CREATES Research Paper New tests for jumps: a threshold-based approach

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1 CREATES Research Paper New tests for jumps: a threshold-based approach Mark Podolskij ad Daiel Ziggel School of Ecoomics ad Maagemet Uiversity of Aarhus Buildig 1322, DK-8 Aarhus C Demark

2 New tests for jumps: a threshold-based approach Mark Podolskij Uiversity of Aarhus ad CREATES Daiel Ziggel Ruhr-Uiversity of Bochum Jue 19, 28 Abstract I this paper we propose a test to determie whether jumps are preset i a discretely sampled process or ot. We use the cocept of trucated power variatio to costruct our test statistics for (i) semimartigale models ad (ii) semimartigale models with oise. The test statistics coverge to ifiity if jumps are preset ad have a ormal distributio otherwise. Our method is valid (uder very weak assumptios) for all semimartigales with absolute cotiuous characteristics ad rather geeral model for the oise process. We fially implemet the test ad preset the simulatio results. Our simulatios suggest that for semimartigale models the ew test is much more powerful the tests proposed by Bardorff-Nielse ad Shephard (26) ad Aït-Sahalia ad Jacod (28). Keywords: Cetral Limit Theorem; High-Frequecy Data; Microstructure Noise; Semimartigale Theory; Tests for Jumps; Trucated Power Variatio. JEL Classificatio: C1, C13, C14. The first author ackowledges fiacial support from CREATES fuded by the Daish Natioal Research Foudatio. The work of the secod author was supported by the Deutsche Forschugsgemeischaft (SFB 475, Komplexitätsreduktio i multivariate Datestrukture). CREATES, School of Ecoomics ad Maagemet, Uiversity of Aarhus, Buildig 1322, DK-8 Aarhus C, Demark. mpodolskij@creates.au.dk. Ruhr-Uiversity of Bochum, Dept. of Probability ad Statistics, Uiversitätstrasse 15, 4481 Bochum, Germay. daiel.ziggel@rub.de.

3 1 Itroductio The last years have see a rapidly growig literature o statistical methods for high frequecy data (see e.g. Bardorff-Nielse ad Shephard (24a,b), Bardorff-Nielse, Graverse, Jacod, Podolskij ad Shephard (26) or Jacod (28)). I ecoometrics price processes are typically modeled by semimartigales, which costitute a atural class of models uder the assumptio of o-leverage (see Delbae ad Schachermayer (1994)). I geeral, semimartigales are càdlàg processes, which ca be writte as a sum of a cotiuous ad a discotiuous compoet. For various applicatios it is importat to be able to separate these two parts based o discrete (high frequecy) observatios. I particular, practitioers wat to decide whether the discretely observed path of a semimartigale is cotiuous or ot. Quite recetly, several methods have bee proposed to test for jumps i semimartigale models. Bardorff-Nielse ad Shephard (26) use the cocept of bipower variatio to costruct a cosistet estimator of the quadratic variatio of the discotiuous part. This estimator is the applied to test whether a semimartigale has jumps or ot. O the other had, Aït-Sahalia ad Jacod (28) compare the power variatio at differet samplig frequecies to costruct a test for jumps. Both tests apply for geeral Itô semimartigales whe additioally the volatility process is also a semimartigale. Some further approaches ca be foud i Jiag ad Oome (25) or i Lee ad Myklad (27). I this paper we propose a threshold-based procedure to test for jumps. Our method is based upo the trucated power variatio which has bee origially itroduced by Macii (21,24) to obtai jump-robust estimates of some fuctioals of the volatility process. We combie this approach with the wild bootstrap idea (see Wu (1986)) to defie a ew class of test statistics. Our test statistics coverge to a stadard ormal distributio whe the semimartigale is cotiuous, whereas they ted to ifiity for semimartigales with o-vaishig jump part. Furthermore, we costruct tests for jumps i semimartigale models with oise, which are ow itesively studied i the ecoometric literature (see e.g. Zhag, Myklad ad Aït-Sahalia (25) or Hase ad Lude (26)). The advatage of our method is twofold. O the oe had, our test procedure applies for all Itô semimartigales ad we require o further assumptios o the volatility process. O the other had, the threshold-based class of statistics has very good fiite sample properties. The power of our tests is much higher compared with the tests of Bardorff-Nielse ad Shephard (26) ad Aït-Sahalia ad Jacod (28), while we also obtai a reasoable approximatio of the level. 1

4 This paper is orgaised as follows. I Chapter 2 we preset the asymptotic results for the threshold-based class of test statistics i the pure semimartigale settig. The theoretical compariso (via local alteratives) with the tests developed by Bardorff- Nielse ad Shephard (26) ad Aït-Sahalia ad Jacod (28) is give i Sectio 3. The costructio of test statistics for semimartigale models with oise is discussed i Sectio 4. Fially, we illustrate the fiite sample performace of our procedure i Sectios 5 ad 6. All proofs are give i the Appedix. 2 The mai settig, the statistical problem ad the ew class of test statistics We cosider a semimartigale (X t ) t of the form X t = X + t a s ds + t σ s dw s + (x1 { x 1} ) (µ ν) + (x1 { x >1} ) µ, (2.1) defied o the filtered probability space (Ω, F, (F t ) t, P ). Here W deotes a oedimesioal Browia motio, a is a locally bouded ad predictable drift term, σ is a adapted ad càdlàg volatility process, µ is a jump measure ad ν is its predictable compesator. Moreover, we make the followig assumptio o the compesator ν: (H) ν is of the form ν(dt, dx) = dt F t (dx) with (1 x 2 )df t (x) beig a locally bouded ad predictable process. We observe the time cotiuous process X over a give iterval [, t] at equidistat time poits t i = i, i = 1,..., [t]. Based o discrete observatios (X i (ω)) i [t] we wat to decide whether the uobserved path (X s (ω)) s [,t] is cotiuous or ot. As it has bee already metioed i Aït-Sahalia ad Jacod (28) we are oly able to make statistical decisios about the particular (uobserved) path (X s (ω)) s [,t]. It is impossible to say whether the semimartigale model allows for jumps, because there is a positive probability that the path (X s (ω)) s [,t] has o jumps although the model (2.1) allows the process X to jump (this is the case for compoud Poisso processes). Cosequetly, we wat to decide to which of the followig two complemetary sets the path (X s (ω)) s [,t] belogs: Ω j t = {ω : s X s (ω) is discotiuous o [,t]} Ω c t = {ω : s X s (ω) is cotiuous o [,t]}. 2 (2.2)

