Estimatio of a oisy subordiated Browia Motio via two-scales power variatios José E. Figueroa-López Departmet of Mathematics, Washigto Uiversity i St. Louis ad Kiseop Lee Departmet of Statistics, Purdue Uiversity May 8, 07 Abstract High frequecy based estimatio methods for a semiparametric pure-jump subordiated Browia motio exposed to a small additive microstructure oise are developed buildig o the two-scales realized variatios approach origially developed by Zhag et al. 005 for the estimatio of the itegrated variace of a cotiuous Itô process. The proposed estimators are show to be robust agaist the oise ad, surprisigly, to attai better rates of covergece tha their precursors, method of momet estimators, eve i the absece of microstructure oise. Our mai results give approximate optimal values for the umber K of regular sparse subsamples to be used, which is a importat tue-up parameter of the method. Fially, a data-drive plug-i procedure is devised to implemet the proposed estimators with the optimal K-value. The developed estimators exhibit superior performace as illustrated by Mote Carlo simulatios ad a real high-frequecy data applicatio. Keywords: Geometric Lévy Models; Kurtosis ad Volatility Estimatio; Power Variatio Estimators; Microstructure Noise; Robust Estimatio Methods. The first author s research is partially supported by the NSF grats DMS-969 ad DMS-6306.
Itroductio I this paper, we develop estimatio methods for a semiparametric subordiated Browia motio SBM, whose samplig observatios have bee cotamiated by a small additive oise alog the lies of the framework of Zhag et al. 005. I additio to a volatility parameter σ, which cotrols the variace of the icremets of the process at regular time itervals, a SBM is edowed with a additioal parameter, hereafter deoted by κ, which accouts for the tail heaviess of the icremets distributio. Therefore, κ determies the proeess of the process to produce extreme icremet observatios. Such a measure is clearly of critical relevace i may applicatios such as to model extreme evets i isurace ad risk maagemet ad optimal asset allocatio i fiace. The models cosidered here are pure-jump Lévy models ad σ is ot the volatility of a cotiuous Itô process. Nevertheless, give that σ is proportioal to the variace of the icremets of the process, it is atural to refer to σ as the volatility parameter of the model. As i the cotext of a regressio model, the additive oise, typically called microstructure oise, ca be see as a modelig artifact to accout for ay deviatios betwee the observed process ad the SBM model. However, i some circumstaces, the oise ca be lik to some specific physical mechaism such as i the case of bid/aks bouce effects i tick by tick tradig cf. Roll 98. At low frequecies the microstructure oise is typically egligible compared to the SBM s icremets, but at high-frequecies the oise is sigificat ad heavily tilts ay estimates that do ot accout for it. The aim is the to develop iferece methods that are robust agaist potetial microstructure oises. The literature of statistical estimatio methods uder microstructure oise has grow extesively durig the last decade. See Aït-Sahalia & Jacod 0 for a recet i depth survey o the topic ad, also, Aït-Sahalia et al. 005, Zhag et al. 005, Hase & Lude 006, Badi & Russell 008, Myklad & Zhag 0 for a few semial works i the area. Most of these works have focused o the estimatio of the itegrated variace of a semimartigale model. However, the problem of traslatig some of the proposed methods ito estimatio methods for semiparametric models cotamiated by additive oise, as it is the case i the preset work, has received much less attetio i the literature, i particular, whe it comes to the estimatio of a kurtosis type parameter. The performaces of some
classical parametric methods i the estimatio of some popular parametric Lévy models have bee aalyzed i a few works such as Seeta 00, Ramezai & Zeg 007, Behr & Pötter 009, ad Figueroa-López et al. 0, but oe of them have icorporated microstructure oise. To motivate our estimatio procedure, we start by cosiderig Method of Momet Estimators MME for σ ad κ, i the absece of microstructure oise. Throughout the remaider of the itroductio, these estimators are respectively deoted by ˆσ,T ad ˆκ,T, where ad T deote the umber of observatios ad the samplig horizo, respectively. MMEs ad related estimators are widely used i high-frequecy data aalysis due to their simplicity, computatioal efficiecy, ad kow robustess agaist potetial correlatio betwee observatios. I order to establish asymptotic bechmarks for the covergece rates of our proposed estimators, we characterize the asymptotic behavior of the MME estimators, both i the absece ad presece of microstructure oise, whe δ = T/, the time spa betwee observatios, shrik to 0 ifill asymptotics ad T log-ru asymptotics. We idetify the order OT, as the rate of covergece of the estimators uder the absece of oise. Hece, a desirable objective is to develop estimators that are able to achieve at least this rate of covergece i the presece of microstructure oise. A asymptotic aalysis of the estimators i the presece of oise allows to show that ˆσ,T ad ˆκ,T 0, as, both of which are stylized empirical properties of high-frequecy fiacial observatios see Sectio 5. below. Furthermore, it is show that δ ˆσ,T δ ˆκ,T coverge to the secod momet ad the excess kurtosis of the microstructure oise, respectively. I order to develop estimators that are robust agaist a microstructure oise compoet, we borrow ideas from Zhag et al. 005 s semial approach based o combiig the realized quadratic variatios at two-scales or frequecies. More cocretely, there are three mai steps i this approach. First, the high-frequecy samplig observatios are divided i K groups of observatios take at a lower frequecy sparse subsamplig. Secod, the relevat estimators say, realized quadratic variatios are applied to each group ad the resultig K poit estimates are averaged. Fially, a bias correctio step is ecessary for which oe typically uses the estimators at the highest possible frequecy. ad 3
A fudametal problem i the approach described i the previous paragraph is how to tue up the umber of subgroups, K, which strogly affects the performace of the estimators. We propose a method to fid approximate optimal values for K uder a white microstructure oise settig. For the estimator of σ, it is foud that the optimal K takes the form K σ := 3 6 Eε + Eε T σ 3,. where ε represets the additive microstructure oise associated to oe observatio of the SBM. Iterestigly, the optimal value. is cosistet, but differet from that proposed by Zhag et al. 005 i the cotext of a cotiuous Itô semimartigale. It is also foud that the mea-squared error MSE of the resultig estimator usig K as above attais a rate of covergece C σ Eε + Eε 3 3 T 3 up to a costat C σ, which, sice T/ 0, shows the surprisig fact that the estimator coverges at a rate of ot, which is faster tha the rate attaied by the MMEs i the absece of oise. For the estimatio of κ, it is foud that the optimal K takes the form Kκ = 5 Var ε ε 5 5,. 3 3 T σ 8 while the mea-squared error of the resultig estimator coverges at the rate of C κ Var ε ε 3 5 3 5 T 5, up to costat C κ. Here, ε ad ε represet the microstructure oise correspodig to two differet observatios of the SBM. I particular, we agai ifer that the resultig estimator attais a better MSE performace tha the plai MME i the absece of oise. I order to implemet the estimators with the correspodig optimal choices of K, we propose a iterative procedure i which a iitial reasoable guess for σ is used to fid K, which i tur is used to improve the iitial guess of σ, ad so forth. The resultig estimators exhibit superior fiite-sample performace both o simulated ad real high-frequecy stock data. I particular, we foud that the estimators are quite stable as the samplig frequecy icreases, whe compared to their MME couterparts, which, as metioed above, coverge to either 0 or for σ or κ, respectively. The optimal value of K proposed i Zhag et al. 005 see Eq. 58 ad 63 therei lacks the term Eε i the umerator.
