Estimatio of a oisy subordiated Browia Motio via two-scale power variatios José E. Figueroa-López Departmet of Statistics, Purdue Uiversity ad Kiseop Lee Departmet of Mathematics, Uiversity of Louisville May 7, 05 Abstract High frequecy based estimatio methods for a pure-jump subordiated Browia motio exposed to a small additive microstructure oise are developed buildig o the two-scale realized variatios approach developed by Zhag et al. 005 for the estimatio of a cotiuous Itô process. The proposed estimators are show to be robust agaist the oise ad to attai better rates of covergece tha those of stadard 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 crucial tue-up parameter of the method. Fially, a two-step data-drive procedure is devised to implemet the proposed estimators with the optimal K-value. The superior fiite-sample performace ad empirical robustess of the resultig estimators are 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 grat DMS-4969.
Itroductio I this paper, we develop estimatio methods for a subordiated Browia motio SBM, whose samplig observatios have bee cotamiated by a small additive oise ε alog the lies of the framework of Zhag et al. 005. I additio to a volatility type parameter σ, which accouts for the variace of a time series of the process icremets, a SBM is edowed with a additioal parameter, 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 risk maagemet ad optimal asset allocatio. The models cosidered here are pure-jump Lévy models ad σ is ot the volatility of a cotiuous compoet. Nevertheless, give that σ is proportioal to the variace of a time series of log-returs, 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 ε, typically called microstructure oise, ca be see as a modelig artifact to accout for the deviatio of the actual price formatio process from the SBM. I some circumstaces, the oise ca also be give some cocrete iterpretatio based o the actual tradig mechaism such as i the case of bid/aks bouce effects cf. Roll 984. At low frequecies the microstructure oise is relatively egligible compared to the SBM s observatios but at high-frequecy 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, for istace, Aït-Sahalia et al. 005, Zhag et al. 005, Hase & Lude 006, Badi & Russell 008, Myklad & Zhag 0, to metio just a few. Amog these, the problem of estimatig the itegrated variace has received a great deal of attetio. However, to the best of our kowledge, there is almost o paper that focus o the parametric estimatio of the volatility ad kurtosis parameters of a pure-jump Lévy model i the presece of a high-frequecy oise compoet as it is the case i this paper. The empirical performace of some stadard parametric methods, i the absece of a microstructure oise, has bee studied i a few works such as ad Seeta 004, Ramezai & Zeg 007, Behr & Pötter 009, ad Figueroa-López et al. 0.
We begi our aalysis with a derivatio of approximate Method of Momet Estimators MME for σ ad κ, respectively deoted by ˆσ,T ad ˆκ,T throughout the remaider of the itroductio. 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 fact, momet type estimators are arguably the oly possible estimators that ca efficietly be applied for high-frequecy data due to the extremely high computatioal ad umerical burde associated with the volume of such data. Oce the MMEs have bee itroduced, we aalyze the behavior of the estimators whe ifill asymptotics ad whe T log-ru asymptotics both i the absece ad presece of microstructure oise. We idetify the order OT, whe T, as the rate of covergece of the estimators i the mea-squared error sese uder the ideal situatio of absece of oise. Hece, the goal is to develop estimators that are able to achieve at least this rate of covergece i the presece of microstructure oise. The asymptotic aalysis of the estimators i the presece of oise allows to qualitatively characterize the behavior of ˆσ,T ad ˆκ,T. I particular, we foud that ˆσ,T ad ˆκ,T 0, as, both of which are stylized facts validated usig real high-frequecy observatios see Sectio 6.3 below. Furthermore, deotig δ the time spa betwee observatios, it turs out that δ ˆσ,T ad δ ˆκ,T coverge to the secod momet ad the excess kurtosis of the microstructure oise, respectively. The latter properties will be useful to devise oise-robust estimators for σ ad κ. I order to develop estimators that are robust to 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, the idea cosists of the three steps. First, we break up the high-frequecy samplig observatios 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. A fudametal problem i the approach described i the previous paragraph is how 3