5 2.1 Realised trucated power variatio To costruct a ew class of test statistics we use the cocept of realised power variatio ad realised trucated power variatio. Recall that the realised power variatio of the process X is give by [t] V (X, p) t = p 2 1 i X p, (2.3) with i X = X i X i 1. It is well-kow (see, for istace, Bardorff-Nielse, Graverse, Jacod, Podolskij ad Shephard (26)) that V (X, 2) t P [X] t = t i=1 σ 2 sds + s t X s 2, (2.4) where X s = X s X s, whereas V (X, p) t P µ p t σ s p ds o Ω c t o Ω j t (2.5) whe p > 2 (µ p = E[ u p ] with u N(, 1)). The realised trucated power variatio, origially proposed by Macii (21,24), is give by [t] V (X, p) t = p 2 1 i X p 1 { i X c ϖ }, (2.6) i=1 where c > ad ϖ (, 1/2). The threshold give i (2.6) elimiates the icremets i X which are affected by jumps, while the icremets i X are (asymptotically) ot iflueced by the threshold whe there are o jumps o the iterval [ i 1, i ]. Cosequetly, V (X, p) t is robust to jumps, i.e. V (X, p) t t P µ p σ s p ds (2.7) for ay p 2 (this is a straightforward extesio of the results preseted i Jacod (28) ad Cot ad Macii (27)). Moreover, uder a further assumptio o the activity of the jump part of X (ad o the parameter ϖ), the efficiecy of V (X, p) t is the same as the efficiecy of V (X, p) t (see agai Jacod (28)). The results of (2.4), (2.5) ad (2.7) suggest to use the statistic V (X, p) t V (X, p) t 1) for p 2 to decide whether the process X jumps or ot. However, the (or V (X,p) t V (X,p) t derivatio of the distributio theory (o Ω c t) for the above statistics turs out to be a difficult task. 3

6 2.2 New class of test statistics Ispired by the wild bootstrap procedure (see Wu (1986)) we itroduce exteral (i.e. idepedet of F) positive i.i.d. radom variables (η i ) 1 i [t] with E[η i ] = 1 ad E[ η i 2 ] <, ad defie a ew class of test statistics by T (X, p) t = p 1 2 [t] i=1 ) i X (1 p η i 1 { i X c ϖ }, p 2. (2.8) The choice of the distributio of η crucially iflueces the level ad power performace of the test statistic T (X, p) t. distributio of η. I the ext sectio we will explai how to choose the All processes are ow defied o a caoical extesio (Ω, F, (F t ) t, P ) of the origial filtered probability space (Ω, F, (F t ) t, P ), which also supports the radom variables (η i ) 1 i [t]. I what follows we will itesively use the cocept of stable covergece. Recall that a sequece (Y ) is said to coverge towards Y F-stably i law (Y F st Y ) whe the weak covergece (Y, Z) D (Y, Z) holds for ay F-measurable variable Z. This is obviously a slightly stroger mode of covergece tha covergece i law (see Reyi (1963), Aldous ad Eagleso (1978) or Jacod ad Shiryaev (23) for more details o stable covergece). The ext theorem demostrates the stable limit of T (X, p) t o Ω c t ad Ω j t. Theorem 1 Assume that coditio (H) holds ad E[ η i 2+δ ] < for some δ >. For ay p 2 ad ay t >, we obtai the followig results: (i) O Ω c t we have T (X, p) t F st t Var[η i ]µ 2p σ s p dw s, (2.9) where W is a ew Browia motio, defied o the extesio (Ω, F, (F t) t, P ) of the probability space (Ω, F, (F t ) t, P ), which is idepedet of F. (ii) O Ω j t we have T (X, p) t P. (2.1) Proof: see Appedix. Note that the limitig radom variable i (2.9) is mixed ormal with F-coditioal variace give by the expressio t ρ 2 (p) t = Var[η i ]µ 2p σ s 2p ds. (2.11) 4

7 By (2.7) we obtai a jump-robust estimate of ρ 2 (p) t, i.e. ρ 2 (p) t = Var[η i ]V (X, 2p) t Fially let us defie the stadardized statistics P ρ 2 (p) t. S(p) t = T (X, p) t, Ŝ(p) t = T (X, p) t. (2.12) ρ(p) t ρ(p) t By the properties of stable covergece we obtai the followig corollary. Corollary 1 Assume that coditio (H) holds ad E[ η i 2+δ ] < for some δ >. For ay p 2 ad ay t >, we obtai the followig results: (i) O Ω c t we have S(p) t F st U, Ŝ(p) t F st U, (2.13) where U is a stadard ormal radom variable, defied o the extesio (Ω, F, (F t) t, P ) of the probability space (Ω, F, (F t ) t, P ), which is idepedet of F. (ii) O Ω j t we have S(p) t P, Ŝ(p) t P. (2.14) Usig agai the properties of stable covergece ad applyig Corollary 1 we deduce that P (Ŝ(p) t > c 1 α Ω c t) α, P (Ŝ(p) t > c 1 α Ω j t) 1, where c 1 α is the (1 α)-quatile of a stadard ormal distributio. 2.3 The choice of the distributio of η i Here we use the motivatio from Sectio 2.1. As we have already metioed before it is atural to use the statistic V (X, p) t V (X, p) t for p 2 to decide whether the process X jumps or ot. Sice the distributio theory for the afore-metioed statistic is ot available we require a small perturbatio of the icremets. Therefore we suggest to sample (η i ) 1 i [t] from the followig distributio P η = 1 2 (δ 1 τ + δ 1+τ ), (2.15) where δ stads for the Dirac measure. We propose to choose the costat τ relatively small, e.g. τ =.1 or.5. Note that for small values of τ our class of statistics T (X, p) t 5