The rest of the paper is orgaized as follows. I Sectio, we give the model ad the estimatio framework. Sectio 3 itroduces the method of momet estimators. Their i-fill ad log-ru asymptotic behavior are aalyzed i Sectio 3.. Sectio itroduces the estimators for σ ad κ that are robust to a microstructure oise compoet together with bias corrected versios of these with optimal selectio of K. Sectio 5 shows the fiitesample performace of the proposed estimators via simulatios as well as their empirical robustess usig real high-frequecy trasactio data. Fially, the proofs of the paper are deferred to the Appedix. The model ad the samplig scheme I this sectio, we itroduce the model used throughout the paper. We cosider a subordiated Browia motio of the form X t = σw τ t + θτ t + bt,. where σ, κ > 0, θ, b R, W := {W t} t 0 is a stadard Browia motio, ad {τ t } t 0 := {τt; κ} t 0 is a idepedet subordiator i.e., a o-decreasig Lévy process satisfyig the followig coditios: i Eτ t = t, ii Varτ t = κt, iii Eτ j <, j =,..., 8.. The first coditio is eeded for idetifiability purposes, while the secod oe allows to iterpret κ as a measure of the excess kurtosis. The coditio.-iii is imposed so that X t admits fiite momets of sufficietly large order. I fiacial applicatios, X is ofte iterpreted as the log-retur process X t = logs t /S 0 of a risky asset with price process {S t } t 0. I that case, τ plays the role of a radom clock aimed at icorporatig variatios i busiess activity through time. It is well kow that the process X is a Lévy process see, e.g, Sato 999. Hereafter, ν will deote the Lévy measure of X, which cotrols the jump behavior of the process i that νx, x + dx measures the expected umber of jumps with size ear x per uit time. Two prototypical examples of. are the Variace Gamma VG ad the Normal Iverse Gaussia NIG Lévy processes, which were proposed by Carr et al. 998 ad 5
Bardorff-Nielse 998, respectively. I the VG model, τt; κ is Gamma distributed with scale parameter β := κ ad shape parameter α := t/κ, while i the NIG model τt; κ follows a Iverse Gaussia distributio with mea µ = ad shape parameter λ = /tκ. As see from the formulas for their momets see 3. below, the model s parameters have the followig iterpretatio:. σ dictates the overall variability of the process icremets or, i fiacial terms, the log returs of the asset; i the symmetric case θ = 0, σ is the variace of log returs divided by the time spa of the returs;. κ cotrols the kurtosis or the tail s heaviess of the log retur distributio; i the symmetric case θ = 0, κ is the excess kurtosis of log returs multiplied by the time spa of the returs; 3. b is a drift compoet i the caledar time;. θ is a drift compoet i the busiess time ad cotrols the skewess of log returs; Throughout the paper, we also assume that the log retur process {X t } t 0 is sampled durig a time iterval [0, T ] at evely spaced times: t i, = t i := iδ, i =,...,, where δ := T..3 This samplig scheme is sometimes called caledar time samplig c.f. Oome 006. Uder the assumptio of idepedece ad statioarity of icremets, we have at our disposal a radom sample i X := X iδ X i δ, i =,...,,. of size of the distributio of X δ. I real markets, high-frequecy log returs exhibit certai stylized features, which caot be accurately explaied by efficiet models such as.. There are differet approaches to model these features, widely termed as microstructure oise. Microstructure oises may come from differet sources, such as clusterig oises, o-clusterig oises such as bid/ask bouce effects, ad roudoff errors cf. Campbell et al. 997, Zeg 003. I what follows, we adopt a popular approach due to Zhag et al. 005, where the et effect of 6
the market microstructure is icorporated as a additive oise to the observed log-retur process: X t := Xt := X t + ε t,.5 where {ε t } t 0 is assumed to be a cetered process, idepedet of X. I particular, uder this setup, the log retur observatios at a frequecy δ are give by X i := X iδ X i δ = i X + ε i,δ,.6 where ε i,δ := ε iδ ε i δ ca be iterpreted as the cotributio of the microstructure oise to the observed icremet X. i I the simplest case, the oise {ε t } t 0 is a white oise; i.e., the variables {ε t } t 0 are idepedet idetically distributed with mea 0. It is well kow ad ot surprisig that stadard statistical methods do ot perform well whe applied to high-frequecy observatios if the microstructure oise is ot take ito accout. A stadig problem is the to derive iferece methods that are robust agaist a wide rage of microstructure oises. I Sectio, we proposed a approach to address the latter problem, borrowig ideas from the semial two-scales correctio techique of Zhag et al. 005 applied to Method of Momet Estimators MME. Before that, we first itroduce the cosidered MMEs ad carry o a simple ifill asymptotic aalysis of the estimators both i the absece ad presece of the microstructure oise. 3 Method of Momet Estimators The Method of Momet Estimators MME are widely used to deal with high-frequecy data due to their simplicity, computatioal efficiecy, ad kow robustess agaist potetial correlatio betwee observatios. For the geeral subordiated Browia model.-., the cetral momets ca easily be computed i closed forms as µ X δ := EX δ = θ + bδ, µ X δ := VarX δ = σ + θ κδ, µ 3 X δ := EX δ EX δ 3 = 3σ θκ + θ 3 c 3 τ δ, 3. µ X δ := EX δ EX δ = 3σ κ + 6σ θ c 3 τ + θ c τ δ + 3µ X δ, where, hereafter, c k Y := i k d k du l E e iuy k 7, u=0
represets the k-th cumulat of a r.v. Y. For the VG model, c 3 τ, c τ = κ, 6κ 3, while for the NIG model, c 3 τ, c τ = 3κ, 5κ 3. Throughout, we assume that θ = 0 or, more geerally, that θ is egligible compare to σ see Remark 3. below for further discussio about this assumptio. The assumptio that θ = 0 allows us to propose tractable expressios for the MME of the parameters σ ad κ as follows: σ X := δ ˆµ, X, κ X := δ 3 ˆµ, X ˆµ,X δ, 3. where hereafter ˆµ k, X represets the sample cetral momet of k th order as defied by ˆµ k, X := i X X k, k, X := i= i= i X = log S T S 0. 3.3 We ca further simplify the above statistics by omittig the terms of order Oδ = O/ i particular, we leave out the term δ i 3. ad X i sample momets of 3.3: ˆσ X := T [X, X], ˆκ X := δ 3 i= i X i= i X = 3 T [X, X] T [X,, 3. X] where above we have expressed the estimators i terms of the realized variatios of order ad, which hereafter are defied by [X, X] := i= i X, [X, X] = i X. Remark 3. I the case that θ << σ i.e., θ is egligible relative to σ, we ca see the estimators 3.-3. as approximate Method of Momet Estimators. The assumptio of θ 0 has bee suggested by some empirical literature e.g., Seeta 00, who i turs cites Hurst et al. 997. Usig MME ad MLE ad itraday high-frequecy data, this was also validated by Figueroa-López et al. 0 for NIG ad VG models. Uder the latter two frameworks, we ca perform a simple experimet to assess this assumptio. From the formulas for µ ad µ 3 i 3. as well as the formula for c 3 τ i the NIG ad VG cases, we have that µ 3 X δ µ X δ θ κ θ κ, i= 8