to determie the umber of subgroups, K, which strogly affects the performace of the estimators. We propose a method to fid approximate optimal values for K. For the estimator of σ, it is foud that the optimal K takes the form 6Eε Kσ 4 := +Eε 3 3,. T σ 4 where ε represets the additive microstructure oise associated to oe observatio of the SBM. 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 4 +Eε 3 3T 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 = 4 5Varε ε 4 5 5,. 3 3 4 T 4 σ 8 while the mea-squared error of the resultig estimator coverges at the rate of C κ Var ε ε 43 5 3 5 T 5, up to costat C κ. Here, ε ad ε represet the microstructure oise associated 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 aiterative procedure i which aiitial 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 superior fiitesample ad empirical performace of the resultig estimators are illustrated by simulatio ad real high-frequecy stock data. I particular, the estimators do t exhibit the lack of robustess as the samplig frequecy icreases as their MME couterparts. 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 4. Sectio 5 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 6 shows the fiitesample performace of the proposed estimators via simulatios as well as their empirical 4
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 model for the price process {S t } t 0 of a risky asset. Cocretely, give some costats σ,κ > 0,θ,b R, the log retur process X t := logs t /S 0 of the asset is assumed to take the form X t = σwτ t +θτ t +bt,. where W := {Wt} t 0 is a Wieer process 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 3.. The coditio.-iii is imposed so that X t admits fiite momets of arbitrary order. I the formulatio., τ plays the role of a radom clock aimed at icorporatig variatios i busiess activities through time. It is well kow that the resultat process X is a Lévy process see, e.g, Sato 999. Hereafter, ν will deote the Lévy measure of X. 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 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 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; 5
3. b is a drift compoet i the caledar time; 4. 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 := 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 =,...,,.4 of size of the desity of X δ. I real markets, log returs at high-frequecy exhibit certai stylized features which caot be accurately explaied by efficiet models such as.4. 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 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 i X := X iδ X i δ = X iδ X i δ + εiδ ε i δ =: i X + ε i,,.6 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 6
icorporated i the derivatio of the estimators. A stadig problem is the to derive iferece methods that are robust agaist a wide rage of microstructure oises. I Sectio 5, we proposed a approach to address the latter problem, borrowig ideas from the semial two-scale 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 are give i closed forms as follows: 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 4 X δ := EX δ EX δ 4 = 3σ 4 κ+6σ θ c 3 τ +θ 4 c 4 τ δ +3µ X δ, where, hereafter, c k Y := d k i k du le e iuy k, u=0 represets the k-th cumulat of a r.v. Y. For the VG model, c 3 τ,c 4 τ = κ,6κ 3, while for the NIG model, c 3 τ,c 4 τ = 3κ,5κ 3. Throughout the rest of the paper, we assume that θ = 0, which is cosistet with extesive empirical evidece suggestig that the skewess of high-frequecy data is egligible. To assess the latter assertio, we cosider the sample cetral momet of third order, ˆµ 3,, divided by δ = T/. Accordig to 3., θ will be close to 0 whe ˆµ 3, /δ is small. The results are collected i Table. Note that c 3 τ = Eτ 3 = 0 u 3 ν τ du > 0, where ν τ is the Lévy measure of τ. Thus, µ 3 X δ /δ θ 3 c 3 τ θ 3 κ 3/ by Jese s iequality. 7
δ 5 sec 0 sec 30 sec mi 5 mi 0 mi 30 mi INTEL.7 0 4.8 0 4.5 0 4.6 0 4.9 0 4.8 0 4.9 0 4 CVX 4.8 0 6 3.4 0 6 9.5 0 7 3. 0 5.3 0 5 4.4 0 6 3. 0 5 CSCO 7.9 0 5 8.9 0 5 8.8 0 5 7.6 0 5 5.8 0 5 6.3 0 5 7.3 0 5 PFE 9.5 0 4 9. 0 4 9.6 0 4 9.9 0 4 7.9 0 4 7. 0 4 9.6 0 4 Table : Computatio of the sample cetral momet of third order, ˆµ 3,, divided by δ for differet stocks based o high-frequecy data durig the year of 005 T = 5 days. The simplifyig assumptio θ = 0 allows us to fid tractable expressios for the MME of the parameters σ ad κ as follows: σ X := δ ˆµ, X, κ X := δ 3 3 ˆµ, X ˆµ4, X where ˆµ k, X represets the sample cetral momet of k th order as defied by, 3. ˆµ k, X := i X X k, k, X := i X = log S T S 0. We ca further simplify the above statistics by eglectig the terms of order Oδ = O/ as follows: ˆσ X := T [X,X], ˆκ X := δ 3 i X4 i X = 3 T [X,X] 4 T, 3.3 [X,X] where above we have expressed the estimators i terms of the realized variatios of order ad 4: [X,X] := ix, [X,X]4 = ix 4. Note that, i the geeral case θ 0, we ca see the estimators 3.-3.3 as approximate Method of Momet Estimators up to first order. We ow proceed to show some simple i-fill with fixed T asymptotic properties of the estimators i 3.-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. 3.4 8
