CENTRE FOR ECONOMETRIC ANALYSIS

CENTRE FOR ECONOMETRIC ANALYSIS CEA@Cass http://www.cass.city.ac.uk/cea/idex.html Cass Busiess School Faculty of Fiace 106 Buhill Row Lodo EC1Y 8TZ Testig for Oe-Factor Models versus Stochastic Volatility Models Valetia Corradi ad Walter Distaso CEA@Cass Workig Paper Series WP CEA 08-2004

Testig for Oe-Factor Models versus Stochastic Volatility Models Valetia Corradi Quee Mary, Uiversity of Lodo Walter Distaso Uiversity of Exeter November 2004 Abstract This paper proposes a testig procedure i order to distiguish betwee the case where the volatility of a asset price is a determiistic fuctio of the price itself ad the oe where it is a fuctio of oe or more possibly uobservable factors, drive by ot perfectly correlated Browia motios. Broadly speakig, the objective of the paper is to distiguish betwee a geeric oe-factor model ad a geeric stochastic volatility model. I fact, o specific assumptio o the fuctioal form of the drift ad variace terms is required. The proposed tests are based o the differece betwee two differet oparametric estimators of the itegrated volatility process. Buildig o some recet work by Badi ad Phillips 2003 ad Bardorff-Nielse ad Shephard 2004a, it is show that the test statistics coverge to a mixed ormal distributio uder the ull hypothesis of a oe factor diffusio process, while diverge i the case of multifactor models. The fidigs from a Mote Carlo experimet idicate that the suggested testig procedure has good fiite sample properties. Keywords: realized volatility, stochastic volatility models, oe-factor models, local times, occupatio desities, mixed ormal distributio JEL classificatio: C22, C12, G12. We are grateful to Karim Abadir, Carol Alexader, James Davidso, Marcelo Ferades, Nour Meddahi, Peter Phillips ad the semiar participats to the 2004 SIS coferece i Bari, Uiversity of Exeter ad Uiversità di Padova for very helpful commets ad suggestios. The authors gratefully ackowledge fiacial support from the ESRC, grat code R000230006. Quee Mary, Uiversity of Lodo, Departmet of Ecoomics, Mile Ed, Lodo, E14NS, UK, email: v.corradi@qmul.ac.uk. Uiversity of Exeter, Departmet of Ecoomics, Streatham Court, Exeter EX4 4PU, UK, email: w.distaso@ex.ac.uk.

1 Itroductio I fiace the dyamic behavior of uderlyig ecoomic variables ad asset prices has bee ofte described usig oe-factor diffusio models, where volatility is a determiistic fuctio of the level of the uderlyig variable. 1 Sice determiig the fuctioal form of such diffusio processes is particularly importat for pricig cotiget claims ad for hedgig purposes, several specificatio tests have bee proposed, withi the class of oe-factor models. Examples iclude Aït-Sahalia 1996, who compares the parametric desity implied by a give ull model with a oparametric kerel desity estimator. He rejects most of the commoly employed models ad argues that rejectios are maily due to oliearity i the drift term. 2 Similar fidigs to those of Aït-Sahalia 1996 have bee also provided by Stato 1997 ad Jiag 1998. Durham 2003 also rejects most of the popular models; i his case rejectios are maily due to misspecificatio of the volatility term. I particular, he fids implausibly high values for the elasticity parameter i the Costat Elasticity of Variace CEV model, implyig violatio of the statioarity assumptio. Badi 2002 applies fully oparametric estimatio of the drift ad variace diffusio terms, based o the spatial methodology of Badi ad Phillips 2003, ad fids that the drift term is very close to zero over most of the rage of the short term iterest rate. Therefore, rejectios of a give model seem to be due to failure of the mea reversio property rather tha to oliearity i the drift term. Qualitatively similar fidigs are obtaied by Coley, Hase, Luttmer ad Scheikma 1997, usig geeralized method of momets tests based o the properties of the ifiitesimal geerator of the diffusio. 3 Most of the papers cited above have suggested testig ad modelig procedures which are valid uder the maitaied hypothesis of a oe-factor diffusio data geeratig process. Hece, the eed of testig for the validity of the whole class of oe-factor models. This is the objective of the paper. Uder miimal assumptios, the paper proposes a testig procedure i order to distiguish betwee the case i which the volatility process is a determiistic fuctio of the level of the uderlyig variable ad the oe i which it is a fuctio of oe or more 1 Although i the fiacial literature there is a somewhat widespread cosesus about the fact that stock prices are better characterized by multifactor stochastic volatility models, short term iterest rates are still ofte modeled as a oe-factor diffusio process, i which volatility is a determiistic fuctio of the level of the variable see e.g. Vasicek, 1977, Brea ad Schwartz, 1979, Cox, Igersoll ad Ross, 1985, Cha, Karolyi, Logstaff ad Saders, 1992, Pearso ad Su, 1994. 2 Aït-Sahalia 1996 does ot reject a geeralized versio of the Costat Elasticity of Variace model. His results have bee revisited by Pritsker 1998, who poits out the sesitivity of Aït-Sahalia s test to the degree of depedecy i the short iterest rate process. 3 See also the comprehesive review o estimatio of oe-factor models by Fa 2003. 1

possibly uobservable factors, drive by ot perfectly correlated Browia motios. With a slight abuse of termiology, the former class of models is referred to as oe-factor models ad the latter as stochastic volatility models. 4 I particular, the paper compares geeric classes of oe-factor versus stochastic volatility models, without makig assumptios o the fuctioal forms of either the drift or the variace compoet. If the ull hypothesis is ot rejected, the oe ca use the differet testig ad modelig procedures metioed above, based o the maitaied hypothesis of a oe-factor diffusio geeratig process. Coversely, if the ull hypothesis is rejected, the oe has to perform model diagostics withi the class of stochastic volatility models, usig for example the efficiet method of momets e.g. Cherov, Gallat, Ghysels ad Tauche, 2003, or geeralized momet tests based o the properties of the ifiitesimal geerator of the diffusio see e.g. Corradi ad Distaso, 2004. For example, oe ca test the validity of multi factor term structure models, suggested by e.g. Duffie ad Sigleto 1997, Dai ad Sigleto 2000, 2002. The suggested test statistics are based o the differece betwee a kerel estimator of the istataeous variace, averaged over the sample realizatio o a fixed time spa, ad realized volatility. The ituitio behid the chose statistic is the followig: uder the ull hypothesis of a oe-factor model, both estimators are cosistet for the uderlyig itegrated volatility; uder the alterative hypothesis the former estimator is ot cosistet, while the latter is. More precisely, buildig o some recet work by Badi ad Phillips 2003 ad Bardorff-Nielse ad Shephard 2004a, it is show that the statistics weakly coverge to mixed ormal distributios uder the ull hypothesis ad diverge at a appropriate rate uder the alterative. The derived asymptotic theory is based o the time iterval betwee successive observatios approachig zero, while the time spa is kept fixed. As a cosequece, the limitig behavior of the statistic is ot affected by the drift specificatio. Also, o statioarity or ergodicity assumptio is required. The proposed testig procedure is derived uder the assumptios that the uderlyig variables are observed without measuremet error ad that the geeratig processes belog to the class of cotiuous semimartigales. Therefore, the provided tests are ot robust to the presece of either jumps or market microstructure effects; more precisely, whe either of the two occur, the test teds to reject the ull hypothesis, eve if the volatility process is a determiistic fuctio of the uderlyig variable. However, as the test is computed over a fiite time spa, oe ca first test for the hypotheses of o jumps ad o microstructure effects, ad the perform the suggested testig procedure over a time spa i which either of the hypotheses above is rejected. 4 I the stochastic volatility literature, ofte by oe-factor model oe meas a model i which volatility is a fuctio of a sigle stochastic factor, drive by a Browia motio ot perfectly correlated with the oe drivig the uderlyig ecoomic variable or the asset price. 2