assumig that, as it is usually the case, θ. Therefore, σ θ κ µ X δ δ µ 3 X δ. The followig table reports the values of ˆµ,X δ ˆµ 3, X for a few stocks. Thus, for istace, the value of for miute INTEL data suggests that σ is at least times larger tha θ κ ad thus, we ca assume that µ X δ σ δ. Oe ca do a similar aalysis to justify that µ X δ 3σ κδ. δ 5 sec 0 sec 30 sec mi 5 mi 0 mi 30 mi INTEL 8. 57 6.7.7 3 CVX 36.8 303.8 8706.9.9 5.0 3.5 75.3 CSCO 587.5 55.8 9. 77.5 67. 5.3 37.6 PFE 7.8. 0.7 7.89 7.67 7.63 8.5 Table : Computatio of ˆµ X δ ˆµ 3 X for differet stocks based o high-frequecy data durig the year of 005 T = 5 days. 3. Simple ifill properties i the absece of oise We ow proceed to show some i-fill with fixed T asymptotic properties of the estimators i 3.-3.. As above, i the sequel we assume that θ = 0 ad eglect Oδ = O/ terms. I that case, it is easy to see that Eˆσ = E σ = σ + O, Var ˆσ = Var 3σ σ κ = T + O. 3.5 From the above formulas, we coclude the ot surprisig fact that, o a fiite time horizo, ˆσ is ot a mea-squared cosistet estimator for σ, whe the samplig frequecy icreases, but the MSE is of order O/T, as T. A aalysis of the bias ad variace of ˆκ ad κ is more complicated due to the oliearity of the sample kurtosis. However, we ca deduce some iterestig features of its ifill asymptotic behavior. First, we have lim P ˆκ = lim P κ = 3 T t T X t T t T X t =: ˆκT, 3.6 9
where above X t = X t X t is the jump size of X at time t ad the summatios are over the radom coutable set of times t for which X t 0. The limit 3.6 follows from the well-kow formula k P i= Xiδ X i δ t T X t k, as, valid for ay k ad a pure-jump Lévy process X. Furthermore, the covergece of the correspodig momets also holds true sice 0 δ ˆµ, /ˆµ, δ = T <, ad, thus, lim Eˆκ = lim E κ = Eˆκ T ad lim Var ˆκ = lim Var κ = Var ˆκ T. 3.7 The followig result, whose proof is give i the Appedix, expads the expectatio ad variace of ˆκ T above ad shows that the MSE of ˆκ T is OT, as T. Propositio 3. Let X be a geeral Lévy process with Lévy measure ν. Let c i := c i X be the i th cumulat of X, κ := c /3c, ad suppose that x i νdx < for ay i. The, as T, E ˆκ T = κ + 3c c 6 c T + OT, 3.8 3c E ˆκ T κ c 8 c c c 6 + c = c T + OT. 3.9 9c 5 3. Properties of the MME uder microstructure oise I this part we characterize the effects of a microstructure oise compoet ito the asymptotic properties of the MME itroduced above. The results for the case of the volatility estimators are classical ad their proofs are give oly for the sake of completeess. The results for the estimators of the kurtosis parameter κ are ot hard to get either but are less kow. We adopt the setup itroduced at the ed of Sectio, uder which the observed log-returs are give by i X := X iδ X i δ = X iδ X i δ + εiδ ε i δ =: i X + ε i,. 3.0 Furthermore, throughout we assume that, for each, ε i, i satisfies the followig mild assumptio, for ay positive iteger k : ε i, k i= P m k ε,, for some m k ε R. 3. 0
Obviously, the previous assumptio covers the microstructure white-oise case, where ε t t 0 are i.i.d., i which case m k ε := E ε, k. Note that ε is ot required to be idepedet of the process X ad, furthermore, we oly eed for X to be a pure-jump semimartigale. Let us first describe the ifill asymptotic behavior of the estimators for σ, itroduced i 3.-3., but based o the oisy observatios: σ X := δ X i X, ˆσ X := [ X, X] T = δ i= X i. 3. For future referece, let us state the followig simple result that follows from applyig Cauchy s iequality, the coditio 3., ad the fact that i= i X m P s T X s m. Lemma 3.3 For arbitrary itegers m ad k 0, i X m ε i, k P 0, as. 3.3 i= We are ow ready to aalyze the asymptotic behavior of the estimators i 3.. The followig result gives the i-fill asymptotic behavior of ˆσ X ad σ X. Propositio 3. Both estimators ˆσ X ad σ X admit the decompositio i= ˆσ X = A + B, σ X = à + B where the r.v. s above are such that lim P A = lim P à = T X s, s T lim P δ B = m ε, lim P δ B = m ε m ε. Proof. We oly give the proof for σ := σ X. The proof for ˆσ X is idetical. First ote that σ = δ i= i X X + δ =: à + B, + B,. i= ε i, ε + δ i X X ε i, ε The term à coverges to T s T X s, as, sice i= i X s T X s ad X = O P /. Clearly, 3. implies that δ B, = i= ε i, ε i=
coverges to m ε m ε, i probability, whe. Also, usig Lemma 3.3, δ B, = i= i X ε i, X ε goes to 0 i probability. Next, let us cosider the estimators for κ itroduced i 3.-3., but applied to the oisy process X: κ X = δ 3 ˆµ, X ˆµ, X 3, ˆκ X := T 3 [ X, X] [ X, X] The followig result states that, for large, the above estimators behave asymptotically as δ C, for some costat C, depedig o the ergodic properties of the microstructure oise. Propositio 3.5 There exist o-zero costats C ad C such that, as,. ˆκ δ X P C, κ δ X P C. 3. Proof. We oly give the proof for κ := κ X. The proof for ˆκ X is similar. First, observe that ˆµ, X = i X X + i= ε i, ε + i= i X X ε i, ε. By Lemma 3.3, the first ad third terms o the last expressio above ted to 0 i probability, while the secod term coverges to C 0 := m ε m ε by 3.. Similarly, ˆµ, X = 3 l=0 i= i X l X l ε i, ε l + i= ε i, ε, ad, agai, by Lemma 3.3, all the terms i the first summatio above ted to 0 i probability, while the secod term therei coverges to C := m ε m 3 εm ε+6m εm ε 3m ε, i light of our assumptio 3.. Therefore, the secod limit i 3. follows with C := C /3 C 0. Remark 3.6 As a cosequece of the proof, it follows that, if m ε = 0, the C = C = m ε 3 m ε. I particular, if the microstructure oise ε t t 0 i.5 is white-oise, the the costat coicides with the excess kurtosis, E ε /3 E ε, of the radom variable ε := ε ε. i=