From the above formulas, we coclude the ot surprisig fact that ˆσ is ot a measquared cosistet estimator for σ, whe the samplig frequecy icreases. 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 usig some well kow properties of Lévy processes. For reader s coveiece, we recall those. First, if x x k νdx < for k, the E X t k < for ay t > 0 see, e.g., Sato 999 ad, furthermore, lim t 0 t E Xt k = x k νdx+υ {k=}, 3.5 where ν deotes the Lévy measure of X ad υ is the variace of the Browia compoet of X see, e.g., Lemma. i Asmusse & Rosiński 00. Secod, for ay k, k P Xiδ X i δ [X,X] T := t t T X k +υ T {k=} 3.6 as, where X t = X t X t is the jump size of X at time t ad where the above summatio is over the coutable radom set of times t for which X t 0. It is coveiet to express the above limit process i terms of the jump measure M of X, which is defied by Mu,v] C := #{t u,v] : X t C}. I light of the Lévy-Itô decompositio of X cf. Sato, 999, Sectio 9, M is a Poisso radom measure o R + R\{0} with mea measure EMdt dx = dtνdx. Furthermore, the Poisso itegral T 0 fxmdt,dx := t T f X t iswell-defied if, foristace, fx νdx < e.g., seetheorem0.5ikalleberg x 997. I view of 3.6, we first ote the followig limit, as, lim P ˆκ = lim P κ = T T 0 x 4 Mdt,dx 3 T T 0 := ˆκ T. 3.7 x Mdt,dx The covergece of the correspodig momets also holds true sice 0 δ ˆµ 4, /ˆµ, δ = T <, which follows from the triagle iequality. Thus, lim Eˆκ = lim E κ = Eˆκ T ad lim Varˆκ = lim Var κ = Var ˆκ T. 3.8 9
I order to aalyze the limit values i 3.8, let us ote that, by the Strog Law of Large Numbers, ˆκ T T x νdx 3 x νdx = c 4X 3c X = κ, which suggests that ˆκ has a small bias ad variace whe the time horizo T is large. I fact we have the followig result, which gives a explicit estimate of the bias ad measquared error of the statistic κ T. The result is valid for a geeral pure-jump Lévy process X with fiite momets of sufficietly large order ad its proof is give i Appedix A: Propositio 3. Let c i := c i X be the i th cumulat of X, κ := c 4 /3c, ad suppose that x i νdx < for ay i. The, as T, Eˆκ T = κ+ 3c 4 c 6 c T +OT, 3.9 3c 4 E ˆκ T κ c 8 c 4c 4 c 6 +4c = 4 c T +OT. 3.0 9c 5 4 Properties of the MME uder microstructure oise I this part we study how a microstructure oise compoet affects the estimators itroduced i the previous sectio. I tur, such aalyses will help us to develop bias correctio techiques i the subsequet sectio. 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,. 4. Furthermore, throughout we assume that, for each, ε i, i have idetical distributio with zero mea ad fiite momets of ay order. Moreover, the followig mild assumptios are imposed for ay positive iteger k : i ε i, k P m k ε,, for some m k ε R; 4. ii lim supe ε i, k <. 4.3 Let us remark that the previous assumptios ot oly cover the microstructure white-oise case, where ε t t are i.i.d., but also block depedet sequeces, where ε i, ad ε j, are 0
assumed to be idepedet, wheever i j k, for some fixed positive iteger k. Also, ote that, uder the white oise case, m k ε := E ε, k. Let us first aalyze the ifill asymptotic behavior of the estimators for σ, defied aalogously to 3.-3.3, but based o the oisy observatios: σ X := δ i X X, ˆσ X := [ X, X] T = δ i X. 4.4 The followig simple result is eeded i the sequel. Lemma 4. For each positive iteger m ad k i Xm ε i, k P 0, as. Proof. The secod momet of i Xm ε i, k / ca be clearly writte as E X m E ε, k + E ix m E jx m E ε i, ε j, k. i j The first term above is clearly O δ due to 3.5 ad 4.3. The secod term above ca be bouded i absolute value by E Xm E ε i, k, ad thus, it is agai O δ due to 3.5 ad 4.3. By Markov s iequality, i Xm ε i, k coverges to 0 i probability. We are ow ready to aalyze the asymptotic behavior of the estimators i 4.4. Our first result shows that, whe is large, both ˆσ X ad σ X asymptotically behave like T s T X s + δ B, for some costat B. I that case, for large T, the estimators will asymptotically behave like σ + δ B. Propositio 4. Both estimators ˆσ X ad σ X admit the decompositio ˆσ X = A +B, σ X = Ã + B such that, as, lim P A = lim P Ã = 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. Let us first ote that σ = i δ X X + ε i, ε = i δ X X + ε i, ε + δ δ =: à + B, + B,. The term à coverges to T X = O P /. Clearly, 4. implies that δ B, = i X X ε i, ε s T X s as i light of 3.6 ad the fact that ε i, ε P m ε m ε. Also, usig Lemma 4., δ B, = ix ε i, X ε goes to 0 i probability. Let us cosider the estimators for κ itroduced i 3.-3.3, but applied to the oisy process X: κ X = δ 3 ˆµ 4, X 3 ˆµ, X, ˆκ X := T 3 [ X, X] 4. [ 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 4.3 There exist o-zero costats C ad C such that, as, δ ˆκ X P C, δ κ X P C. Proof. We oly give the proof for κ := κ X. The proof for ˆκ X is idetical. Clearly, 3 δ κ +3 = i i X X 4 X X =: N D 4.5 Observe that D = ix X + ε i, ε + ix X ε i, ε.