8 is quite close to (V (X, p) t V (X, p) t ). This feature esures a very good power performace of our test statistics. O the other had the symmetry of the distributio of η i aroud 1 is resposible for a reasoable level approximatio of our test. This is partially justified by the followig propositio. Propositio 2 Assume that X t = σw t. The, for ay p 2, it holds that P (Ŝ(p) t x) = Φ(x) + O( 1 ), (2.16) where Φ deotes the stadard ormal distributio. Notice the absece of the term of order 1/2 o the right-had side of (2.16). This meas that we have a secod-order refiemet. 3 Compariso with other test procedures via local alteratives I this sectio we discuss the behaviour of the statistic T (X, p) t uder local alteratives ad compare it with the behaviour of the tests proposed by Bardorff-Nielse ad Shephard (26) ad Aït-Sahalia ad Jacod (28). Let us briefly recall the ideas of these tests. i.e. Bardorff-Nielse ad Shephard (26) propose to use the (1, 1)-bipower variatio, V (X, 1, 1) t = [t] 1 i=1 i X i+1x, (3.1) to costruct a test for jumps. Ideed, uder assumptio (H), V (X, 1, 1) t is robust to jumps of the process X (see e.g. Aït-Sahalia ad Jacod (28)) ad it holds that V (X, 1, 1) t P µ 2 1 The authors propose to use the simple statistic t σ 2 sds. (3.2) T BS (X) t = (V (X, 2) t µ 2 1 V (X, 1, 1) t ) (3.3) to decide whether the process X has jumps or ot. To show a stable cetral limit theorem (o Ω c t) the followig assumptio is required: (V) The volatility process σ is itself a semimartigale with absolute cotiuous characteristics ad it does ot vaish o [, t]. 6

9 O Ω c t, uder assumptio (V), it holds for ay t > T BS (X) t κqq t F st U, (3.4) where U is defied as i Corollary 1, κ = π2 4 + π 5 ad QQ t is give by [t] 3 QQ t = µ 4 1 i X i+1x i+2x i+3x. O the other had T BS (X) t coverges to ifiity o Ω j κqq t t. i=1 Remark 1 I fact, Bardorff-Nielse ad Shephard (26) propose to use the ratio statistic T BS,r (X) t = (1 µ 2 1 V (X, 1, 1) t V (X, 2) t )/ κ max(qq t /(µ 2 1 V (X, 1, 1) t ) 2, 1/t) (3.5) to test for jumps. The above statistic turs out to have better fiite sample properties. However, it has the same behaviour as T BS (X) t uder local alteratives. Aït-Sahalia ad Jacod (28) compare V (X, p) t (with p > 3) at differet samplig frequeces to costruct a test for jumps. I particular, they aalyze the behaviour of the statistic O Ω c t, uder assumptio (V), it holds for ay t > T AJ (X) t = ( /2 2V (X, 4) ) t 2. (3.6) V (X, 4) t T AJ (X) t ˆV t where U is defied as i Corollary 1 ad ˆV t F st U, (3.7) is give by ˆV t = κ V (X, 8) t (V (X, 4) t ) 2 with κ = O Ωj t Aït-Sahalia ad Jacod (28) also showed the stable covergece of the statistic (3.6) whe 2 is replaced by 1 i the defiitio of T AJ (X) t. Now we cosider local alteratives of the form X () t = X + t a s ds + t σ s dw s + γ J t, where J t is a compoud Poisso process ad γ is some sequece with γ. We obtai the followig theorem. 7

10 Theorem 3 For ay t > we have the followig results. (i) Cosider the assumptios of Theorem 1 ad set γ = p 1 2p. If ϖ < p 1 2p t T (X (), p) F st t Var[η i ]µ 2p σ s p dw s + J s p for ay p 2 ad t >. (ii) Cosider the assumptio (V) ad set γ = 1 4. It holds T BS (X () ) F st t Ut c + J s 2, s t s t where Ut c is the stable limit of T BS (X () ) t whe J = give by U c t = κ t σ 4 sds U, it holds where U is defied as i Corollary 1 (see Bardorff-Nielse ad Shephard (26) for the proof i the cotiuous case [i.e. whe J = ]). (iii) Cosider the assumptio (V) ad set γ = 3 8. It holds T AJ (X () ) t F st U c t + 2 s t J s 4 3 t σ4 sds where U c t is the stable limit of T AJ (X () ) t whe J = give by κ t µ U 8 c t = σ8 sds t U, µ 4 σ4 sds where U is defied as i Corollary 1 (see Aït-Sahalia ad Jacod (28) for the proof i the cotiuous case). Proof: see Appedix., Notice that the rate at which our class of test statistics ucovers local alteratives is varyig betwee γ = 1/4 (for p = 2) ad γ = 1/2 (for p ). I this respect T (X (), p) t outperforms T BS (X () ) t for p > 2, while T (X (), p) t outperforms T AJ (X () ) t for p > 4. However, the mai reaso for a better power performace of our test statistic T (X, p) t (see the simulatio results) is differet. At moderate samplig frequecies the power of the test crucially depeds o the robustess properties of statistic V (X, p) t, which is implicitly used to costruct T (X, p) t. Oce the threshold i the defiitio of V (X, p) t ucovers a jump it is immediately elimiated by the idicator fuctio. The test statistics proposed by Bardorff-Nielse ad Shephard (26) ad Aït-Sahalia ad Jacod (28) do ot have this property. 8