Robust Method of Momets Estimators I this sectio, we adapt the so-called two-scales bias correctio techique of Zhag et al. 005 to develop estimators for σ ad κ that are robust agaist microstructure oises. Roughly, their approach cosists of three mai igrediets: sparse subsamplig, averagig, ad bias correctio. Let us first itroduce some eeded otatio. Let Ḡ := {t 0, t,..., t } be the complete set of available samplig times as described i.3. For a subsample G = {t i,..., t im } with i i m ad a atural l N, we defie the l th -order realized variatio of the process X over G as [ X, X] m G l = j=0 Xt ij+ Xt ij l. Next, we partitio the grid Ḡ ito K mutually exclusive regular sub grids as follows: G i := G i,k := {t i, t i +K, t i +K,..., t i +i K}, i =,..., K, with i := i,k := [ i + /K]. As i Zhag et al. 005, the key idea to improve the estimators itroduced i 3. cosists of averagig the relevat realized variatios over the differet sparse sub grids G i, istead of usig oly oe realized variatio over the complete set Ḡ. Hece, for istace, for estimatig σ, we shall cosider the estimator ˆσ := ˆσ,K := K K i= T i,k [ i G X, X],. where T i,k := t i +i K t i = Kδ i. The estimator. is costructed by averagig estimators of the form ˆσ X i 3. over sparse sub-grids. The above estimator correspods to the so-called secod-best estimator i Zhag et al. 005. This estimator ca be improved i two ways. First, by correctig the bias of the estimator ad, secod, by choosig the umber of sub grids, K, i a optimal way. We aalyze these two approaches i the subsequet two subsectios. At this poit it is coveiet to recall that we are assumig the subordiated Browia motio model. with θ = 0. For simplicity, we also assume that b = 0, which wo t affect much what follows sice we are cosiderig high-frequecy type estimators ad, thus, the cotributio of the drift is egligible i that case. Regardig the microstructure oise, 3
we assume that the oise process {ε t } t 0 appearig i Eq..5 is a cetered statioary process with fiite momets of arbitrary order, idepedet of X. Furthermore, we assume that, for ay l N, E [ ε ] ] l δ lim E l [ ε δ = 0;. δ,δ 0 δ δ where hereafter ε δ deotes a radom variable with the same distributio as ε t,δ := ε t+δ ε t, which does ot deped o t. Note that. implies the existece of a costat m l ε R such that lim E [ ε ] l δ = ml ε..3 δ 0 The simplest case is the white oise, whe the variables {ε t } t 0 are idepedet idetically distributed. I that case, { ε iδ,δ } i follows a statioary Movig Average MA process with E ε iδ,δ = 0 ad E ε iδ,δ = E ε.. Bias corrected estimators I order to deduce the bias correctio, we first adopt the white oise case, where {ε t } t 0 are i.i.d. I that case, the distributio of ε δ does ot deped o δ. A radom variable with this distributio is deoted ε. We start by devisig bias correctio techiques for the estimator.. Clearly, from 3. ad the idepedece of the oise ε ad the process X, we have: E ˆσ,K = σ + E ε K K i= i = σ + E ε.. T i,k Kδ The relatio. shows that the bias of the estimator diverges to ifiity whe the time spa betwee observatio δ := T/ teds to 0. To correct this issue, first ote that. also implies that E δ ˆσ, = σ δ + E ε E ε..5 Hece, a atural bias-corrected estimator would be ˆ σ := ˆ σ,k := ˆσ,K Kδ ˆµ, ε,.6 where ˆµ, ε := δ ˆσ,. However, from. with K =, we have: E ˆ σ = σ + E ε KT K σ + E ε T = K K σ,
which implies that ˆ σ is ot truly ubiased. Nevertheless, the above relatioship yield the followig ubiased estimator for σ : ˆ σ,k := K K ˆ σ,k = K K i= T i,k [ i G X, X] [ X, X]Ḡ..7 K T The estimator.7 correspods to the small-sample adjusted First-Best Estimator of Zhag et al. 005. Propositio. Uder a cetered statioary oise process {ε t } t 0 idepedet of X, E ˆ σ,k = σ + E ε Kδ E ε δ. δ K I particular, ˆ σ,k is a asymptotically ubiased respectively, ubiased estimator for σ uder the coditio. respectively, a white microstructure oise settig. We ow devise approximate bias-corrected estimators for κ. I order to separate the problem of estimatig κ ad σ, i this part we assume that σ is kow. I practice, we have to replace σ with a accurate estimate such as the estimator.7. Let us start by cosiderig the mea of the statistic K K i= T i,k [ X, X which is the aalog of.. To this ed, we use the fact that E X δ + ε = 3σ κδ + 6σ E ε δ + E ε + 3σ δ, which is a easy cosequece of 3. ad the idepedece of the oise ε ad X. I that case, we have K i G E [ X, X] = 3σ κ + 6σ E ε + E ε + 3σ Kδ,.8 K T i,k Kδ i= This idetifies the estimator ˆκ,K := 3σ K K i= T i,k [ ] G i, i G X, X] Kδ,.9 as a ubiased estimator for κ i the absece of microstructure oise. However, as with the estimate of σ, the bias of the above estimate blows up whe δ 0 due to the third 5
term i.8. To correct this issue we eed a estimate for E ε, which ca be iferred from the followig limit lim E δ [ X, X]Ḡ = E ε,.0 T which is a easy cosequece of.8 with K =. Together with.5, these two suggests the followig estimate: where ˆ κ := ˆκ,K σ ˆµ, ε ˆµ, ε := δ [ X, X]Ḡ T, ˆµ, ε := δ T 3σ Kδ ˆµ, ε,. [ X, X]Ḡ.. However, as with the estimator ˆ σ above, the above estimator is oly asymptotically ubiased for large ad K. The followig result provides a ubiased estimator for κ based o the realized variatios of the process o two scales. The proof follows from. ad.8 ad is omitted. Propositio. Let ˆ κ := 3σ K K i= T i,k [ i G X, X] [ X, X]Ḡ σ K δ. [ X, 3σ X]Ḡ.3 K T The, uder a white microstructure oise idepedet of X, ˆ κ is a ubiased estimator for κ. Furthermore, for a geeral cetered statioary oise process, we have E ˆ κ K = κ + E ε σ K Kδ E E ε ε δ + Kδ E ε δ, 3σ δ K which shows that ˆ κ is asymptotically ubiased uder coditio... Optimal selectio of K I this part, give a specified fuctio bk,, T, O u bk,, T meas that there exists a costat c, idepedet of K,, ad T, such that O u bk,, T cbk,, T, for all K,, ad T. We also assume the white-oise case where the microstructure oise {ε t } t 0 are cetered i.i.d. r.v. s. 6
A importat issue whe usig the two-scales procedure described i the previous sectio is the selectio of the umber of subclasses, K. A atural approach to deal with this issue cosists of miimizig the variace of the relevat estimators over all K. This procedure will yield a optimal K for the umber of subclasses. Let us first illustrate this approach for the estimator ˆσ,K give i.. The ext result, whose proof is give i Appedix A., gives the variace of ˆσ,K. Theorem.3 The estimator. is such that Var ˆσ,K σ K = 3 + Eε K T + σ 3 + 3σ κ T + σ 3K + 8σ Eε KT K K + O u + O u + O u. T KT. Remark. As a cosequece of., for a fixed arbitrary K ad a high-frequecy/loghorizo samplig setup T ad δ = T / 0, a sufficiet asymptotic relatioship betwee T ad δ for the estimator ˆσ,K to be mea square cosistet is that δ T. If K is chose depedig o ad T, as we ited to do ext, the feasible values K := K,T must be such that K,T / 0 ad /K,T T 0 as T ad δ = T/ 0. Now, we are ready to propose a approximately optimal K. To that ed, let us first recall from. that the bias of the estimator is Together.-.5 implies that MSE ˆσ,K Bias ˆσ,K σ K = 3 + σ 3 + 3σ κ T K K + O u + O u T = Eε T K..5 + σ 3K + 8σ Eε KT + O u. KT + Eε K T + Eε T K.6 Our goal is to miimize the MSE with respect to K whe is large. Note that the oly term that is icreasig i K is σ K/3, while out of the terms decreasig i K, the term Eε /T K is the domiat whe is large. It is the reasoable to cosider oly these two terms leadig to the approximatio : MSE ˆσ K σ K 3 + T K Eε =: MSE ˆσ K..7 7