The rest of this paper is orgaized as follows. I Sectio 2, the testig procedure is outlied ad the relevat limit theory is derived. Sectio 3 reports the fidigs from a Mote Carlo exercise, i order to assess the fiite sample behavior of the proposed tests. Cocludig remarks are give i Sectio 4. All the proof are gathered i the Appedix. p d I this paper,, ad a.s. deote respectively covergece i probability, i distributio ad almost sure covergece. We write 1 { } for the idicator fuctio, ϖ for the iteger part of ϖ, I J for the idetity matrix of dimesio J ad Z MN, to deote that the radom variable Z is distributed as a mixed ormal. 2 Testig for Oe-Factor vs Stochastic Volatility Models 2.1 Set-Up As discussed above, our objective is to device a data drive procedure for decidig betwee oefactor diffusio models ad stochastic volatility models, uder miimal assumptios. We cosider the followig class of oe-factor diffusio models dx t = µx t dt + σ t dw 1,t ad the followig class of stochastic volatility models σ t = σ X t 1 dx t = µx t dt + σ t dw 1,t σt 2 = gf t df t = bf t dt + σ 1 f t dw 2,t, 2 where f t is typically a uobservable state variable drive by a Browia motio, W 2,t, possibly but ot perfectly correlated with the Browia motio drivig X t, thus allowig for possible leverage effects. The models i 1 ecompass the class of parametric specificatios aalyzed by Aït-Sahalia 1996, ad they also allows for geeric oliearities. The models i 2 iclude the square root stochastic volatility of Hesto 1993, the Garch diffusio model Nelso, 1990, the logormal stochastic volatility model of Hull ad White 1987 ad Wiggis 1987, ad are also related to the class of eigefuctio stochastic volatility models of Meddahi 2001. Note that f t may be a multidimesioal process, thus allowig for multifactor stochastic volatility processes. Also, the oe-factor model may be possibly ested withi the stochastic volatility model, i the sese that we ca allow for the specificatio σt 2 = σ 2 X t gf t. Aderse ad Lud 1997 ad Durham 2003 3

propose to exted the differet oe-factor models by addig a stochastic volatility term, ad suggest models i which volatility depeds o both the level of the uderlyig variable ad a latet factor, drive by a differet Browia motio. 5 I particular, it should be stressed that i our procedure we compare geeric classes of oe factor versus stochastic volatility models, without ay fuctioal form assumptio o either the drift or the variace term. We state the hypothesis of iterest as H 0 : σ 2 t = σ 2 X t, a.s. versus the alterative H A : σt 2 = g f t, a.s. where ω Ω +, 1 0 g f s g X s ds 0 ad Pr Ω + = 1, with Ω + Ω, ad Ω deotes the probability space o which f t, X t are defied. Thus, uder the ull hypothesis the volatility process is a measurable fuctio of the retur process X t. O the other had, uder the alterative, the volatility process is a measurable fuctio of a possibly uobservable process f t. I the paper, we simply require that the occupatio desities of the observable process X t ad of the possibly uobservable factor f t do ot coicide. I fact, if they do coicide, the the itegrated volatility process would be almost surely the same uder both hypotheses. Fially, ote that the case of σ 2 t = σ 2 X t gf t falls uder the alterative hypothesis, while the case of a costat variace falls uder the ull. I the sequel, we assume that we have data recorded at two differet frequecies, over a fixed time spa, which for sake of simplicity, but without loss of geerality, is assumed equal to 1. 6 More specifically, we assume to have ad m observatios, with m, so that the discrete samplig iterval is equal respectively to 1/ ad 1/m. The proposed test statistics are based o Z,m,r = m 1 SX 2 i/ RV m,r, 3 where r 0, 1], S 2 X i/ = 1 j=1 1 { X j/ X i/ <ξ } X j+1/ X j/ 2 1 j=1 1 { X j/ X i/ <ξ } 4 5 Aderse ad Lud 1997 fid that the iclusio of a stochastic volatility compoet i a square root model helps the elasticity parameter to fall i the statioary regio. Durham 2003 fids that, although the additio of a secod factor icreases the likelihood, it has very little impact as to what cocers bod pricig. 6 I Sectio 3, reportig the results of the simulatio study, we will cosider a time spa equal to five days. 4

ad RV m,r = m 1r j=1 Xj+1/m X j/m 2. 5 Note that S 2 X i/ is a oparametric estimator of the volatility process evaluated at X i/ ; Flores- Zmirou 1993 has established cosistecy ad the asymptotic distributio of a scaled versio of 4 whe the variace process follows 1. 7 Recetly, S 2 X i/ has bee used by Badi ad Phillips 2003, i the cotext of fully oparametric estimatio of diffusio processes; their asymptotic theory is based o both the time spa goig to ifiity ad the discrete iterval betwee successive observatios goig to zero. This is because they are iterested i the joit estimatio of the drift ad variace diffusio terms. 8 Coversely, our objective is to distiguish betwee the cases i which volatility is a measurable fuctio of the observable process, ad the oe i which it depeds o some other state variable. Therefore we remai silet about the drift term, ad we oly cosider asymptotic theory i terms of the discrete iterval approachig zero. I fact, o a fiite time spa the cotributio of the drift term is asymptotically egligible. Notice that S 2 X i/ is a cosistet estimator of the istataeous variace oly uder the ull hypothesis. Therefore, also its average over the sample realizatio of the process o a fiite time spa, 1/ S 2 X i/, is a cosistet estimator of itegrated volatility oly uder the ull hypothesis. RV m,r, which is kow as realized volatility, has bee proposed as a measure for volatility cocurretly by Aderse, Bollerslev, Diebold ad Labys 2001, Aderse, Bollerslev, Diebold ad Ebes 2002 ad Bardorff-Nielse ad Shephard 2002. The properties of realized volatility have bee extesively aalyzed by Bardorff-Nielse ad Shephard 2002, 2004a,b, Aderse, Bollerslev, Diebold ad Labys 2003, Bardorff-Nielse, Graverse ad Shephard 2004 see also Aderse, Bollerslev, Meddahi, 2004a,b, ad Meddahi, 2002, 2003. Realized volatility is a model free estimator of the quadratic variatio of the processes defied i 1 ad 2, ad is cosistet for the itegrated daily volatility uder both hypotheses. Bardorff-Nielse ad Shephard 2004a have show that a scaled ad cetered versio of RV m,r weakly coverges to a mixed ormal distributio whe the log price process follows a cotiuous semimartigale, a result which we will use i the proof of our Theroem 1. The reaso why we use two differet sample frequecies i the 7 The estimator S 2 X i/ has bee also used by Corradi ad White 1999 i order provide a test for the correct specificatio of the variace process, regardless of the drift specificatio. Withi the class of oe-factor models, a more geeral test, also allowig for time o-homogeeity, has bee suggested by Dette, Podolskij ad Vetter 2004. 8 Badi ad Phillips 2003 cosider a slightly modified versio of S 2 X i/, with a geeric kerel K replacig the idicator fuctio. See also Jiag ad Kight 1997. 5

