Time-Varying Periodicity in Intraday Volatility. Torben G. Andersen, Martin Thyrsgaard and Viktor Todorov. CREATES Research Paper

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1 ime-varyig Periodicity i Itraday Volatility orbe G. Aderse, Marti hyrsgaard ad Viktor odorov CREAES Research Paper Departmet of Ecoomics ad Busiess Ecoomics Aarhus Uiversity Fuglesags Allé 4 DK-8210 Aarhus V Demark oekoomi@au.dk el:

2 ime-varyig Periodicity i Itraday Volatility orbe G. Aderse Marti hyrsgaard Viktor odorov Jauary 9, 2018 Abstract We develop a oparametric test for decidig whether retur volatility exhibits timevaryig itraday periodicity usig a log time-series of high-frequecy data. Our ull hypothesis, commoly adopted i work o volatility modelig, is that volatility follows a statioary process combied with a costat time-of-day periodic compoet. We first costruct time-of-day volatility estimates ad studetize the high-frequecy returs with these periodic compoets. If the itraday volatility periodicity is ivariat over time, the the distributio of the studetized returs should be idetical across the tradig day. Cosequetly, the test is based o comparig the empirical characteristic fuctio of the studetized returs across the tradig day. he limit distributio of the test depeds o the error i recoverig volatility from discrete retur data ad the empirical process error associated with estimatig volatility momets through their sample couterparts. Critical values are computed via easyto-implemet simulatio. I a empirical applicatio to S&P 500 idex returs, we fid strog evidece for variatio i the itraday volatility patter drive i part by the curret level of volatility. Whe market volatility is elevated, the period precedig the market close costitutes a sigificatly higher fractio of the total daily itegrated volatility tha is the case durig low market volatility regimes. JEL classificatio: C51, C52, G12. Keywords: high-frequecy data, periodicity, semimartigale, specificatio test, stochastic volatility. Aderse s ad odorov s research is partially supported by NSF grat SES Departmet of Fiace, Kellogg School, Northwester Uiversity, NBER ad CREAES. CREAES, Departmet of Ecoomics ad Busiess Ecoomics, Aarhus Uiversity. Departmet of Fiace, Kellogg School, Northwester Uiversity. 1

3 1 Itroductio Stock returs have time-varyig volatility ad this has importat theoretical as well as practical ramificatios. Most existig work o volatility has modeled it as a statioary process. However, there is both theoretical see, e.g., [1, 25] ad empirical evidece see, e.g., [2, 3] for the presece of itraday periodicity i volatility. o illustrate this pheomeo, o Figure 1, we plot the average level of the S&P 500 idex retur volatility as a fuctio of time-of-day. As see from the figure, the itraday periodic compoet of volatility is otrivial. Ideed, the average volatility at the market close is about three times the average volatility aroud luch :00 10:00 11:00 12:00 13:00 14:00 15:00 ime of Day CS Figure 1: Itraday Volatility Periodicity for the S&P 500 Idex. he plot presets smoothed estimates of the average time-of-day volatility, ormalized by the tradig day volatility. Details regardig the costructio of the series are provided i Sectio 5. High-frequecy data is icreasigly used, as it offers very sigificat efficiecy gais for measurig ad forecastig volatility, see, e.g., [5]. he proouced periodic patter 2

4 exhibited i Figure 1 has strog implicatios regardig the appropriate methodology for studyig volatility usig the itraday retur data. he usual approach i the literature assumes that the time-of-day compoet of volatility is costat across days ad the stadardizes the high-frequecy returs by the correspodig estimates, see, e.g., [2, 3], [9] ad [35] amog may others. However, this oly aihilates the itraday volatility compoet from the returs, if the latter remais time-ivariat. he goal of the curret paper is to test this ull hypothesis withi a geeral oparametric settig. Moreover, if the ull hypothesis is rejected, we provide techiques that ca help idetify the sources of variatio i the itraday periodic compoet. he statistical aalysis is coducted usig a log spa of high-frequecy retur data. he major challeges i desigig the test stem from the fact that volatility is ot directly observed ad both the statioary ad periodic compoet of volatility ca chage over the course of the day. We take advatage of the log time spa as well as the short distace betwee the itraday observatios to circumvet these latecy problems. We first estimate the average periodic compoet of volatility from the high-frequecy returs. he we stadardize the returs with these estimated time-of-day volatility compoets. Uder the ull hypothesis, this studetizatio of the returs is sufficiet to aihilate the periodic volatility compoet. herefore, the studetized high-frequecy returs should have the exact same distributio regardless of time-of-day, if the ull is true. I cotrast, this is violated uder the alterative hypothesis, where the studetized retur distributio is give by a covolutio of the distributios for the statioary volatility compoet ad the stadardized time-varyig periodic volatility compoet. As a result, the distributio of the studetized returs depeds o the time-of-day, whe the periodic volatility compoet varies over time. Give the discrepacy i the distributioal properties of the studetized returs uder 3

5 the ull ad alterative hypotheses, our test statistic is desiged to measure the distace betwee the studetized retur distributio for differet parts of the tradig day. I particular, we rely o a weighted L 2 orm of the differece i the real parts of the empirical characteristic fuctios of the studetized returs. We use oly the real parts of the empirical characteristic fuctios, because the high-frequecy returs are approximately coditioally ormally distributed, whe coditioig o the iformatio at the begiig of the retur iterval. Uder the ull hypothesis, the limitig distributio of our test statistic depeds o the error i recoverig volatility from the returs as well as the empirical process error associated with estimatig populatio momets of volatility usig their sample couterparts. he cotributio of the first of these errors to the limitig distributio is a distictive feature of our test, settig it apart from other estimatio ad testig problems ivolvig joit i-fill ad log-spa asymptotics for high-frequecy retur data, where this error is egligible asymptotically. he reaso for the added complicatio is that, due to the ature of the testig problem at had, we oly use a limited umber of high-frequecy returs per day i formig the statistic. Hece, we caot derive the limitig distributio assumig that volatility, effectively, is observed, which is a coveiet simplificatio i determiig the asymptotic properties of existig joit i-fill ad log-spa iferece procedures. As a cosequece, the limit distributio of our test statistic is o-stadard, but its quatiles are readily evaluated through simulatio. We implemet our ew testig procedure o high-frequecy retur data for the S&P 500 idex. Eve after excludig tradig days comprisig scheduled macroecoomic aoucemets, our test rejects the ull hypothesis of a time-ivariat itraday periodicity i volatility. Additioal aalysis shows that a sigificat driver of the variatio i the periodic volatility compoet is the cocurret level of volatility, as proxied by the VIX volatility 4

6 idex at the market ope. Upo separatig the tradig days ito regimes of low, medium, ad high volatility, accordig to the level of the VIX, we fid that our test rejects sigificatly less o these three subsamples. Specifically, whe volatility is elevated, the period before the market close cotributes a substatially higher fractio of the total itegrated daily volatility compared with regimes featurig lower volatility. Our paper is related to several strads of earlier work. First, there is a large literature o detectig ad modelig periodicity i discrete time series. Examples iclude [14], [20], [22], [27], [32], [34] ad [36]. Secod, there is a sizable literature that estimates assumed costat itraday volatility patters. his icludes empirical work by [2, 3], [21], [23] ad [38]. he costat itraday periodicity is further explicitly modeled or accouted for i papers estimatig volatility models ad detectig jumps, such as [9], [13], [17] ad [28]. [24] study the commoality i the itraday periodicity across may assets. Fially, [15] assume that the statioary volatility compoet is costat withi the day ad test whether the periodic volatility compoet ca explai the full dyamic evolutio across each day. We reiterate that a commo feature of the above literature is the assumed ivariace for the periodicity of volatility, while the goal of the curret paper is to test this uderlyig assumptio of existig work, ad to explore potetial sources of deviatio from this hypothesis. hird, [4] cosider testig for chages i the periodic compoet of volatility at a specific kow poit i time ad i a parametric volatility settig both for the statioary ad periodic compoets. Ulike that paper, the aalysis here is fully oparametric ad we ca test for chages i the periodic compoet, which ca happe at ukow times ad be stochastic. Fourth, our paper is related to a volumious statistical literature that tests for the equality of two distributios by usig a weighted L 2 distace betwee the associated empirical characteristic fuctios. Applicatios of this approach for testig depedece betwee two variables ca be foud i, e.g., [7, 8], [16], [19] ad 5

7 [33]. he empirical characteristic fuctio has also bee used to study serial depedece i time series by [26] ad to test for Gaussiaity i statioary time series by [18]. he major differece betwee this strad of the literature ad the curret paper is that the variables, whose distributios are compared, are ot directly observable ad eed to be filtered from the data, ad this filterig procedure affects the limitig distributio of the test statistic. he rest of the paper is orgaized as follows. Sectio 2 presets the settig ad itroduces the statistics for assessig stochastic time-variatio i the periodicity of itraday volatility. I Sectio 3 we derive the asymptotic limit theory for these statistics, ad we the apply it for developig a feasible testig procedure for whether the itraday volatility patter is time-ivariat. Sectio 4 summarizes the results obtaied from a large-scale simulatio study, ad Sectio 5 presets a empirical implemetatio of the testig procedure. Sectio 6 cocludes. All formal assumptios ad proofs are deferred to a Supplemetary Appedix. 2 Setup ad Estimatio of Periodic Volatility he log price process X is defied o some filtered probability space Ω, F, F t t 0, P. Cosistet with the absece of arbitrage, it follows a Itô semimartigale of the form, dx t = a t dt + σ t dw t + x µdt, dx, 1 where a t is the drift, W t a Browia motio, σ t the diffusive stochastic volatility, µ the coutig measure for jumps i X with compesator b t dt F dx, where b t is a càdlàg process ad F : R R +. Our mai focus is the stochastic volatility compoet. Beyod the customary statioary part, we assume it cotais a periodic compoet with a cycle R 6