By Lemma 4. ad 3.6, the first ad third terms o the last expressio above ted to 0 i probability, while the secod term coverges to m ε m ε by 4.. Regardig the umerator i 4.5, this ca be decomposed as follows: N = ix X 4 + 4 + 4 ix X 3 ε i, ε + 6 i X X ε i, ε 3 + ε i, ε 4. ix X ε i, ε Agai,byLemma4.ad3.6,allthetermsitheexpressioabovetedto0iprobability except the last term that coverges to m 4 ε 4m 3 εm ε + 6m εm ε 3m4 ε. Therefore, as, δ κ P m 4 ε 4m εm 3 ε+6m εm ε 3m 4 ε 3m ε m ε =: C. Remark 4.4 As a cosequece of the proof, it follows that, if m ε = 0, the C = C = m 4 ε 3m ε. I particular, if the microstructure oise ε t t i.5 is white-oise, the the costat coicides with the excess kurtosis, E ε 4 /3E ε, of the radom variable ε := ε ε. 5 Robust Method of Momets Estimators I this sectio, we adapt the so-called two-scale 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 m [ X, X] G l = j=0 Xt ij+ Xt ij l. 3
Next, we partitio the grid Ḡ ito K mutually exclusive regular sub grids as follows: G i := Gi,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 3.3 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 T [ Gi X, X], 5. which is costructed by averagig estimators of the form ˆσ X i 4.4 over sparse subgrids. 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 motiomodel.withθ = 0. Forsimplicity, wealsoassumethatb = 0, whichwo taffect much what follows sice we are cosiderig high-frequecy type estimators ad, thus, the cotributio of the drift is egligible i that case. Furthermore, hereafter we assume that the oise process {ε t } t 0 appearig i Eq..5 is white oise; i.e., the variables therei are idepedetly idetically distributed with mea 0. I particular, the oises of the icremets, ε i, := ε iδ ε i δ, follow a statioary Movig Average MA process with E ε i, = 0 ad E ε i, = Eε. For simplicity, i the sequel ε ad ε deote variables with the same distributio as ε i, ad ε t, respectively. 5. Bias corrected estimators We start by devisig bias correctio techiques for the estimator 5.. Clearly, from 3. ad the idepedece of the oise ε ad the process X, we have: E ˆσ,K = σ +E ε KT K i = σ +E ε K +. 5. KT 4
The relatio 5. 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, recall from Propositio 4. that ˆm, ε := δ ˆσ, = [ X, X]Ḡ as. Hece, a atural bias-corrected estimator would be P m ε = E ε, 5.3 ˆ σ := ˆ σ,k := K K T [ Gi X, X] K + [ X, X]Ḡ KT. 5.4 However, from 5. with K =, E ˆ σ = σ +E ε K + K + σ +E ε = KT K T +K σ, K which implies that ˆ σ is ot truly ubiased but oly asymptotically ubiased whe ad K. Nevertheless, the above relatioships yield the followig ubiased estimator for σ : ˆ σ,k := K +K ˆ σ,k := +K K T [ Gi X, X] K + [ X, X]Ḡ T+K. 5.5 The estimator 5.4 correspods to the small-sample adjusted First-Best Estimator of Zhag et al. 005. We ow devise approximate bias-corrected estimators for κ. I order to separate the problem of estimatig κ ad σ, let us assume that σ is kow. I practice, we will have to replace σ with a accurate estimate such as the estimator 5.5. The aalog of the estimator 5. for κ is give by ˆκ := ˆκ,K := 3σ 4 K K T [ Gi TK K + X, X] 4 5.6 which is actually a ubiased estimator for κ i the absece of the microstructure oise ε see 5.7 below. I the presece of microstructure oise, the bias of 5.6 diverges as the frequecy of observatios icreases. Ideed, from 3. ad the idepedece of the oise ε ad X, let us first ote that EX δ + ε 4 = 3σ 4 κδ+6σ E ε δ+e ε 4 +3σ 4 δ. Therefore, 5
for a subsample G = {t i,...,t im }, with i i m, K E T Gi [ X, X] 4 = 3σ 4 κ+6σ E ε K + + K TK E ε 4 +3σ 4TK K +. 5.7 It is ow clear that, for a fixed K, the bias of the estimator ˆκ diverges to as T/ 0. The above formula also suggests the estimator ˆ κ := 3σ 4 K where K T [ X, Gi X] 4 σ ˆm K +, ε 3σ 4 TK ˆm TK K + 4, ε, 5.8 ˆm 4, ε := [ X, X]Ḡ 4, 5.9 which coverges to E ε 4. 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 is direct ad is omitted. Propositio 5. Let ˆ κ := 3σ 4 +K K T [ Gi X, X] 4 X]Ḡ σ[ X, T K K. + The, ˆ κ is a ubiased estimator for κ. K + [ X, 3σ 4 X]Ḡ 4 5.0 T+K 5. Optimal selectio of K A importat issue whe usig the two-scale procedure 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 5.. The ext result, whose proof is give i Appedix A., gives the variace of ˆσ,K. 6
Theorem 5. The estimator 5. is such that Var ˆσ,K 4σ 4 K = 3 + 4Eε4 K T + 4σ4 3 + 3σ4 κ T + σ4 3K + 8σ Eε KT K K +O +O +O, T KT 5. wherehereafter OaK,,Tforafuctio ak,,tmeasthatoak,,t bak,,t for a absolute costat b idepedet of K,, ad T. Remark 5.3 As a cosequece of 5., for a fixed arbitrary K ad a high-frequecy/loghorizo samplig setup δ 0, T, a sufficiet asymptotic relatioship betwee T ad δ := T/ for the estimator ˆσ,K to be mea square cosistet is that δt or, equivaletly, /T 0. If K is chose depedig o ad T, 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 5. that the bias of the estimator is Bias ˆσ,K = Eε K + TK = Eε TK K Eε. 5. TK Together 5.-5. implies that MSE ˆσ,K 4σ 4 K = 3 + 4σ4 3 + 3σ4 κ T + σ4 3K + 8σ Eε KT K K +O +O +O. T KT + 4Eε4 K T + 4 Eε T K 5.3 Our goal is to miimize the MSE i K whe is large. Note that the oly term that is icreasigik is4σ 4 K/3,whileoutofthetermsdecreasigiK,theterm4 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 4σ 4 K 3 + 4 T K Eε =: MSE ˆσ K. 5.4 The right-had side i the above expressio attais its miimum at the value: 6Eε K = 3. 5.5 T σ 4 7
