Goodness-of-fit testing for fractional diffusions. Mark Podolskij and Katrin Wasmuth. CREATES Research Paper

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1 Goodess-of-fit testig for fractioal diffusios Mark odolskij ad Katri Wasmuth CREATES Research aper Departmet of Ecoomics ad Busiess Aarhus Uiversity Bartholis Allé 1 DK-8 Aarhus C Demark oekoomi@au.dk Tel:

2 Goodess-of-fit testig for fractioal diffusios Mark odolskij Katri Wasmuth April 16, 212 Abstract This paper presets a goodess-of-fit test for the volatility fuctio of a SDE drive by a Gaussia process with statioary ad cetered icremets. Uder rather weak assumptios o the Gaussia process, we provide a procedure for testig whether the ukow volatility fuctio lies i a give liear fuctioal space or ot. This testig problem is highly o-trivial, because the volatility fuctio is ot idetifiable i our model. The uderlyig fractioal diffusio is assumed to be observed at high frequecy o a fixed time iterval ad the test statistic is based o weighted power variatios. Our test statistic is cosistet agaist ay fixed alterative. Keywords: cetral limit theorem, goodess-of-fit tests, high frequecy observatios, fractioal diffusios, stable covergece. JEL Classificatio: C1, C13, C14. 1 Itroductio I this paper we cosider a stochastic differetial equatio dx t = a(t, X t dt + σ(t, X t dg t, X = x R, drive by a Gaussia process (G t t with statioary ad cetered icremets. Our aim is to derive a goodess-of-fit test for the volatility fuctio σ 2 : [, 1] R R + o the basis of discrete high frequecy observatios X i, i =,..., [1/ ], of the path (X t t [,1]. The otio of high frequecy refers to the ifill asymptotics settig, i.e.. More precisely, we will propose a procedure for testig whether the ukow volatility fuctio σ 2 lies i a liear subspace geerated by give volatility fuctios σ 2 k : [, 1] R R+, 1 k d. Mark odolskij gratefully ackowledges fiacial support from CREATES fuded by the Daish Natioal Research Foudatio. Departmet of Mathematics, Heidelberg Uiversity, INF 294, 6912 Heidelberg, Germay, m.podolskij@ui-heidelberg.de. Departmet of Mathematics, Heidelberg Uiversity, INF 294, 6912 Heidelberg, Germay, KatriWasmuth@aol.com. 1

3 Goodess-of-fit testig for fractioal diffusios 2 This type of goodess-of-fit testig has bee studied i the literature for the case of classical diffusios that are drive by a stadard Browia motio (see e.g. [6]; we also refer to [1] for similar testig problems uder a differet samplig scheme. Compared to the classical case, we will have to deal with additioal o-trivial theoretical challeges. First of all, uder our mild assumptios o the drivig Gaussia process G, the volatility fuctio σ 2 is ot idetifiable. This fact ca be easily see by observig that a multiplicatio of the volatility fuctio σ 2 by a costat ca ot be distiguished from the multiplicatio of G by the same costat. Secodly, the resultig test statistic should be robust to the presece of the drift process a (i.e. the limit theory should ot deped o a, because the drift a ca ot be estimated o a fixed time iterval [, 1]. The third theoretical problem is comig from the fact that the (stable cetral limit theorems for usual power variatio statistics, which will be the mai tool for hypothesis testig, oly hold for a subrage of smoothess parameters of G, while we would like to obtai a valid testig procedure for all smoothess parameters of G. Our test statistic is a fuctio of various weighted power variatios based o the secod order differeces of X. Higher order differeces is a key istrumet for obtaiig valid cetral limit theorems for all smoothess parameters of G, but more importatly they also isure the robustess to the drift process a (i cotrast to the first order differeces. The secod crucial issue of our test statistic is the self-scalig property, which makes it idepedet of the ukow variace of dg t. This solves the o-idetifiability problem that we metioed above. Our paper is orgaised as follows. I sectio 2 we preset the mai assumptios o the processes a, σ 2 ad G, which esure the existece of the uique solutio of the above SDE ad the validity of the asymptotic results. I sectio 3 we explai the testig problem ad costruct a distace measure, which eables us to make statistical decisios. I sectio 4 we derive our test statistic ad show its asymptotic mixed ormality. Sectio 5 is devoted to proofs. 2 Model ad assumptios We cosider a oe-dimesioal fractioal diffusio model of the form X t = x + t a(s, X s ds + t σ(s, X s dg s, t [, 1], (2.1 defied o a probability space (Ω, F, with x R. Here the fuctio a : [, 1] R R represets the drift, σ : [, 1] R R is the o-vaishig volatility fuctio ad G = (G t t [,1] is a Gaussia process with statioary ad cetered icremets. To esure the existece of the secod itegral i (2.1 (i the Riema-Stieltjes sese we eed to assume that the process G is Hölder cotiuous of order bigger tha 1/2. More precisely, we impose structural assumptios o the covariace kerel of the Gaussia driver G. Let R deote the variace fuctio of the icremets of G, i.e. R(t = E[ G t+s G s 2 ], t, (2.2