8 spaig oe uit of time. Specifically, σ 2 t = σ 2 t f t,t t for some statioary process σ t ad time-of-day fuctio f : N + [0, 1] R + with f t,0 = f t,1, where the time uit is oe day. I the stadard settig, adopted i most curret work, f is determiistic ad depeds oly o the time-of-day, t t. I fact, high-frequecy data is icreasigly used as it offers very sigificat efficiecy gais for measurig ad forecastig volatility, see e.g., [5].. However, it is plausible that the periodic compoet might vary with the cocurret level of the statioary compoet of volatility, as well as the occurrece of evets such as prescheduled macroecoomic aoucemets ad, more geerally, ay shifts i the orgaizatio ad operatio of the fiacial markets. he goal of the curret paper is to test whether the time-of-day periodic compoet of volatility chages over time. he iferece will be based o discrete observatios of the process X at equidistat times 0, 1, 2,...,, where the iteger represets the time spa, ad the iteger idicates the umber of times we sample withi a uit iterval. We deote the legth of the samplig iterval by = 1/ ad the high-frequecy icremets of X by t,κx = X t 1+ κ / X t 1+ κ 1/, for t N + ad κ 0, 1]. he asymptotic settig ivolves ad, where, ituitively, the icreasig samplig frequecy assists i the oparametric idetificatio of the level of stochastic volatility from discrete observatios of X, ad the log time spa allows us to separate the statioary ad periodic compoets of volatility. Our estimate for the time-of-day compoet of volatility is give by, f κ = π 2 t,κx t,κ X 1 {A t,κ }, A t,κ = { t,κx v t,κ X v }, 2 for v = α ϖ with ϖ 0, 1/2 ad α > 0. Uder appropriate coditios, fκ coverges i probability to Ef t,κ σ 2 t+κ, for t N +. herefore, up to the costat Eσ 2 t, f κ provides a estimate for the periodic compoet of volatility, whe the latter is time-ivariat. 7

9 We test for ivariace of the itraday compoet of volatility by comparig the distributio of estimates for volatility deseasoalized by f κ over differet parts of the tradig day. Uder the ull hypothesis, these distributios are idetical, while they differ uder the alterative. he iferece for the distributio of volatility at differet parts of the day will be based o the result i [35] that the real part of the empirical characteristic fuctio of the high-frequecy icremets i X is a estimate for the Laplace trasform of stochastic volatility. herefore, we itroduce, L κu = 1 2u cos t,κx/ f κ, u R +, 3 ad, as show i the ext sectio, L κ coverges i probability i a fuctioal sese to, [ L κ u = E e u ft,κ σ2 t+κ / E[ft,κ σ2 t+κ] ], for t N + ad κ 0, 1]. 4 3 estig for ime-ivariat Periodicity of Volatility We proceed with the formal asymptotic results for L κ, which i tur will allow us to costruct a feasible test for detectig time-varyig itraday volatility periodicity. 3.1 Ifeasible Limit heory Our results will be based o the fuctio L κu i u, ad the fuctioal covergece results below take place i the Hilbert space L 2 w, { } L 2 w = f : R + R fu 2 wu du <, 5 R + for some positive-valued cotiuous weight fuctio w with expoetial tail decay. As usual, we deote the ier product ad the orm o L 2 w by, ad, respectively. Covergece i probability for L κ is established i the followig theorem. 8

10 heorem 1. Suppose Assumptios 1-3 i the Supplemetary Appedix hold with K = {κ}, for some κ 0, 1], ad ϖ [ 1, 1. he, as ad, we have, 8 2 L κ P L κ. 6 he ituitio behid the above result is the followig. First, fκ is a estimate of Ef t,κ σt+κ. 2 Secod, over small time itervals, we have t,κx σ κ t 1+ t,κw. From here, the result i heorem 1 follows by a Law of Large Numbers. heorem 1 requires both ad, but imposes o restrictio o their relative rate of growth. We emphasize that the above result is fuctioal, i.e., we recover the Laplace trasform L κ as a fuctio of u. As is well kow, the Laplace trasform of a positive-valued radom variable uiquely idetifies its distributio. herefore, ay differeces i L κ for differet times-of-day differet values of κ must stem from time variatio i the periodic compoet of volatility. I this case, studetizig the high-frequecy icremets by the time-of-day estimate f κ will ot be eough to elimiate the itraday periodic compoet. We ext derive a Cetral Limit heorem CL for the differece i L κ, for two differet values of κ, uder the ull hypothesis. heorem 2. Suppose Assumptios 1-3 i the Supplemetary Appedix hold with K = {κ, κ } ad f t,κ f κ costat time-of-day periodicity for t N +. Let ϖ [ 1, 2 ]. he, 8 5 for ay κ, κ 0, 1], as ad with 0, we have, L κ L κ L N0, K, 7 where K is a covariace itegral operator characterized by, Khz = kz, u hu wu du, R + h L 2 w, 8 9

11 with kerel kz, u = j= E [d 1zd j+1 u] d t u = cos 2uσ 2t 1+κZ t u L u π 2 ad, cos 2uσt 1+κ 2 Z t σ 2 t 1+κ Z t Z t σ 2 t 1+κ Z t Z t, 9 for {Z t }, { Z t }, {Z t} ad { Z t} beig sequeces of idepedet stadard ormal radom variables defied o a extesio of the origial probability space ad idepedet of F. he limit result has some otable features. First, the rate of covergece is cotrolled by the time spa of the data. he limit result requires 0, that is, we sample slightly faster tha we icrease the time spa of the data. his is a stadard coditio i joit i-fill ad log-spa asymptotic settigs. It esures that certai biases associated with measurig volatility from discrete observatios o X are egligible. Secod, L κ L κ differece of fuctios of icremets over differet parts of the day. is based o the Cosequetly, the asymptotic covariace operator K depeds oly o the autocovariace of the differetial betwee trasforms of volatility at differet times-of-day. As a result, the persistece i d t u is typically small, eve if σ 2 t cotais a very persistet statioary compoet. o illustrate, suppose σ 2 s is costat durig the day, i.e., for s [t 1, t] ad t N +. he we have Ed 1 zd j+1 u = 0 for j 0. he implicatio is that, eve if volatility is highly persistet which is true empirically, we do ot require a large time spa for reliable recovery of L κ L κ. his is ulike the situatio, where oe seeks to recover L κ ad L κ separately, as the precisio of those estimates will be compromised by strog volatility persistece. hird, the asymptotic limit i heorem 2 reflects two sources of error. associated with ucoverig the latet stochastic variace σ 2 t he first is from high-frequecy data. he secod is the empirical process error capturig the deviatio of sample averages for 10

12 trasforms of volatility from their ucoditioal meas. his is ulike most existig joit i-fill ad log spa asymptotic limit results, i which the error from recoverig the latet volatility is asymptotically egligible. he reaso is that here, ulike i previous work, we do ot itegrate fuctios of the high-frequecy data over the full tradig day, but rather rely o oly a fixed umber of high-frequecy icremets each day. he mai error i measurig volatility from high-frequecy returs of X stems from the icremets of the Browia motio over the small samplig itervals. We allow for these icremets to be correlated with the iovatios of σ 2, that is, the so-called leverage effect is accommodated. Nevertheless, sice the legth of the high-frequecy itervals shriks asymptotically, this depedece has a asymptotically egligible effect o the limit result i heorem 2. Hece, for the purposes of the CL of L κ L κ, our asymptotic settig becomes equivalet to coductig iferece from observatios of Z t fκ σt 1+κ, 2 Z t fκ σt 1+κ 2 ad t N + Z t f κ σt 1+κ 2, Z t f κ σ 2t 1+κ, where {Z t }, { Z t }, {Z t} ad { Z t} are i.i.d. sequeces t N + of stadard ormals idepedet from the volatility process. his situatio mirrors some features of the CL for measurig quatities associated with the jump part of X, such as their quadratic variatio, see, e.g., [29]. I that case, oly the icremets of the Browia motio over the itervals cotaiig the jumps drive the asymptotics. I our case, because we study time-of-day volatility patters, we similarly rely o oly a fiite umber of highfrequecy icremets per day. Ulike the high-frequecy aalysis of jumps, however, we also have ad, cosequetly, we have a additioal source of error drivig the CL, amely the empirical process error associated with the recovery of ucoditioal momets of volatility from the correspodig sample averages. Give heorems 1 ad 2, our test statistic is quite ituitive. It is give by the weighted squared differece of the estimates for the volatility Laplace trasforms over the two distict 11

13 periods across the tradig day, S, κ, κ = L κ L κ 2 R + L κu L κ u 2 wu du, κ, κ 0, 1]. 10 he asymptotic behavior of S, κ, κ uder the ull hypothesis follows directly from the CL i heorem 2. It is stated formally i Corollary 1. Corollary 1. Uder the coditios of heorem 2, we have, S, κ, κ L Zκ, κ, where Zκ, κ is a weighted sum of idepedet chi-squared distributios with oe degree of freedom, defied o a extesio of the origial probability space ad idepedet from F. he weights are give by the eigevalues of the covariace operator K, defied i heorem 2. Whe the alterative hypothesis is true, i.e., whe the time-of-day periodic compoet of volatility varies over time, the L κ ad L κ differ ad, from heorem 1, we coclude that S, κ, κ diverges to ifiity. 3.2 Feasible Iferece ad Costructio of the est he feasible versio of our test statistic will be based o the limit results i heorem 2 ad Corollary 1. For implemetatio, we eed to obtai a estimate of the covariace operator K from the data. o this ed, we first costruct the feasible couterpart of d t u give by d t, u = d κ κ t,u d t,u with, 2u d κ t,u = cos t,κx/ f κ + u L κu e 0.5 π 2 t,κx t,κ X f κ 1 {A t,κ }, 11 12

14 for u R + ad κ 0, 1] ad with, L κu = 1 2u t,κ X si t,κx/ f κ 1 { 2u f t,κ X v }. 12 κ I defiig d κ t,u, we impose a small sample correctio by usig u L κu e 0.5 istead of u L κu. his is because we have sup u R ul κu e 1, so it follows that the above correctio has o asymptotic effect. Give d κ t,u, the feasible kerel-type estimator of the covariace operator is give by, K f s = k s, ufu wu du, 13 R + where k is give by, k u, s = 1 ad Γ is the liear operator defied by, Γ d t, s = h0 d t, s + d t, u Γ d t, s, 14 j=1 j h dt j, s + B d t+j, s, 15 with the covetio that d t, s = 0 if t 0 or t >, ad h is a kerel which satisfies the regularity coditios give i the Supplemetary Appedix. o coduct feasible iferece for S, κ, κ, we require estimates for the eigevalues of the operator K. Sice K is a Hilbert-Schmidt operator, it follows that the eigevalues of K coverge to zero. he limitig distributio of S, κ, κ depeds o all the eigevalues of the covariace operator K. However, sice the eigevalues coverge to zero, it is atural to approximate the distributio of Zκ, κ through estimates for oly the p largest oes, where p is a sequece of positive itegers that asymptotically diverge to ifiity. his is what we do below. 13

