Taesuk Lee (2011) Rate-optimal Test for Jumps in Diffusion Processes.pdf

8/13/2019 Taesuk Lee (2011) Rate-optimal Test for Jumps in Diffusion Processes.pdf

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Rate-optimal Tests for Jumps in Diusion Processes

Taesuk Leea and Werner PlobergerbaDepartment of Economics, University of Auckland

bDepartment of Economics, Washington University in St. Louis

January 2011

Abstract

Suppose one has given a sample of high-frequency intra-day discrete observations of a

continuous-time random process (e.g. stock market data) and wants to test for the presence

of jumps. We show that the power of any test of this hypothesis depends on the frequency

of observation. In particular, we show that if the process is observed at intervals of length

1=n and the instantaneous volatility of the process is given by t, at best one can detect

jumps of height no smaller than tp

2log(n)=n. We construct a test which achieves this

rate in the case for diusion-type processes.

Keywords : High Frequency Data, Jump, Likelihood Test.

0 Further author information:

Taesuk Lee: Lecturer, E-mail: [email protected]

Werner Ploberger: Thomas H. Eliot Distinguished Professor, E-mail: [email protected]

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1 Introduction

Continuous diusion models have provided a simple, exible, and powerful tool to analyze eco-

nomic and nancial data since the time before high frequency data become available. Because

data are only observed at discrete times, we do not have full information on the trajectory of the

process. Therefore we may have modeling errors due to the discreteness of the observations, butthis problem can be signicantly mitigated with high frequency data which is now available with

technological progress.

High frequency data, however, generate their own challenges: We are not sure that the data

generating process is continuous; there may be jumps. Since continuous diusion models do not

capture jumps, researchers must know whether the data contain jumps or not. Furthermore, many

datasets (such as when returns are measured over short intervals - say 1 to 5 minutes) contain

some contamination commonly called "market microstructure noise". Our aim in this paper is to

propose an optimal test for the null hypothesis of continuous diusion models against an alterna-

tive hypothesis of jump diusion models while allowing for the presence of market microstructure

noise in the data. Though several tests (Barndor-Nielsen and Shephard (2006), hereafter BNS,

Ait-Sahalia and Jacod (2009), hereafter AJ and Lee and Mykland (2008). hereafter LM) were

already introduced, their power properties were not explicitly considered. In this paper, we de-

rive a rate-optimal test valid under fairly general assumptions about the data generating process.

Furthermore we compare power of our test and that of other competing tests.

2 Local Power Bound

As the null, we consider the usual (i.e., purely continuous) diusion model:

dXt = tdt + tdWt, (1)

where W = (Wt : t 2 [0; 1)) is a Wiener process and t and t are non-anticipating randomprocesses fullling the usual requirements of Ito-calculus. Later on we will impose additional

assumptions on and:(For example, we will assume them to be smooth to a certain extent, so

that the processXt specied in equation(1) has certain "nice" properties.)

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As the alternative, we want to consider jumps present in the diusion model :

dXt = tdt + tdWt+ Jtdt,

whereJt is a non-zero random variable whose absolute value species the jump size and t is a

counting process governing whether there is a jump or not at time t.

The problem, however, is that in empirical practice, we cannot observe the entire process.

Instead, it is observed only at discrete times t = i=n; where n is a positive integer denoting the

"sample size" and

0 i n:

The rst test for this testing problem - and still the "gold standard" for all tests was developed

by Barndor-Nielsen and Shephard (2002). Since the problem of detecting jumps is of enormous

practical importance, further research has occurred in this eld. An alternative test was developed

by Ait-Sahalia and Jacod (2009), and an informal "testing procedure" was provided by Lee and

Mykland (2008).

To the best of our knowledge, none of these contributions, however, discussed power of the

tests. In this paper, we establish the following two results.

