Model checks for the volatility uder microstructure oise Mathias Vetter ad Holger Dette Ruhr-Uiversität Bochum Fakultät für Mathematik 4478 Bochum Germay email: mathias.vetter@rub.de; holger.dette@rub.de FAX: +49 34 3 4 559 July, 9 Abstract We cosider the problem of testig the parametric form of the volatility for high frequecy data. It is demostrated that i the presece of microstructure oise commoly used tests do ot keep the preassiged level ad are icosistet. The cocept of preaveragig is used to costruct ew tests, which do ot suer from these drawbacks. These tests are based o a Kolmogorov or Cramér-vo-Mises fuctioal of a itegrated stochastic process, for which weak covergece to a coditioal Gaussia process is established. The ite sample properties of a bootstrap versio of the test are illustrated by meas of a simulatio study. AMS Subect Classicatio: 6M, 6G, 6P Keywords ad phrases: goodess-of-t test, microstructure oise, stable covergece, parametric bootstrap, heteroscedasticity Itroductio The volatility is a popular measure of risk i ace with umerous applicatios icludig the costructio of optimal portfolios, hedgig ad pricig of optios. Therefore estimatig ad ivestigatig the volatility ad its dyamics is of particular importace i applicatios ad umerous models have bee proposed for this purpose see e.g. Black ad Scholes 6], Vasicek 5], Cox et al. 8], Hull ad White 5] ad Hesto 4] amog may others]. Because the misspecicatio of the form of the volatility ca lead to serious cosequeces i the subsequet
Model checks for the volatility data aalysis umerous authors recommed to use goodess-of-t tests for the postulated model see e.g. Ait-Sahalia ], Corradi ad White 9], Dette et al. ], Dette ad Podolski ] amog others]. The literature o statistical iferece i this cotext ca be divided ito two classes depedig o the type of available data. The rst class of goodess-of-t tests ca be used, whe the available data cosists of discrete observatios of the process sampled at time poits,, 3,...,, where > is xed ad. The other class of tests addresses the situatio of high frequecy data, where discretely observed data of the price process is available at time,,,..., = T, where T is xed ad which meas that for a icreasig sample size. I the preset paper we cosider the case of high frequecy data, where - i priciple - for a icreasig sample size iformatio about the whole path of the volatility would be available. However, i cocrete applicatios the situatio is much more complicated because of the presece of microstructure oise, which is usually existet i high frequecy data. This additioal oise is caused by may sources of the tradig process such as discreteess of observatios see e.g. Harris 9], ]], bid-ask bouces or special properties of the tradig mechaism see e.g. Black 5] or Amihud ad Medelso 4]]. While microstructure oise has bee take ito accout for the costructio of estimators of the itegrated volatility ad other related quatities see e.g. Zhag et al. 7], Jacod et al. 7] or Podolski ad Vetter ], ]], properties of goodess-of-t tests i this cotext have ot bee ivestigated so far i the literature. Cosider for example the problem, where the process Z t } t,] is observed at the time poits /, /,...,. Uder the assumptio that Z t = X t with dx t = σ t dw t. Dette ad Podolski ] proposed to reect the hypothesis of a costat diusio coeciet i., i.e. H : σ t = σ t, X t = σ, wheever T = sup t,] t Z Z k k t Z Z k k Z Z k k > c α,. where c α deotes the α quatile of the supremum of a Browia Bridge. Now cosider the situatio, where microstructure oise is preset, which is usually modeled by a additioal additive compoet, that is Z i = X i + U i, i =,...,.3 where U i i =,..., } deotes a triagular array of radom variables with mea ad variace ω. I Table we show the ite sample behaviour of the test. for the hypothesis of a costat volatility if σ t = σ t, x = θ + θx ote that the case θ = correspods to the ull hypothesis. We observe that the test keeps its preassiged level oly i the case where ω is rather small. I most cases the omial level is clearly uderestimated. O the other had, the test is ot able to detect ay alterative. A ituitive explaatio for this behaviour
Model checks for the volatility 3 is that i the presece of microstructure oise the variaces of the diereces Z k Z k domiated by the term ω. This leads to icosistet estimates of the itegrated volatility as poited out i Zhag 7]. More precisely, a straightforward calculatio shows that uder microstructure oise the statistic T shows the same asymptotic behavior as the the statistic sup which coverges weakly to t,] t U U k k t U U k k U U k k λ sup B t, t,] are,.4 o matter if the ull hypothesis is valid or ot. Here B t deotes a Browia bridge ad λ = EU k/ /ω 4 ]. This meas that i the presece of microstructure oise the test. has asymptotic level α if ad oly if λ =. I all other cases the test does ot keep its preassiged level. Moreover, because the asymptotic properties uder ull hypothesis ad alterative are the same, the test is ot cosistet. INSERT TABLE HERE] The preset paper is devoted to the problem of costructig a cosistet asymptotic level α test for a geeral parametric form of the volatility i the presece of microstructure oise. I Sectio ad 3 we preset the basic model ad itroduce a stochastic process which ca be used to test parametric hypotheses about the form of the volatility i models with microstructure oise. For this purpose we use the cocept of pre-averagig, which was itroduced i Podolski ad Vetter ] ad exteded i several other papers see e.g. Jacod et al. 7] or Podolski ad Vetter ]] i the cotext of volatility estimatio. Our mai results are preseted i Sectio 4, where we prove stable covergece of two stochastic processes which will form the basis of the proposed ew tests for the parametric form of the volatility. The ew tests ca detect alteratives covergig to the ull hypothesis with a rate /4 ad therefore achieve the optimal rate of covergece i problems of this type see Gloter ad Jacod 3]]. Sectio 5 deals with the problem of testig oliear hypotheses for the volatility. Roughly speakig, this situatio ca be reduced to the liear case usig stadard argumets from oliear regressio models see Seber ad Wild 4]], but there appear iterestig diereces i the asymptotic distributio of the process, if the ull hypothesis is ot satised. I Sectio 6 we ivestigate the ite sample properties of a bootstrap versio of the ew tests ad ivestigate the eect of microstructure oise i the cotext of goodess-of-t testig. I particular, it is demostrated that the ew tests based o the cocept of pre-averagig provide a satisfactory solutio to the problem of checkig model assumptios i the presece of microstructure oise. Fially, all proofs of the results ad techical details are preseted i a Appedix.