11 4 The oise case I this sectio we preset a extesio of our theory to oisy semimartigales. Assume first that the semimartigale X give i (2.1) is defied o some filtered probability space (Ω, F, (F t ) t, P ). However, we do ot directly observe the process X, but a process Z which is cotamiated by the oise. The modellig of the oise process is adapted from Jacod, Li, Myklad, Podolskij ad Vetter (27). More precisely, we cosider the process Z, observed at time poits i/, i =, 1,..., [t], which is give as Z t = X t + ɛ t, (4.1) where ɛ t s are the errors which are, coditioally o X, cetered ad idepedet. This ca be formally costructed as follows. For ay t, cosider a trasitio probability Q t (ω, dz) from (Ω, F t ) ito IR. We edow the space Ω 1 = IR [, ) with the product (Borel) σ-field F 1 ((ɛ t ) t is regarded as the caoical process o this space). The probability measure Q(ω, dω 1 ) is give as a product t Q t (ω, ). The filtered probability space (Ω, F, (F t ) t, P ), o which the process Z lives, is defied as Ω = Ω Ω 1, F = F F 1, F t = s>t F s F 1 s, P (dω, dω 1 ) = P (dω )Q(ω, dω 1 ). } (4.2) Furthermore, we assume that zq t (ω, dz) = X t (ω ), ad αt 2 (ω ) = E[Zt 2 F ](ω ) Xt 2 (ω ) is càdlàg. (4.3) Fially, we defie the process N t (q) = z q Q t (ω, dz). (4.4) Remark 2 Typical examples of a process Z which satisfies the above costructio ad coditio (4.3) are the followig. (i) (Additive i.i.d. process) Whe Z i = X i + ɛ i where (ɛ i ) i is a i.i.d. process with expectatio ad variace α 2, coditio (4.3) is obviously satisfied. (ii) (Additive i.i.d. process + roudig) Cosider the process Z i [X i + ɛ i ] = γ, γ, 9

12 where γ >, (ɛ i ) i is as i (i) ad has a U([, γ]) distributio. The ({ αt 2 = γ 2 Xt } { Xt } 2 ), γ γ ad coditio (4.3) is fulfilled (here {x} deotes the fractioal part of x). Sice the true process X is cotamiated by oise we eed to pre-filter the data. For this purpose we use the method which has bee proposed i Jacod, Li, Myklad, Podolskij ad Vetter (27) ad Podolskij ad Vetter (28) (see also Podolskij ad Vetter (26)). First, we choose a sequece k of itegers, which satisfies k = θ + o( 1 4 ) (4.5) for some θ >, ad a ozero real-valued fuctio g : IR IR, which fulfills the followig coditios (i) g vaishes outside of (, 1) (ii) g is cotiuous ad piecewise C 1 (iii) Its derivative g is piecewise Lipschitz. We associate with g the followig real valued umbers ψ 1 = 1 Furthermore, we defie the quatity (g (s)) 2 ds, ψ 2 = Z i = k 1 j=1 1 (g(s)) 2 ds. (4.6) ( j ) g k i+jz. (4.7) Next, we choose the costats c > ad ϖ (, 1/4). Fially, we itroduce exteral (i.e. idepedet of F) positive i.i.d. radom variables (η i ) i [t] with E[η i ] = 1 ad E[ η i 2 ] <, ad defie a class of test statistics by [t] k +1 T oise (Z, p) t = (p 2)/4 i= Z i p (1 η i 1 { Z i c ϖ } Note that T oise (X, p) t has the same structure as T (X, p) t. ), p 2. (4.8) All processes are defied o a caoical extesio (Ω, F, (F t ) t, P ) of the origial filtered probability space (Ω, F, (F t ) t, P ), which also supports the radom variables (η i ) i [t]. 1

13 Remark 3 If V is a cotiuous semimartigale or a oise process costructed as above, the V i = O p ( 1/4 ) (see e.g. Jacod, Li, Myklad, Podolskij ad Vetter (27) for more details). This explais the coditio o ϖ, i.e. ϖ (, 1/4). with To formulate the theoretical results we eed to itroduce the followig sets Fially, we defie the statistic Ω c t = Ω,c t Ω 1 ad Ω j t = Ω,j t Ω 1 Ω,c t = {ω s X s (ω ) is cotiuous o [, t]}, Ω,j t = {ω s X s (ω ) is discotiuous o [, t]}. ad set [t] k +1 Γ(p) t = Var[η i ] (p 2)/2 i= Z i 2p 1 { Z i c ϖ } (4.9) S oise (p) t = T oise (Z, p) t. (4.1) Γ(p) t The mai result of this sectio is the followig theorem. Theorem 4 Assume that coditio (H) holds, E[ η i 2+δ ] < for some δ > ad the process N t (q) defied i (4.4) is locally bouded for some q > ay t >, we obtai the followig results: 2p. For ay p 2 ad 1 4ϖ (i) O Ω c t we have S oise (p) t F st U, (4.11) where U is a stadard ormal radom variable, defied o the extesio (Ω, F, (F t) t, P ) of the probability space (Ω, F, (F t ) t, P ), which is idepedet of F. (ii) O Ω j t we have S oise (p) t P. (4.12) Proof: see Appedix. Now we deduce from Theorem 4 that P (S oise (p) t > c 1 α Ω c t) α, P (S oise (p) t > c 1 α Ω j t) 1, where c 1 α is the (1 α)-quatile of a stadard ormal distributio. 11