15 Our estimates for the eigevalues of K will be based o its estimate K. By costructio, k is a degeerate kerel ad, thus, it has oly fiitely may o-zero eigevalues. Furthermore, the rage of K is spaed by d 1, u,..., d, u, so the eigefuctios are of the form ψ j u = 1 β d j,t t, u, for a sequece of coefficiets {β j,t } ad j = 1,...,. he estimated eigevalues are the obtaied by solvig the followig equatio, K ψj u = λ j ψj u, j = 1,...,. 16 Solvig for these eigevalues is equivalet to fidig the eigevalues of the matrix C, whose i, j th elemet equals c ij dj, u Γ d i, u wu du, for i, j = 1,...,. he the = 1 R + eigevalues of C, deoted λ 1,..., λ, are atural estimators of λ 1,..., λ. Based o these estimated eigevalues, we costruct the followig approximatio of the limitig distributio i Corollary 1, Ẑ κ, κ = p i=1 λ i χ 2 i, 17 where {χ 2 i } i 1 deotes the sequece of χ 2 1 distributed radom variables from Corollary 1. he followig theorem shows that the limitig distributio of our test statistic ca be approximated by Ẑ κ, κ. heorem 3. Suppose Assumptios 1-4 i the Supplemetary Appedix hold with K = {κ, κ } ad f t,κ f κ costat time-of-day periodicity for t N +. Let ϖ [ 1 4, 3 8] ad, with 0. Suppose B ad p such that, B 2 0 ad p B B We the have, Ẑ κ, κ Zκ, κ P

16 he sequece B cotrols the umber of lags of d t, u we use i the costructio of our estimator for the covariace operator K. he choice of B aturally depeds o the persistece of the uderlyig series. I stadard time series applicatios, see, e.g., [6], it typically takes values like B = O 1/3 or B = O 1/5. Give our earlier discussio, d t u will display limited persistece, ad we ca therefore reliably estimate K with oly a relatively small umber of lags icluded i the costructio of K. he secod coditio i equatio 18 puts a upper boud o the rate of growth of p which, we recall, cotrols the umber of the largest eigevalues of K icluded i costructig Ẑ κ, κ. Because the smallest eigevalues of K are estimated with less precisio, we determie the upper boud o p via the magitude of the error K K. his error, i tur, stems from the samplig error i iferrig K as well as the bias due to usig oly B autocovariaces of d t, u ad their smoothig with the kerel h i the costructio of K. As we later documet, the eigevalues of K typically die out very fast ad, hece, the test has very limited sesitivity with respect to the choice of p. With the feasible approximatio Ẑ κ, κ of Zκ, κ, we are ow ready to formally defie our test. For some κ, κ 0, 1] with κ κ, the ull ad alterative hypotheses are give by, H 0 : {L κ = L κ } ad H A : {L κ L κ }, 20 where the equality ad iequality are to be uderstood i the L 2 w sese. Defie ext, cv α, κ, κ = Q 1 α Ẑ κ, κ F, 21 where Q α Z deotes the α-quatile of a geeric radom variable Z. cv, α κ, κ is computed umerically usig the estimated eigevalues { λ i } i=1,...,p ad the simulatio of a sequece of i.i.d. χ 2 1 distributed radom variables. We the have the followig result. 15

17 Corollary 2. Suppose Assumptios 1-4 i the Supplemetary Appedix hold with K = {κ, κ } ad the sequeces B ad p satisfy coditio 18. he test defied by the critical regio { S, κ, κ > cv, α κ, κ } has asymptotic size α uder the ull ad asymptotic power oe uder the alterative, i.e., P S, κ, κ > cv, α κ, κ H0 α, P S, κ, κ > cv, α κ, κ HA Extesios Averagig Multiple ime-of-day Itervals Oe atural extesio is to compare the average Laplace trasforms of volatility over two distict sets of time-of-day itervals. his has the beefit of reducig the measuremet error, ad hece icreases the power of the test. Of course, the averagig igores the potetial differeces i the Laplace trasforms that we average. herefore, this procedure is most advatageous for itervals i which the periodic volatility compoet is similar, eve if it is time-varyig. his is aturally satisfied for adjacet itervals durig the tradig day, e.g., eighborig five-miute itervals withi oe hour. I fact, sice the periodic compoet of volatility is assumed to be Hölder of order 1/2 see Supplemetary Appedix, the distace betwee the Laplace trasforms of the deseasoalized volatilities over high-frequecy itervals withi eighborhoods of asymptotic size of order o1/ of the time-of-day κ ad κ will be o1/. herefore, the averagig of the Laplace trasforms over these blocks of high-frequecy data will cotiue to provide a valid test for the ull hypothesis of equality betwee L κ ad L κ, but with more power relative to the origial test i Corollary 2. We formalize this extesio of our test without a formal proof, as it follows straightforwardly from our earlier results. For simplicity, we restrict attetio to the case where 16

18 the umber of high-frequecy itervals, over which averagig is performed, remais fixed, that is, it does ot icrease with the samplig frequecy. he locatio of the elemets i the two sets o 0, 1] may chage with the samplig frequecy but deviates from fixed poits o 0, 1] by terms which are o1/. Deote two disjoit fiite sets of umbers i 0, 1] by K ad K. he typical example of such a set, K, takes the form { } K = κ, κ +1,..., κ +k, for some fixed iteger k 1. his correspods to usig several high-frequecy icremets located i the viciity of κ durig the tradig day. We the defie, L Ku = 1 K κ K L κ u, 23 with K deotig the cardiality of the set K. he test statistic is ow geeralized to, S, K, K = L K L K We defie the couterpart of d κ t,u by, d K t,u = 1 K κ K d κ t,u. 25 he extesio of the test is the based o the critical regio { S, K, K > cv α, K, K }, where cv α, K, K = Q 1 α Ẑ K, K F with Ẑ K, K costructed from d K t,u ad d K t,u, exactly as Ẑ κ, κ is costructed from d κ t,u ad d κ t,u Icorporatig Additioal Iformatio We ca further exted the aalysis by cosiderig coditioig iformatio, L κ,bu = 1 2u 1 {Bt 1 } cos t,κx/ f κ,b, 26 17

19 for where {B t } t N+ f κ,b = π 2 t,κx t,κ X 1 {A t,κ B t 1 }, 27 is a sequece of F t -adapted radom sets. Provided appropriate ergodicity ad mixig coditios hold, L κ,b u coverges i probability to, L κ,b ] = E [e uft,κσ2 t+κ / E[ft,κσ2 t+κ 1 {B t 1 }] 1Bt 1, for t N +, 28 ad the CL of heorem 2 cotiues to apply with d t u replaced by, [ d B t u = 1 {Bt 1 } cos 2uσ 2t 1+κZ ] t cos 2uσt 1+κ 2 Z t 1 {Bt 1 } u L u π σ 2 2 t 1+κ Z t Z t σt 1+κ 2 Z t Z t. 29 We ca similarly defie S, B κ, κ ad d B t,u from S, κ, κ ad d t, u, ad the coduct tests o the basis of S, B κ, κ ad critical regios costructed exactly as i Corollary 2. We omit formal proofs of these extesios, as they follow directly from our results i Sectios 3.1 ad 3.2. he above geeralizatio may be used to estimate Laplace trasforms coditioal o specific evets, e.g., level of volatility or the occurrece of a prescheduled aoucemet. his eables us to ivestigate potetial sources of variatio i the periodic compoet of volatility. 4 Simulatio Study I this sectio, we assess the fiite sample properties of the proposed test through a Mote Carlo study. We rely o the followig two-factor affie jump-diffusio with a itraday 18

20 periodic volatility compoet, dv i t X t = X 0 + t 0 Ṽ s dw s + = κ i θ V i t dt + ξ i N t s=1 V i t Z s, Ṽ t = f t,t t V 1 t + V 2 t, db i t, i = 1, 2, 30 where W, B 1 ad B 2 are idepedet stadard Browia motios, N t is a Poisso process with itesity λ J, ad Z t is ormally distributed with mea zero ad variace σ 2 j. his represetatio captures the mai features of the U.S. equity market idex. I accordace with [11], we fix the model parameters as follows, κ 1, κ 2, θ, ξ 1, ξ 2, λ J, σ 2 j = , , , , , 0.2, 0.19, o explore the size of the test uder the ull hypothesis, f t,κ f κ costat time-ofday periodicity, we set f κ equal to the average time-of-day effect obtaied i our empirical applicatio, displayed i Figure 1. Uder the alterative hypothesis, f t,κ varies with t. Cosistet with our empirical fidigs i Sectio 5, we let f t,κ be a fuctio of the statioary compoet of volatility, V t, for ivestigatig the power of the test. hus, we stipulate, f κ, l if V t Q 0.25 V t, f t,κ = fκ m, if V t Q 0.25 V t, Q 0.75 V t, t N +, 31 fκ h, if V t Q 0.75 V t, where f l κ, f m κ, ad f h κ equal our empirical estimates for the time-of-day periodic volatility compoet after coditioig o whether, at the start of the tradig day, the VIX volatility idex a optio-based idicator of future volatility belogs to the respective empirical VIX quatile, amely 0, Q 0.25 V IX], Q 0.25 V IX, Q 0.75 V IX ad [Q 0.75 V IX,. Specifically, we compute f i,b/ i=1 f i,b o the real data for each of the three regios B above, ad the apply a Nadaraya-Watso kerel regressio with a Gaussia kerel ad badwidth correspodig to a five-miute widow, to obtai estimates for the stadardized 19