4 Goodess-of-fit testig for fractioal diffusios 3 which is idepedet of s sice G has statioary icremets. Below, the fuctios L R, L R (4 : R > R are assumed to be cotiuous ad slowly varyig at, f (k deotes the k-th derivative of a fuctio f ad H (1/2, 1. We assume that the followig coditios are satisfied: (i R(t = t 2H L R (t. (ii R (4 (t = t 2H 4 L R (4(t. (iii There exists a costat b (, 1 such that lim sup sup L R (4(y <. x L R (x y [x,x b ] We remark that coditio (i implies that G is Hölder cotiuous of all orders smaller tha H (1/2, 1. Of course, the whole set of coditios is ot required to guaratee the existece of a uique solutio of the SDE (2.1, but we do require assumptios (i-(iii for the cetral limit theorems that are preseted i sectio 4 (cf. [3]. Notice that coditio (i idicates that the local behaviour of the Gaussia process G is similar to the local behaviour of the fractioal Browia motio with Hurst parameter H (1/2, 1. This fact will be reflected i the cetral limit theorems preseted i sectio 4. Now, the results of [9] imply the existece of a uique solutio of the SDE (2.1, where the secod itegral i (2.1 is defied pathwise i the Riema-Stieltjes sese, give the followig set of coditios is satisfied: (iv The fuctio a(t, is Lipschitz, uiformly i t [, 1]. The fuctio a(, x is Hölder cotiuous of order γ H, uiformly i x R. Furthermore, it satisfies the growth coditio a(t, x C x + a (t for some C > ad a L p ([, 1] for some p > 1 1 H. (v The fuctio σ(t, is Lipschitz, uiformly i t [, 1], ad σ is cotiuously differetiable i x. Furthermore, there exist costats < α, β 1 with β > 1/2 ad α > 1 H 1 such that: (a xσ(t, is locally Hölder cotiuous of order α, uiformly i t [, 1], (b σ(, x ad xσ(, x are Hölder cotiuous of order β, uiformly i x R. The assumptios (iv ad (v iclude some coditios that are required to prove our asymptotic results; hece, they are slightly stroger tha cosidered i [9]. Furthermore, uder coditios (iv-(v, the uique solutio of (2.1 satisfies β X C 1 δ ([, 1] a.s., (2.3 for ay δ (1 H, mi{ 1 2, α, 1+β } (see agai [9]. Throughout this paper we assume that, for ay k N ad ay t 1,..., t k >, the radom vector (X t1,..., X tk has a Lebesgue desity. This techical coditios esures the ivertibility of the matrices Σ ad Σ defied by (3.6 ad (4.3, respectively.