Iterestigly eough, the value above coicides with the optimal K proposed i Zhag et al. 005 see Eq. 8 therei. Pluggig 5.5 i 5.3 ad, sice δ = T/ 0, it follows that MSE ˆσ K = 4 3 3 3 Eε 3 σ 8 3 T 3 +3κσ 4 T +ot. 5.6 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 5.5. Its proof is give i Appedix A.. Propositio 5.4 The estimator 5.5 is such that Var ˆ σ,k = 4σ 4 K 3 + 4Eε4 +Eε T K +O +O +O. 5.7 K 3 T TK As before, the previous result suggests to fix K so that to miimize the first two leadig terms i 5.7. Such a miimum is give by K = 3 6Eε 4 +Eε T σ 4 3, 5.8 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 5.7, the resultat estimator attais the MSE: MSE ˆ σ K = 4 3 3 3 Eε 4 +Eε 3 σ 8 3 3 T 3 +ot. 5.9 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 3.4. Now, we proceed to study the optimal selectio problem of K for the estimator 5.6 for κ. As with ˆσ,K, we first eed to aalyze the variace of the estimator. Theorem 5.5 The estimator 5.6 is such that Varˆκ,K = 64 T K 3 T K +O. 5.0 5 3 3 The optimal value of K proposed i Zhag et al. 005 lacks the term Eε 4 i the umerator. 8
Now, we propose a method to choose a value of K that approximately miimizes the MSE of the estimator ˆκ,K. Let us first recall from 5.7 that the bias of the estimator ˆκ,K is Biasˆκ,K = Eˆκ,K κ = E ε 4 K + + E ε = E ε 4 +l.o.t. 5. TKσ 4 σ TKσ4 where l.o.t. mea lower order terms. Together, 5.0-5. implies that MSEˆκ,K = 64 5 T K 3 + E ε 4 +l.o.t. 5. 3 T K σ8 As with the estimator ˆσ,K, it is the reasoable to select K so that the leadig terms of the MSE are miimized. The aforemetioed miimum is reached at Pluggig 5.3 i 5., it follows that 5E ε K3 4 = 5. 5.3 96T 4 σ 8 MSE ˆκ 3 K 3 = 45 5 3 3 5 E ε 4 6 5 σ 4 5 T 5 +o T 5, whoserateofcovergece to0isslower thatherateofo T /3 attaiedbytheestimator ˆσ K Ḟially, we cosider the ubiased estimator for κ itroduced i Propositio 5.. Below, l.o.t. refers to lower order terms. Theorem 5.6 The estimator 5.0 is such that where eε = Varε ε 4. Var ˆ κ 64T K 3,K = + 5 3 9σ 8 T Keε+l.o.t., 5.4 The two terms o the right-had side of 5.4 reach their miimum value at K 4 = 4 5 5 5eε. 5.5 76T 4 σ 8 After pluggig K 4 i 5.4, we obtai that MSE ˆκ K 4 = 8 5 5 5 3 9 5 eε 3 5 σ 4 5 3 5 T 5 +o T, 9
whichagai, sicet/ 0,impliesthatMSE ˆκ K 4 = ot. Theaforemetioedresult should be compared to 3.0, which essetially says that the estimator ˆκ K 4 has better efficiecy tha the cotiuous-time based estimator ˆκ T, obtaied by makig i the estimators ˆκ ad κ see 3.7. It is worth poitig out here that oe ca devise a cosistet estimator for eε i light of the relatioships [ X, X]Ḡ 4 P Eε ε 4, [ X, X]Ḡ 8 P Eε ε 8. 5.6 6 Numerical Performace ad Empirical Evidece I the sequel, we cosider a Variace Gamma VG model with white Gaussia microstructureoise. Thevariaceoftheoiseε t isdeotedby sothattheoiseofthei th icremet, ε i,, is N0,. Other parameters are set as: σ = 0.0, κ = 0.3, ad = 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. 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 5.8 or 5.5, 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 superior fiite-sample ad empirical performace of the resultig estimators are illustrated by simulatio ad a real high-frequecy data applicatio. For briefess, i the sequel we shall make use of the followig otatio 6m Km,σ := T σ 4 6. Estimators for σ 3, K m,m 4,σ := 3 6m4 +m T σ 4 We compare the fiite sample performace of the followig estimators:. The estimator ˆσ,K optimal value K 3. give i 5. with K determied by a suitable estimate of the give i 5.5. As show i Propositio 4., a cosistet estimator 0
for m ε = E ε = Eε = is provided by ˆm, ε := δ σ suggests the followig cosistet estimate for Eε : = [ X, X]Ḡ /, which ˆ := Êε := [ X, X]Ḡ. 6. The oly missig igrediet for estimatig 5.5 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 ˆ,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 ˆ,ˆσ. Fially, we set ˆσ := ˆσ, ˆK. We cosider the bias-corrected estimator ˆ σ,k itroduced i 5.5, 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 is defied aalogously to ˆK, but replacig ˆσ with ˆσ. 3. Fially, we also cosider the estimator ˆ σ,k itroduced i 5.5 but usig the optimal value K i 5.8. Cocretely, we set ˆσ 3 = ˆ σ, ˆK with ˆK := ˆ, K ˆ,σ 0, where σ 0 is a iitial reasoable value for σ ad ˆ is a cosistet estimator for Eε 4. Next, we improve the estimate of ˆσ 3 by settig ˆσ 3 := ˆ σ, with, ˆK ˆK := K ˆ, ˆ,ˆσ 3. 6. To estimate Eε 4, we use 5.9. Cocretely, as show i the proof of Propositio 4.3, the statistics ˆm 4, ε := [ a cosistet estimate for Eε 4 is give by X, X]Ḡ 4 / coverges to E ε 4 = Eε 4 +6Eε. Therefore, ˆ := Êε4 := [ X, X]Ḡ 4 3. Êε 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. 5.6 ad 5.9.