1. Clearly, when n, the number of observations increases, we should expect our test to have

"better" power. In particular, we want to consider the power against local alternatives for the

jumping processes. So we consider for eachn the alternative

J(n)t =cn

where we assume that cn is a sequence converging to zero, and assume the process t remains

(uniformly) bounded. (So we assume there is only a maximum number of jumps). Let" >0 be

arbitrary. Then we show that - even if we know that t = - it is impossible to construct tests

that have nontrivial power against alternatives with

cn= p2(1 ")log npn (2)2. As a second result, we analyze a class of tests very similar to Lee and Mykland (2008) so

that (even in the general case)

cn= t

p2(1 + ")log np

n :

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the power of the test converges to one. So in a certain way, our tests attain the "optimal rate".

The main dierence of our tests to the ones in Lee and Mykland (2008) is a dierent construction

of the estimator for the volatility. We only use information from narrow time intervals, which

might be useful when the volatility varies a lot.

This is an advantage over the classical BNS or AJ tests: their local alternatives shrink with

the ordern1=4 (or - in the case of AJ - with the order ofn1=2+1=p, wherep is a positive number

determining the test statistic. Moreover, our test is able to deal with simple mis-specication due

to microstructure).

Let us rst deal with our rst assertion. Let assume we even deal with the simplest case,

namelyt = 0and t = 1, so our underlying processXt is a Wiener process. Then let us assume

that - under the alternative - we only have one jump, and the time of the jump is distributed

uniformly in the interval[0; 1] :We rst will show that even under this rather ideal conditions we

will be unable to construct tests with nontrivial power if the cn are following (2).

Theorem 1 We want to test the null of Xt being a Wiener process Wt ,of known variance,

against the alternative of

Xt = Wt+ cnIf tg ;

whereis an independent random variable following an uniform distribution. Suppose we observe

the processXt only at the time points0; 1=n; 2=n; :::1: Supposecn follows (2) (or is smaller then

this bound). Then it is impossible to construct nontrivial tests.

Proof. We did assume the variance of the Wiener process W to be known. Without loss

of generality, we can assume this variance to be 1. Let Pn be the probability measure of

(X0; X1=n; X2=n;:::;X1) under the null, and Qn be the measure under the alternative. Let the

zi be dened as

zi=

Xi=n X(i1)=np

n

Then we can easily see that the zi are i.i.d. standard normal, and that

dQndPn

= 1

n

nXi=1

exp

(cnp

n)zi 12

(cnp

n)2

Since each of theziis standard normal, the expectation of each exp

(cnp

n)zi 12(cnp

n)2

equals

one.

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Since we are interested in small" >0 (the smaller we choose ", the bigger are our jumps), we

can maintain the assumption that

" 0;

2 "2(1 ") < 1:

Our assumption (3), namely that " 2:

So the limit of the Fn lies between1 and2:Hence we can nd constants , with

2< < < 1:

So that for all but nitely many n;

< Fn< : (15)

Without loss of generality we can assume that (15) holds for all n. Let us now analyze the

integrand in (10):1 exp(y) y

y yFn exp

(log y)

2

4(1 ")log n

(16)

The rst factor,

(y) =1 exp(y) y

y

is easily seen to be uniformly bounded for all nonnegative real y . So there exists an M that

j(y)j M

for all nonnegativey: Using the power series representation for exp(), one can easily see that(:)is an analytic function for all y and

(0) = 0:

Since any analytic function has derivatives bounded on any compact set, we may conclude that

for0 y 1;j(y)j Cy

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for some universal constantC: The third factor is easily seen to have absolute value smaller than

1.

Let us now distinguish two cases: For y 1,

1 exp(y) yy

yFn exp( (log y)2

4(1 ")log n) Cy1+:

Since > 2, Z 10

1 exp(y) yy

yFn exp( (log y)2

4(1 ")log n)dy (17)

CZ 10

y1+dy= C 1

2 + :

Fory 1;

1 exp(y) yy yFn exp (log y)24(1 ")log n My

and since < 1; Z11

1 exp(y) yy

yFn exp

(log y)

2

4(1 ")log n

dy

MZ

1

1

ydy= M 11 +

:

Hence we may conclude thatZ10

1 exp(y) yy

yFn exp

(log y)

2

4(1 ")log n

dy

remains uniformly bounded in n. Therefore for alls;

limn!0

R10

1exp(y)yy y

Fn exp( (log y)24(1")logn)dy

p2(1 ")log np

2n"2

4(1") s "