5 Goodess-of-fit testig for fractioal diffusios 4 3 The test problem We assume that the process X defied i (2.1 is observed at time poits t i = i, i.e. the high frequecy observatios X, X, X 2,..., X [1/ ] are give ad. For d 1, let σ1 2,..., σ2 d : [, 1] R R be kow fuctios that satisfy the assumptio (v from sectio 2. We assume that the fuctios σ1 2,..., σ2 d are liearly idepedet o the sets of the form [, 1] [a, b], for all a < b, ad deote by V the vector space geerated by σ1 2,..., σ2 d, i.e. V = spa{σ1, 2..., σd 2 }. (3.1 Our mai aim is to decide, o the basis of the uderlyig observatios X i, whether the ukow volatility fuctio σ 2 is a liear combiatio of the kow fuctios σ 2 1,..., σ2 d. More precisely, the ull hypothesis is give by H : σ 2 (t, X t = d λ k σk 2 (t, X t for some λ k s ad a.s. all t [, 1], (3.2 k=1 while the alterative is defied as the complemet hypothesis. We remark that ay possible test procedure ca oly give pathwise coclusios. For istace, whe H holds for the observed path X(ω it does ot mea that it remais true for a differet path X(ω. Furthermore, H does ot imply the idetity σ 2 (t, x = d k=1 λ kσk 2(t, x for some (λ 1,..., λ d ad a.s. all (t, x [, 1] R. Our test procedure will be based o the estimatio of the L 2 -distace betwee the vector space V ad the volatility fuctio σ 2. For this purpose we itroduce the followig radom scalar product: for ay fuctios f, g : [, 1] R R we defie f, g := 1 f(s, X s g(s, X s ds. (3.3 We also set f := f, f ad deote by V f the distace betwee the vector space V ad the fuctio f, i.e. V f := if{ v f : v V}. Now, we ca reformulate the ull hypothesis ad the alterative as H : V σ 2 =, H 1 : V σ 2 >, (3.4 where the vector space V is defied i (3.1. The stadard argumets from the Hilbert space theory imply the idetity ( V σ 2 2 = σ 2 2 σ1, 2 σ 2,..., σd 2, σ2 Σ 1(, σ1, 2 σ 2,..., σd 2, σ2 (3.5 Σ = ( σk 2, σ2 l, (3.6 1 k,l d

6 Goodess-of-fit testig for fractioal diffusios 5 where v deotes the traspose of the vector v. I cotrast to the work of [6], who followed a similar approach for the goodess-of-fit testig for classical diffusios (that are drive by a Browia motio, the distace measure V σ 2 2 is ot the right quatity to estimate i the fractioal diffusio case. The mai reaso lies i the fact that the objects σ 2 ad σ 2 k, σ2, 1 k d, ca ot be idetified, because our assumptios are ot sufficiet to fully idetify the variace fuctio R. I order to obtai a feasible estimator, we defie a self-scalig measure T as T = V σ2 2 σ 2 2, (3.7 which turs out to be a crucial trasformatio of the origial distace measure V σ 2. We will see that the radom measure T ca be cosistetly estimated without a precise kowledge of the fuctio R. We also have the appealig property T 1, which describes the deviatio from the ull hypothesis T = i percet (i cotrast, it is ot clear whe the quatity V σ 2 ca be idetified as small. = 1, which correspods to the homoscedas- Remark 3.1 For the simple case d = 1, σ1 2 ticity testig, we obtai the idetity ( 1 σ2 (s, X s ds T = 1 1 σ4 (s, X s ds 2. 4 The testig procedure I this sectio we costruct a cosistet, asymptotically mixed ormal, estimator T of the radom variable T defied i (3.7. Our pla is to costruct empirical aalogues of the quatities σ 2 2, σk 2, σ2, σk 2, σ2 l, 1 k, l d, prove their asymptotic mixed ormality ad apply the delta-method. I the high frequecy settig (weighted power variatios based o icremets of X i X (i 1 are kow to be cosistet estimators of itegrated powers of volatility. However, i our framework of fractioal diffusios, it makes more sese to use higher order differeces. We set i,2x = X i 2X (i 1 + X (i 2, τ 2 = E[ i,2g 2 ].