21 periodic compoet of volatility coditioal o the value of the VIX idex. he resultig periodic volatility compoets are displayed o Figure :00 10:00 11:00 12:00 13:00 14:00 15:00 Figure 2: Periodic Volatility Compoets used i the Mote Carlo. he dashed lie correspods to f l κ, the dotted lie to f m κ, ad the solid lie to f h κ. I the Mote Carlo we set = 77, correspodig to samplig every five miutes across a 6.5 hour tradig day ad discardig the first 5-miute iterval. Give the imprecisio associated with evaluatio of our test statistic for high values of u, we trucate the itegral i equatio 10 at u max, which is set to satisfy 1 i=1 L i u max = he weight fuctio, w, correspods to the desity of a ormal distributio with mea zero ad variace such that u max wudu = Next, we use the followig data-drive trucatio for the jumps, v = 3.5 BV t 1 RV t 1 3/8, where BV ad RV are the bipower variatio see, e.g., [10] ad realized volatility estimators defied as, BV t = π 2 t,i 1X t,ix, RV t = i=2 t,ix 2, t N RV t is a measure of the total quadratic variatio of X over [t 1, t], while BV t is a jump- 20 i=1

22 robust couterpart, estimatig the diffusive compoet of the retur variatio, t t 1 σ2 sds. For estimatio of the covariace operator, we set B = 1/5 ad we use the Bartlett kerel for h. Fially, the critical values of the test are calculated o the basis of 10,000,000 simulatios for the χ 2 1 radom variables appearig i Ẑ κ, κ. Exactly as i the empirical applicatio, we perform the test over itervals of 30 miutes, i.e., K i equatio 23 equals the fractio of the tradig day represeted by half a hour. he Mote Carlo results uder the ull hypothesis are give i able 1. We otice the margial sesitivity with respect to the umber of eigevalues icluded i the computatio of the critical values beyod p = 2. Similarly, the performace of the test is remarkably similar for differet sample sizes,, ad empirical rejectio rates are very close to the omial level of the test p able 1: Mote Carlo Results uder the Null Hypothesis, f t,κ f κ. he table reports empirical rejectio rates of the test of omial size 0.05 usig 1, 000 simulatios. K ad K correspod to 8: ad 12: , respectively. urig to the power of the test, we provide simulatio results for the alterative 21

23 hypothesis i able 2. We ote that the power of the test depeds o which time itervals are compared. his is ot surprisig give that the time variatio i the periodic volatility compoet differs substatially across time-of-day, as depicted i Figure 2. he largest discrepacies i the periodic compoet across volatility regimes occur towards the ed of the tradig day, ad our test picks this up, eve for moderate sample sizes. he test struggles more with idetifyig time variatio i the periodic compoet of volatility i the morig versus the middle of the day, because the margial distributio of volatility is less distict across those times-of-day. Furthermore, while power declies slightly as the umber of eigevalues icluded i calculatig the critical values icreases, the discrepacies i power, lookig beyod the secod eigevalue, are small. Fially, as expected, the power icreases as the sample size grows. 8: vs 12: : vs 14: p p able 2: Mote Carlo Results uder the Alterative Hypothesis, f t,κ f κ. he table reports empirical rejectio rates for the test at omial size 0.05 usig 1000 simulatios. 22

24 5 Empirical Applicatio Our empirical aalysis is based o high-frequecy data for the E-mii S&P 500 futures cotract, spaig the period Jauary 1, 2005, till Jauary 30, After removig partial tradig days from the sample, we ed up with a total of 2, 516 days. Each day, we sample every five miutes over the period CS, which geerates 77 returs per day. For part of the aalysis, we also make use of the VIX volatility idex, recorded at the start of each tradig day. I the implemetatio of the test, the trucatio, the weight fuctio ad the tuig parameters of K are set exactly as i the Mote Carlo study. I additio, as i the simulatio study, we implemet the test over half hour itervals with the exceptio of the first period, which spas a iterval of 20 miutes 8:40-9:00 CS. able 3 reveals that the ull hypothesis is rejected, except whe the test ivolves itervals which are very close withi the tradig day. he failure of the test to reject for adjacet periods is ot mechaical, as the respective estimates for the empirical Laplace trasforms, L K ad L K, have oly a mior overlap i terms of the uderlyig high-frequecy data maily a fivemiute iterval which is due to the staggerig of returs i the computatio of fκ for κ K ad κ K. Istead, this empirical fidig is a maifestatio of the fact that, although there is a time-variatio i the periodic compoet of volatility, it is quite similar for adjacet itervals. Overall, these results provide strog evidece that the itraday periodicity i volatility is time varyig. Oe possible explaatio for the overwhelmig rejectio of the ull hypothesis is that the itraday volatility patter is differet for days with scheduled release of macroecoomic ews. here are umerous such aoucemets durig the tradig hours. We focus o the release of ews from the Federal Ope Market Committee FOMC, which are regu- 23

25 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 8:40-9: :00-9: :30-10: :00-10: :30-11: :00-11: :30-12: :00-12: :30-13: :00-13: :30-14: :00-14: able 3: Ucoditioal est Results. he table reports the results from the ucoditioal test of Sectio 3.2 over the period 1 Jauary, 2005 to 30 Jauary, Critical values are computed usig p = 3. op row idicates the begiig of each half-hour iterval. larly scheduled for 1pm CS every six weeks. Other oteworthy aoucemets durig tradig hours iclude the ISM Maufacturig ad No-Maufacturig Idices as well as the Cosumer Setimet report. Ureported results show that these releases have a much smaller impact o the itraday volatility patter tha the FOMC aoucemet. Hece, for brevity, we oly aalyze the latter here. I total, we have 96 FOMC aoucemets i our sample, ad we label these FOMC days. Figure 3 depicts estimates for the periodic volatility compoet o FOMC ad o-fomc days. he estimates for the periodic compoet o o-fomc days are almost idetical to those for the full sample displayed o Figure 1, while the correspodig estimates o FOMC days display a sharp icrease immediately after the aoucemet. his elevatio i volatility is accompaied by heighteed 24

26 tradig volume, as diverse groups of ivestors assess the impact of the ews for asset prices, ad the ecoomy more geerally :00 10:00 11:00 12:00 13:00 14:00 15:00 ime of Day CS Figure 3: Itraday Volatility Periodicity with ad without FOMC Aoucemets. he figure plots smoothed f i,b/ i=1 f i,b with Gaussia kerel ad badwidth of 5-miute iterval for B beig FOMC days dashed lie ad o-fomc days solid lie. Give this evidece, we coducted our test for costat periodicity i volatility excludig the FOMC days. he test results are very similar to those for the whole sample, reported i able 3, ad, importatly, the strog rejectio of the ull hypothesis is preserved results ot reported to coserve space. I summary, scheduled macro aoucemets caot explai the variatio i the periodic compoet of volatility. Additioal i-depth aalysis of the sources of variatio i the itraday periodic volatility compoet is outside the scope of the curret paper. Noetheless, we illustrate how our approach facilitates direct exploratio of this questio. he basic ratioale from ecoomic theory cocerig the observed itraday U-shape i volatility is as follows. he opeig hours represet a price discovery phase where overight ews arrivals ad large customer 25

27 orders submitted to differet dealers eed to be aalyzed ad processed by the agets i the market. Heterogeeous asset positios ad beliefs, asymmetric iformatio, ad diverse orders iteract to geerate elevated volatility, but ofte oly moderately high volume. he latter is due to the fact that large orders ted to be broke up ad processed throughout the tradig day to avoid excessive price pressure. he typical icetive scheme for order executio relies o the average trade price achieved for the order relative to some metric like the volume-weighted average price VWAP across the tradig day. As a cosequece, risk-averse dealers will prefer to trade later i the day, whe the iitial bulk of ews ad the directio of the order flow have bee absorbed ito the price, ad the price impact typically is lower. Risk aversio will iduce agets to postpoe some trades util later i the day, uless they are based o short-lived iformatio that must be acted o quickly before others do so or before the iformatio becomes public ad prices adjust. Sice the tedecy to postpoe a fractio of the o-iformatioal trades will be commo across dealers there, aturally, will be a cocetratio of uiformed order flow towards the ed of the day. Recogizig this feature of the market dyamic, the price impact per trade will be low towards the ed of the tradig day. As a result, we expect to see highly elevated tradig accompaied by some icrease i volatility i the fial hour of regular tradig. his lie of reasoig further implies that a period of elevatio i the statioary compoet of volatility should push the itesity of tradig ad the retur volatility further back towards the ed of the tradig day. Cosequetly, we ow explore whether the level of volatility affects the shape of the itraday volatility patter. As a proxy for the latet retur volatility at the start of the tradig day, we rely o the value of the VIX volatility idex. I the left pael of Figure 4, we plot the estimated itraday volatility patter i high- ad low-volatility regimes. Specifically, we idetify the high volatility regime as the set of days i the sample i which 26

28 the VIX idex at market ope is betwee its 75th ad the 95th quatiles across the sample period. Similarly, the low volatility regime is the set of days i which the VIX idex at market ope is betwee the 5th ad the 25th quatiles. We exclude days of very low ad very high VIX values below the 5th ad above the 95th quatiles to guard agaist the effect of extreme outliers. From Figure 4 we see that the two itraday patters are roughly idetical aroud oo, differ substatially aroud the opeig ad close, with the periodic compoet i the low volatility regime beig almost flat towards the market close as opposed to its steep couterpart i the high volatility regime. O the right pael of Figure 4, we plot the ratio of the estimated time-of-day effects i the two volatility regimes relative to the oe based o the whole sample. As see from the plot, the periodic compoet i the high volatility regime is very close to the average oe. his is because the high volatility regime cotributes, i relative terms, more tha the low volatility regime to the estimatio of the average time-of-day periodic compoet. O the other had, the differece i the average estimate for the periodic compoet of volatility ad the oe recovered i the low volatility regime is substatial, particularly i the period before market close. his implies that the periodic compoet of volatility will be severely overstated durig periods of low volatility, whe relyig o the usual procedure of stadardizig returs by the average estimates for the time-of-day effect. he exploratory aalysis above does suggest that the level of volatility is a importat source of variatio i the itraday volatility patter. We ow test formally whether the depedece of the itraday volatility patter o the volatility regime ca explai the high rejectio rates of our test eve o o-fomc days by icorporatig the additioal iformatio for the VIX idex ad followig the procedure i Sectio 3.3. o accout for the fact that the Laplace trasforms have bee shifted dowward, we adjust the choice of u max so that it accurately reflects the effective sample size. Formally, we set u max = u max /adj 2, 27