δ ˆσ ˆσ ˆσ ˆσ ˆσ 3 ˆσ 3 5 mi mi 30 sec sec Mea 0.074333 0.00666 0.099858 0.0988843 0.0999695 0.099964 Std Dev 0.00068548 0.00434344 0.00079454 0.00479476 0.0008839566 0.0007044640 MSE 7.995654e-06.74604e-06 6.35694e-07.5698e-06 7.83885e-07 4.96843e-07 Mea 0.088498 0.006693 0.0995456 0.098484 0.099737 0.00004 Std Dev 0.00064839 0.00060565 0.0007468549 0.0060905 0.0007887707 0.0006469303 MSE 8.7433e-06.57774e-06 5.598574e-07.37075e-06 6.95e-07 4.8547e-07 Mea 0.093765 0.00755 0.0998865 0.0993685 0.0000009 0.000709 Std Dev 0.0006537998 0.000690 0.00075576 0.00497640 0.00078558 0.000636466 MSE 9.0579e-06.6939e-06 5.649076e-07.35945e-06 5.6794e-07 4.05330e-07 Mea 0.09604 0.007658 0.0998938 0.09940 0.000040 0.000068 Std Dev 0.000634697 0.000546469 0.000785086 0.004567 0.000639388 0.00059739 MSE 9.66839e-06.6987e-06 5.308377e-07.306553e-06 4.0886e-07 3.56838e-07 Table : Sample meas, stadard deviatios, ad mea-squared errors for differet estimator of σ = 0.0 based o 000 simulatios. 6. 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 5.6 with σ replaced with the estimate ˆσ 3 i Eq. 6. ad K determied by a suitable estimate of the optimal value K 3 give i 5.3. To estimate E ε 4, we used the statistic i the first limit i Eq. 5.6.. The ubiased estimator ˆ κ defied i 5.0 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 5.0 replacig σ with ˆσ 3, but ow the value of K is give by 5.5. We replace σ therei with ˆσ 3, while to estimate eε = Varε ε 4, we exploit the limits i 5.6. 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 ˆσ 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 0.99840 0.9967835 0.5748966 0.98936 0.9686684 Std Dev 0.783893 0.836394 0.097904650 0.573096 0.59975870 0.075809358 MSE.08994e-0 3.358543e-0 9.586563e-03 9.9953e-0.56455e-0 5.755750e-03 δ = 30 sec δ = sec Mea 0.58784 0.999056 0.967773 0.573787 0.904678 0.945534 Std Dev 0.6799873 0.63678990 0.06934758 0.6874998 0.65066890 0.066836990 MSE.05064e-0.6793e-0 4.89465e-03.04499e-0.733795e-0 4.496860e-03 Table 3: Sample meas, stadard deviatios, ad mea-squared errors for differet estimator of κ = 0.3 based o 000 simulatios. 6.3 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 4.-4.3, 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 ˆ defied i 6., the estimator ˆσ,K defied i 5. with K =, the estimator ˆ σ,k defied i 5.5 with K = ˆK as give i 6., the estimator ˆκ,K defied i 5.6 with K =, ad fially the estimator ˆ κ,k defied i 5.0 with K = ˆK 4 as give i 5.5. 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 4-??, the estimators ˆ σ ad ˆ κ do ot exhibit the drawbacks of the estimators ˆσ ad ˆκ at high frequecies. As a cosequece of the empirical results therei, for istace, Itel s stock exhibits a aualized volatility σ of about 0.04 5 = 0. per year, while its excess kurtosis icreases with /δ at a rate of about 0.5. I 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 3
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. ˆ ˆσ, ˆ σ, ˆK ˆκ, ˆ κ, ˆK 4 0 mi 0.00988 0.0373969 0.03565 0.77846688 0.645084939 0 mi 0.00584536 0.0399567 0.03833 0.589344904 0.7708959 5 mi 0.005404 0.04394983 0.035377 0.495378704 0.76830688 mi 0.000558856 0.05590867 0.0365998 0.3499494734 0.79349570 30 sec 0.00043675 0.06494093 0.039405766 0.879954 0.687574045 0 sec 0.000348354 0.06858945 0.0459630 0.56680373 0.657549576 0 sec 0.0007869 0.08560843 0.04574963 0.833444 0.5993405 5 sec 0.0007435 0.003806 0.0478887 0.08457006 0.4987667343 Table 4: Estimatio of the parameters σ ad κ of a subordiated Browia motio with microstructure oise for INTC Itel stock. ˆ ˆσ, ˆ σ, ˆK ˆκ, ˆ κ, ˆK 4 0 mi 0.0030884 0.0443934 0.047933 3.55809339 3.665645436 0 mi 0.0067865 0.0486633 0.039679 3.33040039 4.96333 5 mi 0.00394 0.058049 0.03758805 3.3955939 4.45884370 mi 0.00058559 0.06437 0.048960.885849749 3.0747770 30 sec 0.000437978 0.073003060 0.047847384.009477905.539989978 0 sec 0.0003733763 0.08063435 0.04958930.88909947.3587546 0 sec 0.0003068 0.0067063 0.050440707.039570694.394987 5 sec 0.00054700 0.04644060 0.0539585 0.555478783.3750789809 Table 5: Estimatio of the parameters σ ad κ of a subordiated Browia motio with microstructure oise for PFE Pfeizer stock. 4
A Proofs A. Proof of Propositio 3.. We shall eed the followig result that ca be show easily from the momet geeratig fuctio for Poisso itegrals: Lemma A. Suppose that M is a Poisso radom measure with mea measure m defied o a ope domai of R d 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 < for k =,...,5, the E k Mf = mf k, for k =,3, E 4 Mf = 3mf + mf 4, ad E 5 Mf = 0mf mf 3 + mf 5. Similarly, E k Mg Mf = mgf k ad E 3 Mg Mf = mgf 3 +3mf mgf. Lemma A. Let M be the Poisso jump measure of a Lévy process X with Lévy measure ν ad let Mdx,dt := Mdx,dt νdxdt be the correspodig compesated measure. Also, suppose that f is such that fx k νdx < for some k. The, for ay T, there exists a costat A k f such that E T T 0 fx Mdx,dt k A k ft k/. Proof. If T is a positive iteger, the result is a direct cosequece of the followig iequality see Lemma 5.3. i Bickel & Doksum 00: E Z µ Z k C k E Z k k/, A. where Z = Z i, µ Z = EZ, ad {Z i } i are i.i.d. such that E Z k <. For geeral T, let [T] the iteger part of T. The, we ca write T T 0 fx Mdx,dt = T [T] 0 fx Mdx,dt+ T T [T] fx Mdx,dt. Hece, E T T 0 fx Mdx,dt k k E [T] [T] 0 [T] T fx Mdx,dt k + k T k E T [T] 0 fx Mdx,dt k 5