7 Goodess-of-fit testig for fractioal diffusios 6 The empirical aalogues of σ 2 2, σk 2, σ2, σk 2, σ2 l, 1 k, l d, are defied as σ 2 2 = [1/ ] 3τ 4 i,2x 4, (4.1 σ 2 k, σ2 = τ 2 [1/ ] [1/ σk 2, σ2 l ] = i= σ 2 k,(i 2 i,2x 2, σ 2 k,i σ 2 l,i, where we use a shorthad otatio σ 2 k,s = σ2 k (s, X s. I our settig usig secod order differeces has the followig crucial advatages: (a The cetral limit theorem preseted below holds for all H (1/2, 1 while it would hold oly for H (1/2, 3/4 if we would use the stadard icremets of X; this effect of higher order differeces is kow i the literature i the pure Gaussia framework (see e.g. [8], (b More importatly, the cetral limit theorem is ot iflueced by the presece of the drift fuctio a, which would ot be true i the case of the stadard icremets of X (this is a less kow fact. Fially, we defie σ 2 2 ( σ1 2, σ2,..., σd 2, σ2 T = σ 2 2 Σ = Σ 1 ( σ 2 1, σ2,..., σ 2 d, σ2, (4.2 ( σk 2, σ2 l. (4.3 1 k,l d Remark 4.1 The statistics defied i (4.1 are ot feasible, because the ormalisig costat τ is ukow! However, it is easy to see that the test statistic T is feasible due to the self-scalig property. For istace, whe d = 1 ad σ1 2 = 1 we obtai the idetity ( [1/ ] 2 3 i,2 X 2 T = 1, i,2 X 4 which obviously does ot deped o the ukow costat τ. As it has bee metioed i e.g. [5], uder the assumptio (i of sectio 2, it holds that corr( 1,2G, 1+j,2G ρ(j = (j + 22H + 4(j + 1 2H 6j 2H + 4 j 1 2H j 2 2H, 2 (4 2 2H as, ad ρ = ρ H is the correlatio fuctio of the secod order fractioal oise ( i,2 BH i 1 with Hurst parameter H. Notice that ρ(j j 2H 4 as j, which implies that j=1 ρ(j < for all H (1/2, 1.

8 Goodess-of-fit testig for fractioal diffusios 7 I the ext theorem we demostrate a multivariate stable cetral limit theorem. Recall that a sequece of radom variables Y o (Ω, F, coverges stably i law towards Y st (Y Y, which is defied o the extesio (Ω, F, of the origial probability space (Ω, F,, iff lim E[f(Y Z] = E [f(y Z] for all bouded radom variables Z ad all bouded, cotiuous fuctios f (see e.g. [2] for more details. We call Y mixed ormal with mea ad coditioal covariace matrix V (Y = MN(, V whe, coditioally o F, Y has a ormal distributio with mea ad covariace matrix V. The ext results maily follow from the theory preseted i [3], [4] ad [5]. Theorem 4.2 Assume that the coditios (i-(v from sectio 2 hold. The we obtai the covergece σ 2 2 σ 2 2, σk 2, σ2 σk 2, σ2, σk 2, σ2 l σk 2, σ2 l. (4.4 Furthermore, we deduce the stable covergece σ 2 2 σ 2 2 σ1 2, σ2 σ1 2, σ2 1/2. σ 2 d, σ2 σ 2 d, σ2 st MN d+1 (, where V s R (d+1 (d+1 is a symmetric matrix defied as V 11 s = α H σ 8 s, V 1,k+1 s = β H σ 2 k,s σ6 s, 1 k d, 1 V k+1,l+1 s = γ H σ 2 k,s σ2 l,s σ4 s, 1 k, l d, ad the costats α H, β H ad γ H are give by α H = 1 9 β H = 1 3 ( 96 + γ H = {48ρ 4 (k + 144ρ 2 (k}, k=1 ( k=1 ρ 2 (k. k=1 ρ 2 (k, V s ds, (4.5 Fially, it holds that 1/2 (Σ Σ. (4.6