2(1")

= 0:

Now we can combine this result with (13) and (9) and conclude that

limn!0

R10

1exp(y)y y

Fn exp( (log y)24(1")logn)dyp2(1 ")log np2n "

2

4(1") s "

2(1")

= 1:

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We can now use this limit to characterizeSn dened in (8):

limn!1

Sn1p2

np2(1")logn

1s Cn

np2(1 ")log np2n "

2

4(1") s "

2(1")

o (18)= lim

n!1Sn

n(2")2=4(1") ns "

2(1")1

oCn= 1:

Since the Cn were dened as

Cn= n (2")2

4(1") s(2")2(1") exp

(log s)

2

4(1 ")log n

;

we can easily see that the denominator of the expression in (18) converges to one. But this implies

that

limn!1

Sn= 1;

which is exactly (6), the result we wanted to prove. Hence the Laplace transforms

E

exp

sdQn

dPn

of the density ratios converge to exp(s); which is the Laplace transform of a measure concen-trated in1.

We therefore have shown that the density ratios

dQndPn

converge in distribution to a constant, namely 1 (Theorem 2, p. 431, Feller (1971)). Therefore

they converge in probability to this constant, too. Hence for an arbitrary >0;

Pn

dQndPn 1> ! 0:

Now letAnbe a sequence of events. Then we have

(1 )Pn(An) Pn

dQndPn

1

>

< Qn(An)

< (1 + )Pn(An) + PndQndPn 1> :

Since was arbitrary, we can conclude that

Pn(An) Qn(An) ! 0:

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SinceAn is an arbitrary sequence of events, we can conclude that the total variation between Pn

andQn converges to zero, hence for all measurable functions'n with0 'n 1we haveZ 'ndPn

Z 'ndQn! 0:

But this is exactly what we wanted to show: For every sequence of tests, the power under the

null (Pn) is the same as under the alternative (Qn).

Now we want to present a test statistic, for which we will show that the local power of the

test statistic attains this bound. The preceding result indicates that for xedcn our best statistic

is an exponential sum of the zi. We should, however, keep in mind that our zi are increments

over shorter and shorter time intervals. In order to consider relevant alternatives, we might be

interested in alternatives wherecnp

nbecomes "large". In this case, the test statistic gives more

and more increasing inuence to bigger values. Continuing with this line of reasoning, it maybe a good idea to look mainly at the "largest" absolute values of the increments of Xt: The

standard theory of diusion processes guarantees that, when divided by t, these increments

are approximately normal. Next, because we do not knowt, we have to estimate it. Ast is

time-varying, a moving average of the squares of the increments seems to be a natural candidate

estimator. This leads us to propose the following test statistic: Dene (for an arbitraryn) the

returnri = ri;n by

ri = ri;n =

Xi=n X(i1)=n

Then choose an integer ` (the "length of the window") and reject the null whenever

n= supi

r2i(r2i1+ r

2i2+ ::r

2i`)=`

is "too large". This is quite analogous to the test statistics of Lee and Mykland (2008): We

standardize the returns by an estimator for 2 (t; Xt). We use the usual quadratic estimator

instead of the bipower estimator. One might argue that jumps could unduly aect the properties

of our estimator. We think, however, that the much simpler form is justied, essentially in part

for the following two reasons:

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1. We assume that the jumps are separated events:Beforethe rst jump occurs, our estimator

for2 (t; Xt)is not inuenced by it.

2. We only use a window of length ` to estimate 2 (t; Xt). So a jump will only inuence a

small number of estimated values of2 (t; Xt). Our test would get only distorted if we had

two (or more) jumps within an interval of length `=n, an event whose probability we assume

converges to zero.

The main reason, however, for using this specic estimator is convenience. Specically, only

Lemma 6 is essential to prove the rate-optimality of our proposed test. We think analogous results

will hold for more general classes of estimators.

3 Critical Values and Local Power of Our Test

Letzi; i= 1; : : ;nbe a sequence of independent, identically distributed standard normal random

variables. Assume that for each n we have given an ` = `(n), and let us denote by Fi thealgebra generated byzi; zi1; ::Then let us dene

wi =`

Xj=1 z2ij;

b2i =wi=`and

i = z2i =b2i :

For the computation of the critical values the following lemma is very helpful.