computatio of S 2 X i/ ad RV m,r will become clear i the ext subsectio. I the sequel we shall eed the followig assumptio. Assumptio 1. a σ ad µ, defied i 1, satisfy local Lipschitz ad growth coditios. Therefore, for ay compact subsets M uder the ull hypothesis ad J uder the alterative hypothesis of the rage of the process X t, there exist costats K M 1, KM 2, KM 3, KM 4, KJ 1 ad KJ 2, such that, x, y M ad x, y J, σx σy K M 1 x y, σx 2 K2 M 1 + x 2, µx µy K4 M x y, µx µy K J 2 x y ad xµx K M 3 1 + x 2, x µx K J 1 1 + x 2. b σ 1 ad b, defied i 2, satisfy local Lipschitz ad growth coditios. Therefore, for ay compact subset L of the rage of the process f t, there exist costats K L 1, KL 2, KL 3 K L 4, such that, p, q L, σ 1 p σ 1 q K L 1 p q, σ 1 p 2 K L 2 1 + p 2, bp bq K L 3 p q ad ad pbp K L 4 1 + p 2. c µ, σ ad g are cotiuously differetiable. Assumptio 1a states local Lipschitz ad growth coditios for the drift term uder both hypotheses ad for the variace term uder the ull hypothesis. Assumptio 1b states local Lipschitz ad growth coditios for the variace term uder the alterative. Assumptios 1ab esure the existece of a uique strog solutio uder both hypotheses see e.g. Chug ad Williams, 1990, p.229. Sice we are studyig the diffusio processes over a fixed time spa, we do ot eed to impose more demadig assumptios, such as statioarity ad ergodicity. 9 9 Note that Badi ad Phillips 2001, 2003 allow the time spa to approach ifiity, ad the require the diffusio to be ull Harris recurret. 6

2.2 Limitig Behavior of the Statistic We ca ow establish the limitig distributio of the proposed test statistics based o Z,m,r, defied i 3, for both the cases where = m ad m/ 0, as m,. Theorem 1. Let Assumptio 1 hold. Uder H 0, ia if, as, m, ξ 1, ξ ad for ay arbitrarily small ε > 0, 1/2+ε ξ 0, ad if m =, the, poitwise i r 0, 1 Z,r d Z r MN 0, 2 σ 4 a L Xr, a L X 1, a L X r, a da L X 1, a, 6 where Z,r Z,,r ad 1 1 r L X r, a = lim ψ 0 ψ σ 2 1 a {Xu [a,a+ψ]}σ 2 X u du 0 deotes the stadardized local time of the process X t. ib Defie Z = max j=1,...,j Z,rj ad Z = max j=1,...,j Z rj, where 0 < r 1 <... < r j 1 < r j <... < r J < 1, for j = 1,..., J, with J arbitrarily large but fiite. If, as, m, ξ 1, ξ, ad, for ay ε > 0 arbitrarily small, 1/2+ε ξ 0, ad if m =, the Z d Z, with Z r1 Z r2. Z rj MN 0, V r 1, r 1 V r 1, r 2... V r 1, r J V r 2, r 1 V r 2, r 2... V r 2, r J...... V r J, r 1 V r J, r 2... V r J, r J, 7 where r, r, V r, r = V r, r = 2 σ 4 a L Xmir, r, a L X 1, a L X mir, r, a da. L X 1, a ic If, as, m, ξ 1 0, the, ξ ad ξ 2 0, ad, for ay ε > 0 arbitrarily small, m/ 1 ε Z,m,r d ZM r MN 0, 2 σ 4 a L X r, ada. 7

id Defie Z,m = max j=1,...,j Z,m,rj ad ZM = maxj=1,...,j ZMrj, where 0 < r1 <... < r j 1 < r j <... < r J < 1, for j = 1,..., J, with J arbitrarily large but fiite. If, as, m, ξ 1, ξ ad ξ 2 0, ad, for ay ε > 0 arbitrarily small, m/ 1 ε 0, the d ZM, Z,m with ZM r1 ZM r2. ZM rj MN 0, V Mr 1, r 1 V Mr 1, r 2... V Mr 1, r J V Mr 2, r 1 V Mr 2, r 2... V Mr 2, r J...... V Mr J, r 1 V Mr J, r 2... V Mr J, r J, 8 where that, r, r, V Mr, r = V Mr, r = 2 σ 4 a L X mir, r, ada. ii Uder H A, if, as, m, ξ 1 poitwise i r 0, 1],, ξ ad ξ 2 0, ad if m/ π 0, the, Pr ω : 1 Z,m,r ω ςω 1, m where ςω > 0 for all ω Ω +, where Ω + is defied as i the statemet of H A. Notice that, as show i the proof i the Appedix, uder the alterative hypothesis, ad i the case where f t is a oe-dimesioal process, the domiat term of the proposed statistic is a scaled versio of the absolute value of the differece betwee the local times of X t ad f t. If istead f t is a multidimesioal process, the the multivariate local time aalogue of the L f 1, a used i Theorem 1 is ot defied, but it ca still be iterpreted as a occupatio desity of the multivariate diffusio f t see e.g. Gema ad Horowitz, 1980 ad Badi ad Moloche, 2001. Therefore, i both cases, there exists a almost surely strictly positive radom variable ς, such that 1/ m Z m,,r ς, with probability approachig oe. The followig Corollary cosiders the case where r = 1, i.e. whe we use the whole spa of data i costructig the test statistic. Corollary 1. Let Assumptio 1 hold. Uder H 0, if, as, m, ξ 1, ξ ad ξ 2 0, ad, for ay ε > 0 arbitrarily small, m/ 1 ε 0, the d MN 0, 2 σ 4 a L X 1, ada. Z,m,1 8