9 Goodess-of-fit testig for fractioal diffusios 8 Now, we wat to apply the delta-method. For a ivertible matrix A R d d ad x R d+1, we defie the fuctio g(x; A := x 1 (x 2,..., x d+1 A 1 (x 2,..., x d+1 x 1. The the idetity T T = g( σ 2 2, σ 2 1, σ 2,..., σ 2 d, σ2 ; Σ g( σ 2 2, σ 2 1, σ 2,..., σ 2 d, σ2 ; Σ holds. Notice that the matrices Σ ad Σ are ivertible a.s., because the fuctios σ1 2,..., σ2 d are liearly idepedet o the sets of the form [, 1] [a, b] for all a < b, ad the radom vector (X t1,..., X tk has a Lebesgue desity for all k N ad all t 1,..., t k >. Usig (4.6 we deduce that T T = g( σ 2 2, σ 2 1, σ 2,..., σ 2 d, σ2 ; Σ g( σ 2 2, σ 2 1, σ 2,..., σ 2 d, σ2 ; Σ + o ( 1/2. This decompositio will imply the mixed ormality of 1/2 (T T by the delta-method for stable covergece. However, to obtai a feasible test statistic we still eed to estimate the coditioal covariace matrix V = 1 V sds. This cosists of two subproblems: (a Estimatio of the parameter H (1/2, 1 ad (b Estimatio of itegrated products of various volatility fuctios. The first estimatio problem is solved via a chage-offrequecy method H := 1 2 log 2 ( i=4 X i 2X (i 2 + X (i 4 2, X i 2X (i 1 + X (i 2 2 which compares oe realised measure at two differet frequecies (log 2 deotes the logarithm with basis 2. Ideed, sectio 4.3 from [5] shows that H H. (4.7 Notice that the coefficiets α H, β H ad γ H are cotiuous i H, which implies that α H α H, β H β H, γ H γ H. Now, we are able to costruct a empirical (symmetric aalogue of the matrix V = 1 V sds: V 11 V 1,k+1 V k+1,l+1 = α H 15τ 8 = β H 15τ 6 [1/ ] = γ H 3τ 4 [1/ ] i,2x 8, [1/ ] σ 2 k,(i 2 i,2x 6, 1 k d, σ 2 k,(i 2 σ 2 l,(i 2 i,2x 4, 1 k, l d.

10 Goodess-of-fit testig for fractioal diffusios 9 Before we preset the ext theorem, we itroduce a shorthad otatio g = g( σ 2 2, σ 2 1, σ 2,..., σ 2 d, σ2 ; Σ, g = g( σ 2 2, σ 2 1, σ 2,..., σ 2 d, σ2 ; Σ. The ext result follows directly from Theorem 4.2 ad the delta-method for stable covergece (see e.g. ropositio 2.5 i [1]. Theorem 4.3 Assume that the coditios (i-(v from sectio 2 hold. The we obtai the followig results: (a It holds that 1/2 (T T st MN (, gv g with V = 1 V sds. (b We obtai the covergece i probability V 1/2 (T T g V g V ad d N (, 1. (4.8 As we have already metioed, part (a of Theorem 4.3 follows from Theorem 4.2 ad the delta-method for stable covergece. The stadard cetral limit theorem i (4.8 follows agai from the properties of stable covergece, part (a of Theorem 4.3 ad V V, g g. Remark 4.4 The statistic g V g is ideed feasible, i.e. it does ot deped o the ormalisig costat τ. For istace, whe d = 1 ad σ1 2 = 1 we deduce the idetities ( [1/ ] 2 i,2 X 2 Σ = 1, g = V = α H 15τ 8 β H 15τ 6 τ ( 2 3τ 4 i,2 X 4 2, i,2 X 8 β H 15τ 6 i,2 X 6 γ H 3τ 4 2 τ 2 3τ 4 i,2 X 2, i,2 X 4 i,2 X 6 i,2 X 4 which implies that ( [1/ ] 2 g V g 9 i,2 X 2 6 [1/ ] i,2 = ( [1/ ] 2, X 2 i,2 X 4 i,2 X 4 α H 15 β H 15 i,2 X 8 β H 15 i,2 X 6 γ H 3 ( [1/ ] 2 9 i,2 X 2 ( [1/ ] 2, i,2 X 4 i,2 X 6 i,2 X 4 6 [1/ ] i,2 X 2 i,2 X 4,

11 Goodess-of-fit testig for fractioal diffusios 1 Fially, we defie our test statistic as S = 1/2 T. (4.9 g V g The ext propositio follows directly from part (b of Theorem 4.3. ropositio 4.5 Assume that the coditios (i-(v from sectio 2 hold. We reject the ull hypothesis H : V σ 2 = if S > u 1 α, where u 1 α deotes the (1 α quatile of N (, 1. The it holds that H (S > u 1 α α, H1 (S > u 1 α 1. I other words, the test statistic S has asymptotic level α ad it is cosistet agaist ay fixed alterative. 5 roofs We cocetrate o the proof of Theorem 4.2 as all other theorems follow from this result. The justificatio of Theorem 4.2 relies o a combiatio of various methods preseted i [3], [4] ad [5], which we sketch below. Let us defie X t = X + t σ s dg s, (5.1 which correspods to the process X with a (recall the otatio σ s = σ(s, X s. (a We start by cosiderig the process X istead of the origial process X. The theory preseted i [3] ad [5] implies that the approximatio i,2x σ (i 2 i,2g does ot ifluece the limit behavior. The paper [3] oly deals with power variatios of the first order differeces of the process X, while [5] cosiders higher order differeces of Browia semi-statioary processes, which do ot have the form (5.1. However, the secod order differeces of both processes are well approximated by the same expressio σ (i 2 i,2g (we refer to (5.5 ad (5.1 i [5] for more details. For this reaso we ca rely o the methods of [3], [4] ad [5] from ow o, eve though the papers [4] ad [5] do ot deal with the process X directly.