Model checks for the volatility 4 Testig parametric hypotheses for the volatility Suppose that the process X = X t t is deed o some appropriate ltered probability space Ω, F, F t t,], P ad admits the represetatio X t = X + a s ds + σ s dw s,. where W = W t t is a stadard Browia motio ad the drift process a ad the volatility process σ satisfy some weak regularity coditios, which will be specied later. Furthermore, we assume that the process ca be observed at discrete poits o a xed time iterval, say, ]. Various assumptios o the structure of the volatility process have bee proposed i the literature, typically depedig o the acial asset, whose price process is modeled by X. Amog such models, a large class ivolves the case where σ is deed to be a local volatility process, thus merely a fuctio of time ad state see e.g. Black ad Scholes 6], Vasicek 5], Cox et al. 8], Cha et al. 7], Ait-Sahalia ] or Ah ad Gao 3] amog may others]. Because a appropriate modelig of the volatility is of particular importace for the costructio of portfolios, hedgig ad pricig, may authors poit out that the postulated model should be validated by a appropriate goodess-of-t test see e.g. Ait-Sahalia ] or Corradi ad White 9]]. I several cases the hypothesis for the parametric form of the volatility is liear ad oe has to cosider the followig two situatios: or H : σ t = σ t, X t = H : σ t = σt, X t = d θ i σi t, X t a.s.. i= d θ i σ i t, X t a.s.,.3 i= where the fuctios σ,..., σ d or σ,..., σ d are kow ad the parameters θ,..., θ d or θ,..., θ d are ukow. Other models ivolve volatility fuctios, where the parameters eter oliearly see Ait-Sahalia ]] ad the correspodig hypotheses will be cosidered later i Sectio 5, because the basic cocepts are easier to explai i the liear cotext. Let us focus o the problem raised i. for the momet, as the testig problem i.3 ca be treated i the same way. Dette ad Podolski ] proposed to costruct a test statistic usig a empirical versio of the stochastic process N t := σ s d = } θ mi σ s, X s ds,.4
Model checks for the volatility 5 where θ mi = θ mi,..., θ mi T := argmi θ R d d σ s d θ σ s, X s } ds. = Thus, oe uses the L distace to determie the best approximatio to the ukow volatility process σ by a liear combiatio of the give fuctios σ,..., σd. It ca easily be see that the ull hypothesis i. is equivalet to N t = t, ] a.s., ad a well-kow result from Hilbert space theory see Achieser ]] implies that where B t = σ s ds ad B i t = N t = B t B T t D C,.5 σ i s, X s ds for i =,..., d, D ad C deote a d d-matrix ad a d-dimesioal vector, respectively, with D i = σ i s, X s σ s, X s ds ad C i = σ s σ i s, X s ds. Note that these quatities deped o the particular path of the process. I practice, oe does ot observe the etire path of the diusio process X = X t t ad it is therefore ecessary to dee a empirical versio based o appropriate estimators for the quatities i.5. Let us briey discuss the solutio to the problem i the case, where the diusio process X = X t t ca be observed at the discrete times t,i = i i without further restrictios. Based o the decompositio above, Dette ad Podolski ] propose to dee a empirical versio Ñ t = B t B T t D C pluggig i appropriate estimators for the ukow quatities. Quite aturally, oe uses a Riema approximatio of each itegral, where oe chooses X k X k as a local estimate for σ k. Thus, D i = C i = σi k, X σ k k, X for i, =,..., d,.6 k σ i k, X X k k X k for i =,..., d,
Model checks for the volatility 6 ad the quatities B t ad B t = B t,..., B d t T are give by B t := t X k X k, Bi t := t σi k, X for i =,..., d..7 k I this cotext oe ca prove a stable cetral it theorem for the process Ñt N t t with the optimal rate of covergece, from which oe may costruct test statistics of Cramér-vo- Mises or Kolmogorov-Smirov type. For example, if d =, σt, X t =, the hypothesis. reduces to the hypothesis of costat volatility cosidered i the itroductio. To be precise, we have D =, C = X X k k, B t = t X X k k, ad B t = t t ad we obtai the process i the Kolmogorov-Smirov statistic., which coverges stably to the supremum of a Browia bridge see Dette ad Podolski ]]. However, as poited out i the itroductio diusio processes observed at high frequecy are cotamiated by microstructure eects such as roudig or bid-ask bouces. I particular, i the presece of microstructure oise the correspodig test for the hypothesis. does ot keep its asymptotic level ad is ot cosistet. Thus a modicatio of the correspodig test statistics is ecessary, which will be discussed i the followig sectios. 3 Assumptios ad deitios I the case of microstructure oise it is less obvious how to estimate the ukow quatities i.5, basically for two reasos: Oe has to to d a local estimator for the ukow volatility fuctio σt which has to be doe i the oiseless framework as well, but becomes more complicated i this settig, ad oe eeds a estimator for the path X t itself, which caot be observed directly. We solve both questios by applyig the idea of pre-averagig, which was itroduced i Podolski ad Vetter ] ad exteded i several other papers see e.g. Jacod et al. 7] or Podolski ad Vetter ]] i the cotext of volatility estimatio. Let us start with some basic assumptios. Sice we are dealig with microstructure oise, we have to dee a secod process Z = Z t t, which is coected to the uderlyig Ito semimartigale X through the equatio Z t = X t + U t for some oise process U. Eve though we assume i the followig that the observatio times are give by t,i = i for i, it will be coveiet to dee the observed process ad thus the oise process as well, eve though it will typically ot be measurable i time for ay t. For this purpose we use a similar settig as i Jacod et al. 7]. We cosider for each t i, ] a probability measure Q t ω, dz, which correspods to the trasitio from X t ω to the observed process Z t o R. Thus, it is atural to dee the space of observatios Ω = R,], equipped with its product Borel-σ-eld F ad the probability