The right-had side i the above expressio attais its miimum at the value: 6Eε K 3 =..8 T σ Iterestigly eough, the value above coicides with the optimal K proposed i Zhag et al. 005 see Eq. 8 therei. Pluggig.8 i.6 ad, sice δ = T/ 0, it follows that MSE ˆσ K = 3 3 3 Eε 3 σ 8 3 T 3 + 3κσ T + ot..9 I particular, the above expressio shows that, i the presece of a microstructure oise compoet, the rate of covergece reduces from OT to oly OT /3 ad, furthermore, that the covergece is worst whe σ, Eε, ad κ are larger. The followig result gives a estimate of the variace of the ubiased estimator.7. Its proof is give i Appedix A.. Propositio.5 The estimator.7 is such that Var ˆ σ,k σ K = 3 + Eε + Eε + O T K u + O u + O K 3 T u..0 T K As before, the previous result suggests to fix K so that to miimize the first two leadig terms i.0. Such a miimum is give by K = 3 6 Eε + Eε T σ 3,. which is similar but ot idetical to the aalog optimal K proposed i Zhag et al. 005 see Eq. 58 & 63 therei. After pluggig K i.0, the resultat estimator attais the MSE: MSE ˆ σ K = 3 3 3 Eε + Eε 3 σ 8 3 3 T 3 + ot.. Iterestigly eough, sice T/ 0, the estimator ˆ σ K attais the order ot, which was ot achievable by the estimators ˆσ K, eve i the absece of microstructure oise, or by the stadard estimators itroduced i Sectio 3 see 3.5. Now, we proceed to study the optimal selectio problem of K for the estimator.9 for κ. As with ˆσ,K, we first eed to aalyze the variace of the estimator. The optimal value of K proposed i Zhag et al. 005 lacks the term Eε i the umerator. 8
Theorem.6 The estimator.9 is such that Var ˆκ,K = 6 T K 3 + O 5 3 u T K 3..3 We are ow ready to propose a method to choose a value of K that approximately miimizes the MSE of the estimator ˆκ,K. Let us first recall from.8 that the bias of the estimator ˆκ,K is Together,.3-. imply that Bias ˆκ,K = E ˆκ,K κ = E ε T Kσ + E ε.. σ MSE ˆκ,K = 6 T K 3 + E ε + h.o.t.,.5 5 3 T K σ8 where h.o.t. mea higher order terms. It is the reasoable to select K so that the leadig terms of the MSE are miimized. The aforemetioed miimum is reached at Pluggig.6 i.5, it follows that 5E ε K3 5 =..6 96T σ 8 MSE ˆκ 3 K 3 = 5 5 3 3 5 E ε 6 5 σ 5 T 5 + o T 5, whose rate of covergece to 0 is slower tha the rate of O T /3 attaied by the estimator ˆσ K Ḟially, we cosider the ubiased estimator for κ itroduced i Propositio.. As above, h.o.t. refers to higher order terms. Theorem.7 The estimator.3 is such that where eε = Var ε ε. Var ˆ κ 6 T K 3,K = + eε + h.o.t.,.7 5 3 9σ 8 T K The two terms o the right-had side of.7 reach their miimum value at K = 5 5eε 5..8 76T σ 8 9
After pluggig K i.7, we obtai that MSE ˆ κ K = 8 5 5 5 3 9 5 eε 3 5 σ 5 3 5 T 5 + o T, which agai, sice T/ 0, implies that MSE ˆ κ K = ot. result should be compared to 3.9, which essetially says that the estimator ˆ κ K The aforemetioed has better efficiecy tha the cotiuous-time based estimator ˆκ T, obtaied by makig i the estimators ˆκ ad κ see 3.6. It is worth poitig out here that oe ca devise a cosistet estimator for eε usig the relatioships i [ X, X]Ḡ P E ε ε, ii [ X, X]Ḡ 8 P E ε ε 8..9 Remark.8 It is atural to woder if some types of cetral limit theorems are feasible for the estimators cosidered here. I spite of the fact that we are cosiderig a Lévy model, whose icremets are idepedet, the estimators caot be writte i terms of a row-wise idepedet triagular array. For istace, cosider the estimator ˆσ,K for σ itroduced i. ad, for simplicity, assume that T i,k = T, which asymptotically is satisfied, ad absece of microstructure oise. It ca be show that K [X, X] Gi = K X ti+k X ti, K K whose terms are correlated. i= i=0 5 Numerical Performace ad Empirical Evidece I this sectio, we propose a iterative method to implemet the estimators described i the previous sectio, with the correspodig optimal choices of K. The mai issue arises from the fact that i order to accurately estimate σ, we eed to choose K as i. or.8, which precisely depeds o what we wat to estimate, σ. So, we propose to start with a iitial reasoable guess for σ to fid K, which i tur is the used to improve the iitial guess of σ, ad so forth. The fiite-sample ad empirical performace of the resultig estimators are illustrated by simulatio ad a real high-frequecy data applicatio. For briefess, i what follows we will make use of the followig otatio 6m Km 3, σ :=, K T σ m, m, σ := 6 m + m 3 3. T σ 0
For the simulatio portio of this sectio, we cosider a Variace Gamma VG model with white Gaussia microstructure oise. The variace of the oise ε t is deoted by ϱ so that the oise of the i th icremet, ε i,, is N 0, ϱ. Other parameters are set as: σ = 0.0, κ = 0.3, ad ϱ = 0.005. The time uit here is a day. I particular, the above value of σ correspods to a aualized volatility of 0.0 5 = 0.3. 5. Estimators for σ We compare the fiite sample performace of the followig estimators:. The estimator ˆσ,K give i. with K determied by a suitable estimate of the optimal value K give i.8, as described ext. As show i Propositio 3. ad.5, a cosistet ad ubiased estimator for Eε = E ε / is give: ˆϱ := Êε := [ X, X]Ḡ. 5. The oly missig igrediet for estimatig.8 is a iitial prelimiary estimate of σ, which we will the proceed to improve via ˆσ,K. Cocretely, we propose the followig procedure. First, we evaluate the estimate ˆK := K ˆϱ, σ 0, where σ 0 is a iitial reasoable value for the volatility. Secod, we estimate σ via ˆσ := ˆσ, ˆK. Next, we use ˆσ to improve our estimate of K by settig ˆK := K ˆϱ, ˆσ. Fially, we set ˆσ := ˆσ, ˆK. We cosider the bias-corrected estimator ˆ σ,k itroduced i.7, with a value of K give by ˆK as defied i the poit above. We deote this estimator ˆσ. We also aalyze a iterative procedure similar to that i item, but usig ˆσ. Cocretely, we set ˆσ = ˆ σ, ˆ K, where ˆ K := K ˆϱ, ˆσ. 3. Fially, we also cosider the estimator ˆ σ,k itroduced i.7 but usig a estimate of the optimal value K as defied i Eq... Cocretely, we set ˆσ 3 = ˆ σ, ˆK with ˆK := Kˆϱ, ˆϖ, σ 0, where σ 0 is a iitial reasoable value for σ ad ˆϖ is a cosistet estimator for Eε. Next, we improve the estimate of ˆσ 3 by settig ˆσ 3 := ˆ σ, ˆK, with ˆK := K ˆϱ, ˆϖ, ˆσ 3. 5.