For the first term o the right-had side above, we apply iequality A.. For the secod term, ote that by Burkholder-Davis-Gudy iequality see Protter 004, T [T] k E fx Mdx,dt t E sup fx Mdx,dt k BkE k f xmdx,dt t 0 0 0 k/. This suffices to obtai the iequality of the lemma. Proof of Propositio 3.. Usig the idetity ad the otatio +x = k i i+x i + k x k +x k ++kx, A. ˆµ T k := T i=0 T we have the followig decompositio: { } Eˆκ T = 3c X E ˆµ T 4 + ˆD = T 0 x k Mdt,dx, ˆDT := ˆµT c X, { 3c X E + 3 E { =: L T +R T. } ˆµ T 4 ˆD T +3ˆD T 4ˆD T 3 +5ˆD T 4 6ˆD T 5 } ˆµ T 4 ˆµ T 7+6ˆD T ˆD6T LetusfirstaalyzetheresidualtermR T usigthefollowigeasycosequece ofthetriagle iequality: ˆµ T 4 / = / X T / s 4 s T T / s T X s = T / µ T. A.3 Thus, sice 7+6ˆD 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 ca be see to be c 4 X 3c X c 6X 3c 3 X T + c 4 X c 4 X T +OT. 6
The last two term of L T ca be see to be OT from Lemma A. ad Cauchy iequality. Ideed, Eˆµ T 4 ˆD 4 T c 4 X EˆD T 4 + c 4 X E Kc T + E ˆµ T 4 c 4 X 4 ˆµ T 4 c 4 X ˆµ T c X E which is OT i light of Lemma A.. We fially obtai that ˆµ T c X 8 /, Eˆκ Eˆκ T = c 4X 3c X c 6X 3c 3 X T + c 4 X c 4 X T +OT. I order to show the boud for the variace, we use agai A. to get ˆκ T = ˆµT 4 3c X ˆD T +3ˆD T 4ˆD T 3 + ˆµ T 4 3 5+4ˆD T ˆD4 T. The, ˆµ T ˆκ T c 4X 3c X = ˆµ T 3c 4 c 4 X ˆµT 4 X 3c X ˆD T + ˆµT 4 c X ˆD T 4ˆµT 4 3c X ˆD T 3 + 3 ˆµ T 4 ˆµ T 5+4ˆD T ˆD4 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: { } ˆµ 9c 4 X E T 4 c 4 X + 4 { } 9c 4 X E ˆµ T 4 ˆD T. 4 { } 9c 4 X E ˆµ T 4 c 4 X ˆµ T 4 ˆD T Subtractig c 4 X from ˆµ T 4 ithe secod adthird terms above, ad usig agailemmas A. ad A., we ca check that the above expressio ideed coicides with the expressio i 3.0. A. Proofs of Sectio 5. Proof of Theorem 5.. Clearly, Var ˆσ,K = K T i<j K Cov [ X, Gi X],[ 7 X, X] Gj + K T K Var [ Gi X, X]. A.4
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 Cov Xti +K Xt i, Xt j +K Xt j + j Cov Xti +K Xt i +K, Xt j +K Xt j = i CK +i jδ + j Cj 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 +4CovSU,SV = Var S +EVCov S,S +EUCov S,S +4EUEVVarS. Fially, usig that EU = EV = 0 as well as the momet formulas i 3., Cδ = VarS is give by Cδ = σ 4 δ +3σ 4 κδ. Usig the previous formula together with the fact that i /K <, the first term i A.4, which we deote A, ca be computed as follows: A = K K T + K K T = K K KT i<j K i<j K σ 4 j i δ +3σ4 κj iδ σ 4 K +i j δ +3σ 4 κk +i jδ +R 3σ 4 κkδ + 3 σ4 KK δ +R, where R is such that R 4K KT 3σ 4 κkδ + 3 σ4 KK δ. A.5 8
Now, we cosider the secod term i A.4, which we deote B. Each variace term of B ca be writte as B i := Var [ X, = i q=0 i + Gi X] Var Xti +q+k Xt i +qk 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 4 δ +3σ 4 κδ +8σ E ε δ +E ε +E ε 4 Cov Xt+δ Xt, Xv Xt+δ = E ε 4 E ε, valid for ay t < t+δ < v, we get B i = i σ 4 Kδ +3σ 4 κkδ +8σ E ε Kδ + i E ε 4 +E ε. Therefore, usig that K i = K +, B = A.6 K + K T σ 4 Kδ +3σ 4 κkδ +8σ E ε Kδ +4E ε 4 KT E ε 4 E ε. Puttig together A ad B above, Var ˆσ,K K = 3σ 4 κkδ K KT + 3 σ4 KK δ +R K + + σ 4 Kδ K T +3σ 4 κkδ +8σ E ε Kδ +4E ε 4 E ε 4 E ε. A.7 KT Recallig that δ = T/ ad usig A.5, we get the expressio 5.. Proof of Propositio 5.4. Let a.k := Var ˆ σ,k = a,k Var ˆσ,K +b,k Var K ad b +K,K := K+. Clearly, T+K [ X, X]Ḡ a,k b,k Cov ˆσ,K,[ X, X]Ḡ 9
From the expressios i Eqs. A.6-A.7, we have Var [ X, X]Ḡ = σ 4 δ +3σ 4 κδ +8σ E ε δ + E ε 4 +E ε 3σ 4 κkδ + 3 σ4 KK δ +R Var ˆσ,K K = K KT K + + σ 4 Kδ K T +3σ 4 κkδ +8σ E ε Kδ +4E ε 4 E ε 4 E ε. KT To compute the last covariace, let us first ote that Cov ˆσ,K,[ X, X]Ḡ = KT K Cov [ Gi X, X],[ X, X]Ḡ Each covariace term o the right had side above is give by B i = i q=0 =: KT K B i. 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 +e i r=0 Cov XtK Xt 0, Xt r+ Xt r, r=0 A.8 where above e i deote the umber of subitervals i { [t i +qk,t i +q+k ] } i which itersect the ed poits 0 ad T. Note that K e i =. Now, we use the followig formulas, which ca be directly computed Cov Xv Xu, Xv Xu = σ 4 v u +3κσ 4 v u, u < u < v < v Cov Xt Xs, Xu Xt = Eε 4 Eε, s < t < u. We the get B i = i {Kσ 4 δ +3κσ 4 δ +Eε 4 Eε } e i Eε 4 Eε. Next, usig that K i = K + ad K e i =, Cov ˆσ,K,[ X, X]Ḡ = K + T Puttig together the previous relatioships, Var ˆ σ,k 4σ 4 K = 3 + 4Eε4 +Eε +O T K σ 4 δ +3κσ 4 δ + K TK Eε4 Eε. 30 +O +O. K 3 T TK