Model checks for the volatility 7 measure P ω, dω, which is the product t,] Q t ω, to esure some sort of coditioal idepedece of the oise variables. Z t t is the give as the caoical process o Ω, F, P with the atural ltratio F t = σz s ; s t. The ltered probability space Ω, F, F t t,], P, o which both processes X ad Z live, is the deed as Ω = Ω Ω, F = F F, F t = s>t F s F s, P dω, dω = P dω P ω, dω. This settig allows for quite geeral forms of oise; however, we restrict ourselves to the case of i.i.d. oise, thus the trasitio probability Q t ω, dz depeds o ω oly through z X t ω ad has the form Q t ω, dz = k z X t ω dz, where k is a desity with bouded support. Furthermore, we assume that the momet coditios EU t ] =, EU t ] = ω, EU 4 t ] < 3. hold. I order to itroduce the pre-averaged statistics we have to dee some further quatities. First, we choose a sequece m, such that } 3. m = κ + o 4 3.3 for some κ >, ad a ozero real-valued fuctio g : R R, which vaishes outside of the iterval,, is cotiuous ad piecewise C ad has a piecewise Lipschitz derivative g. We associate with g ad the followig real valued umbers ad fuctios: g = g m, g = g g+, ψ = g s ds, ψ = gs ds s, ] φ s = s g ug u s du, φ s = gugu s du s i, =, : Φ i = φ isφ s ds 3.4 Furthermore, we dee for a arbitrary process V the radom variables V = V, V = V V, V k = m = g k+v. 3.5 Typically, we have V = X or Z ad for these processes V k ca be represeted as V k = k+m k g s k dv s with g s = m = g, ]s, 3.6
Model checks for the volatility 8 where we use the covetio b a c du s = cu b U a for arbitrary costats a, b ad c. Fially, we set ˆX k = m Z m k+. 3.7 = As poited out before, we eed additioal assumptios o the process X as well as o the give basis fuctios i. ad.3, respectively. Sice the coditios o σi ad σ i are similar, we will restrict ourselves to the rst case oly. It is required that the fuctios σ,..., σd are liearly idepedet ad that each σ i is twice cotiuously dieretiable. Moreover, we assume that E detd β ] < 3.8 for some β >. Regardig the various processes i X, the assumptios are as weak as possible whe testig for.. We simply have to esure that the process i. is well-deed, which follows if we assume that a is locally bouded ad predictable ad that σ is càdlàg. see Jacod ad Shiryaev 8] or Revuz ad Yor 3]]. Whe workig with.3 we propose additioally that the true volatility process σ is almost surely positive ad that is has a represetatio of the form. as well, amely that it satises σ t = σ + a s ds + σ s dw s + v s dv s, where a, σ ad v are adapted càdlàg processes, with a also beig predictable ad locally bouded, ad V is a secod Browia motio, idepedet of W. 4 Goodess-of-t tests addressig microstructure oise The two estimators of iterest are Z k ad ˆX k, which are both local averages of the oisy data, but with slightly dieret ituitios behid them. For the latter oe, the lterig applies to the observatios directly, ad it is easy to see that such a procedure reduces the impact of the oise variables aroud time k ad still provides iformatio about the latet price X, sice k the path of X does ot uctuate too much. For Z k, the averagig happes o the icremets rather tha o the prices, but due to the assumptios o g the iterpretatio is similar: oe reduces the oise eects, but keeps iformatio about the icremets of X. We start with the costructio of a test for the hypothesis. agai. Local estimators for σ ca be obtaied from Z k, but it is well kow that this quatity is ot a ubiased estimate it cotais a itrisic bias due to the oise variables U ad has a dieret stochastic order
Model checks for the volatility 9 tha the icremets X k X k i the o-oise case. Thus, we dee ˆω := i Z, i= which is a cosistet estimator for ω, see Zhag et al. 7]. Mimicig the procedure from the o-oise case preseted i Sectio, we set as well as ad ˆD i := Ĉ i := m κψ σ i k, ˆX k m ˆB t := κψ ˆB i t := We dee at last the process σ k, ˆX k for i, =,..., d, 4. σi k, ˆX k Z k ψ κ ˆω t m t m Z k ψ κ ˆω for i =,..., d, 4. 4.3 σi k, ˆX k for i =,..., d. 4.4 ˆN t = ˆB t ˆB T t ˆD Ĉ, 4.5 which turs out to be a appropriate estimate of the process N t } t,] deed i.4. Our rst result species the asymptotic properties of the process A t} t,] with A t = 4 ˆNt N t. 4.6 Theorem If the assumptios stated i the previous sectios are satised, the process A t t,] deed i 4.6 coverges weakly i D, ] to a mea zero process At t,]. Coditioally o F the itig process is Gaussia, ad its ite dimesioal distributios coicide with the coditioal with respect to F ite dimesioal distributios of the process γ V IV t} Bt T D gv, X V where V U, ], ad γ s ds Bt T D } γ s gs, X s ds, 4.7 t,] gv, X V = σ V, X V,..., σ dv, X V T 4.8 γs = 4 Φ ψ κσ 4s σ + Φ sω κ + Φ ω 4. 4.9 κ 3
Model checks for the volatility Note that the rate of covergece 4 is optimal for this problem, sice it is already optimal for the estimatio of Bt eve i a parametric settig cf. Gloter ad Jacod 3]]. I order to costruct a test statistic based o Theorem we have to dee a appropriate estimator for the coditioal variace of the process At} t,], which is give by s t = γs ds Bt T D γsgs, X s ds + Bt T D γsgs, X s g T s, X s ds D B t. Obviously, we use ˆB t ad ˆD as the empirical couterparts for B t ad D. I order to obtai estimates for the other radom elemets of s t, we dee Γ k = 4 Φ 3 κ ψ 4 + 4 κ 3 Φ ψ Z k 4 + Φ ψ ψ 3 8 κ Φ ψ 3 Φ ψ Z k ˆω + Φ ψ ψ 4 as a local estimator for the process γ ad observe that see Jacod et al. 7]] ψ 4 ˆω 4 ĝ t := ĝ i t = ĝ i = t m t m Γ k P Γ k σ i k Γ k σ i k γ s ds, ˆX k, ˆX k P σ k γ s σ i s, X s ds, ˆX k P γ s σ i s, X s σ s, X s ds. Isertig these estimators i the correspodig elemets of s t that is gives the cosistet estimator, ŝ t = ĝ t ˆB T t ˆD ĝt + ˆB T t ˆD Ĝ ˆD ˆBt, 4. where ĝt = ĝ t,..., ĝ d t T ad Ĝ = ĝ i d i,=. A cosistet test for the hypothesis. is ow obtaied by reectig the ull hypothesis for large values of Kolmogorov-Smirov or Cramér-va-Mises fuctioal of the process } /4 ˆNt. ŝ t t,] I priciple a similar approach ca be used to costruct a test for the hypothesis.3. However, i this case thigs chage cosiderably. Quite aturally, Dette ad Podolski ] restate this hypothesis as M t = t, ] a.s.,