14 Remark 4 As i Sectio 2 we suggest to geerate the exteral variables (η i ) 1 i [t] from the distributio P η = 1 2 (δ 1 τ + δ 1+τ ) (4.13) for relatively small values of τ (e.g. τ =.1 or.5). 5 The choice of the threshold I this sectio we ivestigate how to choose the threshold i our test statistic. A sesible choice of c ad ϖ is crucial for the fiite sample performace of our test. 5.1 Semimartigale model Although the asymptotic results are valid for all c > ad ϖ (, 1/2), it is very importat for the fiite sample performace to choose both costats i a reasoable way. We have to esure that the threshold is sharp eough to detect ad elimiate jumps, while icremets of the cotiuous part should ot be affected by the threshold. Here we preset a easy but effective way to determie c ad ϖ. First, we compute a robust estimator for the itegrated volatility t σ2 sds. A suitable estimator for this quatity is give by µ 2 1 V (X, 1, 1) t (see (3.2)). Next, we choose c = 2.3 V (X, 1, 1) t, where the quatity 2.3 is approximately the 99%-quatile of the stadard ormal distributio. Notice that V (X, 1, 1) t represets the average level of volatility. Therefore, we expect that the most icremets of the cotiuous part do ot exceed the threshold c 1/2. The costat ϖ obviously cotrols the rate of covergece of the threshold. This meas, the bigger ϖ is the faster coverges the threshold to zero. Cosequetly, jumps become faster elimiated for large values of ϖ. O the other had too large values of ϖ icrease the probability to declare a icremets of the cotiuous part as a jump. As a balace betwee this effects we suggest to use ϖ = Semimartigale model with oise The choice of the costats become more ivolved i the semimartigale model with oise. Notice the strog depedece betwee eighbored Z i s. Cosequetly, there is a high probability of elimiatig more tha oe summad if the threshold classifies a big icremet of the cotiuous part as a jump. Therefore, we have to choose the threshold very carefully. 12

15 We suggest the followig procedure. First, we choose where ad c = 2.3 BT (1, 1) t = 1 2 µ 2 1 BT (1, 1) t + ψ 1 θ NV t, (5.1) [t] 2k +1 i= Z i Z i+k NVt = V (Z, 2, ) t. 2 The structure of c is similar to the case without oise. By the results of Podolskij ad Vetter (28) we have that µ 2 1 BT (1, 1) t + ψ 1 θ NV t P O the other had the quatity Z i ψ 1 θ α2 i t (θψ 2 σ 2 s + ψ 1 θ α2 s)ds. is asymptotically distributed as 1/4 N(, θψ 2 σ 2 i + ]. Cosequetly, whe Z has o ) whe Z does ot jump o the iterval [ i, i+k jumps the most quatities Z i should ot exceed the threshold c 1/4 (if the processes σ 2 ad α 2 are ot very volatile). By applyig the same ituitio as for pure semimartigale models we recommed to use ϖ = Simulatio results I this sectio we ivestigate the performace of the differet test statistics i fiite samples. First, we compare our test statistic Ŝ(p) t with p = 2 ad p = 4 with the test statistics T BS (X) t, T BS,r (X) t of Bardorff-Nielse ad Shephard (26) ad the test statistic T AJ (X) t proposed by Aït-Sahalia ad Jacod (28). After that we ivestigate the behavior of the test statistic S oise (p) t (with p = 2) i semimartigale models with oise. We assume that all processes live o the iterval [, 1]. We cosider two differet cotiuous semimartigale models. The first model is a Browia semimartigale with costat volatility σ = 2, i.e. X t = 2W t. (6.1) Secod, we cosider a two factor model. It is specified by the stochastic differetial equatio dx t = µdt + σ t dw t (6.2) with σ t = exp(β + β 1 τ t ), dτ t = ατ t dt + db t, corr(dw t, db t ) = ρ. 13

16 The parametes are chose as µ =, β =.3125, β 1 =.125, α =.25 ad ρ =.3. The two factor model with the afore-metioed parameters has bee adapted from Bardorff-Nielse, Hase, Lude ad Shephard (26) ad Podolskij ad Vetter (26). I this model the level of the volatility process σ 2 varies from 1.4 to 2.1. Furthermore, we geerate the exteral i.i.d. sequece (η i ) 1 i from the distributio (2.15) with τ =.5. We cosider three differet types of jump models: (i) Oe jump with a fixed jump size (ii) Two jumps with fixed jump sizes (iii) Three jumps with radom N(, a 2 )-distributed jump sizes All jump times are idepedet ad U([, 1])-distributed. Moreover, we adapt the jump size(s) to the particular model to make it comparable with the magitude of the volatility process σ. To study the performace of the test statistic S oise (p) t defied i (4.1) for semimartigales with oise, we cosider a i.i.d. model for the oise process ɛ, which is assumed to be idepedet of the semimartigale. These radom variables (ɛ i ) are geerated accordig to a N(,.5 2 ) distributio. Moreover, we use g(x) = (mi(x, 1 x)) + ad θ = 1/3 as proposed i Jacod, Li, Myklad, Podolskij ad Vetter (27). We did 1 simulatio rus for each model. The simulatio results are reported i Tables Level performace We start with the pure semimartigale models. We compare the level performace of test statistics for differet levels (α = 1%, 2.5%, 5%, 1%, 25%) ad differet sample sizes ( = 1, 2, 5, 1, 3, 1). The simulated level results are listed i Table 1 (for the costat volatility model (6.1)) ad Table 5 (for the two factor model (6.2)). We observe that the test statistics (3.4), (3.5) proposed by Bardorff-Nielse ad Shephard (26) ad our test statistics Ŝ(2) t ad Ŝ(4) t ted to overestimate the true level, while the testig procedure (3.7) proposed by Aït-Sahalia ad Jacod (28) uderestimates it. The particular performace of the tests depeds o the sample size. While the ratio statistic T BS,r (X) t of Bardorff-Nielse ad Shephard (26) ad the test statistic T AJ (X) t of Aït-Sahalia ad Jacod (28) yield better results for small sample sizes, our test statistics Ŝ(2) t ad Ŝ(4) t have the best performace for = 1 ad larger (Ŝ(4) t is slightly better tha Ŝ(2) t ). However, all test statistics perform rather well. 14