. 10 The theoretical results derived above provide a ufeasible limit theory, sice the variace Thus, for r = 1, the statistic has a mixed ormal limitig distributio for m/ 0 as m, compoets have to be estimated. A cosistet estimator of the stadardized local time is give by L X, r, a = 1 1 2ξ Sa 2 1 { Xi/ a <ξ }. Thus a estimator of 2 σ 4 a L Xr, a L X 1, a L X r, a da, 9 L X 1, a i.e. of the quatity resultig i Theorem 1 part ia, is give by where 2 1 σ 4 a = L X, r, a LX, 1, a L X, r, a σ a 4 L X, 1, a 1 1 4 { X i/ a <ξ } 2 X i+1/ X i/ 1 1. { X i/ a <ξ } da, 10 I order to implemet the estimator i 10, we eed to choose the iterval of itegratio, = 1, 2. Now, if we choose too small, the we may ru the risk of gettig a icosistet estimator of the term i 9. O the other had, if we choose too large, the for some a, L X, r, a ad L X, 1, a would be very close to zero, ad the estimator i 10 will result i a ratio of two terms approachig zero. Of course, whe computig 10 we ca exclude all a for which, say, L X, 1, a δ, where δ 0 as. However, devicig a data-drive procedure for choosig δ is ot a easy task. I order to avoid this problem, we istead propose below a upper boud for the critical values of the limitig distributio i Theorem 1, parts ia ad ib. I fact, ote that almost surely, 2 2 σ 4 a L Xr, a L X 1, a L X r, a da L X 1, a σ 4 al X r, ada 2 r 0 σ 4 X s ds, where the last equality above follows from Lemma 3 i Badi ad Phillips 2003. Now, Bardorff-Nielse ad Shephard 2002 have show that 3 10 Whe m = ad r = 1, the statistic coverges to zero i probability. r 4 p Xi+1/ X i/ σsds, 4 11 0 9

where σs 4 = σ 4 X s uder H 0 ad σs 4 = g 2 f s uder H A ; i other words the estimator defied i 11 is cosistet for the true itegrated quarticity uder both hypotheses ad therefore provides a estimator of the upper boud of the term i 9. O the other had, we shall provide correct asymptotic critical values for the limitig distributio i Theorem 1, parts ic ad id ad i Corollary 1. I order to obtai asymptotically valid critical values ad to make the limit theory derived i Theorem 1 part id feasible, we will use a data-depedet approach. For s = 1,..., S, where S deotes the umber of replicatios, let d s 1/2 m,r 1 Ĉ m r 1, r 1 Ĉ m r 1, r 1 Ĉ m r 1, r 1 Ĉ m r 1, r 1 η s 1 d s d s m,r 2 Ĉ m r 1, r 1 Ĉ m r 2, r 2 Ĉ m r 2, r 2 Ĉ m r 2, r 2 η s m,r = =.., 12..... 2. d s m,r J Ĉ m r 1, r 1 Ĉ m r 2, r 2... Ĉ m r J, r J η s J where m 1r j Ĉ m r j, r j = 2 m 4 X 3 i+1/m X i/m is a cosistet estimator of twice the itegrated quarticity ad, for each s, η s 1 η s 2... η s J is draw from a N0, I J. The compute max ds j=1,...,j m,r, repeat this step S times, ad costruct the empirical distributio. As S, the empirical distributio of max ds j=1,...,j will coverge the distributio of a radom variable defied as 0, MN 2 max j=1,...,j σ 4 a L X r j, ada. Therefore a asymptotically valid critical value for the limit theory i Theorem 1 part id will be give by CVα S s, which deotes the 1 α quatile of the empirical distributio of max j=1,...,j d m,r j, computed usig S replicatios. Give the discussio above, CVα S will provide a upper boud for the critical values of the limitig distributio derived i Theorem 1, part ib. The implied rules for decidig betwee H 0 ad H A are outlied i the followig Propositio. Propositio 1. Let Assumptio 1 hold. a Let S. Suppose that as, m, ξ 1, ξ ad, for ay ε > 0 arbitrarily small, 1/2+ε ξ 0. If m =, the do ot reject H 0 if m,r Z CVα S ad reject otherwise. This rule provides a test with asymptotic size smaller tha α ad asymptotic uit power. 10

b Let S. Suppose that, as, m, ξ 1, ξ ad ξ 2 0, ad, for ay ε > 0 arbitrarily small, m/ 1 ε 0; the do ot reject H 0 if Z,m CVα S ad reject otherwise. This rule provides a test with asymptotic size equal to α ad asymptotic uit power. As metioed above, our test is desiged to compare two classes of models, amely the oefactor diffusio models ad the stochastic volatility models, regardless of the specificatio of the drift term. Therefore, if for example model 1 is augmeted by addig aother factor ito the drift term see e.g. Hull ad White, 1994, our test will still fail to reject the ull hypothesis cosidered, because the drift term is, over a fixed time spa, of a smaller order of probability tha the diffusio term ad so is asymptotically egligible. 2.3 Market Microstructures ad jumps The asymptotic theory derived i the previous subsectio relies o the fact that the uderlyig process is a cotiuous semi-martigale. However, some recet fiacial literature has poited out the effects of possible jumps ad market microstructure error o realized volatility see e.g. Bardorff-Nielse ad Shephard, 2004c,d, Corradi ad Distaso, 2004, Aderse, Bollerslev ad Diebold, 2003 for jumps, ad Aït-Sahalia, Myklad ad Zhag, 2003, Zhag, Myklad ad Aït- Sahalia, 2003, Badi ad Russell, 2003, Hase ad Lude, 2004 for microstructure oise. We begi by aalyzig the cotributio of large ad rare jumps. Suppose that the geeratig process i 1 is augmeted by a jump compoet, dx t = µx t dt + dz t + σ t dw 1,t, where σ t = σx t, ad z t is a pure jump process. The test statistics based o Z,m,r are ot robust to the presece of jumps. The ituitive reaso is that jumps have a differet impact o the two compoets of the statistics, amely 1 S 2 X i/ ad RV m,r. I fact, i the presece of jumps, RV m,r coverges to the itegrated volatility process plus the sum of the squared magitudes of the jumps see Bardorff-Nielse ad Shephard, 2004c. Coversely, 1 S 2 X i/ coverges to itegrated volatility plus the weighted sum of the squared magitudes of the jumps, where the weights deped o the local time of X t. Broadly speakig, a 11