To estimate Eε, we use.. Cocretely, as show i the proof of Propositio 3.5 ad also i Eq..0, the statistics ˆm, ε := [ Eε + 6 Eε. Therefore, a cosistet estimate for Eε is give by ˆϖ := Êε := [ X, X]Ḡ 3 Êε. X, X]Ḡ / coverges to E ε = The sample mea, stadard deviatio, ad mea-squared error MSE based o 000 simulatios are preseted i the Table. Here, we take T = 5 days ad σ 0 0.063, which correspods to a aualized volatility of. As expected, the estimator ˆσ exhibits a oticeable bias ad that this bias is corrected by ˆσ. However, ˆσ 3 is much more superior to other cosidered estimators, which is cosistet with the asymptotic results for the mea-squared errors described i Eqs..9 ad.. δ ˆσ ˆσ ˆσ ˆσ ˆσ 3 ˆσ 3 5 mi mi 30 sec sec Mea 0.07333 0.00666 0.099858 0.098883 0.0999695 0.09996 Std Dev 0.0006858 0.0033 0.000795 0.007976 0.0008839566 0.0007060 MSE 7.99565e-06.760e-06 6.3569e-07.5698e-06 7.83885e-07.9683e-07 Mea 0.08898 0.006693 0.099556 0.0988 0.099737 0.0000 Std Dev 0.0006839 0.00060565 0.00076859 0.0060905 0.0007887707 0.000669303 MSE 8.733e-06.5777e-06 5.59857e-07.37075e-06 6.95e-07.857e-07 Mea 0.093765 0.00755 0.0998865 0.0993685 0.0000009 0.000709 Std Dev 0.0006537998 0.000690 0.00075576 0.009760 0.00078558 0.00063666 MSE 9.0579e-06.6939e-06 5.69076e-07.3595e-06 5.679e-07.05330e-07 Mea 0.0960 0.007658 0.0998938 0.0990 0.00000 0.000068 Std Dev 0.00063697 0.0005669 0.000785086 0.00567 0.000639388 0.00059739 MSE 9.66839e-06.6987e-06 5.308377e-07.306553e-06.0886e-07 3.56838e-07 Table : Sample meas, stadard deviatios, ad mea-squared errors for differet estimators of σ = 0.0 based o 000 simulatios. 5. Estimators for κ We compare the fiite sample performace of the followig three estimators, which are respectively deoted by ˆκ, ˆκ, ˆκ 3.. The estimator ˆκ,K give i.9 with σ replaced with the estimate ˆσ 3 i Eq. 5. ad K determied by a estimate of the optimal value K 3 give i.6 obtaied by replacig σ ad E ε with ˆσ 3 ad Eq..9-i, respectively.
. The ubiased estimator ˆ κ defied i.3 with the same value of K as the previous item. As before, we replace σ by the estimator ˆσ 3. 3. Agai, the ubiased estimator ˆ κ i.3 replacig σ with ˆσ 3, but ow the value of K is give by.8. We replace σ therei with ˆσ 3, while to estimate eε = Var ε ε, we exploit the limits i.9. The sample mea, stadard deviatio, ad mea-squared error MSE based o 000 simulatios are preseted i Table 3. Here, we take T = 5 days ad σ 0 = 0.063. As expected, the estimator ˆκ 3 has much better performace tha ay other estimator therei. ˆκ ˆκ ˆκ 3 ˆκ ˆκ ˆκ 3 δ = 5 mi δ = mi Mea 0.5777957 0.9980 0.9967835 0.578966 0.98936 0.968668 Std Dev 0.783893 0.83639 0.09790650 0.573096 0.59975870 0.075809358 MSE.0899e-0 3.35853e-0 9.586563e-03 9.9953e-0.5655e-0 5.755750e-03 δ = 30 sec δ = sec Mea 0.5878 0.999056 0.967773 0.573787 0.90678 0.9553 Std Dev 0.6799873 0.63678990 0.0693758 0.687998 0.65066890 0.066836990 MSE.0506e-0.6793e-0.8965e-03.099e-0.733795e-0.96860e-03 Table 3: Sample meas, stadard deviatios, ad mea-squared errors for differet estimator of κ = 0.3 based o 000 simulatios. 5.3 Rate of Covergece Aalysis I this sectio we study the rates of covergece of the stadard errors of the ubiased estimators ˆ σ,k ad ˆ κ,k as defied by Eqs..7 ad.3, whe K is chose accordig to the optimal values. ad.8, respectively. I particular, we wat to assess our claim that the covergece rates of the estimator s variaces are faster tha T. To this ed, we plot log Var ˆ σ,k,t agaist logt for T s ragig from moths to years ad eight itraday samplig frequecies δ see left pael i Figure. We also show the best liear fit for each plot. Here, Var ˆ σ,k,t represets the sample variace of the estimator ˆ σ,k,t computed by Mote Carlo usig 00 simulatios. I Table, we also report the 95% cofidece itervals for the slopes of the best liear fits secod colum i the table. 3
It is apparet that the liear fit is very good, which idicates that Var ˆ σ,k,t T β, for large T ad some β > 0, ad furthermore, the slope s estimates idicate that the covergece rate of Var ˆ σ,k,t is slightly better tha T the average rate is T.03. We also perform the same aalysis for the estimator ˆσ 3, as described i Sectio 5., which is desiged to be a data-drive proxy of the oracle estimator ˆ σ,k,t. The results are show i the right pael of Figure ad the third colum of Table. The average covergece rate of Var ˆσ 3 is T.05. Note that the CI s idicate that the slope is sigificatly differet tha i almost all cases. We carry out the same aalyses for the estimators for κ. The graphs of log Var ˆ κ,k,t ad log Var ˆκ3 agaist logt are show i Figure. The CI s for the slope of the best liear fits are show i Table last two colums. The average covergece rate of the variace of ˆ κ,k,t is T.5, while the average covergece rate of the variace of ˆκ 3 is T.8. δ log Var ˆ σ,k log Var ˆσ 3 log Var ˆ κ,k log Var ˆκ3 5 sec.036 ± 0.05.03 ± 0.07.3 ± 0..9 ± 0.7 0 sec.053 ± 0.06.00 ± 0.06.7 ± 0.5.9 ± 0.7 30 sec.03 ± 0.05.058 ± 0.06. ± 0.38.97 ± 0.3 mi.03 ± 0.03.03 ± 0.03.35 ± 0.58.96 ± 0.78 0 mi.00 ± 0.06 0.998 ± 0.0.086 ± 0.99.9 ± 0.5 0 mi.05 ± 0.030.073 ± 0.06.056 ± 0.099.68 ± 0.87 30 mi.08 ± 0.036.076 ± 0.09 0.93 ± 0.77.056 ± 0.3 hr.0 ± 0.00.053 ± 0.03. ± 0.05.07 ± 0.33 Table : 95% CI s for the slope of the liear regressio fit of log Var sigma Estimator agaist logt for T {m, 3m,..., m}, ad log Var kappa Estimator agaist logt for T {m, 3m,..., m}. 5. Empirical study We ow proceed to aalyze the performace of the proposed estimators whe applied to real data. As it was explaied above ad was theoretically verified by Propositios 3.-3.5,