17 Now we cosider the oisy semimartigale. The correspodig results for S oise (2) t are reported i Tables 9 ad 1. We observe that the asymptotic theory starts to work for relatively large sample sizes, i.e. for = 9, 16, 25. It is ot surprisig, because the semimartigale process is corrupted by oise, so we expect a slower speed of covergece. Quite iterestigly, the performace of S oise (2) t looks rather good for very small sample sizes. However, this issue is due to the fact that differet fiite sample effects seem to elimiate each other as it has bee reported i Podolskij ad Vetter (26). 6.2 Power performace We start with the o-oise case. The cotiuous part of the semimartigale is geerated accordig to the models (6.1) ad (6.2). We add to the cotiuous part the followig jump processes: (i) oe jump with the jump size.4 for (6.1) ad.26 for (6.2), (ii) two jumps with jump sizes.8 ad.8 for (6.1) ad.26 2 /2 ad.26 2 /2 for (6.2), (iii) three jumps with N(,.16 )-distributed jump sizes for (6.1) 3 ad with N(,.262 )-distributed jump sizes for (6.2). All jump times are idepedet 3 ad U([, 1])-distributed. Notice that the quadratic variatio of the jump is kept (approximately) costat (.16 for model (6.1) ad.26 2 for model (6.2)). The correspodig power performace is reported i Tables 2-4 ad 6-8. The results are strikig. Our test statistics Ŝ(2) t ad Ŝ(4) t yield by far the best power performace for all models. More precisely, our method detects the jumps at relatively small sample frequecies (i.e. = 5, 1), whereas the testig procedures of Bardorff-Nielse ad Shephard (26) ad Aït-Sahalia ad Jacod (28) start to work at quite high sample frequecies (i.e. = 3, 1). Besides, the results show that it is more difficult to fid small jumps tha oe big jump (which is ot surprisig). Fially, let us cosider the semimartigale model with oise. We geerate the same semimartigale processes as described above. The power performace of the test statistic S oise (2) t is preseted i Tables 9 ad 1. We observe that the jumps are much harder to detect i models with oise. This is due to the slower covergece rate of the threshold. Cosequetly, much more data poits are required to detect jumps. Our test yields good results for the case of oe big jump whe the sample size is rather high (i.e. = ). If the jumps are small it takes extremely large samples to ucover them. Nevertheless, the power performace seems to be quite reasoable sice we cosider oisy observatios of semimartigales. 15

18 7 Appedix 7.1 Proofs By stadard trucatio techique (see e.g. Bardorff-Nielse, Graverse, Jacod, Podolskij ad Shephard (26)) we ca assume w.l.o.g. that the processes a ad σ are bouded. We deote all costats which appear i the proofs by C or by C p whe they deped o a additioal parameter p. I the followig we will ofte use the iequality E[ i X c l ] C l l 2, l >, (7.1) where X c deotes the cotiuous part of X, which is deduced by the Burkholder iequality. Proof of Theorem 1: (i) Assume first that the process X has o jumps o the iterval [, t], i.e. we are o the set Ω c t. It suffices to show that S(p) t F st U (see (2.13) of Corollary 1). The, by the properties of stable covergece, we also obtai Note that by (7.1) we have T (X, p) t F st t Var[η i ]µ 2p σ s p dw s. p 1 2 [t] i=1 [t] i X p η i 1 { i X >c ϖ } C l p 1 2 +lϖ for ay l >. Choosig l > 1 2(ϖ 1/2) T (X, p) t = p 1 2 [t] i=1 i=1 we obtai the approximatio i X p+l η i = O P ( l(ϖ 1/2)+1/2 ) i X p ( 1 η i ) + o P (1) =: T (X, p) t + o P (1). From Theorem 2 i Podolskij ad Ziggel (27) we deduce that P ( T (X, p) t ρ(p) t ) P x F Φ(x), where Φ is the distributio fuctio of a stadard ormal variable ad ρ(p) t is defied i (2.11). Set Y = T (X, p) t ρ(p) t 16

19 ad cosider a F-measurable variable Z. The we obtai as E [1 {Y x,z z}] = E [1 {Z z} P (Y x F)] Φ(x)P (Z z) = P (U x, Z z), where U is defied i (2.13). It follows by defiitio that S(p) t F st U ad we are doe. (ii) By the results of Aït-Sahalia ad Jacod (28) we obtai uder assumptio (H), for ay p 2, [t] i=1 i X p (1 η i 1 { i X c ϖ } ) P s t X s p. Hece, o Ω j t we have T (X, p) t P, which completes the proof of Theorem 1. Proof of Propositio 2: Recall that X = σw ad the distributio of η is give by (2.15). As i the proof of Theorem 1 we obtai the approximatio ( ) p 1 [t] 2 Ŝ(p) i=1 i X p 1 η i t = + O Var[ηi ]V (X, 2p) P ( 1 ) =: S(p) t + O P ( 1 ) t o Ω c t. For the coditioal cumulats of S(p) t we deduce the followig idetities k 1 := plim E[ S(p) t F] =, k 3 := ( plim E[( S(p) t ) 3 F] 3E[( S(p) t ) 2 F]E[( S(p) t ) F] + 2(E[( S(p) ) t ) F]) 3 =, where the secod idetity follows from the fact that η has a symmetric distributio (aroud 1). Usig a stadard Edgeworth expasio (see Hall (1992), p. 48) we coclude that ( ) P S(p) t x F = Φ(x) + R (x), where R (x) satisfies E[ R (x) ] = O( 1 ). By takig the expectatio we obtai ( ) P S(p) t x = Φ(x) + O( 1 ), 17