Model checks for the volatility where ad θ mi = θ mi,..., M t := σ s d = mi θ d T := argmi θ R d } θ mi σ s, X s ds 4. σ s d θ σ s, X s } ds. = Obviously, we have a aalogous represetatio as i.5, amely M t = R t R T t Q S, where R t = σ s ds ad R i t = σ i s, X s ds for i =,..., d, ad Q ad S are a d d-matrix ad a d-dimesioal vector, respectively, with Q i = σ i s, X s σ s, X s ds ad S i = However, a appropriate deitio of a empirical versio of the form σ s σ i s, X s ds. ˆM t = ˆR t ˆR T t ˆQ Ŝ requires some less obvious modicatios, because local estimators for σ s are more dicult to obtai i this settig. Usig a pre-averaged estimator of the form Z k agai causes a itrisic bias, but due to the absolute value istead of the square as i the previous settig its correctio turs out to be impossible at the optimal rate. However, it has bee argued i Podolski ad Vetter ] that usig i 3.3 a sequece of a larger magitude tha reduces the impact of the oise terms i Z k. This modicatio makes iferece about σ s possible, though resultig i a worse rate of covergece. To be precise, we x some δ > ad choose l 6 such that l +δ = ρ + o 4 + δ for some ρ >. Usig the sequece l istead of m, we dee all quatities from 3.4 to 3.7 i the straightforward way. Next we set Ŝ i = µ ρψ l 3 4 + δ σ i k, ˆX k Z k for i =,..., d, ad ˆR t = µ ρψ t l 3 4 + δ Z k,
Model checks for the volatility where µ deotes the rst absolute momet of a stadard ormal distributio. Moreover, it is atural to use the followig estimators ˆR t = ˆRˆt,..., ˆR d t T ad ˆQ = ˆQ i d i,=, for the quatities R t ad Q: ad Fially, we dee ˆQ i = ˆR i t := l t l σ i k, ˆX k for ay t, ] ad obtai the followig result. σ i k, ˆX k for i =,..., d σ k, ˆX k for i =,..., d. B t = 4 δ ˆMt M t 4. Theorem If the assumptios stated i the previous sectios are satised, the process B t t,] deed i 4. coverges weakly i D, ] to a mea zero process Bt t,]. Coditioally o F the itig process is Gaussia, ad its ite dimesioal distributios coicide with the coditioal with respect to F ite dimesioal distributios of the process γ } V IV t} Rt T Q ḡv, X V γ s ds Rt T Q γ s ḡs, X s ds, 4.3 t,] where V U, ], ḡv, X V = σ V, X V,..., σ d V, X V T ad γ s = ρξ σs, Ξ = fu = π µ φ s ξs ds, ξs = f, ψ u arcsiu + u. The estimatio of the coditioal variace of the process Bt} t,] r t = γ s ds Rt T Q γ sḡs, X s ds + Rt T D γ sḡs, X s ḡ T s, X s ds Q R t. becomes easier i this cotext. With the otatio Γ k = +δ Ξ Z ψ µ k,
Model checks for the volatility 3 we have ĥ t = ĥ i t = ĥ i = t l t l Γ k P γ s ds Γ k σ i k Γ k σ i k, ˆX k, ˆX k P γ s σ k, ˆX k σ i s, X s ds P γ s σ i s, X s σ s, X s ds ad cosequetly a cosistet estimator ˆr t for the coditioal variace is give by ˆr t = ĥt ˆR T t ˆQ ĥt + ˆR T t ˆQ Ĥ ˆQ ˆRt, 4.4 where ĥt = ĥt,..., ĥdt T ad Ĥ = ĥi d i,=. A cosistet test for the hypothesis.3 is ow obtaied by reectig the ull hypothesis for large values of the Kolmogorov-Smirov or Cramér-va-Mises fuctioal of the process } /4 δ/ ˆMt. ˆr t Note that oe kows from previous work that it is either ecessary to dee X to be a Ito semimartigale with cotiuous paths as i. or to model the oise terms U as beig idepedet ad idetically distributed to obtai similar results as i Theorem ad. I fact, for a uderlyig Ito semimartigale exhibitig umps oe ca use bipower-type estimators as discussed i Podolski ad Vetter ] i order to dee a estimator closely related to ˆB t. Moreover, it has bee argued i Jacod et al. 7] that eve for a oise process with càdlàg variace depedig o ω a similar theory as preseted i this paper applies. t,] 5 Noliear hypotheses I this sectio we briey discuss the case of a oliear hypothesis H : σ t = σ t, X t = σ t, X t, θ, 5. where θ Θ R d deotes the ukow parameter. Uder suitable coditios o the parameter space Θ, H ca be restated as N t = t, ] a.s., where the process N t } t,] is deed by N t = B t B t θ := } σs σ s, X s, θ ds.
Model checks for the volatility 4 Here, θ is the parameter correspodig to the best L -approximatio of σs by the parametric class, that is θ = argmi θ Θ gθ, where gθ = A aalogue of the process ˆN t itroduced i 4.5 is give by where ˆB t is deed by 4.3, ad ˆB t ˆθ = σ s σ s, X s, θ} ds. ˆN t = ˆB t ˆB t ˆθ, 5. t m ˆθ = argmi θ Θ g θ, where g θ = σ k, ˆX k, ˆθ, 5.3 m s k σ k, ˆX k, θ} 5.4 s k = Z κψ k ψ κ ˆω. 5.5 From similar argumets as i the proof of Theorem 3 i the Appedix we see that B t θ ˆB t ˆθ = σ t, X t, θ σ t, X t, ˆθ } ds + o p 4. Assumig the commo regularity coditios for oliear regressio see Gallat ] or Seber ad Wild 4]] θ is the uique miimum of g ad attaied at a iterior poit of Θ. It is easy to see that ˆθ θ i probability i this case, ad thus we ca assume that ˆθ satises g ˆθ =. This implies that = g ˆθ = g θ + g θˆθ θ ˆθ θ = g θ g θ for a suitable choice of θ. Moreover, = = g θ = m s k m s k m θ σ k, ˆX k θ σ k, ˆX k s k σ k, ˆX k, θ, θ θ=θ, θ θ=θ σs } θ σ k, ˆX k, θ θ=θ σ s, X s, θ θ σ s, X s, θ θ=θ θ σ s, X s, θ θ=θ ds + o p 4, ds + o p 4
Model checks for the volatility 5 where the last equality follows from the deitio of θ. Thus, the quatity g θ has a similar structure as the term Ĉ C i the liear case, ad i particular it is of order O p 4 as well. Furthermore, we have θ θ i probability, ad thus it ca be assumed that g θ is positive deite ad that the dierece g θ g θ is small. We coclude that ˆθ θ = O p 4, ad thus B t θ ˆB t ˆθ = = θ σ s, X s, θ θ=θ θ σ s, X s, θ θ=θ T ds ˆθ θ + o p 4 Furthermore, the d d dimesioal matrix g θ takes the form g θ = ST S m T ds g θ g θ + o p 4. s k σ k, ˆX k, θ }H k, where the m d matrix S is give by S = θ σ k, ˆX k, θ θ=θ,..., m ad H k deotes the Hessia H k = θ σ k, X, θ k θ=θ. Agai, a similar calculatio as give i the Appedix shows that where the d d matrix g θ = g θ = g θ + O p 4, θ σ s, X s, θ T θ=θ θ σ s, X s, θ θ=θ ds } σs σ s, X s, θ θ σ s, X s, θ θ=θ ds is positive deite. Note that the secod term i this sum vaishes, whe either the hypothesis is liear sice the Hessia is zero or the ull hypothesis is valid sice σ s equals σ s, X s, θ. I these cases the matrix g θ takes precisely the same form as D i the liear settig. I ay case, g θ is of order O p, ad thus we ed up with the represetatio B t θ ˆB t ˆθ = θ σ s, X s, θ θ=θ T ds g θ g θ + o p 4,