jump occurrig at time j/ has a larger effect o the compoet 1 S 2 X i/ if there are may observatios i the eighborhood of X j/. However, sice our test is carried over a fixed time spa, we ca pretest for the presece of o jumps, followig for example Bardorff-Nielse ad Shephard 2004c,d; they proposed a test based o the properly scaled differece betwee realized volatility ad bipower variatio, which is a cosistet estimator of itegrated volatility i the presece of large ad rare jumps i the log price process. If the ull hypothesis is ot rejected, we ca apply our methodology. Huag ad Tauche 2004 also suggest a variety of Hausma type tests for jumps ad fid evidece of a relatively small umber of jumps i the log price process. A similar fidig is reported by Aderse, Bollerslev ad Diebold 2003. As for the presece of microstructure effects, suppose that the observed price of a asset ca be decomposed ito X j/m = Y j/m + ɛ j/m. Here ɛ j/m is iterpreted as a oise capturig the market microstructure effect. The cotributio of the microstructure oise o realized volatility has already bee aalyzed i a series of recet papers see e.g. Aït-Sahalia, Myklad ad Zhag, 2003, Zhag, Myklad ad Aït-Sahalia, 2003, Badi ad Russell, 2003 ad Hase ad Lude, 2004. For example, if the microstructure oise has a costat variace, i.e. idepedet of the samplig iterval, the m 1 RV m,r p 2rν where ν deotes the variace of the microstructure oise see Zhag, Myklad ad Aït-Sahalia, 2003. As for 1 SX 2 i/, due to the discreteess of the measuremet error compoet, the behavior of ξ 1 1 j=1 1 { X j/ X i/ <ξ } is ot easy to assess. Therefore, our procedure will ot be valid if the log price process is cotamiated by microstructure oise. Similarly to the case of large ad rare jumps, it is possible to pretest the series uder ivestigatio for the absece of microstructure oise. I fact, Awartai, Corradi ad Distaso 2004 have suggested a simple test for the ull hypothesis of o market microstructure, based o the appropriate scaled differece betwee two realized volatility measures costructed over differet samplig frequecies. 11 We ca the apply our procedure over a time spa for which either the ull hypothesis of o jumps or the ull hypothesis of o microstructure oise has bee rejected. 11 Awartai, Corradi ad Distaso 2004 also propose a specificatio test of the ull hypothesis of microstructure oise with costat variace. See also Bardorff-Nielse ad Shephard 2004c for a alterative model of the market microstructure oise, where the variace of the oise is allowed to deped o the samplig frequecy of the data. 12

3 A Simulatio Experimet I this sectio, the small sample performace of the testig procedure proposed i the previous sectio will be assessed through a Mote-Carlo experimet. Uder the ull hypothesis, we cosider a versio of the Cox, Igersoll ad Ross 1985 model with a mea revertig compoet i the drift, dx t = κ + µx t dt + η X t dw 1,t. 13 We first simulate a discretized versio of the cotiuous trajectory of X t uder 13. We use a Milstei scheme i order to approximate the trajectory, followig Pardoux ad Talay 1985, who provide coditios for uiform, almost sure covergece of the discrete simulated path to the cotiuous path, for give iitial coditios ad over a fiite time spa. I order to get a very precise approximatio to the cotiuous path, we choose a very small time iterval betwee successive observatios 1/5760; moreover, the iitial value is draw from the gamma margial distributio of X t, ad the first 1000 observatios are the discarded. We the sample the simulated process at two differet frequecies, 1/ ad 1/m, ad compute the differet test statistics. I particular, the time spa has bee fixed to five days ad five differet values have bee chose for the umber of itradaily observatios, ragig from 144 correspodig to data recorded every te miutes to 1440 correspodig to data recorded every miute. Therefore, the total umber of observatios rages from T = 720 to T = 7200, where T deotes the fixed time spa expressed i days. Also, the experimet has bee coducted for six T.7 T.75 T.8 T.9 T.95 differet values for m amely /T, /T, /T, /T, /T ad the the limitig case m =. ad The process is repeated for a total of 10000 replicatios. Results are reported for two test statistics, amely 1r j 1 Z,m = max m S j=1,...,j 2 X i/ RV m,rj Z,m,1 = 1 1 m S 2 X i/ RV m,1. Uder the coditios stated i Theorem 1, we kow that for m/ 0, Z,m d ZM = max ZMrj, j=1,...,j ad for m =, Z d Z = max Zrj, j=1,...,j 13

where the vectors ZM r1 ZM r2... ZM rj ad Z r1 Z r2... Z rj are defied respectively i 8 ad 7. I the simulatio experimet, J = 16, with r startig from r 1 =.15 ad the icreasig by.05 util r 16 =.85. The critical values defied i 12 have bee obtaied with S = 1000. Similarly, uder the coditios stated i Corollary 1, we have that for m/ 0, d Z,m,1 MN 0, 2 σ 4 a L X 1, ada The empirical sizes at 5% ad 10% level of the tests discussed above are reported i Table 1, for κ = 0, η = 1, µ =.8, ξ = 10/13. The results for differet values of the parameters eeded to geerate 13 ad the badwidth ξ display a virtually idetical patter ad therefore are omitted for space reasos. Ispectio of the Table reveals a overall good small sample behaviour of the cosidered test statistics. The reported empirical sizes are everywhere very close to the omial oes, with a slight tedecy to uderreject for the test based o Z,m. The zeros appearig i the rows whe = m are ot surprisig; i fact, whe usig the statistic Z, the critical values used i the simulatio exercise are just a upper boud of the true oes, ad therefore oe should expect a udersized test. Uder the alterative hypothesis, the followig model has bee cosidered, dx t = κ + µx t dt + η exp σt 2 1 ρ 2 dw 1,t + ρdw 2,t dσ 2 t = κ 1 + µ 1 σ 2 t dt + η 1 σ 2 t dw 2,t. 14 A discretized versio of 14 has bee simulated usig a Milstei scheme as above, with κ 1 = 1, η 1 = 1, µ 1 =.2. The, usig the obtaied values of σ 2 t, the series for X t has bee geerated, with ρ = 0 ad keepig the remaiig parameters at the values used to geerate X t uder 13. The fidigs for the power of the tests based o Z,m ad Z,m,1 are reported i Table 2. The experimet reveals that the proposed tests has good power properties. The test based o Z,m is more powerful tha the oe based o Z,m,1 ; this is ot surprisig, give that Z,m is specifically costructed to highlight the differeces betwee the local times of X t ad f t. I fact, i the case of Z m, the term drivig the power is max r r 0 L Xr, a L f r, a da, which is i geeral larger tha 1 0 L X1, a L f 1, a da, the term drivig the power of Z. Also, the power of the test based o Z,m is geerally icreasig i ad m, as oe should expect. I some cases, however, the power remais costat or eve decreases whe m approaches amely, the cases whe = 144, 288, 576; this is due to the fact that, whe = m, we are ot usig the correct critical values for the test, but just a upper boud, ad this may decrease the resultig power of the test. 14

4 Cocludig remarks This paper provides a testig procedure which allows to discrimiate betwee oe-factor ad stochastic volatility models. Hece, it allows to distiguish betwee the case i which the volatility of a asset is a fuctio of the asset itself ad therefore the volatility process is Markov ad predictable i terms of its ow past, ad the case i which it is a diffusio process drive by a Browia motio, which is ot perfectly correlated with the Browia motio drivig the asset. The suggested test statistics are based o the differece betwee a kerel estimator of the istataeous variace, averaged over the sample realizatio o a fixed time spa, ad realized volatility. The ituitio behid is the followig: uder the ull hypothesis of a oe-factor model, both estimators are cosistet for the true uderlyig itegrated daily volatility; uder the alterative hypothesis the former estimator is ot cosistet, while the latter is. More precisely, we show that the proposed statistics weakly coverge to well defied distributios uder the ull hypothesis ad diverge at a appropriate rate uder the alterative. The derived asymptotic theory is based o the time iterval betwee successive observatios approachig zero, while the time spa is kept fixed. As a cosequece, the limitig behavior of the statistic is ot affected by the drift specificatio. Also, o statioarity or ergodicity assumptio is required. The fiite sample properties of the suggested statistic are aalyzed via a small Mote Carlo study. Uder the ull hypothesis, the asset process is modelled as a versio of the Cox, Igersoll ad Ross 1985 model with a mea revertig compoet i the drift. Thus, volatility is a square root fuctio of the asset itself. Uder the alterative, the asset ad volatility processes are geerated accordig to a stochastic volatility model, where volatility is modelled as a square root diffusio. The empirical sizes ad powers of the proposed tests are reasoably good across various m/ ratios. 15