Covergece Rate Aalysis for the "Oracle" Estimator of Sigma Covergece Rate Aalysis for the Estimator of Sigma logvarestimator of Sigma -5.5-5.0 -.5 -.0-3.5-3.0 -.5 logvarestimator of Sigma -5.5-5.0 -.5 -.0-3.5-3.0 -.5.0.5 5.0 5.5 6.0.0.5 5.0 5.5 6.0 logt Figure : Regressio Aalysis of log Var ˆ σ,k,t agaist logt left pael ad log Var ˆσ 3 right pael for T { m, 3 m,..., m}, ad δ = 5 sec Red, δ = 0 sec Blue, δ = 30 sec Brow, δ = mi Gree, δ = 0 mi Purple, δ = 0 mi Orage, δ = 30 mi Pik, ad δ = hr Grey. The sample variace is computed based o 00 simulatios. logt Covergece Rate Aalysis for the "Oracle" Estimator of kappa Covergece Rate Aalysis for the Estimator of kappa logvarestimator of kappa -6.0-5.5-5.0 -.5 -.0-3.5 logvarestimator of kappa -6.0-5.5-5.0 -.5 -.0-3.5 5.6 5.7 5.8 5.9 6.0 6. 6. 5.6 5.7 5.8 5.9 6.0 6. 6. logt logt Figure : Regressio Aalysis of log Var ˆ κ,k,t agaist logt left pael ad log Var ˆκ3 right pael for T { m, 3 m,..., m}, ad δ = 5 sec Red, δ = 0 sec Blue, δ = 30 sec Brow, δ = mi Gree, δ = 0 mi Purple, δ = 0 mi Orage, δ = 30 mi Pik, ad δ = hr Grey. The sample variace is computed based o 00 simulatios. 5
traditioal estimators are ot stable as the samplig frequecy icreases. Ideed, ˆσ ad σ both diverge to while ˆκ ad κ coverge to 0, as. The objective is to verify that the proposed estimators do ot exhibit the aforemetioed behaviors at very high-frequecies. We cosider high-frequecy stock data for several stocks durig 005, which were obtaied from the NYSE TAQ database of Wharto s WRDS system. For briefess ad illustratio purposes, we oly show Itel INTC ad Pfeizer PFE. For these, we compute the estimator ˆϱ defied i 5., the estimator ˆσ,K defied i. with K =, the estimator ˆ σ,k defied i.7 with K = ˆK as give i 5., the estimator ˆκ,K defied i.9 with K =, ad fially the estimator ˆ κ,k defied i.3 with K = ˆK as give i.8. I the case of ˆκ,, we used σ = ˆσ,. Both ˆσ, ad ˆκ, represet the estimators without ay techique to alleviate the effect of the microstructure oise. As oe ca see i Tables 5-6, the estimators ˆ σ ad ˆ κ do ot exhibit the drawbacks of the estimators ˆσ ad ˆκ at high frequecies. As a coclusio of the empirical results therei, we deduce that Itel s stock exhibits a aualized volatility σ of about 0.0 5 = 0. per year, while its excess kurtosis icreases with /δ at a rate of about 0.5 see item above Eq.. for the iterpretatio of κ. By compariso, eve though the volatility of Pfizer s stock is just slightly larger about 0.05 5 = 0.3, its excess kurtosis icreases at a rate of about.3 with /δ, showig much more riskiess due to the much heavier tails of its retur s distributio. This example illustrates the importace of cosiderig a parameter which measures the tail heaviess of the retur distributio ad ot oly its variace. 6
ˆϱ ˆσ, ˆ σ, ˆK ˆκ, ˆ κ, ˆK 0 mi 0.00988 0.0373969 0.03565 0.7786688 0.6508939 0 mi 0.0058536 0.0399567 0.03833 0.589390 0.7708959 5 mi 0.0050 0.039983 0.035377 0.9537870 0.76830688 mi 0.000558856 0.05590867 0.0365998 0.399973 0.7939570 30 sec 0.0003675 0.069093 0.03905766 0.87995 0.6875705 0 sec 0.0003835 0.0685895 0.059630 0.56680373 0.65759576 0 sec 0.0007869 0.0856083 0.057963 0.833 0.599305 5 sec 0.000735 0.003806 0.078887 0.0857006 0.98766733 Table 5: Estimatio of the parameters σ ad κ of a subordiated Browia motio with microstructure oise for INTC Itel stock. ˆϱ ˆσ, ˆ σ, ˆK ˆκ, ˆ κ, ˆK 0 mi 0.003088 0.0393 0.07933 3.55809339 3.6656536 0 mi 0.0067865 0.086633 0.039679 3.3300039.96333 5 mi 0.0039 0.05809 0.03758805 3.3955939.588370 mi 0.00058559 0.0637 0.08960.8858979 3.077770 30 sec 0.00037978 0.073003060 0.078738.00977905.539989978 0 sec 0.0003733763 0.0806335 0.0958930.8890997.358756 0 sec 0.0003068 0.0067063 0.0500707.03957069.39987 5 sec 0.0005700 0.06060 0.0539585 0.55578783.3750789809 Table 6: Estimatio of the parameters σ ad κ of a subordiated Browia motio with microstructure oise for PFE Pfeizer stock. A Proofs A. Proof of Propositio 3.. We shall eed the followig stadard result that ca easily be show usig the momet geeratig fuctio for Poisso itegrals see, e.g., Cot & Takov, 00, Chapter : Lemma A. Suppose that M is a Poisso radom measure o a ope domai of R d with mea measure m ad let Mf = fzm mdz deote the itegral of f with respect the compesated radom measure M = M m. If m f k := fz k mdz <, 7
for k =,..., 5, the E Mf k = mf k, for k =, 3, E Mf = 3mf + mf, ad E Mf 5 = 0mf mf 3 + mf 5. E Mg Mf 3 = mgf 3 + 3mf mgf. Similarly, E Mg Mf k = mgf k ad Lemma A. Let M be the jump measure of a Lévy process X with Lévy measure ν i.e., Ms, t B := #{u s, t : X u B}, for ay s < t ad B BR d, ad let Mdt, dx := Mdt, dx dtνdx be the correspodig compesated measure. Also, suppose that f is such that fx k νdx < for some k. The, there exists a costat A k f such that, for ay T, E T T Proof. Throughout the proof, let M s,t f := t s 0 k fx Mdt, dx A k ft k/. fx Mdt, dx ad let [T ] be the iteger part of T. We eed the followig classical iequality see Bickel & Doksum, 00, Lemma 5.3.: E Z µ Z k C k E Z k k/, A. where Z = i= Z i, µ Z = EZ, ad {Z i } i are i.i.d. such that E Z k <. First, ote that E T M 0,T f k k E [T ] T [T ] M 0,[T ] f + k T E k M[T ],T f k. For the first term o the right-had side above, we apply A. with Z i := M i,i f, which are i.i.d. because M is a Poisso radom measure. For the secod term, we apply Burkholder-Davis-Gudy iequality see Protter 00 to get, T k E fx Mdt, dx BkE k f xmdt, dx [T ] This completes the proof. Proof of Propositio 3.. 0 Throughout the proof, M deotes the jump measure of the Lévy process X; i.e., Ms, t B := #{u s, t : X u B}, for ay s < t ad B BR. I particular, let us ote that M is Poisso radom measure with mea measure dtνdx ad t T X t l = t 0 x l Mdt, dx. Let also Mdt, dx := Mdt, dx dtνdx be the correspodig compesated measure. Let us start by otig the idetity + x = k i i + x i + k x k k + + kx, A. + x i=0 8 k/.