Model checks for the volatility 6 ad the asymptotics are drive by g θ. Cosequetly, it follows from the proof of Theorem i the Appedix that i the case of testig a oliear hypothesis of the form 5., the process ˆN t N t } t,] exhibits a similar asymptotic behavior as i the liear case, that is 4 ˆNt N t } t,] = At} t,], where coditioally o F the itig process is Gaussia, ad its ite dimesioal distributios coicide coditioally o F with the ite dimesioal distributios of the process γ V IV t} γ s ds θ σ s, X s, θ θ=θ θ σ s, X s, θ θ=θ T ds g θ T ds g θ θ σ V, X V, θ θ=θ T γ s θ σ s, X s, θ T } θ=θ ds, t,] where the costat γ u is deed i 4.9. We ally ote agai that, i the case of a xed alterative ad a oliear ull hypothesis, this expressio has a dieret structure tha the correspodig term i Theorem. 6 Simulatio study We have idicated i the itroductio that the origial test for a costat volatility from the oise-free model loses its asymptotic properties i the presece of oise. Usurprisigly, for a smaller variace of the oise variables, the data look more like observatios from a cotiuous semimartigale ad thus the test statistics behaves roughly i the same way as before, provided that the sample size is ot too large. O the other had, for a large variace of the error terms these are domiatig, ad thus the whole procedure breaks dow eve for small sample sizes. The same problem arises if the variace of the error is small but the sample size is large see the discussio i the itroductio. We start with a further example simulatig the level of the bootstrap test proposed by Dette ad Podolski ] for a parametric hypothesis, assessig its quality for various sample sizes ad dieret variaces ω. INSERT TABLE HERE] Precisely, we have used the bootstrap test i Dette ad Podolski ] for testig the hypothesis H : σ t, x = θx, where bt, x =.x. The results are obtaied from simulatio rus ad 5 bootstrap replicatios ad displayed i Table for various sample sizes ad stadard deviatios ω of the oise process. We observe that for = 56 ad a small stadard deviatio of ω =. the test does roughly keep its asymptotic level, whereas it caot be used at all whe the variace becomes larger. Moreover, eve if the variace is small but the sample size is icreased, the test does ot keep its pre-assiged level see the results for ω =. ad = 4 i Table. Roughly speakig, we observe from these ad similar simulatio
Model checks for the volatility 7 results that there is o eed for usig tests, which address the problem of microstructure oise, if both the variace of the oise terms ad the sample sizes i our example 56 are small. O the other had, it is kow from empirical research that it is ot realistic to assume extremely large values of ω, but the sample size for high frequecy data is usually much larger tha 56. Cosequetly, i may applicatios tests igorig the presece of microstructure will either keep their pre-assiged level or be cosistet, ad the applicatio of testig procedures addressig the problem of microstructure oise is strictly recommeded. I the followig sectio we illustrate the ite sample properties of a bootstrap versio of the Kolmogorov-Smirov test based o the processes ivestigated i Sectio 4 ad 5. Sice the stochastic order of i Z is basically determied by the maximum of ad ω which are the orders of i X ad i U, respectively, we kept ω =.4 xed i order to have comparable results for dieret sample sizes. The regularisatio parameters κ ad ρ were set to be / each. All simulatio results preseted i the followig paragraphs are based o simulatio rus ad 5 bootstrap replicatios if the bootstrap is applied to estimate critical values. For all testig problems discussed below we have ot used exactly the statistics ˆN t ad ˆM t, but related versios accoutig for ite sample adustmets. Followig Jacod et al. 7], where it has bee show that ite sample correctios improve the behaviour of the estimate ˆB t ad presumably of Ĉ as well substatially, we have replaced the quatities ψ i ad Φ i i 3.4 by certai umbers ψi ad Φ i, which costitute the "true" quatities for ite samples, but are replaced by their its ψ i ad Φ i i the asymptotics. See Jacod et al. 7] for details. 6. Testig for homoscedasticity I the problem of testig for homoscedasticity the itig process At t,] i 4.7 has a extremely simple form, whe the ull hypothesis of a costat process σ t t,] holds. I fact, the ite dimesioal distributios of the process At t,] coicide with the ite dimesioal distributios of the process γiv t} t} t,] for V U, ], which meas that At t,] is a rescaled Browia bridge. Thus we obtai the weak covergece A t ŝ t t,] D B t t,], 6. where B t t,] is a stadard Browia bridge. Of course, this result ca be used to costruct a Kolmogorov-Smirov or a Cramér-vo-Mises test, ad we have ivestigated the properties of the Kolmogorov-Smirov test for dieret sample sizes, where the oise satises U N, ω ad the drift fuctio is agai give by bt, x =.x. A similar test ca be costructed usig
Model checks for the volatility 8 Theorem, but the correspodig results are omitted for the sake of brevity as the rate of covergece i this case becomes worse. INSERT TABLE 3 HERE] I Table 3 we preset the simulated level of the Kolmogorov-Smirov test usig the critical values from the asymptotic distributio. It ca be see that the asymptotic level of the test is slightly uderestimated. This eect becomes less visible for a larger sample size, but eve the it is still apparet. Note that these digs are i lie with previous simulatios o oisy observatios ad it is likely that they are due to the fact the rate of covergece for most testig problems is oly 4, ust as i our case. 6. Testig geeral hypotheses For a geeral ull hypothesis i., the distributio of the itig process At t,] depeds o the path of the uderlyig semimartigale X t t,] ad o the volatility σ t t,], ad thus we caot use it directly for the calculatio of critical values. For this reaso we propose the applicatio of the parametric bootstrap i order to obtai simulated critical values. First we compute the global estimators ˆω ad ˆθ = ˆD Ĉ as well as each 4 ˆNt ad ŝ t from the observed data. Uder the ull hypothesis N t equals zero, ad thus it is ituitively clear that the ull hypothesis has to be reected for large values of the stadardised Kolmogorov-Smirov statistic Y = sup t,] I a secod step, we geerate bootstrap data where Z = X Z i i 4 ŝ t ˆNt., i =,..., =,... β, + U, the X are realisatios of the process i. with b s ad σs = σ s, X s = d ˆθ k σk s, X s correspodig to the ull hypothesis ad each U i is ormally distributed with mea zero ad variace ˆω. Usig these data, we calculate the correspodig bootstrap statistics Y ad use these to compute the quatiles of the bootstrap distributio. Fially, the ull hypothesis is reected if Y is larger tha the α-quatile of the bootstrap distributio. INSERT TABLE 4 HERE] I order to ivestigate the approximatio of the omial level we cosider the hypothesis of costat volatility ad the hypothesis H : σ t, x = θx. The data is geerated uder the ull hypothesis with drift fuctio bt, x =.x ad the reectio probabilities are depicted i Table 4. These results show that the bootstrap approximatio works well eve for a small. I particular, we see that i the case of homoscedasticity the exact asymptotic test usig