Table 1: Actual sizes of the tests based o Z,m,r for differet values of m ad Z,m Z,m,1 5% omial size 10% omial size 5% omial size 10% omial size = 144 m = 20 0.03 0.07 0.05 0.07 m = 27 0.03 0.07 0.03 0.07 m = 38 0.04 0.08 0.04 0.09 m = 74 0.04 0.08 0.05 0.10 m = 103 0.03 0.07 0.04 0.09 m = 0.00 0.00 = 288 m = 32 0.03 0.07 0.05 0.12 m = 46 0.03 0.07 0.07 0.10 m = 67 0.03 0.07 0.07 0.10 m = 139 0.03 0.07 0.06 0.09 m = 200 0.03 0.07 0.05 0.12 m = 0.00 0.00 = 576 m = 52 0.02 0.07 0.08 0.11 m = 78 0.02 0.07 0.05 0.10 m = 117 0.03 0.08 0.08 0.13 m = 259 0.03 0.08 0.04 0.08 m = 386 0.03 0.07 0.06 0.11 m = 0.00 0.00 = 720 m = 61 0.03 0.08 0.07 0.10 m = 92 0.03 0.08 0.04 0.09 m = 139 0.03 0.08 0.08 0.13 m = 317 0.03 0.08 0.04 0.09 m = 478 0.02 0.07 0.05 0.10 m = 0.00 0.00 = 1440 m = 100 0.03 0.07 0.07 0.12 m = 156 0.03 0.08 0.08 0.12 m = 243 0.06 0.10 0.07 0.12 m = 592 0.05 0.10 0.07 0.12 m = 923 0.05 0.11 0.08 0.13 m = 0.00 0.00

Table 2: Actual powers of the tests based o Z,m,r for differet values of m ad Z,m Z,m,1 5% omial size 10% omial size 5% omial size 10% omial size = 144 m = 20 0.12 0.22 0.14 0.17 m = 27 0.10 0.14 0.10 0.17 m = 38 0.14 0.18 0.09 0.15 m = 74 0.34 0.42 0.09 0.16 m = 103 0.38 0.44 0.19 0.25 m = 0.42 0.44 = 288 m = 32 0.20 0.26 0.11 0.21 m = 46 0.22 0.36 0.10 0.16 m = 67 0.44 0.48 0.12 0.16 m = 139 0.54 0.56 0.33 0.45 m = 200 0.56 0.64 0.26 0.30 m = 0.54 0.54 = 576 m = 52 0.22 0.26 0.11 0.16 m = 78 0.44 0.54 0.12 0.14 m = 117 0.48 0.54 0.16 0.19 m = 259 0.56 0.70 0.13 0.14 m = 386 0.84 0.88 0.70 0.76 m = 0.76 0.82 = 720 m = 61 0.44 0.52 0.12 0.18 m = 92 0.40 0.54 0.14 0.18 m = 139 0.58 0.66 0.21 0.25 m = 317 0.68 0.72 0.25 0.33 m = 478 0.72 0.82 0.22 0.29 m = 0.76 0.88 = 1440 m = 100 0.50 0.60 0.21 0.29 m = 156 0.55 0.75 0.20 0.24 m = 243 0.65 0.70 0.60 0.62 m = 592 0.90 0.90 0.88 0.89 m = 923 0.90 0.95 0.65 0.70 m = 0.95 0.95

A Proofs Before provig Theorem 1, we eed the followig Lemmas. Lemma 1. Let Assumptio 1 hold. The sup µx s = O a.s. ε/4, s [0,1] sup s [0,1] σ 2 X s = Oa.s. ε/2, for ay ε > 0, arbitrarily small. sup gf s = O a.s. ε/2, s [0,1] A.1 Proof of Lemma 1 We start from the case whe X t follows 1. Defie R l = {if t : X t > l}. Thus, R l is a F t measurable stoppig time. Let X mit,rl = mit,rl 0 mit,rl µx s ds + σ 2 X s dw 1,s. 0 Obviously, for all t R l, X mit,rl = X t. Now let Ω l = {ω : R l > 1} ad l = l = ε/4. Thus, give the growth coditios i Assumptio 1a, X t is a o-explosive diffusio, ad so PrΩ l 1 = 1. By a similar argumet, give Assumptios 1a, 1b, the same holds whe the volatility process follows 2. Therefore, the statemet follows. Lemma 2. Let Assumptio 1 hold. Uder H 0, if, as, ξ, ξ 2 0 ad, for ay ε > 0 arbitrarily small, m/ 1 ε 0, the, poitwise i r, A.2 Proof of Lemma 2 By Ito s formula m S 2 X i/ σ 2 X i/ p 0. = m S 2 X i/ σ 2 X i/ }{{} m A,m,r 1 j=1 1 { X j/ X i/ <ξ } X j+1/ X j/ 2 1 j=1 1 { X j/ X i/ <ξ } σ 2 X i/ 18

= 1 m j=1 1 { X j/ X i/ <ξ } 2 j+1/ j/ Xs X j/ σxs dw 1,s 1 j=1 1 { X j/ X i/ <ξ } }{{} G,m,r 1 m j=1 + 1 { X j/ X i/ <ξ } 2 j+1/ j/ Xs X j/ µxs ds 1 j=1 1 { X j/ X i/ <ξ } }{{} H,m,r 1 m j=1 + 1 { X j/ X i/ <ξ } j+1/ j/ σ 2 X s σ 2 X i/ ds 1 j=1 1. 15 { X j/ X i/ <ξ } }{{} D,m,r Thus, we eed to show that G,m,r, H,m,r ad D,m,r are o P 1. Now, because of Lemma 1, D,m,r m sup σ 2 X s σ 2 X τ X s X τ ξ m sup σ 2 X τ sup X s X τ τ [0,1] X s X τ ξ provided that m 1/2 ε/2 ξ 0. Sice m = o 1 ε, the which approaches zero almost surely. = O m O a.s. ε/2 O a.s ξ = o a.s. 1, 16 O a.s m ε/2 ξ = o a.s 1/2 ξ, As for G,m,r, by the proof of Step 1 of Theorem 1, partia, below, / mg,m,r = G,r coverges i distributio ad so it s O P 1; therefore G,m,r = o P 1, give that m/ 0, as m,. Fially, give the cotiuity of µ, H,m,r m sup s [0,1] µx s sup i/ s 1/ s [0,1] Xs X i/ = mo a.s. ε/4 O a.s. 1/2 log = o a.s. 1. 17 I fact, because of the modulus of cotiuity of a diffusio see McKea, 1969, p.96, sup Xs X i/ = Oa.s. 1/2 log, i/ s i/ s [0,1] ad 1/2 ε+ε/4 1/2 log = 3ε/4 log 0. Therefore, the statemet follows. We ca ow prove Theorem 1. 19