29 :00 10:00 11:00 12:00 13:00 14:00 15:00 ime of Day CS :00 10:00 11:00 12:00 13:00 14:00 15:00 ime of Day CS Figure 4: Itraday Volatility Periodicity ad Volatility. he left pael plots the smoothed values for f i,b/ i=1 f i,b usig a Gaussia kerel ad badwidth of five miutes, for B idicatig high VIX solid lie or low VIX dashed lie. he right pael displays the smoothed ratio 5 f i,b/ f i usig a Gaussia kerel ad badwidth of te miutes, for B idicatig high VIX solid lie or low VIX dashed lie. he low high VIX state correspods to the iterval betwee the 5th ad 25th 75th ad 95th empirical quatiles of the VIX idex. FOMC days are excluded from the computatio. where adj = / obs reflects how much larger the full sample is relative to the oe based o the coditioig iformatio. he results from the tests for the high ad low volatility regimes are reported i able 4 similar results hold also for a media volatility regime. From able 4, we coclude that accoutig for the level of volatility captures a otrivial part of the time variatio i the itraday volatility periodicity. However, cotrollig for the volatility level aloe is clearly ot sufficiet to capture the behavior of volatility durig the first 90 ad the last 30 miutes of the tradig day i the low volatility regime. Determiig what drives the periodicity durig these periods is a importat questio that we leave for future research. 28

30 9:00 9:30 10:00 10:30 11:00 11:30 12:00 12:30 13:00 13:30 14:00 14:30 Low volatility 8:40-9: :00-9: :30-10: :00-10: :30-11: :00-11: :30-12: :00-12: :30-13: :00-13: :30-14: :00-14: High volatility 8:40-9: :00-9: :30-10: :00-10: :30-11: :00-11: :30-12: :00-12: :30-13: :00-13: :30-14: :00-14: able 4: Coditioal est Results. he table reports results from the test of Sectio 3.3 over the period 1 Jauary, 2005 to 30 Jauary, he coditioig set is o-fomc days ad VIX belogig to oe of two states: low betwee 5th ad 25th quatile of its empirical distributio ad high betwee the 75th ad 95th quatile. Critical values are computed usig p = 3. op row idicates the begiig of each half-hour iterval. 29

31 6 Coclusio I this paper we develop a ovel test for decidig whether the itraday periodic compoet of volatility is time-ivariat, usig a log spa of high-frequecy data. he test is based o formig estimates of the average value of the periodic compoet of volatility ad the stadardizig by it the high-frequecy returs. We exploit a weighted L 2 orm of the distace betwee the empirical characteristic fuctios of the studetized high-frequecy returs at differet times of the day to separate the ull ad alterative hypotheses. he aalysis is exteded to allow for testig the hypothesis o coditioig sets which ca aid idetifyig the sources of time variatio i the volatility periodicity. Our empirical applicatio reveals that itraday volatility periodicity chages stochastically over time, with the shifts partially explaied by the level of volatility. Refereces [1] A. Admati ad P. Pfleiderer. A theory of itraday patters: Volume ad price variability. Review of Fiacial Studies, 1:3 40, [2]. G. Aderse ad. Bollerslev. Itraday periodicity ad volatility persistece i fiacial markets. Joural of Empirical Fiace, 4: , [3]. G. Aderse ad. Bollerslev. Deutsche mark-dollar volatility: Itraday activity patters, macroecoomic aoucemets, ad loger ru depedecies. Joural of Fiace, 531: , [4]. G. Aderse,. Bollerslev, ad A. Das. Variace-ratio statistics ad high- 30

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36 Supplemetary Appedix to ime-varyig Periodicity i Itraday Volatility orbe G. Aderse Marti hyrsgaard Viktor odorov December 5, 2017 Abstract his documet cosists of two appedices. Appedix A states the formal assumptios ad itroduces otatio. Appedix B cotais the proofs of all formal results stated i the paper. Aderse s ad odorov s research is partially supported by NSF grat SES Departmet of Fiace, Kellogg School, Northwester Uiversity, NBER ad CREAES. CREAES, Departmet of Ecoomics ad Busiess Ecoomics, Aarhus Uiversity. Departmet of Fiace, Kellogg School, Northwester Uiversity. 1

37 Appedix A: Assumptios ad Notatio I the assumptios ad proofs we will deote with C a positive ad fiite costat that does ot deped o ad, ad ca chage from lie to lie. We will further use throughout the shorthad otatio V t = σt 2. Assumptio 1. We have sup t R+ E a t 8 + sup t R+ E b t 4 + sup t R+ E V t 8 < as well as F R < ad R x 4 F dx <, ad further if t N + if f t,κ > ɛ ad κ 0,1] for some o-radom 0 < ɛ < ɛ <. sup sup t N + κ 0,1] f t,κ < ɛ, A.1 Assumptio 2. he followig smoothess i expectatio coditios hold for 0 < s t: E a t a s 2 + E b t b s 2 + E σ t σ s 2 + E V t V s 2 C t s, A.2 for some positive ad fiite costat C that does ot deped o t ad s. Assumptio 3. For a give fiite set K of umbers i 0, 1], deote with Y t = { f t,κ σ 2 t+κ ad suppose that it is a fuctio of a Markov process {Ỹt} t N+. {Ỹt} t N+ } κ K We the assume that is statioary, ergodic ad α-mixig with α t = ot 16/3 whe t, ad where α t = sup{ P A B PAP B : A G 0, B G t }, for G 0 = σ Ỹs, s 0 ad G t = σ Ỹs, s t. Assumptio 4. he kerel fuctio h used for costructig K satisfies the followig coditios: h : R [ 1, 1], h0 = 1, hx = h x, h is cotiuous ad is further cotiuously differetiable i a eighborhood of zero with the potetial exceptio at zero where h ±0 exist ad are bouded. 2

38 Appedix B: Proofs B.1 Auxiliary Results ad Notatio hroughout the proofs we will assume without loss of geerality that EV t = 1. Furthermore, to improve the readability of the proofs we itroduce the followig otatio: V t,κ = V t 1+ κ 2/, f κ = E f t,κ V t+κ, κ 0, 1], t = 1,...,, B.3 ad we will use similar otatio for f t,κ, ad if the latter does ot deped o t the ull hypothesis for our test, we will further simplify otatio to f κ. We ext deote with X c ad X d the cotiuous ad discotiuous parts of X: X c t = X 0 + t 0 a s ds + With this otatio, we set d t, u = cos 2u Vt,κ t,κw f κ = 1 π 2 t 0 σ s dw s, X d t = t cos 2u Vt,κ t,κ W 0 R xµds, dx. π 2 ul u V t,κ t,κw t,κ W V t,κ t,κ W t,κ W, t,κx c t,κ X c, L κu = 1 fκ = 1 π 2 f t,κv t,κ t,κw t,κ W. B.4 B.5 B.6 si 2u Vt,κ t,κw Vt,κ t,κw. B.7 2u Lemma 1. Suppose Assumptios 1-3 hold with K = {κ, κ } ad let sup t R+ E a t q + sup t R+ Eσ q t < for some q 8. he, for p [1, 4] ad κ 0, 1] ad with ϖ > p, we have E f κ f κ p C p [ q1/2 ϖ+2pϖ 1+p/2+pϖ p+1 2p/q ], B.8 3

39 E f κ f κ p p 2 C q p q, B.9 E f κ f κ 2 C 1, E u L κu ul κu 2 C u 1 1, B.10 for some positive ad fiite costat C that does ot deped o u. Proof. For the first boud i B.8, we make use of the followig algebraic iequality t,κx t,κ X 1 {A t,κ } t,κx c t,κ X c Cv 2 1 { t,κ X c v t,κ Xc v } + t,κx c t,κ X d 1 { t,κ X d 2v } + t,κx d t,κ X c 1 { t,κ X d 2v } + t,κx c t,κ X c 1 { t,κ X c v 2 t,κ Xc v 2 t,κ Xd v 2 t,κ Xd v 2 } + t,κx d t,κ X d 1 { t,κ X d 2v t,κ Xd 2v }. B.11 By Markov iequality ad Burkholder-Davis-Gudy iequality ad takig ito accout the itegrability coditio for a ad σ as well as makig use of F R <, we have E t,κx c q C q/2, P t,κx c v C q1/2 ϖ, P t,κx d 0 C. B.12 From here, by applicatio of Hölder s iequality, we have E E t,κx c p t,κ X d p 1 { t,κ X d 2v } C 1+p/2+pϖ, B.13 E t,κx d p t,κ X c p 1 { t,κ X d 2v } t,κx c p t,κ X c p 1 { t,κ X c v2 t,κ Xc v2 } C 1+p/2+pϖ, B.14 C q 2 q 2pϖ, B.15 E t,κx c p t,κ X c p q 2p p+ 1 { t,κ X d v2 t,κ Xd v2 } q C. B.16 Next, t,κx d t,κ X d is ozero oly if there are jumps i both itervals ad further τ+ by Markov s iequality we have P τ τ R µds, dx 1 CE τ τ+ b τ s ds for ay τ. 4

40 From here, usig successive coditioig, Assumptio 2, the above iequality ad Hölder s iequality, we have E t,κx d p t,κ X d p 1 { t,κ X d 2v t,κ Xd 2v } C 2+2pϖ. B.17 Combiig the above bouds, we get the result i B.8. For the secod boud i B.9, we use the algebraic iequality t,κx c t,κ X c ft,κv t,κ t,κw t,κ W t,κ X c ft,κv t,κ t,κw t,κ X c + t,κ X c ft,κv t,κ t,κ W Vt,κ t,κw. B.18 For r [2, q], by applicatio of Burkholder-Davis-Gudy iequality, iequality i meas ad makig use of the itegrability of a t, σ t as well as the smoothess i expectatio of σ t ad f t, we have E t,κ X c f t,κv t,κ t,κw r C 1+r/2. B.19 From here the result follows by a applicatio of Hölder s iequality raisig the term t,κ X c p or V t,κ t,κw p to power q. For the first of the bouds i B.10, we make use of the decompositio f κ f κ = 1 π 2 ft,κv t,κ t,κw t,κ W π f t,κv t,κ f κ. B.20 Successive applicatio of the Burkholder-Davis-Gudy iequality, give the itegrability coditio for σ t, gives E ft,κv t,κ t,κw t,κ W 2 p C p/2. π B.21 5

