows good performace i fiite samples ad should provide a easy-to-implemet alterative to existig methods. 5. Coclusio. We propose jackkife estimators for efficietly estimatig geeral itegrated volatility fuctioals. Such fuctioals are broadly useful as risk measures or momet coditios i the estimatio of scietific models. The proposed jackkife estimator, alog with a simulatio-based estimator for its asymptotic variace, is easy to implemet i practice. I particular, the cumbersome task of calculatig (a large umber of) partial derivatives is completely avoided. We show the jackkife estimator is semiparametrically efficiet ad exhibits a type of robustess with respect to the choice of local widows, i that the latter ca have both udersmoothig ad oversmoothig behaviors. Ackowledgemets. We would like to thak Tim Bollerslev, Peter Hase, Adrew Patto, Jeffrey Russell, ad George Tauche for helpful discussios. Li s research is partially supported by NSF grat SES Xiu s research is partially supported by the IBM Corporatio Faculty Scholar Fud at the Uiversity of Chicago Booth School of Busiess. Refereces. [1] Aït-Sahalia, Y., J. Fa, ad D. Xiu (21). High-frequecy covariace estimates with oisy ad asychroous data. Joural of the America Statistical Associatio 15, [2] Aït-Sahalia, Y. ad D. Xiu (215). Pricipal compoet aalysis of high frequecy data. Techical report.

22 22 JIA LI, YUNXIAO LIU AND DACHENG XIU [3] Aderse, T. G., T. Bollerslev, F. X. Diebold, ad P. Labys (23). Modelig ad forecastig realized volatility. Ecoometrica 71 (2), pp [4] Bardorff-Nielse, O. E., P. R. Hase, A. Lude, ad N. Shephard (28). Desigig realized kerels to measure the ex post variatio of equity prices i the presece of oise. Ecoometrica 76, [5] Bardorff-Nielse, O. E., P. R. Hase, A. Lude, ad N. Shephard (211). Multivariate realised kerels: Cosistet positive semi-defiite estimators of the covariatio of equity prices with oise ad o-sychroous tradig. Joural of Ecoometrics 162, [6] Bardorff-Nielse, O. E. ad N. Shephard (24). Ecoometric aalysis of realized covariatio: High frequecy based covariace, regressio, ad correlatio i fiacial ecoomics. Ecoometrica 72 (3), pp [7] Bibiger, M., N. Hautsch, P. Malec, ad M. Reiß (214). Estimatig the quadratic covariatio matrix from oisy observatios: Local method of momets ad efficiecy. The Aals of Statistics 42 (4), [8] Carr, P., H. Gema, D. B. Mada, ad M. Yor (22). The fie structure of asset returs: A empirical ivestigatio. Joural of Busiess 75, [9] Clémet, E., S. Delattre, ad A. Gloter (213). A ifiite dimesioal covolutio theorem with applicatios to the efficiet estimatio of the itegrated volatility. Stochastic Processes ad their Applicatios 123, [1] Duffie, D., J. Pa, ad K. J. Sigleto (2). Trasform aalysis ad asset pricig for affie jump-diffusios. Ecoometrica 68, [11] Efro, B. (1982). The jackkife, the bootstrap ad other resamplig plas, Volume 38 of CBMS-NSF Regioal Coferece Series i Applied Mathematics. Philadelphia, Pa.: Society for Idustrial ad Applied Mathematics (SIAM). [12] Hayashi, T. ad N. Yoshida (25). O covariace estimatio of o-sychroously observed diffusio processes. Beroulli 11, [13] Jacod, J., Y. Li, P. A. Myklad, M. Podolskij, ad M. Vetter (29). Microstructure oise i the cotiuous case: The pre-averagig approach. Stochastic Processes ad Their Applicatios 119, [14] Jacod, J. ad P. Protter (212). Discretizatio of Processes. Spriger. [15] Jacod, J. ad M. Rosebaum (213). Quarticity ad Other Fuctioals of Volatility: Efficiet Estimatio. Aals of Statistics 41 (3), [16] Jacod, J. ad M. Rosebaum (215). Estimatio of volatility fuctioals: the case of a widow. I P. Friz, J. Gatheral, A. Gulisashvili, A. Jacquier, ad J. Teichma (Eds.), Large Deviatios ad Asymptotic Methods i Fiace, Volume 11 of Spriger Proceedigs i Mathematics & Statistics. Spriger, Cham. [17] Kalia, I. ad D. Xiu (216). Noparametric estimatio of the leverage effect: A trade-off betwee robustess ad efficiecy. Joural of the America Statistical Associatio. [18] Kristese, D. (21). Noparametric filterig of the realized spot volatility: A kerelbased approach. Ecoometric Theory 26, [19] Li, J., Y. Liu, ad D. Xiu (217). Supplemet to efficiet estimatio of itegrated volatility fuctioals via multiscale jackkife. Techical report, Duke Uiversity ad Uiversity of Chicago. [2] Li, J., V. Todorov, ad G. Tauche (216a). Adaptive estimatio of cotiuous-time regressio models usig high-frequecy data. Joural of Ecoometrics, forthcomig. [21] Li, J., V. Todorov, ad G. Tauche (216b). Iferece theory o volatility fuctioal depedecies. Joural of Ecoometrics 193, [22] Li, J. ad D. Xiu (216). Geeralized method of itegrated momets for high-

23 JACKKNIFE ESTIMATION OF VOLATILITY FUNCTIONALS 23 frequecy data. Ecoometrica 84 (4), [23] Myklad, P. ad L. Zhag (217). Assessmet of ucertaity i high frequecy data: the observed asymptotic variace. Ecoometrica 85 (1), [24] Reiß, M. (211). Asymptotic equivalece for iferece o the volatility from oisy observatios. The Aals of Statistics 39 (2), pp [25] Reault, E., C. Sarisoy, ad B. J. Werker (216). Efficiet estimatio of itegrated volatility ad related processes. Ecoometric Theory. [26] Shephard, N. ad D. Xiu (217). Ecoometric aalysis of multivariate realized QML: Estimatio of the covariatio of equity prices uder asychroous tradig. Joural of Ecoometrics, forthcomig. [27] Zhag, L. (211). Estimatig covariatio: Epps effect ad microstructure oise. Joural of Ecoometrics 16, [28] Zhag, L., P. A. Myklad, ad Y. Aït-Sahalia (25). A tale of two time scales: Determiig itegrated volatility with oisy high-frequecy data. Joural of the America Statistical Associatio 1, Departmet of Ecoomics Duke Uiversity Durham, NC jl41@duke.edu Departmet of Statistics ad Operatios Research Uiversity of North Carolia at Chapel Hill Chapel Hill, NC yuxiao@live.uc.edu Booth School of Busiess Uiversity of Chicago Chicago, IL dacheg.xiu@chicagobooth.edu