A.3 Proof of Theorem 1 ia Z,r = 1 S 2 X i/ σ 2 X i/ }{{} j=1 A,r r 2 Xj+1/ X j/ σ 2 X s ds } {{ } B,r + 1 σ 2 X i/ r σ 2 X s ds. 18 } {{ } C,r The proof of the statemet is based o the four steps below. d Step 1: A,r MN 0, 2 σ4 a L Xr,a 2 L X 1,a da. d Step 2: B,r MN 0, 2 σ4 al X 1, ada. Step 3: Let < A, B > r defie the discretized quadratic covariatio process. plim < A, B > r 2 0 0 σ 4 a L Xr, a 2 da = 0. L X 1, a Step 4: C,r = o P 1. Proof of Step 1: First ote that usig Ito s formula A,r = 1 1 j=1 1 { X j/ X i/ <ξ } 2 X j+1/ X j/ 1 j=1 1 σ 2 X i/ { X j/ X i/ <ξ } = 1 1 j=1 1 { X j/ X i/ <ξ } 2 j+1/ j/ Xs X j/ σxs dw 1,s 1 j=1 1 { X j/ X i/ <ξ } }{{} G,r + 1 1 j=1 1 { X j/ X i/ <ξ } 2 j+1/ j/ Xs X j/ µxs ds 1 j=1 1 { X j/ X i/ <ξ } }{{} H,r 20

+ 1 1 j=1 1 { X j/ X i/ <ξ } j+1/ j/ σ 2 X s σ 2 X i/ ds 1 j=1 1. { X j/ X i/ <ξ } }{{} D,r Now, give Lemma 1, D,r = o a.s. 1, provided that 1/2+ε ξ 0, as. It is immediate to see that H,r is of a smaller order of probability tha G,r. Let < G > r deote the discretized quadratic variatio process of F,r. By a similar argumet as i Badi ad Phillips 2003, pp.271-272, plim < G > r 2 σ 4 a L Xr, a 2 da = 0. L X 1, a Thus, by the same argumet as i the proof of Theorem 3 i Badi ad Phillips 2003, the statemet i Step 1 follows. Proof of Step 2: It follows from Theorem 1 i Bardorff-Nielse ad Shephard 2004a. Proof of Step 3: The discretized covariatio process < A, B > r, = 4 = 2 < A, B > r r r = 2 0 0 = 2 j=1 j=1 j+1/ 2 1 { Xj/ X i/ <ξ } j/ Xs X j/ σxs dw s 1 j=1 1 { X j/ X i/ <ξ } 1 { X j/ X i/ <ξ } σ4 X j/ + o a.s. 1 1 j=1 1 { X j/ X i/ <ξ } 1 { Xu Xa <ξ }σ 4 X u du 1 0 1 { X u X a <ξ }du 1 { u a <ξ}σ 4 ul X r, udu 1 { u a <ξ }L X 1, udu da + o a.s. 1 L X r, ada + o a.s. 1, 19 where the 2 istead of 4 o right had side of 19 comes from Lemma 5.3 i Jacod ad Protter 1998. Alog the lies of Badi ad Phillips 2001, 2003, by the chage of variable we have that u a ξ = z, < A, B > r 21

= 2 = 2 a.s. 2 Proof of Step 4: 1 { u a <ξ }σ 4 ul X r, udu 1 L X r, ada + o a.s. 1 { u a <ξ }L X 1, udu 1 { zξ <ξ }σ 4 a + zξ L X r, a + zξ dz 1 L X r, ada + o a.s. 1 { zξ <ξ }L X 1, a + zξ dz σ 4 a L Xr, a 2 da. 20 L X 1, a C,r = 1 σ 2 X i/ r σ 2 X s ds 0 = 1 σ 2 X i/ = i+1/ i/ i+1/ i/ σ 2 X s ds σ 2 X i/ σ 2 X s ds 21 ad, give the Lipschitz assumptio o σ 2, the last lie i 21 is o P 1 by the same argumet as the oe used i Step 1. Give Steps 1-4 above, it follows that the quadratic variatio process of Z,r is give by = 2 2 σ 4 a L X r, ada + 2 σ 4 a L Xr, a 2 L X 1, a da 4 σ 4 a L Xr, a 2 L X 1, a da σ 4 a L Xr, a L X 1, a L X r, a da. 22 L X 1, a The statemet i the theorem the follows. ib Without loss of geerality, suppose that r < r. By otig that 1 S 2 X i/ = 1 [ 1r ] S 2 X i/ [ 1r ] Xi+1/ X i/ 2 Xi+1/ X i/ 2, with S 2 X i/ = 0 ad X i+1/ X i/ 2 = 0 for i > 1r, the result the follows by the cotiuous mappig theorem. 22

ic The statistic Z,m,r ca be rewritte as m Z,m,r = m S 2 X i/ σ 2 X i/ } {{ } A,m,r m 1r j=1 r 2 Xj+1/m X j/m σ 2 X s ds } {{ } B m,r m + Note that A,m,r = o P 1 by Lemma 2. σ 2 X i/ r m } {{ } C,m,r 0 0 σ 2 X s ds. 23 We first eed to show that C,m,r = o a.s. 1. Give Assumptio 1a, Lemma 1, ad recallig the modulus of cotiuity of a diffusio see McKea, 1969, pp.95-96, m σ 2 X i/ r m σ 2 X s ds 0 = m σ 2 X i/ i+1/ m σ 2 X s ds i/ i+1/ = m σ 2 X i/ σ 2 X s ds i/ m i+1/ i/ σ 2 X i/ σ 2 X s ds m sup σ 2 X s σ 2 X τ m sup s τ 1/ s [0,r] τ [0,r] = mo a.s. ε/2 O a.s. 1/2 log = o a.s. 1, as 1/2 ε/2 1/2 log 0. Thus, Z,m,r = B m,r + o a.s. 1. σ 2 X τ The statemet the follows from the proof of Step 2 i part ia. sup s τ 1/ s [0,r] X s X τ id The statemet follows by the same argumet as the oe used i part ib ad by the cotiuous mappig theorem. 23

ii We will prove the Theorem for the case aalyzed i part ic; i the other cases the proof follows straightforwardly ad is therefore omitted. Uder H A, we have that dx t = µx t dt + σt 2dW 1,t σ 2 t = gf t Poitwise i r, we ca rewrite Z,m,r as df t = bf t dt + σ 1 f t dw 2,t. Z,m,r = = m m + g f i/ S 2 X i/ g m 1r f i/ m m S 2 X i/ g f i/ }{{} m m 1r j=1 E,m,r j=1 r 2 Xj+1/m X j/m g f s ds } {{ } F m,r m + 0 Xj+1/m X j/m 2 g r f i/ m g f s ds. 24 } {{ } L,m,r 0 By the same argumet used i the proof of part ia, Step 4 ad Step 2 respectively L,m,r = o P 1 ad F m,r = O P 1. We ca expad E,m,r as = = E,m,r 1 m j=1 1 { X j/ X i/ <ξ } 2 X j+1/ X j/ 1 j=1 1 g f i/ { X j/ X i/ <ξ } 1 m j=1 1 { X j/ X i/ <ξ } 2 j+1/ j/ Xs X j/ g fs dw 1,s 1 j=1 1 { X j/ X i/ <ξ } }{{} P,m,r 24