41 Applyig Lemma VIII of [2] ad Hölder s iequality ad takig ito accout the itegrability assumptios for σ t as well as for the mixig coefficiet of Y t, we have 2 E ft,κv t,κ f κ C k α k K. B.22 Combiig the above two results, we get the first boud i B.10. he secod oe is approved i a aalogous way. Lemma 2. Suppose the settig of Lemma 1 ad i additio f t,κ f κ costat time-ofday for t N +. Let 0 < ɛ < if κ [0,1] f κ /4 ad assume q 8 ad ϖ 2/q. he, we have k=0 E u L κ u1 { fκ >ɛ} u L κu [ 1 q 4 C u 1 1 ϖ ] 2, B.23 where the positive ad fiite costat C does ot deped o u. Proof. he derivatio is doe o the basis of the followig boud: 2u si t,κ X t,κ X/ f κ 1 { t,κ X v f f κ >ɛ} κ si 2u Vt,κ t,κw Vt,κ t,κw C t,κx1 { t,κ X v } fκ Vt,κ t,κw + C1{ fκ ɛ} + C u Vt,κ t,κw t,κx c + C f κ f κ V t,κ t,κw + uv t,κ t,κw 2, V t,κ t,κw f κ V t,κ t,κw + 1{ t,κ X d >0} B.24 for a positive ad fiite costat C that does ot deped o u. Applyig Cauchy-Schwarz iequality [ E Vt,κ t,κw t,κx c fκ Vt,κ t,κw ] + 1{ t,κ X d >0} C. 6 B.25

42 Applyig Burkholder-Davis-Gudy iequality ad makig use of F R <, we get E t,κx1 { t,κ X v } fκ Vt,κ t,κw C q 11/2 ϖ. B.26 By applicatio of Hölder s iequality as well as the results of Lemma 1 ad usig ϖ 2 q ad q 8, we have [ E f κ fκ Vt,κ t,κw + Vt,κ t,κw 2 ] + 1 { fκ ɛ} Vt,κ t,κw C q ϖ 1. B.27 Combiig the estimates i B.25-B.27 with the boud i B.24 we get the result of the lemma. B.2 Proof of heorem 1 We make the decompositio 2u 2u cos t,κx/ f κ cos ft,κv t,κ t,κw/ f κ = where 3 j=1 χ j t,u, κ, 2u χ 1 2u t,u, κ = cos t,κx/ f κ cos ft,κv t,κ t,κw/ f κ, 2u 2u χ 2 t,u, κ = cos ft,κv t,κ t,κw/ f κ cos ft,κv t,κ t,κw/ f κ, f κ χ 3 t,u, κ = cos 2u f t,κv t,κ t,κw/ B.28 2u cos ft,κv t,κ t,κw/ f κ. he proof will the cosist of aalysis of the separate terms i the decompositio. Usig the iequalities cosx cosy 2 si x y 2 χ 1 t,u, κ 1 { t,κ X d 0 f κ<ɛ} + C u t,κx c x y 2 we ca boud χ1 t, as follows ft,κv t,κ t,κw, B.29 7

43 where ɛ is some costat satisfyig 0 < ɛ < if κ [0,1] f κ /4, ad ɛ ad C do ot deped o u. From here, applyig Burkholder-Davis-Gudy iequality, the smoothess i expectatio assumptio for σ t ad f t, the itegrability assumptios for a t, b t ad σ t ad Lemma 1, we have for q 4 q 2 t,u, κ = O p 1 ϖ 2 1 χ 1. B.30 urig ext to χ 2 t,u, κ, we have the followig boud ote that f is bouded from above χ 2 t,u, κ 1 { fκ<ɛ f + C κ<ɛ} u Vt,κ t,κw f κ f κ, B.31 for a positive ad a fiite costat C that does ot deped o u. From here by applicatio of Hölder s iequality, the itegrability coditios for σ t, the result of Lemma 1, ad sice q 6 ad ϖ 1 q q is the costat of Lemma 1, we have q 3 t,u, κ = O p 1 ϖ 2 1 χ 2. B.32 For χ 3 t,u, κ, we have χ 3 t,u, κ C u 1 ft,κv t,κ t,κw 1 f κ f κ, B.33 for a positive ad a fiite costat C that does ot deped o u. he, by applicatio of Cauchy-Schwarz iequality ad Lemma 1, we have 1 χ 3 t,u, κ = O p 1. Fially, usig Assumptios 1-3 ad applyig Lemma VIII of [2] ad Hölder s iequality, we have [ 2u E cos f t,κv t,κ t,κw/ f κ E e ] 2 uf t+κv t+κ /Ef t+κ V t+κ C, B.34 for a positive ad a fiite costat C that does ot deped o u. 8

44 Combiig the above bouds, we get the cosistecy result of the theorem. For further use, we ote also that χ 1 t,u, κ + χ 2 t,u, κ = o p 1/, B.35 provided 0 ad ϖ q 4 2q 6. Uder this same coditio we also have L κ L κ = O p 1/. B.3 Proof of heorem 2 he proof cosists of two lemmas. Lemma 3. Uder the coditios of heorem 2, we have L κ u L κ u 1 P d t, u 0. B.36 Proof of Lemma 3. We deote 0 < ɛ < if κ [0,1] f κ /4 ad make the followig decompositio 2u cos t,κx/ f κ cos 2u Vt,κ t,κw = where, usig the fact that f t,κ = f κ, we deote 2u t,κx/ f κ cos χ 1 t,u, κ = cos 5 j=1 χ j t,u, κ, 2u fκ Vt,κ t,κw/ f κ, 2u 2u χ 2 t,u, κ = cos fκ Vt,κ t,κw/ f κ cos fκ Vt,κ t,κw/ f κ, χ 3 t,u, κ = 1 2 si fκ 2u Vt,κ t,κw 2u Vt,κ fκ t,κw, fκ B.37 9

45 χ 4 t,u, κ = 1 2 si fκ 2u Vt,κ t,κw 2u Vt,κ fκ t,κw 1 fκ { fκ ɛ}, χ 5 t,u, κ = 1 2 g 2u fκ Vt,κ t,κw ; f 2 κ fκ fκ 1{ f, κ >ɛ} with f κ beig a itermediary value betwee f κ ad fκ, ad further a a 2 a 3a ga; x = cos x 4x si x 3 4x. 5/2 We ca write χ 3 u, κ 1 χ 3 t,u, κ + ul u 1 = u L κu + ul u 1 With this otatio, we fially have L κu L κ u 1 π 2 V t,κ t,κw t,κ W 1 π 2 V t,κ t,κw t,κ W 1. d t, u = 1 j=1,2,4,5 + χ 3 u, κ χ 3 u, κ. χ j t,u, κ χ j t,u, κ B.38 B.39 B.40 he proof cosists of showig the asymptotic egligibility of the terms 1 χj t,u, κ for j = 1, 2, 4, 5 as well as the egligibility of χ 3 u, κ for arbitrary κ 0, 1]. For j = 1, 2, this was already established i the proof of heorem 1 uder the coditio for ϖ of the theorem. For χ 4 t,u, κ, sice ɛ < if κ [0,1] f κ, we have χ 4 t,u, κ C u Vt,κ t,κw f κ fκ 2, B.41 for some positive ad fiite C that does ot deped o u. Similarly, χ 5 t, ca be bouded as follows χ 5 t,u, κ C uvt,κ t,κw 2 + uvt,κ t,κw f κ fκ 2, B.42 10

46 with C as above. herefore, 1 χ 4 t,u, κ 1 + C f κ f κ 2 1 χ 5 t,u, κ Vt,κ t,κw 2 + Vt,κ t,κw, B.43 for some positive ad fiite C that does ot deped o u. From here, sice 1 Vt,κ t,κw 2 + Vt,κ t,κw = O p 1, ad utilizig the result of Lemma 1, we have 1 χ4 t,u, κ + 1 χ5 t,u, κ = o p 1. We are left with χ 3 u, κ. Usig Lemma 1 ad Cauchy-Schwarz iequality, we have E χ 3 u, κ C, B.44 for some positive ad fiite C that does ot deped o u. he asymptotic egligibility of χ 3 u, κ the readily follows. o state the ext lemma, we will eed some additioal otatio which we ow itroduce. We decompose where ξ 1 t,u = cos d t, u = ξ 1 t,u + ξ 2 t,u, 2u Vt,κ t,κw e uv t,κ cos 2u V + ul uv t,κ π 2 t,κ W t,κ W 1 B.45 t,κ t,κ W + e uv t,κ ul uvt,κ π 2 t,κw t,κ W 1 ξ 2 t,u = e uv t,κ e uv t,κ + ul u V t,κ V t,κ., 11

47 Fix a positive iteger l ad deote for t = 1,..., : l 1 ξ 2 t,,l u = E t ξ 2 t+k, u E t 1ξ 2 t+k, u. B.46 k=0 With this otatio we set d t,,l u = ξ t,u ξ t,,l u, B.47 ad deote the differece R,lu = 1 d t, u d t,,l u. B.48 Usig the decompositio we have l k=0 E t ξ 2 t+k, u E t 1ξ 2 t+k, u = R,lu = 1 1 E t ξ 2 t+l, u 1 t=0 l 1 E t ξ 2 t+k, u E t ξ 2 t+k+1, u, k=0 l 1 k=1 t=0 B.49 E ξ 2 +k, u E 0ξ 2 k, u. B.50 Usig the mixig coditio for Y t, the itegrability assumptio for V t, ad Lemma VIII of [2], we have E E t ξ 2 t+k, u 2 Kα 3/8 k E ξ 2 t+k, u 8 1/8. herefore, sice α k = ok 8/3 as k, by Fatou s lemma, the limit d t,, = ξ 1 2 t, + ξ t,, is fiite almost surely, where ad the same holds for ξ 2 t,, := lim l ξ2 t,,l = k=0 R, := lim l R,l = 1 E t ξ 2 t+k, E t 1ξ 2 t+k,, B.51 k=1 E 0 ξ 2 k, E ξ 2 +k,. B.52 12