12 Goodess-of-fit testig for fractioal diffusios 11 (b Usig part (a it ca be deduced that τ p [1/ ] i,2x p 1 m p σ s p ds, p >, with m p = E[ N (, 1 p ]. This result was show i [3] for the first order differeces (see Theorem 2 therei, but the proof remais true for higher order differeces (cf. Theorem 3.1 i [5]. Observig the approximatio of part (a ad followig the lies of the proof i [3] ad [4], we see that the above covergece i probability (ad also the associated cetral limit theorem remais valid if we itroduce weight processes σk 2 as log as they are Lebesgue itegrable. Thus, the covergece i probability stated i (4.4 ad the covergece V V from part (b of Theorem 4.3 are valid for the process X. (c It remais to justify the cetral limit theorem preseted i (4.5. Here we will rely o the multivariate cetral limit theorem for secod order differeces preseted i [5] (see Theorem 3.3 therei. First of all, we eed to show that the volatility process σ s = σ(s, X s is Hölder cotiuous of order > 1/2 to apply the aforemetioed result. But this is obviously true as σ s σ t σ(s, X s σ(t, X s + σ(t, X s σ(t, X t C 1 t s β + C 2 t s 1 δ, for some C 1, C 2 >, due to assumptio (v i sectio 2 ad (2.3. Sice β > 1/2 ad δ ca be chose to satisfy 1 δ > 1/2 (because H > 1/2, we deduce that (σ s s [,1] is Hölder cotiuous of order > 1/2. Now we are able to apply Theorem 3.3 from [5] for the process X. For simplicity we oly preset a joit cetral limit theorem for the vector 1/2 ( σk 2, σ2 σk 2, σ2, σl 2, σ2 σl 2, σ2 with 1 k, l d. Theorem 3.3 i [5] (applied to X states that with 1/2 ( σ 2 k, σ2 σ 2 k, σ2, σ 2 l, σ2 σ 2 l, σ2 st MN 2 (, V 11 s = µσ 4 k,s σ4 s, V 12 s = µσ 2 k,s σ2 l,s σ4 s, V 22 s = µσ 4 l,s σ4 s, 1 V s ds ad µ = lim 1 Var [1/ ] i,2 BH 2 Var( i,2 BH 2, where B H deotes a fractioal Browia motio with Hurst parameter H. A straightforward calculatio shows that µ = γ H. Hece, we obtai (4.5 for the process X. (d Fially, we eed to prove that Theorem 4.2 holds for the origial process X (rather

13 Goodess-of-fit testig for fractioal diffusios 12 tha oly for X. For simplicity, let us oly cosider the first estimator σ 2, σ 2. Sice Theorem 4.2 is valid for the process X, it suffices to show that 1/2 3τ 4 ( [1/ ] [1/ ] i,2x 4 i,2x 4. Let us set A t = t a(s, X sds. The mea value theorem implies that i,2x 4 i,2x 4 C i,2a ( i,2x 3 + i,2x 3 for some C >. Now, due to assumptios (i, (iv, (v from sectio 2 ad (2.3, we obtai for some ε > small eough ad K > : τ 4 K 4H ε, i,2a K 1+H ε, i,2x K H ε, i,2x K H ε. Thus, 1/2 3τ 4 ( [1/ ] [1/ ] i,2x 4 Choosig ε > small eough we deduce the desired result. i,2x 4 K 1/2 5ε. (e We are left to provig the covergece 1/2 (Σ Σ from (4.6. Sice the processes (σ k,s s [,1], k = 1,..., d, are Hölder cotiuous of order η > 1/2 (see step (c, we readily deduce that for all 1 k, l d. 1/2 Σ Σ 1/2 1 σk,s 2 σ2 l,s σ2 k,[s/ ] σl,[s/ 2 ds ] Refereces [1] Aït-Sahalia, Y. (1996: Testig cotiuous time models of the spot iterest rate. Review of Fiacial Studies 9, [2] Aldous, D.J., ad G.K. Eagleso (1978: O mixig ad stability of limit theorems. A. of rob. 6(2, [3] Bardorff-Nielse, O.E., J.M. Corcuera ad M. odolskij (29: ower variatio for Gaussia processes with statioary icremets. Stochastic rocesses ad Their Applicatios 119, [4] Bardorff-Nielse, O.E., J.M. Corcuera ad M. odolskij (211: Multipower variatio for Browia semistatioary processes. Beroulli 17(4,