1 m j=1 + 1 { X j/ X i/ <ξ } 2 j+1/ j/ Xs X j/ µxs ds 1 j=1 1 { X j/ X i/ <ξ } }{{} S,m,r 1 m j=1 + 1 { X j/ X i/ <ξ } j+1/ j/ g fs g f i/ ds 1 j=1 1. 25 { X j/ X i/ <ξ } }{{} Q,m,r Now, P,m,r is O p 1, agai for the same argumet used i the proof of part ia Step 1, ad similarly S,m,r = o P 1, sice it has the same behavior uder both H 0 ad H A. The, we eed to show that Q,m,r, i absolute value, diverges. Now, 1 Q,m,r = 1 1 j=1 1 { X j/ X i/ <ξ } j+1/ j/ g Xs g X i/ ds m 1 j=1 1 { X j/ X i/ <ξ } }{{} T,m,r + 1 1 j=1 1 { X j/ X i/ <ξ } j+1/ j/ g f s g X s ds 1 j=1 1 { X j/ X i/ <ξ } }{{} 1 U,m,r j=1 1 { X j/ X i/ <ξ } g fi/ g Xi/ 1 j=1 1. 26 { X j/ X i/ <ξ } 1 }{{} V,m,r Note that T,m,r is o P 1, by the same argumet as the oe used i part ia, Step 1. As for V,m,r, it ca be rewritte as 1 g fi/ g Xi/ = r 0 r gx s ds gf s ds + O P 1/2, 27 0 where the first term of the right had side of 27 is almost surely differet from 0, give that X s ad f s have differet occupatio desity. Also, i the case i which f s is oe-dimesioal, we have that 1 g fi/ g Xi/ = gal X r, ada gal f r, ada + O P 1/2, where L f r, a resp. L X r, a deotes the stadardized local time of the process f t resp. X t evaluated at time r ad at poit a, that is it deotes the amout of time spet by the 25

process f t resp. X t aroud poit a, over the period [0, 1]. Thus, m g fi/ g Xi/ diverges to either or to, at rate m, provided that L X r, a L f r, a 0 almost surely for all a A, with A havig o-zero Lebesgue measure, that is provided that f t ad X t have differet occupatio desities over a o-egligible set. Fially, U,m,r ca be writte as 1 1 1 ξ j=1 1 { X j/ X i/ <ξ } j+1/ j/ g f s g X s ds. 28 L X 1, X i/ + o P 1 Expadig the sums, 28 ca be rewritte as 1 2/ 1/ g f s g X s ds ξ L X 1, X 1/ + o P 1 1 { X1/ X 2/ <ξ } 2/ 1/ g f s g X s ds + L X 1, X 2/ + o P 1... +... +... + 1 ξ + 1 { X1/ X [ 1r/] <ξ } 2/ 1/ g f s g X s ds L X 1, X 1/ + o P 1 1 { X j/ X 1/ <ξ } j+1/ j/ L X 1, X 1/ + o P 1 g f s g X s ds 1 { Xj/ X 2/ <ξ } j+1/ j/ g f s g X s ds L X 1, X 2/ + o P 1... +... +... + 1 ξ 1 { Xj/ X [ 1r/] <ξ } j+1/ j/ g f s g X s ds L X 1, X 1/ + o P 1 1 { X 1/ X 1/ <ξ } 1 1/ g f s g X s ds L X 1, X 1/ + o P 1 1 { } 1 <ξ g f X 1/ X s g X 2/ 1/ s ds + L X 1, X 2/ + o P 1 1 { } 1 <ξ g f X 1/ X s g X [ 1r/] 1/ s ds... +... L X 1, X 1/ + o P 1. 26

Thus, 1 1 j=1 L X1, X j/ g f j/ g Xj/ + o P 1 sup i L X 1, X i/ 1 1 j=1 1 { X j/ X i/ <ξ } j+1/ j/ g f s g X s ds 1 j=1 1 { X j/ X i/ <ξ } 1 1 j=1 L X1, X j/ g f j/ g Xj/ if i L X 1, X i/ + o P 1. 29 Note that the umerator i the lower ad upper bouds of the iequality i 29 approaches zero if ad oly if L X 1, a L f 1, a = 0 almost surely for all a A, with A havig o-zero Lebesgue measure, or i the multidimesioal case, if X s ad f s have the same occupatio desity, which is ideed ruled out uder the alterative hypothesis. Therefore, 1/ mz,m,r cosists of the sum of two odegeerate radom variables which do ot cacel out each other. Thus, Z,m,r diverges at rate m with probability approachig oe. Therefore, the statemet follows. A.4 Proof of Corollary 1 It follows directly from Theorem 1, part ic. A.5 Proof of Propositio 1 a From equatio 12, it follows that, for r 1 < r 2 <... < r J, d s m,r 1 d s m,r 2 d. d s m,r J MN 0, 2 r 1 0 σ4 X s ds 2 r 1 0 σ4 X s ds... 2 r 1 0 σ4 X s ds 2 r 1 0 σ4 X s ds 2 r 2 0 σ4 X s ds... 2 r 2.. 2 r 1 0 σ4 X s ds 2 r 2 0 σ4 X s ds... 2 r J 0 σ 4 X s ds 0 σ4 X s ds..... Also, ote that 2 r 1 0 σ4 X s ds 2 r 1 0 σ4 X s ds... 2 r 1 0 σ4 X s ds 2 r 1 0 σ4 X s ds 2 r 2 0 σ4 X s ds... 2 r 2 0 σ4 X s ds...... 2 r 1 0 σ4 X s ds 2 r 2 0 σ4 X s ds... 2 r J 0 σ 4 X s ds 27

V r 1, r 1 V r 1, r 2... V r 1, r J V r 2, r 1 V r 2, r 2... V r 2, r J...... V r J, r 1 V r J, r 2... V r J, r J is positive semi-defiite, where the latter matrix above is defied i the statemet of Theorem 1, part ib. Give Theorem 1, part ib, the statemet follows directly. b Immediate from Theorem 1, part id. I both cases, the uit asymptotic power of the proposed tests follows from Theorem 1, part ii. 28

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