Proof of Theorem 5.5. The proof is similar to that of Theorem 5.. First ote that Varˆκ,K = 9σ 8 K T i<j K Cov [ Gi X, X] 4,[ X, X] Gj 4 + 9σ 8 K T K Var [ Gi X, X] 4 =: A+B. A.9 Each covariace i the first term o the right had side above is give by Gi Gj A i,j := Cov [ X, X] 4,[ X, X] 4 = i q=0 j r=0 Cov Xti +q+k Xt i +qk 4, Xt j +r+k Xt j +rk 4 = i Cov Xti +K Xt i 4, Xt j +K Xt j 4 + j Cov Xti +K Xt i +K 4, Xt j +K Xt j 4 = i Cj iδ,k +i jδ,j iδ + j CK +i jδ,j iδ,k +i jδ, where, for ay t,s,s,s 3 > 0, Cs,s,s 3 := Cov Xt+s +s X 4 t, Xt+s +s +s 3 X t+s 4, A.0 which agai ca be proved to be idepedet of t. Cocretely, with the otatio S := X t+s +s X t+s, U := X t+s X t + ε t+s +s ε t, ad V := X t+s +s +s 3 X t+s +s + ε t+s +s +s 3 ε t+s Cs,s,s 3 = Cov S +U 4, S +V 4 = Var S 4 +6 [ EU +EV ] Cov S 4,S +36EU EV Var S +6EU 3 EV 3 VarS where above we used the idepedece of S, U, ad V as well as the fact that EU = EV = ES k = 0 for ay odd positive iteger k. Upo computatio of the relevat momets of U ad V, we get Cs,s,s 3 = Var [ Xs 4 +6 σ s +s 3 +4Eε ] Cov Xs 4,Xs +6 σ s +Eε σ s 3 +Eε Var X s +4 Eε 3 VarXs. A. 3
Note that EX k s = E σw τs k = σ k E W k E τ k/ s = σ k E k/ W k s k/ + for some costat a k,i s. we ow proceed to aalyze each term separately: The cotributio to A due to Var X 4 s : A := K 9σ 8 K T i<j K Var XK+i jδ 4 + K 9σ 8 K T i<j K a k,i s i, Var X 4 j iδ. Usig that VarXt 4 is a polyomial of degree 4 i t with the highest-degree term beig 96σ 8 t 4, A = K 9δ 4 9K T = 9 T K 3 +O 59 3 i<j K T K K +i j 4 + 3. i<j K j i 4 +O K 5 Let us aalyze the cotributio to A due to Var X s. The leadig term is give by: A := 6 K +6 K 9σ 8 K T 9σ 8 K T i<j K i<j K = 643 T K 3 +O 59 3 σ j iδ Var X K+i jδ σ K +i jδ Var X j iδ T K The cotributio to A due to Cov X 4 s,x s is give by: A 3 := 6 K +6 K 9σ 8 K T 9σ 8 K T i<j K i<j K = T K 3 +O 549 3 3 σ j iδ Cov X K+i jδ,x 4 K+i jδ σ K +i jδ Cov X j iδ,x 4 j iδ T K 3 where above we used that CovX s,x 4 s = EX 6 s EX sex 4 s = σ 6 s 3 +l.o.t.. 3
Fially, the cotributio to A due to VarX s will geerate a term of smaller order tha T K 3 / 3. Ideed, A 4 := 4 Eε 3 K 9σ 8 K T = 4 9 Eε 3 σ 6 T. i<j K Var XK+i jδ +Var Xj iδ Puttig together the above relatioships, A = 9 T K 3 + 643 T K 3 + 6 T K 3 59 3 59 3 59 3 = 576 T K 3 T K +O. 59 3 3 T K ++O 3 Now, we cosider the secod term i A.3, which we deote B. Each variace term, B i := Var Gi [ X, X] 4, of B ca be writte as B i = i q=0 i + Var Xti +q+k Xt i +qk 4 Cov Xti +q+k Xt i +qk 4, Xt i +q+k Xt i +q+k 4. q=0 Next, usig argumets similar to those followig A.0, Var Xt+s X t 4 = Var X t+s X t 4 +l.o.t. = 96σ 8 s 4 +l.o.t., A. Cov Xt+s Xt 4, Xt+s +s Xt+s 4 = 36σ 4 Eε s s +l.o.t. valid for ay t,s,s > 0 ad where l.o.t. mea lower order terms. Therefore, B i = i 96σ 8 Kδ 4 +OKδ 3 ad, thus, which shows that B = OT K / 3. Fially, B = 96 K T KT +O. 9 3 Varˆκ,K = 576 T K 3 + 96 59 3 9 K T KT +O 3, which implies the result. 33
Proof of Theorem 5.6. Let a.k := Clearly, Var ˆ κ = a,k Varˆκ,K +b,kvar [ +c,kvar [ X, X]Ḡ a,k c,k Cov K, b +K,K := X, X]Ḡ 4 a,k b,k Cov ˆκ,K,[ X, X]Ḡ 4 +b,k c,k Cov ˆκ,K,[ X, X]Ḡ K+, ad c 3σ 4 T+K,K :=. σ [ X, X]Ḡ 4,[ X, X]Ḡ As i the case of the variace of ˆ σ,k, we are lookig for the terms of the highest power of K ad the terms with the highest power of ad least egative power of K. For Varˆκ,K, the highest power of K is give by 5.0. To fid the highest power of, we recall from the proof of Theorem 5.5 that the variace ca be decomposed ito two terms, called A ad B therei. The term with the highest power i A is due to the term 4 Eε 3 VarX s i A. ad is of order 0. I order to determie the term with the highest power of i B, ote that this will be due to the costat terms of the variace ad covariace i Eqs. A.. These are give by Var Xt+s X t 4 = Var ε ε 4 +h.o.t., A.3 Cov Xt+s Xt 4, Xt+s +s Xt+s 4 = Cov ε ε 4, ε 3 ε 4 +h.o.t. where h.o.t. meas higher order term as powers of s, s, ad s. These terms cotribute to B as follows: B := 9σ 8 K T K Var [ Gi X, X] 4 = 9σ 8 K T dε+h.o.t., wheredε := Varε ε 4 +Cov ε ε 4, ε 3 ε 4. Nowwecosiderb,K Var [ As doe with B, the term with the highest degree i is dε. Clearly, all the terms 9σ 8 T K ic,k [ Var X, X]Ḡ areoflowerordertha/t K. TocomputeCov ˆκ,K,[ X, X]Ḡ 4, let us first ote that Cov ˆκ,K,[ X, X]Ḡ 4 = 3σ 4 KT K Cov [ Gi X, X] 4,[ X, X]Ḡ 4 =: 3σ 4 KT K B i. A.4 X, X]Ḡ 4. 34
Each covariace term o the right had side above is give by B i = i q=0 Cov Xti +q+k Xt i +qk 4, Xt r+ Xt r 4 r=0 = i e i Cov Xti +K Xt i +K 4, Xt r+ Xt r 4 +e i r=0 Cov XtK Xt 0 4, Xt r+ Xt r 4, r=0 where above e i deote the umber of subitervals i { [t i +qk,t i +q+k ] } i which itersect the ed poits 0 ad T. Note that K e i =. Now, it turs out that Cov Xv Xu 4, Xv Xu 4, u < u < v < v A.5 Cov Xt Xs 4, Xu Xt 4 = Cov ε ε 4, ε 3 ε 4 =: gε, s < t < u, where here a b meas lim a /b R\{0}. We the get B i = i gε e i gε+o.t.. Next, usig that K i = K + ad K e i =, Cov ˆκ,K,[ X, X]Ḡ 4 = 3σ 4 K TK gε+o.t. Therefore, the cotributio here is 4 gε Give that c 9σ 8 T K,K is of order, it is ot hard to see that the term a,k c,k Cov ˆκ,K,[ X, X]Ḡ is of a order smaller tha. Fially, cosider the term correspodig to D := Cov [ X, X]Ḡ 4,[ X, X]Ḡ. Note that D = Cov Xtq+ X tq 4, X tr+ X tr q=0 r=0 = Cov Xt X t0 4, X t X t0 +Cov Xt X t0 4, X t X t Cov Xt X t0 4, X t X t Usig A.5, it is clear that D. Hece, b,k c,k Cov [ X, X]Ḡ 4,[ X, X]Ḡ Fially, we obtai that Var ˆ κ 64T K 3,K = + 5 3 which implies the result. 9σ 8 K T dε+ 9σ 8 T K dε 3σ 4 TK. 4 9σ 8 T K gε+l.o.t. 35