Lemma 2 Suppose that

`=o (n) ;

but also

` 2log n:

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Equation (21) is an immediate consequence of Lemma 6, which shows that

P

infb2i M(")2 =Kn nPb2i M(")2 =Kn! 0:To prove that (22) is valid, rst observe that

E[Ifi Kng jFi1] = 2qKnb2i 1:Ifb2i > M(")2 =Kn , we can use inequality (30) to conclude that

log

2

qKnb2i 1 2(1 + ")2 exp Knb2i =2 =q2Knb2i

Hence ( n

Xj=`+1log E[Ifi Kng jFi1] c(1 + ")3

)\

inf

b2i > M(")

2 =Kn

(

2(1 + ")2nX

j=`+1

expKnb2i =2 =q2Knb2i c(1 + ")3) \ infb2i > M(")2 =Kn

Since we already know that P

infb2i > M(")2 =Kn! 1, it is sucient to show thatP

"(2

nXj=`+1

expKnb2i =2 =q2Knb2i c (1 + ")

)#! 1 (23)

Let us now introduce the term Yj by

Yj = 2expKnb2i =2 =q2Knb2i

Then we can easily see that (23) is fullled if

nXj=`+1

Yj! c (24)

in probability. By our denition ofKn, EYj = c=n. Moreover, we know that

b2i is distributed

according to a scaled2

distribution with` degrees of freedom. Hence it is an elementary exerciseto show that E Y2j =O(1=n

2), and that Yj andYk are independent if

jj kj > ` + 1:

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As`=n ! 0, we can easily see that the variance of PYj converges to zero. This establishes (20).Now it is rather easy to establish the claim stated in our lemma: We have to show that

E

"Yi=`+1;::n

Ifi Kng#

! exp(c)

Using (20), it is sucient to show

E

"Yi

Ifi Kng#

! exp(c)

Trivially,

E

Ifi Kng

E(Ifi Kng jFi1)jFi1

= 1:

A straightforward argument, perfectly analogous to the optional sampling theorem, yields

E2666664E"YiIfi Kng#Y

iE[(Ifi Kng jFi1]

3777775 = 1: (25)

According to the denition of ;

(1 + ")2X

j=`+1

Yj logYi

E[Ifi Kng jFi1] (1 ")2X

j=`+1

Yj (26)

and

logYi

E[Ifi Kng jFi1] c(1 + ")3 (27)

Moreover, (20) implies that PhnP

j=`+1 Yj =Pn

j=`+1 Yj

oi! 1. Therefore Pj=`+1 Yj! c as

well. Hence it can be seen that (27) and (26) allow us to deduce from (25) that

exp((1 + ")2c) liminfEYi

Ifi Kng

limsup E

YiIfi Kng exp((1 ")2c):

Now one can easily see that (20) allows us to replace withnin the preceding inequalities, which

proves our lemma.

So far, we have computed the distribution of our test statistic for a very specic case, namely

whent= 0andt = 1. We now have to show that the general case given by (1)can be reduced

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Another alternative test has recently been proposed by Ait-Sahalia and Jacod. This test

is based on the pth power variation of the process Xt and it compares the estimates of thisvariation for dierent time scales.

Denition 5 AJ test statistic(Ait-Sahalia and Jacod (2009))1

bp;kAJ = kp=21 bS(p; k; ) =qbVp;k wherep >3; k 2;bS(p; k; ) =bB (p; k) =bB (p; ) ,bB (p; k)t = n=kX

i=1

jrt+ikjp and

bVp;k is the variance ofbS(p; k; ) under the null.The behavior of both sets of test statistics under our alternatives can easily be analyzed. We

just add a specic jump to one of returns. One can easily see that if the jump is ofo(n1=4),

the dierence between the BNS test statistic under the null and the alternative converges to 0.

Hence their test will be much less powerful against the alternatives we consider. The same is true

for AJ tests: After some calculations, we can see that the corresponding bound is o

n1=2+1=p

.