14 Goodess-of-fit testig for fractioal diffusios 13 [5] Bardorff-Nielse, O.E., J.M. Corcuera ad M. odolskij (211: Limit theorems for fuctioals of higher order differeces of Browia semi-statioary processes. To appear i rokhorov ad Cotemporary robability Theory. [6] Dette, H., M. odolskij ad M. Vetter (26: Estimatio of itegrated volatility i cotiuous time fiacial models with applicatios to goodess-of-fit testig. Scadiavia Joural of Statistics 33, [7] Hall,. ad C.C. Heyde (198: Martigale limit theory ad its applicatio. Academic ress, New York. [8] Istas, J. ad G. Lag (1997: Quadratic variatios ad estimatio of the local Hölder idex of a Gaussia process. A. Ist. Heri oicaré, robabilités et statistiques, 33(4, [9] Nualart, D. ad A. Rascau (22: Differetial equatios drive by fractioal Browia motio. Collect. Math. 53(1, [1] odolskij, M. ad M. Vetter (21: Uderstadig limit theorems for semimartigales: a short survey. Statistica Nederladica 64(3,

15 Research apers : Torbe G. Aderse ad Oleg Bodareko: VIN ad the Flash Crash : Tim Bollerslev, Daiela Osterrieder, Natalia Sizova ad George Tauche: Risk ad Retur: Log-Ru Relatioships, Fractioal Coitegratio, ad Retur redictability : Lars Stetoft: What we ca lear from pricig 139,879 Idividual Stock Optios : Kim Christese, Mark odolskij ad Mathias Vetter: O covariatio estimatio for multivariate cotiuous Itô semimartigales with oise i o-sychroous observatio schemes 212-1: Matei Demetrescu ad Robiso Kruse: The ower of Uit Root Tests Agaist Noliear Local Alteratives 212-2: Matias D. Cattaeo, Michael Jasso ad Whitey K. Newey: Alterative Asymptotics ad the artially Liear Model with May Regressors 212-3: Matt. Dziubiski: Coditioally-Uiform Feasible Grid Search Algorithm 212-4: Jeroe V.K. Rombouts, Lars Stetoft ad Fracesco Violate: The Value of Multivariate Model Sophisticatio: A Applicatio to pricig Dow Joes Idustrial Average optios 212-5: Aders Bredahl Kock: O the Oracle roperty of the Adaptive LASSO i Statioary ad Nostatioary Autoregressios 212-6: Christia Bach ad Matt. Dziubiski: Commodity derivatives pricig with ivetory effects 212-7: Cristia Amado ad Timo Teräsvirta: Modellig Chages i the Ucoditioal Variace of Log Stock Retur Series 212-8: Ae Opschoor, Michel va der Wel, Dick va Dijk ad Nick Taylor: O the Effects of rivate Iformatio o Volatility 212-9: Aastiia Silveoie ad Timo Teräsvirta: Modellig coditioal correlatios of asset returs: A smooth trasitio approach 212-1: eter Exterkate: Model Selectio i Kerel Ridge Regressio : Torbe G. Aderse, Nicola Fusari ad Viktor Todorov: arametric Iferece ad Dyamic State Recovery from Optio aels : Mark odolskij ad Katri Wasmuth: Goodess-of-fit testig for fractioal diffusios