48 Fially, we set ξ 2 t u = e uv t+κ e uv t+κ + ul u V t+κ V t+κ, ad defie ξ 2 t,u. 2 2 ξ t,l u ad ξ t, u from it exactly as we defied 2 2 ξ t,,l u ad ξ t,, u from Lemma 4. Uder Assumptios 1-3 with K = {1}, as ad with 0, we have 1 Proof of Lemma 4. d L t,, N0, K ad R, P 0. By domiated covergece, we have E t 1 dt,, u E d t,, 2 <, ad therefore the array { d t,, } t N+ B.53 = 0 ad is a martigale differece sequece ad we ca apply heorem C of [3] to establish the CL result. I particular, it suffices to show that the followig is true: 1 E t 1 d t,, 2 P racek, B.54 where {e i } i N+ 1 1 E t 1 d t,, 2+ι P 0, for some ι 0, 1, B.55 E t 1 d t,,, e i d P t,,, e j Ke i, e j, i, j N +, B.56 is a orthoormal basis i L 2 w. We have E t 1 ξ 1 t, 2 = E t 1 η u V t,κ, V t,κ 2, B.57 where for two positive costats C 1 ad C 2, we deote η u C 1, C 2 = E η u C 1, C 2 2 with η u C 1, C 2 = cos 2uC1 Z 1 e uc 1 cos 2uC2 Z 2 + e uc 2 π ul u C 1 2 Z 1 Z π 1 1 C 2 2 Z 2 Z B , 13

49 for some idepedet stadard ormal radom variables Z 1, Z 1, Z 2 ad Z 2. From here, we have E t 1 d t,, 2 = E t 1 η u Vt,κ, Vt,κ 2 + ξ t,, E t 1 ξ t,, 1 2 ξ t,,. B.59 Usig successive coditioig, we ca write E t 1 ξ t,u ξ 1 t,, u 2 = E t 1 ξ t,u 1 ι=κ,κ k=0 [ Et 1+ ι ξ 2 t+k, u E ι 2 t 1+ ξ 2 t+k, u]. B.60 For ι = κ, κ ad k 0, we have for some positive ad fiite C: E ξ t,u 1 E ι t 1+ ξ 2 t+k, u E ξ 2 t 1+ ι 2 t+k, u 2 Eξ t,u E 1 2 E ι t 1+ ξ 2 t+k, u E E ι 2 t 1+ 2 = Eξ t,u E 1 2 ξ 2 t+k, u E E ι t 1+ [ C u 1 E C u 1 E ι Eξ 2 1+k, E E ι ξ 2 C u 3/2 1 1/4 E t 1+ ι ξ 2 1+k, u ξ2 1+k+2, 1/4 u ξ2 1+k+2, u2 1+k, u 2 + E ι α 3 16 k ue ι 2 1/4 ξ 2 1+k+2 u, 1/16 E ξ 1,u 2 8 C u 2 1 1/4 ξ 2 t+k, u 2 ξ 2 t+k+2, u 2 ] ξ 2 1+k, u + ξ2 1+k+2 u, α 3 16 k, B.61 where for the first iequality we have made use of Cauchy-Schwarz iequality, for the secod equality we use the statioarity of V t ad hece of its coditioal expectatio, for the third iequality we made agai use of the statioarity of V t as well as the itegrability assumptio 14

50 for V t, for the forth iequality we used Cauchy-Schwarz ad Jese s iequality, ad for the remaiig iequalities, we made use of the itegrability ad smoothess i expectatio coditios for V t, as well as Lemma VIII of [2]. From here, sice by Assumptio 3, α k = ok 16/3 for k, we have E ξ 1 t,u ξ 2 t,, u C u 2 1 1/4, B.62 for some positive ad fiite C that does ot deped o u, ad therefore 1 R E t 1 ξ t,u ξ 1 t,, u 2 wudu = o p 1. B.63 Usig the mixig coditio for Y t, Lemma VIII of [2], Lebesgue s domiated covergece theorem as well as Assumptios 1-2, we have E η u V t,κ, V t,κ 2 η u V t+κ, V t+κ 2 + ξ t,, 2 2 ξ t, 2 2 C, B.64 for some positive ad fiite C that does ot deped o u. Furthermore, give the square itegrability of V t, the assumptio that Ỹt is a Markov process ad hece the coditioal expectatio of a trasformatio of it is a fuctio of the process at the time of the coditioig, ad by a applicatio of a ergodic theorem, we have 1 E t 1 η u V t+κ, V t+κ 2 + ξ 2 t, 2 P E η u V t+κ, V t+κ 2 + ξ t, 2 2, B.65 provided E ξ 2 t, 2 <. he latter is guarateed by the mixig coditio for Y t, use of Lemma VIII of [2] ad Lebesgue s domiated covergece theorem upo makig use of the iequality ξ 2 t,l u 2 l 1 k,p=0 E t ξ 2 t+k u E t 1ξ 2 t+k u E tξ 2 t+pu E t 1 ξ 2 t+pu, B.66 15

51 which i tur implies for some positive ad fiite C that does ot deped o u E ξ 2 t,l u 2 C l 1 α 3/8 k α 3/8 k,p=0 p C. B.67 o show B.54, we therefore eed to show that E η u V κ, V κ 2 + ξ 1, 2 2 = racek. B.68 For this, usig domiated covergece, it suffices to show E η u V κ, V κ 2 + lim E ξ 2 1,l 2 = racek. B.69 l We have E ξ 2 1,l u 2 = = = l 1 k,p=0 l 1 k,p=0 l 1 k,p=0 [ ] E E 1 ξ 2 1+k u E 1 ξ 1+pu 2 E 0 ξ 1+pu 2 [ ] E ξ 2 1+k u E 1 ξ 1+pu 2 E 0 ξ 1+pu 2 E[ξ 2 k ue 0ξ p 2 u] k=1 l k,p=1 E[ξ 2 l 1 [ ] = E[ξ 2 0 u] E ξ 2 0 uξ 2 k u l 1 k=1 E[ξ 2 k ue 0ξ 2 l u] l p=1 E[ξ 2 k ue 0ξ p 2 u] l ue 0 ξ p 2 u], B.70 where for the first equality we made use of successive coditioig ad for the secod iequality we made use of the statioarity of the sequece {ξ 2 t uξ 2 t+su} t 0 ad arbitrary fixed s 0. We ow boud the last two terms i the above iequality. Usig Lemma VIII of [2] ad our itegrability assumptio for V t, we have 1/8 E E 0 ξ p 2 u 2 Cαp 3/8 Eξ 2 0 u 8, p 0, B.71 16

52 for some positive ad fiite C that does ot deped o u. herefore, with C as above, we have l 1 E[ξ 2 k ue l 1 0ξ 2 l u] E E 0 ξ 2 k u 2 E E 0 ξ 2 l u 2 k=1 C l E[ξ 2 l ue 0 ξ p 2 u] p=1 C k=1 1/8 l 1 Eξ 2 0 u 8 3/8 α l α 3/8 k, l p=1 k=1 E E 0 ξ 2 l u 2 E E 0 ξ 2 p u 2 Eξ 2 0 u 8 1/8 α 3/8 l From here, takig ito accout the rate of decay of α k, we have lim E ξ 2 1,l l 2 = race K ku, uwudu, 0 l p=1 α 3/8 p. where the operator K has kerel kz, u = j= E[ d 1 z d j u] ad we deote d t u = e uv t 1+κ e uv t 1+κ ul uv t 1+κ V t 1+κ. B.72 B.73 B.74 From here the result i B.69 ad hece B.54 readily follows. he covergece i B.56 is show aalogously. We are left with establishig B.55. First, usig the itegrability coditio for V t as well as the mixig assumptio for S t ad applyig Lemma VIII of [2], we have E E 0 ξ 2 2+ι k, u 2+ι 1 8 Cαk E ξ 2 2+ι k, u 8 8, B.75 for some costat C that does ot deped o u ad ay ι 0, 6. From here, by iequality i meas, the expoetial decay of the weight fuctio w i the tails, sice α k = ok 8/3, ad by the mootoe covergece theorem, we have for C as above E d t,, 2+ι C, for some ι 0, 1, B.76 17

53 ad from here the result i B.55 follows trivially. We cotiue with the boud for R, u. Usig mootoe covergece, the boud i B.71 above as well as the rate of decay coditio for the mixig coefficiet α k, we have E R, C, B.77 for some positive ad fiite C, ad therefore R, = o p1. Combiig Lemmas 3 ad 4, the result of the theorem follows. B.4 Proof of Corollary 1 Let Y = N0, K, with the operator K give i equatio 8. By the spectral theorem for compact self-adjoit operators see [4] it follows that there exists a complete set of eigefuctios ɛ i i L 2 w ad associated real eigevalues λ 1 λ such that Kɛ i = λ i ɛ i. B.78 Moreover, the eigefuctios form a orthoormal basis for L 2 w. By Parseval s idetity, it the follows that Y 2 = i=1 λ i Y, ɛi λi 2, B.79 heorem 2 implies that Y, ɛ i is ormally distributed with mea zero ad variace Kɛ i, ɛ i. he result the follows upo showig that Cov Y, ɛ i, Y, ɛ j = λ i δ i,j, where δ i,j is Kro- 18

54 ecker s delta. hus, for ay i, j N +, we have [ ] Cov Y, ɛ i, Y, ɛ j = E Y uɛ i uwudu Y uɛ j uwudu R + R [ + ] = E R + Y uy tɛ i uwuɛ j twtdudt R + = E [Y uy t] ɛ i uwuɛ j twtdudt R + R + = ku, tɛ i uwuɛ j twtdudt R + R + = ku, tɛ i uwuɛ j twtdudt R + R + = λ i ɛ i tɛ j twtdt = λ j δ i,j, R + R + where the last equality follows by the defiitio of the eigevalues of K see equatio B.78. B.5 Proof of heorem 3 We will first proof the followig Lemma about the error i the kerel K. Lemma 5. Suppose Assumptios 1-4 hold with K = {κ, κ }, ad with f t,κ f κ costat [ ] 2 time-of-day for t N +. he, for ϖ, q 5, for q the costat of Lemma 1, ad as q 2q 8 ad, we have K K HS = O p B 6 B B 2. B.80 Proof of Lemma 5. We have K K 2 HS = k u, s ku, s 2 wuwsduds. R + R + B.81 19