ad the otatio ˆµ T k := T T 0 x k Mdt, dx, ˆDT := ˆµT c X. I particular, ˆκ T = /3ˆµ T /ˆµ T. The, we have the followig decompositio: { } Eˆκ T = = 3c X E { 3c X E + 3 E { =: L T + R T. ˆµ T ˆµ T + ˆD T ˆµ T ˆD T + 3 ˆD T ˆD 3 T + 5 ˆD T 6 ˆD 5 T ˆµ T 7 + 6 ˆD } T ˆD6T } Let us first aalyze the residual term R T usig the followig easy cosequece of the triagle iequality: ˆµ T / = / X T / s s T T / s T X s = T / µ T. A.3 Thus, sice 7 + 6 ˆD T = + 6 + ˆD T = + 6ˆµ T /c X > 0, we have that 0 R T 7T 3 E ˆD6 T + 6T 3 E ˆD7 T = 7T 6 3c 6 X E ˆµ T 6T c X + 3c 7 X E ˆµ T c X 7. Usig that Eˆµ T = c X ad Lemma A., R T = OT. Similarly, usig Lemma A., the first four terms of L T i.e. those multiplyig ˆD i T up to i = 3 are give by c X 3c X c 6X 3c 3 X T + c X c X T + OT. The last two term of L T ca be see to be OT from Lemma A. ad Cauchy iequality. Ideed, Eˆµ T ˆD T c X E ˆD T + c X E Kc T + E ˆµ T c X ˆµ T c X ˆµ T c X E ˆµ T c X 8 /, 9
which is OT i light of Lemma A.. We fially obtai that Eˆκ Eˆκ T = c X 3c X c 6X 3c 3 X T + c X c X T + OT. I order to show the boud for the variace, we use agai A. to get The, ˆκ T = ˆµT 3c X ˆD T + 3 ˆD T ˆD T 3 + 3 ˆµ T ˆµ T 5 + ˆD T ˆD T. ˆκ T c X 3c X = ˆµ T 3c c X ˆµT X 3c X ˆD T + ˆµT c X ˆD T ˆµT 3c X ˆD T 3 + 3 ˆµ T ˆµ T 5 + ˆD T ˆD T. After expadig the squares, takig expectatios both sides, ad usig Cauchy s iequality together with Lemmas A. ad A., oe ca check that all the terms are at least OT except possibly the followig terms: 9c X E + { } ˆµ T c X { } 9c X E ˆµ T ˆD T. { } 9c X E ˆµ T c X ˆµ T ˆD T Subtractig c X from ˆµ T i the secod ad third terms above, ad usig agai Lemmas A. ad A., we ca check that the above expressio ideed coicides with the expressio i 3.9. A. Proofs of Sectio. Proof of Theorem.3. Throughout we write T i for T i,k. Clearly, Var ˆσ,K = K i<j K Cov [ T i T j Gi X, X], [ X, X] Gj + K K i= T i Var [ Gi X, X]. A. 30
Each covariace i the first term o the right had side above is give by A i,j := Cov [ = i q=0 j r=0 Gi X, X], [ X, X] Gj Cov Xti +q+k Xt i +qk, Xt j +r+k Xt j +rk = i C K + i jδ + j C j iδ, where, for ay u < t < t + δ < v, Cδ := Cov Xt + δ Xu, Xv Xt, which ca be proved to deped oly o δ > 0. More specifically, ote that Cδ = Cov S + U, S + V, where S := Xt + δ Xt, U := Xt Xu + ε t+δ ε u, ad V := Xv Xt + δ + ε v ε t. Next, usig that idepedece of S, U, ad V, Cδ = Var S + Cov S, SV + Cov SU, S + Cov SU, SV = Var S + EV Cov S, S + EUCov S, S + EUEV Var S. Fially, usig that EU = EV = 0 as well as the momet formulas i 3., Cδ = Var S is give by Cδ = σ δ + 3σ κδ. Usig the previous formula together with the fact that j K i j U K ad K j U K for some costat U idepedet of, K, i, T, ad j, the first term i A., which we deote A, ca be computed as follows: A = K 3 T + K 3 T = K K KT i<j K i<j K σ j i δ + 3σ κj iδ σ K + i j δ + 3σ κk + i jδ + R 3σ κkδ + 3 σ KK δ + R, where R is such that R UK KT 3σ κkδ + 3 σ KK δ. A.5 3
Now, we cosider the secod term i A., which we deote B. Each variace term of B ca be writte as B i := Var [ = i q=0 Gi X, X] Var Xti +q+k Xt i +qk i + Cov Xti +q+k Xt i +qk, Xt i +q+k Xt i +q+k. q=0 Next, usig the relatioships Var Xt + δ Xt = σ δ + 3σ κδ + 8σ E ε δ + E ε + E ε Cov Xt + δ Xt, Xv Xt + δ = E ε E ε, valid for ay t < t + δ < v, we get B i = i σ Kδ + 3σ κ Kδ + 8σ E ε Kδ + i E ε + E ε. A.6 Therefore, usig that / i K/ UK / ad / i K / UK 3 / 3, for a costat U idepedet of i, K,, ad T, we have B = C C + R, where C = σ Kδ K T + 3σ κ Kδ + 8σ E ε Kδ + E ε, C = E ε E ε, KT ad R = O u K/C = O u K/C. Puttig together A ad B above, Var ˆσ,K K = 3σ κkδ K KT + 3 σ KK δ + σ Kδ K T + 3σ κ Kδ + 8σ E ε Kδ + E ε E ε E ε + R KT + R. Recallig that δ = T/ ad usig A.5, we get the expressio.. Proof of Propositio.5. Let a K := Var ˆ σ,k = a K Var ˆσ,K + b K Var K ad b K K := [ X, X]Ḡ. Clearly, T K a K b K Cov ˆσ,K, [ X, X]Ḡ A.7 3
From the expressios i Eqs. A.6-A.7, we have Var [ X, X]Ḡ Var ˆσ,K = σ δ + 3σ κδ + 8σ E ε δ + E ε + E ε 3σ κkδ + 3 σ KK δ K = K KT + K + σ Kδ K T + 3σ κ Kδ + 8σ E ε Kδ + E ε E ε E ε + R KT + R. To compute the last covariace, let us first ote that Cov ˆσ,K, [ X, X]Ḡ = K K i= Cov [ T i Gi X, X], [ X, X]Ḡ =: K K i= T i B i. A.8 Each covariace term o the right had side above ca be computed as B i = i q=0 Cov Xti +q+k Xt i +qk, Xt r+ Xt r r=0 = i e i Cov Xti +K Xt i +K, Xt r+ Xt r r=0 + e i Cov XtK Xt 0, Xt r+ Xt r, r=0 where above e i deote the umber of subitervals i the set { [t i +qk, t i +q+k ] } i q=0 which itersect the ed poits 0 ad T. Obviously, K i= e i =. Now, we use the followig formulas: Cov Xv Xu, Xv Xu = σ v u + 3κσ v u, u < u < v < v Cov Xt Xs, Xu Xt = Eε Eε, s < t < u. We the get B i = i {K σ δ + 3κσ δ + Eε Eε } e i Eε Eε. Next, Cov ˆσ,K, [ X, X]Ḡ = { K σ δ + 3κσ δ + Eε Eε } + O u KT KT 33