Model checks for the volatility 9 the weak covergece of Y to the supremum of a stadard Browia bridge is outperformed compare with Table 3. I the case of testig the parametric hypothesis H : σ t, x = x we observe a slight overestimatio of the omial level by the bootstrap test. As a example for testig the hypothesis H deed i.3 we have chose σt, x = θ x ad ivestigated the properties of the aalogues of Y ad Y from above, where we have replaced 4 ˆNt ad ŝ t by 4 δ ˆMt ad ˆr t, respectively. I this case we chose δ =, correspodig to 4 l = O 3 4 ad a rate of covergece 8. Note that i this particular situatio there is o eed for statig the hypothesis i terms of H as it is equivalet to σ t, x = θ x, but evertheless it gives a reasoable impressio o how well the bootstrap approximatio works for testig hypotheses of the form.3. INSERT TABLE 5 HERE] We observe from the results i Table 5 that eve though the rate of covergece i Theorem is worse tha i Theorem, there is o substatial dierece i the approximatio of the omial level by the bootstrap test for both types of hypotheses: The omial level is slightly overestimated, but i geeral the parametric bootstrap yields to a satisfactory ad reliable approximatio of the omial level. Fially, Table 6 cotais the reectio probabilities of the bootstrap test uder the alterative. The ull hypothesis is give by H : σ t, x = θ x ad two alteratives, amely σ t, x = ad σ t, x = + x, ad oe alterative comig from a stochastic volatility model is cosidered. For this case we chose the Hesto model, i.e. X t = X + µ ν s / ds + σ s dw t with ν t = ν + δ α ν s ds + γ ν / s db s, where ν t = σ t ad CorrW, B = η ad the parameters were chose as µ =.5/5, δ = 5/5, α =.4/5, γ =.5/5 ad ρ =.5. INSERT TABLE 6 HERE] We observe from the results depicted i Table 6 that the bootstrap test idicates i all cases that the ull hypothesis is ot satised. It is also remarkable that it is more dicult to detect the alteratives σ t, x = ad σ t, x = + x tha the oe comig from the Hesto model. I the latter case, the reectio probabilities are extremely large eve for a small sample size, i cotrary to the rst two situatios. 7 Appedix: Proof of Theorem ad Before we come to the proof of the two theorems, we start with a typical localisatio argumet, which allows us to assume that several of the quatities ad processes ivolved are bouded. Recall rst that a ad σ are locally bouded by assumptio, from which is follows that X is
Model checks for the volatility locally bouded as well. Thus we ca coclude alog the lies of Jacod 6] that we may assume without loss of geerality that each of these processes is actually bouded. Sice further each σi is cotiuous ad because U has a compact support, we may coclude that both s, X t ad s, ˆX k σ i s, ˆX k for arbitrary s, t, k ad are livig o a compact set, ad thus σ i s, X t ad are also bouded, the latter oe uiformly i. Similar results hold for the rst two derivatives of σ i as well as for ay of the fuctios σ i. Costats are deoted by K throughout this sectio. 7. Some preparatios The proofs of Theorem ad are based o several preiary results, which will be preseted ad proved i this subsectio. We start with two results determiig the rate of covergece of the quatities ˆB i t B i t ad ˆD i D i deed i.7 ad.6, respectively. The followig result esures that the coditioal variace i a it theorem for ˆN t N t will ot deped o ˆB i t ad ˆD i, sice the rate of covergece will be 4. Thus, we will focus i the followig o the behavior of Ĉi ad ˆB t. Theorem 3 Uder the assumptios from Sectio 3 we have ˆB i t B i t = o p 4, for i =,..., d, 7. ˆD i D i = o p 4, for i, =,..., d, 7. where the rst result holds uiformly with respect to t, ]. Proof of Theorem 3: For a proof of the rst estimate 7. we use for a xed idex i the decompositio ˆB i t B i t = t m σi k, ˆX k σi k t m, X + k Regardig the rst term i this sum, ote that σ i k, X k σi s, X s ds. ˆX k X k = m m = U k+ = m U k+ m = + m m + = k+ k X k+ X k σ s dw s + O p, ad thus ˆX k X k = O p 4. Hece a Taylor expasio gives σi k, ˆX k σi k, X = k y σ i k, X ˆX k k X k + y σ i k, ξ k, ˆX k X k
Model checks for the volatility for some radom variables ξ k, with ξ k, X k ˆX k σ y i k, ξ k, is bouded for all k ad which yields t m Hece, with it suces to prove that σi k, ˆX k σi k, X = t m k A k, = m m y σ i k, X U k k+ = X k. As oted before, we have that y σ i k, X ˆX k k + k+ k σ s dw s X k + O p. t m A k, = o p 4. 7.3 However, we have EA k, A l, ] = O for arbitrary k ad l as well as EA k, A k+l, ] = for l m by coditioig o F k+l. This yields E t m ] A k, = t m k=m m l= m EA k, A k+l, ] + O m = O, ad 7.3 follows. For the secod term i the decompositio of ˆBi t B i t it holds that = = t m σi k, X k t t k k k k Sice by assumptio for k σ i k σ i k σ i s, X s ds, X σ k i s, X s ds + O p, X k σi k σ i s, X k, X k σ i s, X k + σi s, X k σi s, X s ds + O p. < K s k ad by a similar expasio as above the claim follows. The result o ˆD i D i ca be show i the same way. The followig result species the covergece of the ite dimesioal distributios of the processes, which are used for the costructio of ˆN D t } t,]. Below we use the otatio G st G to idicate stable covergece of a sequece of radom variables G to a itig variable G, which is deed o a appropriate extesio Ω, F, F t t,], P of the origial probability space Ω, F, F t t,], P. For details o stable covergece see Jacod ad Shiryaev 8].