55 Our goal will be to decompose suitably k u, s ku, s ad boud the secod momets i this decompositio. We first we itroduce some auxiliary otatio. We set γ 0 u, s = 1 γ 0 u, s = Ed 0 ud 0 s, γ k u, s = E[d 1 ud k+1 s + d k+1 ud 1 s], k 1, d t, u d t, s, γ k u, s = 1 k [ d t, u d t k, s + d t+k, s], k 1, ad the correspodig quatities i which d t, u are replaced with d t, u are deoted with γk u, s. With this otatio, we ca decompose B j j k u, s ku, s = γ j u, s h 1 B + j=1 B j=1 j=b +1 j j γ h j u, s γ j u, s. B γ j u, s By coditioig o the sigma algebra of the origial probability space, we have B.82 E[d 1 zd j u] = E[ d 1 z d j u], for j > 1, B.83 where d t u is defied i the proof of heorem 2 ad usig agai the otatio of that proof, we ca write E[d 0 zd 0 u] = Eη z,u V κ, V κ. B.84 From here, usig Lemma VIII of [2], Hölder s iequality ad the fact that E V t 8 <, we have Ed 1 zd j+1 u Cα 3/4 j E d1 z 8 E d 1 u 8 1/8, j 0, B.85 for positive ad fiite C that does ot deped o u, z ad j. Similarly, for k [0, j] by cosiderig separately the cases j k < k ad j k k, we have E d t, ud t j, sd t k ud t j k, s C α j k j E d0, u 8 E d 0, s 8 1/4, B.86 20

56 for positive ad fiite C that does ot deped o u, s, j ad k. Fially, for k j + 1 ad j 0 E d t, ud t j, s γ j u, sd t k ud t j k, s C α k j j E d0, u 8 E d 0, s 8 1/4, B.87 where C is a positive ad fiite costat that does ot deped o u, s, j ad k. Usig the first of the above bouds as well as the cotiuous differetiability of h i a eighborhood of zero, we get B j=1 E[d 1 ud j+1 s] j >B j j γ j u, s h 1 2 B For j = 0, 1,..., 1, we have E γ j u, s γ j u, s 2 C j 2 CB 6 E d1 u 8 E d 1 s 8 1/4, B.88 CB 6 E d1 u 8 E d 1 s 8 1/4. B.89 j αj k k + j 1 αk j j + jα 3/2 j k=0 k=j+1 B.90 E d 0, u 8 E d 0, s 8 1/4, where i the above three bouds C is a costat that does ot deped o j,, u ad s. akig ito accout Assumptio 3, with C as above, we have E γ j u, s γ j u, s 2 C j α j/2 E d0, u 8 E d 0, s 8 1/4. B.91 Hece, usig the boudedess of h ad Assumptio 3, B j j γ h j u, s γ j u, s 2 C B E d0, u 8 E d 0, s 8 1/4, B.92 B j=0 for some positive ad fiite C that does ot deped o u ad s. 21

57 We proceed with the differece γ j u, s γj u, s. If we deote 0 < ɛ < if κ [0,1] f κ /4 { ad ɛ > 2 sup κ [0,1] f κ, the it suffices to aalyze this differece o Ω = ω : f κ, f } κ [ɛ, ɛ], sice PΩ 1 from the results of Lemma 1. We will do so heceforth without further metio. Makig use of sup u R+ ul u e 1, we have for some positive ad fiite C that does ot deped o u: u L κu e 0.5 t,κx t,κ X 1 {A t,κ }/ f κ + ul uvt,κ t,κw t,κ W CV t,κ t,κw t,κ W f κ f κ + C t,κx t,κ X 1 {A t,κ } f κ V t,κ t,κw t,κ W + CV t,κ t,κw t,κ W u L κu e ul u. B.93 herefore, with C as above, we have d t, u d t, u C u 1ζ 1 t,κ ζ κ u+ζ 1 t,κ ζ κ u+ζ 2 t, d t, u Cζ 1 t,κ +ζ 1 t,κ, B.94 where we deote ζ 1 t,κ = Vt,κ t,κw t,κ W + Vt,κ t,κw, ζ 2 t = t,κx t,κ X 1 {A t,κ } f κ V t,κ t,κw t,κ W, ζ κ u = f κ f κ + u L κu e ul u, ad we ote that by applicatio of Lemmas 1 ad 2 ad for q 8 q is the costat of Lemma 1, we have 1 q 4 ζ κ u = O p 1 ϖ 2. B.95 22

58 We ca boud B γ j u, s γ 1 j u, s B d B t, u d t, u d t j, s j=0 j= B 1 B + d t, u B d t j, s d t j, s j= B 1 + B d B t, u d t, u d t j, s d t j, s. j= B B.96 Usig iequality i meas as well as the fact that E V t 4 <, we have 2 1 E ζ 1 B t,αζ t j,β 1 C, j = B,..., 0,..., B, B.97 where α, β = κ, κ. o proceed further, we boud the k-th momets of ζ 2 t. Usig successive coditioig, Hölder s iequality as well as the fact that E b t 4 <, we have k E[ t,κx t,κ X 1 {A t,κ }1 { t,κ X d 0, t,κ Xd 0}] k k2ϖ 1 P t,κx d 0 ad t,κ X d 0 C k2ϖ 1. B.98 Applyig successive coditioig, the smoothess i expectatio coditio for σ t, Hölder s iequality as well as the itegrability coditios for a t, b t ad σ t, we have k E[ t,κx t,κ X 1 {A t,κ }1 { t,κ X d =0, 0}] k C 1+ k 2 2ϖ 1 t,κ Xd, k [1, 2], B.99 k E[ t,κx t,κ X 1 {A t,κ }1 { t,κ X d 0, t,κ Xd =0}] k C 1+ k 2 2ϖ 1, k [1, 2]. B.100 Usig these bouds, Hölder s iequality, the smoothess i expectatio coditio for σ t as well as the itegrability coditios for a t ad σ t, we have E ζ 2 t k C [ 7 4 +k2ϖ 1 1+ k 2 2ϖ 1 q 2k 1 ϖ 2 k 2 23 q k q ], k [1, 2], B.101

59 where q is the costat of Lemma 1. Usig E V t 8 < ad applyig Hölder s iequality, we have E ζ 2 t ζ 1 s,ι C, E ζ 2 t ζ 2 s C, ι = κ, κ, s, t 0, B.102 provided q 4 1 ϖ 1. herefore, 2 2 [ 1 B E ζ 2 t ζ 1 t j,ι B + ζ1 t,ι j= B [ 1 B E ζ 2 t B B j= B ζ 2 t j j= B ζ 2 t j ] ] CB, CB. B.103 B.104 where ι = κ, κ. Combiig these results, we get altogether B j j γ h j u, s γj u, s 2 = O p B 2. B.105 B j=0 Combiig the bouds i B.88-B.89, B.92 ad B.105, we have altogether the result of the lemma. We ca decompose, Zκ, κ Ẑ κ, κ = i=p +1 λ i χ 2 i + p i=1 λ i λ i, χ 2 i. B.106 By assumptio we have that p as. Hece, Parseval s idetity implies that i=p +1 λ iχ 2 i = o p 1. Furthermore, by heorem 4.4 i [1] it follows that herefore, we have Zκ, κ Ẑ κ, κ sup λ j, λ j K K HS. j 1 p i=1 λ i λ i, χ 2 i + o p 1 sup λ j, λ j j 1 p i=1 χ 2 i + o p 1 = o p 1, B.107 B

60 where for the last boud, we made use of the result of Lemma 5 ad the rate coditio for p i the theorem. B.6 Proof of Corollary 2 he result uder the ull hypothesis follows from Corollary 1 ad heorem 3 ad applicatio of portmateau theorem. Uder the alterative hypothesis, oe ca easily show usig the itegrability coditios of the theorem ad usig some of the bouds i the proof of Lemma 5 that we have Ẑ κ, κ = O p B. Furthermore, from the proof of heorem 1, uder the coditios of the theorem, we have L κ L κ 2 = O p. hese two results yield the asymptotic power of oe by takig ito accout that B / 0. Refereces [1] D. Bosq. Liear Processes i Fuctio Spaces. Spriger, [2] J. Jacod ad A.N. Shiryaev. Limit heorems For Stochastic Processes. Spriger-Verlag, Berli, 2d editio, [3] Adam Jakubowski. O Limit heorems for Sums of Depedet Hilbert Space Valued Radom Variables, pages Spriger New York, New York, NY, [4] J. Weidma. Liear operators i Hilbert Spaces. Spriger,

61 Research Papers : Marti M. Adrease, Jes H.E. Christese ad Simo Riddell: he IPS Liquidity Premium : Aastiia Silveoie ad imo eräsvirta: Cosistecy ad asymptotic ormality of maximum likelihood estimators of a multiplicative time-varyig smooth trasitio correlatio GARCH model : Cristia Amado, Aastiia Silveoie ad imo eräsvirta: Modellig ad forecastig WIG20 daily returs : Kim Christese, Ulrich Houyo ad Mark Podolskij: Is the diural patter sufficiet to explai the itraday variatio i volatility? A oparametric assessmet : Marti M. Adrease, Jes H.E. Christese ad Gle D. Rudebusch: erm Structure Aalysis with Big Data : imo eräsvirta: Noliear models i macroecoometrics : Isabel Casas, Eva Ferreira ad Susa Orbe: ime-varyig coefficiet estimatio i SURE models. Applicatio to portfolio maagemet : Hossei Asgharia, Charlotte Christiase, Ai Ju Hou ad Weiig Wag: Log- ad Short-Ru Compoets of Factor Betas: Implicatios for Equity Pricig : Jua Carlos Parra-Alvarez, Olaf Posch ad Mu-Chu Wag: Idetificatio ad estimatio of heterogeeous aget models: A likelihood approach : Adrés Gozález, imo eräsvirta, Dick va Dijk ad Yukai Yag: Pael Smooth rasitio Regressio Models : Søre Johase ad Morte Ørregaard Nielse: estig the CVAR i the fractioal CVAR model : Nektarios Aslaidis ad Charlotte Christiase: Flight to Safety from Europea Stock Markets : ommaso Proietti, Niels Haldrup ad Oskar Kapik: Spikes ad memory i Nord Pool electricity price spot prices : Emilio Zaetti Chii: Forecaster s utility ad forecasts coherece : orbe G. Aderse, Nicola Fusari ad Viktor odorov: he Pricig of ail Risk ad the Equity Premium: Evidece from Iteratioal Optio Markets : orbe G. Aderse, Nicola Fusari, Viktor odorov ad Rasmus. Vareskov: Uified Iferece for Noliear Factor Models from Paels with Fixed ad Large ime Spa : orbe G. Aderse, Nicola Fusari, Viktor odorov ad Rasmus. Vareskov: Optio Paels i Pure-Jump Settigs : orbe G. Aderse, Marti hyrsgaard ad Viktor odorov: ime-varyig Periodicity i Itraday Volatility