20 which completes the proof of Propositio 2. Proof of Theorem 3: First, we itroduce the decompositio X () t = X c t + X j,() t, where X c t deotes the cotiuous part of X () t ad X j,() t = γ J t. (i) Set γ = p 1 2p. Observe that T (X (), p) t = p p 1 2 ) i X () (1 p η i 1 { i X () c ϖ } i I t ( ) i X () p 1 η i 1 { i X () c ϖ } i (It )c with It = {i the process J jumps o [ i 1, i ]}. Note that the first sum is fiite (a.s.), because J is a compoud Poisso process. By Theorem 1 we have ( ) t p 1 2 i X () p F st 1 η i 1 { i X () c ϖ } Var[η i ]µ 2p σ s p dw s, i (It )c ad, sice ϖ < p 1 2p, p 1 2 i X () (1 p η i 1 { i X () c ϖ } i I t Cosequetly, it holds that T (X (), p) t for ay p 2 ad t >. F st ) P s t s t J s p. t Var[η i ]µ 2p σ s p dw s + J s p (ii) Set γ = 1 2. Observe that (recall (7.1)) T BS (X) t = (V (X c, 2) t µ 2 1 V (X c, 1, 1) t + V (X j,(), 2) t ) + o P (1). As above we obtai T BS (X () ) t where U c t is give i Theorem 3. F st Ut c + s t J s 2, (iii) Set γ = 3 8. Sice J has oly fiitely may jumps, we deduce by (7.1) T AJ (X () ) t = ( 2V (X c, 4) /2 ) t V (Xj,(), 2) /2 t V (X c, 4) t V (X c, 4) t + o P (1). 18

21 Cosequetly, we have T AJ (X () ) t F st U c t + 2 s t J s 4 3 t σ4 sds where U c t is give i Theorem 3., Proof of Theorem 4: Sice the process (α 2 t ) defied i (4.3) is supposed to be càdlàg, we ca assume without loss of geerality that (α 2 t ) is bouded. Notice the idetity Z i = k 1 j=1 ( j ) g k i+jz = By Burkholder iequality we obtai k 1 j= ( ( j ) ( j + 1 )) g g k k Z i+j. (7.2) E[ X c i q ] C q/4 (7.3) for all q ad uiformly i i (X c deotes the cotiuous part of X). O the other had, usig the right-had side of (7.2), we deduce the followig iequality for the oise process (whe the process N t (2p) is locally bouded): which holds for ay q < 2p ad uiformly i i. E[ ɛ i q ] C q/4, (7.4) (i) Assume that the process X has o jumps, i.e. we are o Ω c t. Due to iequalities (7.3) ad (7.4) we obtai the approximatios (see the proof of Theorem 1 (i)) ad [t] k +1 T oise (Z, p) t = (p 2)/4 i= [t] k +1 Γ(p) t = Var[η i ] (p 2)/2 Z i p ( 1 η i ) + o P (1) =: T oise (Z, p) t + o P (1) i= ( ) Sice Var T oise (Z, p) t F = Γ(p) t we deduce that Z i 2p + o P (1) =: Γ(p) t + o P (1). ) P (S oise (p) P t x F Φ(x), where Φ is the distributio fuctio of a stadard ormal variable (this follows by the same methods that are used i the proof of Theorem 2 i Podolskij ad Ziggel (27)). The argumets of the proof of Theorem 1 yield the covergece S oise (p) t F st U, 19

22 where U is defied i (4.11), ad we are doe. (ii) By the results of Podolskij ad Vetter (28) (see the proof of Lemma 1 therei) we obtai uder assumptio (H), for ay p 2, ad 1 k [t] k +1 i= Hece, o Ω j t we have Z i p (1 η i 1 { Z i c ϖ } Γ(p) t t P Var[η i ]µ 2p S oise (p) t ) P 1 g(u) p du ( ψ 2 θσs 2 + ψ ) 1 pds. θ α2 s P, s t X s p, which completes the proof of Theorem 4. Refereces [1] Aït-Sahalia Y., Jacod J. (28) Testig for jumps i a discretely observed process, Aals of Statistics, forthcomig. [2] Bardorff-Nielse O.E., Graverse S. E., Jacod J., Podolskij M., Shephard N. (26) A Cetral Limit Theorem for Realized Power ad Bipower Variatios of Cotiuous Semimartigales, i Y. Kabaov, R. Lipster, J. Stoyaov, eds, From Stochastic Calculus to Mathematical Fiace: The Shiryaev Festschrift, Spriger-Verlag, pp [3] Bardorff-Nielse O.E., Shephard N. (24a) Power ad bipower variatio with stochastic volatility ad jumps, Joural of Fiacial Ecoometrics, 2, [4] Bardorff-Nielse O.E., Shephard N. (24b) Ecoometric Aalysis of of Realized Covariatio: High Frequecy Based Covariace, Regressio, ad Correlatio I Fiacial Ecoomics, Ecoometrica, Vol. 72, No. 3, [5] Bardorff-Nielse O.E., Shephard N. (26) Ecoometrics of testig for jumps i fiacial ecoomics usig bipower variatio, Joural of Fiacial Ecoometrics, 4, 1-3. [6] Cot R., Macii C. (28) Noparametric tests for aalyzig the fie structure of price fluctuatios. Workig paper. [7] Delbae, F., Schachermayer, W. (1994). A geeral versio of the fudametal theorem of asset pricig. Mathematische Aale 3,