Model checks for the volatility Theorem 4 Dee for ay xed t,..., t k, ] the k + d k + d matrix Σ t,...,t k s, X s = γ s ls, X s l T s, X s where ls, X s =,t ]s,...,,tk ]s, g T s, X s T by 4.8 ad 4.9, respectively. The we have ad the vector gs, X s ad γ s are deed 4 ˆB t B t,..., ˆB t k B t k, Ĉ C,..., Ĉd C d T Dst Σ t,...,t k s, X s dw s, where W is aother Browia motio, which is idepedet of the σ-algebra F. Proof of Theorem 4: Observe rst that Ĉi ca be decomposed as follows: Ĉ i = κψ m σ i k, X k Z k ψ κ ω + ψ m σ κ i k ψ, X ω ˆω k + κψ + ψ κ ψ m m σi k, ˆX k σi k, X Z k ψ k κ ω σi k, ˆX k σi k, X ω ˆω. k Sice ω ˆω = O p, the secod ad the fourth term i this sum are of the same order. Moreover, we d from similar argumets as give i the proof of Theorem 3 that the third term is of order o p 4 ad thus asymptotically egligible as well. Therefore we are left to focus o F i = κψ m σi k, X Z k ψ k κ ω. Due to the depedece structure of the summads i F i it will be coveiet to use a "smallblocks-big-blocks"-techique as i Jacod et al. 7] i order to prove Theorem 4. To this ed we choose a iteger p, which later will go to iity, ad partitio the observatios ito several subsets: We dee b k p = kp + m ad c k p = kp + m + pm ad dee p to be the largest iteger k such that c k p m holds, which gives the idetity p =. 7.4 p + m
Model checks for the volatility 3 Moreover, we use the otatio i p = p + pm, ad itroduce for each k p ad ay p the followig radom variables: Gk, p = σ κψ i b kp, X b k p Gk, p = σ κψ i c kp The remaider terms are gathered i Gp 3 = κψ m =i p σ i i p, X c k p c k p =b k p b k+ p =c k p Z ψ κ ω, Z ψ κ ω., X Z ip ψ κ ω. Note that each of these quatities depeds o i, although it does ot appear i the otatio. The mai ituitio behid these quatities is that the terms Gk, p are deed o ooverlappig itervals, which meas that the itervals o which each Z withi Gk, p lives are disoit from ay Z withi ay other Gl, p. This is suciet to esure some type of coditioal idepedece, which will be used i order to prove Theorem 4. The variables Gk, p ad Gp 3 are llig the gaps betwee Gk, p ad Gl, p ad ca be show to be asymptotically egligible. A importat tool will be the followig decompositio of Z. We set Vs +s = g u a u du + +s g u σ u dw u, ad obtai from the represetatio of X as i 3.6 ad by a applicatio of Ito's formula X = Thus, +m Vs g s a s + gs +m σ s ds + Vs g s σ s dw s. Z = X + U + X U = + +m +m + U V s g s a s ds + +m g s σ s ds + U + U +m g s σ s dw s =: V s +m g s σ s dw s g s a s ds 6 D l, 7.5 l=
Model checks for the volatility 4 where the last idetity dees the quatities D l For b k p < c k p we itroduce further Dk,, p = σ b k p Dk,, p 6 = σ b k p +m U +m i a obvious maer. +s g u dw u g s dw s, g s dw s as approximatios for the quatities D ad D 6. Additioally, we set where Y k, p = Fially, we dee κψ χp k = Hk, p = σ i b kp c k p =b k p E, X b k Y k, p, p Dk,, p + Dk,, p 6 + D 4 ψ } κ ω. 7.6 sup s,t b k p, c k p F a s a t + σ s σ t bkp ] ]. We start with two auxiliary results which specify the asymptotic properties of F i ad prove the rst assertio i detail. Lemma We have sup 4 p Gk, p + Gk, p + Gp 3 C i k= p k= } Hk, p =. Proof of Lemma : The proof goes through a rather large umber of steps ad makes extesive use of the decompositio i 7.5. We will show rst that the iuece of the radom variables D ad D 5 withi Gk, p is asymptotically egligible, that is p sup 4 k= σ i b kp, X b k p c k p =b k p D + D 5 =. 7.7 Completely aalogous results hold for the correspodig results o Gk, p ad Gp 3 as well. For a proof of 7.7, assume without loss of geerality that b k p < c k p, ad thus we have the decompositio D = D + D with D = D = +m +m +s +s g u a u du g s a s ds, g u σ u dw u g s a s ds.
Model checks for the volatility 5 Obviously, we have E D oly. Usig the decompositio D = a b k p + +m F b k p +m +s ] K, which allows us to focus o the secod term +s g u σ u dw u g s ds g u σ u dw u F b k p g s a s a b k p ds, the martigale property of a stochastic itegral with respect to Browia motio ad the Cauchy-Schwarz iequality we derive that ] E D K 3 4 χp k. Thus with the otatio δk, p = c k p =b k p D, we coclude that ] E δk, p K p 4 χp k. For the same reasos we have ] E δk, p F bk p = F b k p c k p,l=b k p E D D l F b k p ] K p, ad with k > l it follows E σ i b kp Fially, we obtai, X b k p σ i b lp, X b l δl, p p E δk, p F b k p ]} K p χp k. p E 4 = + K p k= p k>l k= E E σ i b kp σ 4 i b kp p p + σ i b kp k>l, X b k p c k p =b k p D ], X b k δk, p p, X b k p σ i b lp p Eχp k]. ], X b l δk, p p δl, p From Lemma 5.4. i Jacod et al. 7] it follows that p Eχp k ] = for ay p, which gives that the rst term i the sum 7.7 coverges to. For a proof of a correspodig ]
Model checks for the volatility 6 statemet for the secod term, we dee δk, p 5 = c k p =b k p ad obtai from the idepedece of X ad U that ] E δk, p 5 = ad E F b k p Hece, a stadard martigale argumet gives p E 4 k= σ i b kp D 5 δk, p 5 F bk p, X b k δk, p p 5 ] ] K p. K p, which ishes the proof of 7.7. The ext step is devoted to the aalysis of the term D. We prove as well as p sup 4 k= σ i b kp p sup 4 sup 4 k=, X b k p σ i c kp σ i i p c k p =b k p, X ip D Dk,, p = 7.8 b k+ p, X c k p m =i p =c k p D =, 7.9 D =. 7. Set b k p < c k p agai ad observe the decompositio D = D + D, where From we coclude +m D = D = E D F b k p ] +m +s +s g u a u du g s σ s dw s, g u σ u dw u g s σ s dw s. = ad E D D l p sup E 4 k= σ i b kp, X b k p c k p =b k p F b k p ] K 3 ] D =
Model checks for the volatility 7 from a similar martigale argumet as i the previous paragraph ad may thus focus o D. We have ] ] E D = ad E D D l K, F b k p F b k p thus 7. follows easily. For 7.9, ote that E b k+ p =c k p D ] K, which gives recall the deitio of p, b k p ad c k p p k= E σ 4 i c kp, X c k p b k+p =c k p ] D K p = K p, which coverges to zero as p teds to iity. We are thus left to prove p sup 4 k= σ i b kp, X b k p c k p =b k p This time, we have ED Dk,, p F b k p ] = ad Thus E D Dk,, p p E 4 = p k= K K 3 k= E p σ i b kp σ 4 i b kp k=,l=b k p p c k p c k p k=,l=b k p D l Dk, l, p F bk p, X b k p, X b k p c k p =b k p c kp =b k p D Dk,, p =. ] K χp k. D Dk, } ], p D Dk, ], p E D Dk,, p D l Dk, ] l, p χp k K p p k= E χp k ]. With a similar argumet as i the proof of 7.7 we are doe. Provig that D 6 ca be replaced by Dk,, p 6 works aalogously, thus we ish the proof of Lemma showig + σ i c kp sup 4 κψ b k+ p, X c k p =c k p p k= D 3 + σ i b kp m =i p, X b k p c k p =b k p } D 3 C i =. D 3 7.
Model checks for the volatility 8 We start with the followig propositio: + b k+ p c k p sup 4 σ i c kp p k= c k p b k p, X c k σ p s ds σ i b kp + ip, X b k σ p s ds 7. σ i i p }, X σ ip s ds C i =. As i the proof of Theorem 3 we obtai where = = c k p b k p c k p b k p c k p b k p σ i s, X s σ i b kp σ i s, X s σ i s, X b k p y σ i s, X b k p, X b k σ p s ds + σ i s, X b k p σ i b kp s p σ u dw u σ m s ds + O p b k p, X b k σ p s ds p =: δ k, p 3 + δ k, p m 3 + O p, 7.3 δ k, p 3 = σ 3 b k p c k p b k p s y σ i s, X b k p b k p dw u ds ad δ k, p 3 is deed implicitly by equatio 7.3. From ] E δ k, p 3 = ad E we coclude F b k p sup p E k= δ k, p 3 F bk p ] δ k, p 3 =. For δ k, p 3 we have E δ k, p 3 F b k p ] K p 3 3 4 χp k p sup 4 k= ] E δ k, p 3 sup K p 3 ] K p 3 3 as usual, thus p k= ] E χp k =. The correspodig results for the other summads i 7. ca be show aalogously.
Model checks for the volatility 9 To ish the proof of Lemma we have to show + σ i c kp + σ i i p sup 4 p k=, X c k p, X ip κψ κψ σ i b kp b k+ p =c k p m =i p, X b k p D 3 D 3 κψ c k p b k+ p ip c k p =b k p D 3 σs ds } σs ds =. c k p b k p σs ds The last term i the sum is egligible. For the other terms we x k for a momet ad observe that c k p b k+ p D 3 = h,p s b kp σs ds, 7.4 κψ with Thus, κψ =b k p b k+ p =c k p h,p s = h,ps, m h,p s = h,ps, m h,ps = κψ D 3 = m = b k p b k+ p+m c k p s + h,p m, pm s + h 3,ps m + i= g i, + s, h,p = m gi = + O, κψ h 3,ps = p 4 k= σ i b kp κψ i= m = m i=+ g i c k p, X b k h p,p b k p+m Other itegrals tha those betwee b kp+m b k p+m b k p + σ i c k p σ i b kp, + h,p s c kp σs ds 7.5 s + h 3,ps pm, m s ad s. s b kp ad c kp, X b k p, X c k p h 3,p s,, p+m σs ds K 4. occur i the followig way: h,p s c k p s b kp } σs ds,
Model checks for the volatility 3 where the rst term ad the secod term i the itegrad come from 7.4 ad 7.5, respectively. A similar result holds for the itegral from b kp+m to c kp. By deitio, we have h,p s b kp + h 3,p s c k p = h,p for b kp 4 s b kp+m, ad hece it is eough to prove that p k= b k p+m b k p h 3,p s b kp σ i b kp, X b k p σ i c kp, X c k σ p s ds coverges to zero i the usual way. Agai, this follows from a Taylor expasio ad a similar argumet as i the rst part of the proof of 7.. Lemma We have sup 4 F i p } Gk, p + Gk, p + Gp 3 =. Proof of Lemma : Without loss of geerality is suces to show sup 4 p c k p k= =b k p k= From aother Taylor expasio we have σ i s, X s σ i b kp thus we are left to prove sup 4 p c k p k= =b k p, X b k p σ i s, X s σ i b kp = y σ i b kp y σ i b kp, X b k Z p ψ κ ω =. s, X b k p b k p pm σ u dw u + O p, s, X b k σ p u dw u Z ψ b k p κ ω =. However, this result follows from similar argumets as i the proof of Lemma. Note that we have completely aalogous results for a decompositio of ˆB t Bt. Thus, we ed up with sup 4 ˆB p t Bt p sup 4 Ĉ i C i k= k= σ i b kp Y k, p c k p t} } }, X b k Y k, p p =, 7.6 =,
Model checks for the volatility 3 where Y k, p was deed i 7.6. Sice EY k, p F b k p ] = p κ γ b k p EY k, p F b k p ] = as i Jacod et al. 7], we coclude p E Y k, p c k p k= p k= p k= E E Y k, p c k p Y k, p σ i b kp t i t F b } k p t i} σ i b kp, X b k p ] =, X b k p σ b kp γ s,ti t ]s ds F b k p, X b k p ] = = ] F b k p + o p ad γ s,ti ]s σ s, X s ds γ s σ i s, X s σ s, X s ds Theorem 4 follows ow from Theorem IX 7.8 i Jacod ad Shiryaev 8], sice the missig coditios ca be show i the same way as i Jacod et al. 7]. 7. Proof of Theorem The covergece of the ite dimesioal distributios follows from the delta method for stably covergig sequeces, sice we have 4 ˆNt N t,..., N tk N tk T D st Y Σ t,...,t k s, X s dw s, where the k d + k-dimesioal matrix Y has the form Y = I k k Y, Bt T D Y =.. Bt T k D A straightforward calculatio shows that the coditioal covariace coicides with the coditioal covariace of the ite dimesioal distributios of the process deed i 4.. Thus we are left to prove the tightess of the process 4 ˆN t N t. We have the uiform decompositio 4 ˆNt N t = 4 ˆB t B t + 4 B T t D Ĉ C + o p ad will prove the tightess of each of the two sequeces o the right had side separately. To this ed, we use Theorem VI. 4.5 i Jacod ad Shiryaev 8], which says i a special case that a family of processes X t t livig o the same probability space Ω, F, F t t, P is tight, as log as the followig two coditios are satised:
Model checks for the volatility 3 i For all ɛ > there exists some N ad K > such that P sup Xt > K < ɛ 7.7 t for all >. ii For all ɛ > we have sup η sup R,S T ;R S R+η where T deotes the set of all stoppig times bouded by. P X R X S > ɛ =, 7.8 For the rst sequece ote that 7.7 ad 7.8 follows easily from Theorem 4, sice it yields the stable covergece 4 ˆB t Bt Dst t γ s dw s, ad the process γ t is bouded. The proof of the tightess of the secod sequece is slightly more ivolved. Note rst that Cramér's rule gives D = add/ detd, where add deotes the adoit matrix of D. From the boudedess of the fuctios σi we coclude that each etry of add is bouded as well, ad thus 3.8 yields E D i β ] < K for all i ad ad some β >. Moreover, we have Bt i < K uiformly i t, ad usig Markov's ad Hölder's iequality we coclude for ay ɛ > ad arbitrary i, d: ] P 4 D i Ĉ C > K K β β 8 E D i β Ĉ C β ] K β ] E D i β E 4 Ĉ C β. From the proof of the previous theorem we kow that the latter expectatio is bouded uiformly i as well. Thus, for all ɛ > there exists some K > such that P 4 B T t D Ĉ C > K < ɛ, sup t for all >. This gives 7.7. Note for the same reasos that sup P 4 η η D i Ĉ C > ɛ = for all ɛ >, ad sice we have BR i Bi S K η for all such stoppig times R, S with R S R + η 7.8 follows ad we are doe. 7.3 Proof of Theorem For most parts the proof works i the same way as the oes for the precedig results. However, sice x x is ot dieretiable, we caot use Ito's formula to obtai a decompositio of Z k