23 [8] Hall P. (1992) The Bootstrap ad Edgeworth Expasio, Spriger-Verlag, New York. [9] Hase P.R., Lude A. (26) Realized variace ad market microstructure oise. Joural of Busiess ad Ecoomic Statistics 24, [1] Jacod J. (28) Asymptotic properties of realized power variatios ad related fuctioals of semimartigales. Stochastic Process. Appl. 118 (4), [11] Jiag G.J., Oome R.C. (25) A ew test for jumps i asset prices, The Uiversity of Warwick, Warwick Busiess Schoo, Techical Report. [12] Lee S.S., Myklad P.A. (27) Jumps i fiacial markets: a ew oparametric test ad jump dyamics, Review of Fiacial Studies, forthcomig. [13] Macii C. (21) Disetaglig the jumps of the diffusio i a geometric jumpig Browia motio, Giorale dell Istituto Italiao degli Attuari LXIV, [14] Macii C. (24) Estimatio of the characteristics of the jumps of a geeral Poisso-diffusio model, Scadiavia Actuarial Joural, 1, [15] Podolskij M., Vetter M. (26) Estimatio of volatility fuctioals i the simultaeous presece of microstructure oise ad jumps. Workig paper. [16] Podolskij M., Vetter M. (28) Bipower-type estimatio i the oisy diffusio settig. Workig paper. [17] Podolskij M., Ziggel D. (27) Bootstrappig bipower variatio, Tech. rep., Ruhr- Uiversität Bochum. [18] Wu C.F.J. (1986) Jackkife, bootstrap ad other resamplig methods i regressio aalysis. Aals of Statistics 14, [19] Zhag L., Myklad P.A., Ait-Sahalia Y. (25) A tale of two time scales: Determiig itegrated volatility with oisy high frequecy data. Joural of the America Statistical Associatio 1 (472),

24 7.1 Simulatio Results T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 1: This table shows the level performace for the model (6.1). 22

25 T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 2: This table shows the power performace for a jump-diffusio process. The process is geerated accordig to the model (6.1) plus oe jump with the jump size.4. 23

26 T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 3: This table shows the power performace for a jump-diffusio process. The process is geerated accordig to the model (6.1) plus two jumps with jump sizes.8 ad.8. 24

27 T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 4: This table shows the power performace for a jump-diffusio process. The process is geerated accordig to the model (6.1) plus three N(,.16 3 )-distributed jumps. 25

28 T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 5: This table shows the level performace for the model (6.2). 26

29 T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 6: This table shows the power performace for a jump-diffusio process. The process is geerated accordig to the model (6.2) plus oe jump with the jump size

30 T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 7: This table shows the power performace for a jump-diffusio process. The process is geerated accordig to the model (6.2) plus two jumps with jump sizes.26 2 /2 ad.26 2 /2. 28

31 T BS (X) t -1% T BS (X) t -2.5% T BS (X) t -5% T BS (X) t -1% T BS (X) t -25% T BS,r (X) t -1% T BS,r (X) t -2.5% T BS,r (X) t -5% T BS,r (X) t -1% T BS,r (X) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(4) t -1% Ŝ(4) t -2.5% Ŝ(4) t -5% Ŝ(4) t -1% Ŝ(4) t -25% T AJ (X) t -1% T AJ (X) t -2.5% T AJ (X) t -5% T AJ (X) t -1% T AJ (X) t -25% Table 8: This table shows the power performace for a jump-diffusio process. The process is geerated accordig to the model (6.2) plus three jumps with N(, )-distributed jump sizes. 29

32 Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Table 9: This table shows the level (upper pael) ad power (lower paels) performace for the model (6.1) which is corrupted by oise. First, we added oe jump with jump size.4 (secod pael). The we added two jumps with jump sizes.8 ad.8 (third pael) ad three N(,.16 3 )-distributed jumps (fourth pael). 3

33 Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Ŝ(2) t -1% Ŝ(2) t -2.5% Ŝ(2) t -5% Ŝ(2) t -1% Ŝ(2) t -25% Table 1: This table shows the level (upper pael) ad the power (lower paels) performace for the model (6.2) which is corrupted by oise. First, we added oe jump with the jump size.26 (secod pael). The we added two jumps with jump sizes.26 2 /2 ad.26 2 /2 (third pael) ad three N(, )-distributed jumps. (fourth pael). 31

34 Research Papers : Ole E. Bardorff-Nielse, José Mauel Corcuera, Mark Podolskij ad Jeaette H.C. Woerer: Bipower variatio for Gaussia processes with statioary icremets 28-22: Mark Podolskij ad Daiel Ziggel: A Rage-Based Test for the Parametric Form of the Volatility i Diffusio Models 28-23: Silja Kiebrock ad Mark Podolskij: A Ecoometric Aalysis of Modulated Realised Covariace, Regressio ad Correlatio i Noisy Diffusio Models 28-24: Matias D. Cattaeo, Richard K. Crump ad Michael Jasso: Small Badwidth Asymptotics for Desity-Weighted Average Derivatives 28-25: Mark Podolskij ad Mathias Vetter: Bipower-type estimatio i a oisy diffusio settig 28-26: Marti Møller Adrease: Esurig the Validity of the Micro Foudatio i DSGE Models 28-27: Tom Egsted ad Thomas Q. Pederse: Retur predictability ad itertemporal asset allocatio: Evidece from a bias-adjusted VAR model 28-28: Frak S. Nielse: Local polyomial Whittle estimatio coverig ostatioary fractioal processes 28-29: Per Frederikse, Frak S. Nielse ad Morte Ørregaard Nielse: Local polyomial Whittle estimatio of perturbed fractioal processes 28-3: Mika Meitz ad Petti Saikkoe: Parameter estimatio i oliear AR-GARCH models 28-31: Igmar Nolte ad Valeri Voev: Estimatig High-Frequecy Based (Co-) Variaces: A Uified Approach 28-32: Marti Møller Adrease: How to Maximize the Likelihood Fuctio for a DSGE Model 28-33: Marti Møller Adrease: No-liear DSGE Models, The Cetral Differece Kalma Filter, ad The Mea Shifted Particle Filter 28-34: Mark Podolskij ad Daiel Ziggel: New tests for jumps: a thresholdbased approach