Despite the fact that both tests have "low" power against "our" alternatives, it should be noted

that there are situations where the BNS and AJ tests have large advantages over our test. For

instance, the relevant alternative could be the occurrence of many jumps in the sample. Assume

there is not only one jump, but many. So let us assume that we haveL jumps of size J (rather

evenly distributed), and let us assume that the number of intervals between successive jumps is

always greater than 1. Then it is easily seen from the denition of BNS statistics that the tests

are consistent (i.e., its power converges to 1) if

pnLJ2 ! 1:

1 ForbVp;k; they suggest two estimators:bVcp;k= nM(p;k) bA(2p;n)tbA(p;n)2t ;eVcp;k= nM(p;k) eA( pp+1 ;2p+2;n)teA( pp+1 ;p+1;n)2t whereM(p; k) = 1m2p

kp2 (1 + k)m2p+ k

p2 (k 1)m2p 2kp=21mk;p;

mp= E(jZ1jp) = 1=22p=2 p+12 ;mk;p= EhjZ1jp Z1+ pk 1Z2pi,

Ziiid N(0; 1) ;bA (p;n)t = 1p=2nmp Pi=1 jniXjp 1 fjniXj $n g, $ 2 (0; 1=2) ;eA (r;q;n)t = 1qr=2nmqr Pi=1qj=1 ni+j1Xr ;niX= XinX(i1)n

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An analogous result holds for AJ tests. These results are easily explainable by taking into account

that both test statistics are constructed from sums: The contributions of several small jumps can

cumulate, whereas our test does not allow for this. So one should employ their tests not as

tests against the alternative of the occurrence of a single jump, but as tests against Levy-type

alternatives. It might be worthwhile to investigate their power of the test against alternatives of

this specic type.

5 Simulations

First, we consider a very simple ideal process for returns:

ri=n =

Z i=n

(i

1)=n

dpt =

Z i=n

(i

1)=n

dWt+

Z i=n

(i

1)=n

Jdt with2 = 0:513 (Model 1)

Second, the stochastic volatility model of Barndor-Nielsen, Hansen, Lunde, and Shephard

(2008) is considered.

ri=T =

Z i=T(i1)=T

dpt=

Z i=T(i1)=T

tdt +

Z i=T(i1)=T

tdWt+ Jd (t) , (Model 2)

log i=T

log (i1)=T = Z

i=T

(i1)=T(0+ 1t) dt;

i=T (i1)=T =Z i=T(i1)=T

tdt + dBt; and

Corr (dWt; dBt) = :

We set = 0:03; 1= 0:125; = 0:025; = 0:3 and 0 = 21

2 :

Third, we considered the stochastic volatility model devised by Barndor-Nielsen and Shep-

hard (2002).

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dp (t) = (s) W(ds) + Jd (t) with (Model 3)

2 (t) =Xk

2k(t)

2k(t) =

Z t

0

k 2k(s) k ds + Z t

0

!kk(s) dBk(s) ;

k = pk0;X

pk = 1:

For the one factor model, we set 0 = 0:513; 1 = 1:44; p1 = 1; and!21 = 2:1. For the two

factor model, we set 0 = 0:509; 1 = 0:0429; 2 = 3:74; p1 = 0:218; p2 = 1 p1. We consider(!21; !

22) = (0:0169; 5:2978) :

We track the performance of four test statistics : Lee-Ploberger (LP), Barndor-Nielsen &

Shephard(BNS), Ait-Sahalia & Jacod(AJ), and Lee-Mykland (LM). We assume J

N(0; 2c

)and

consider three cases for jump sizes : no jump(2c = 0), 20% jump(2c = 0:20(s)), and2log (n) =n

jump(2c = 2log (n) =n 0(s)). We impose one jump at a random time. The sample sizes con-sidered are 72, 288, 1440, 2880, 8640; these sample sizes correspond to sampling interval lengths

of 20 minutes, 5 minutes, 1 minute, 30 seconds, and 10 seconds, respectively, over a 24-hour

trading day. We repeat this simulation 5,000 times. Table 1 - 6 show the results of small simula-

tion experiments varying the size of jumps. In those tables, we compare the empirical rejection

probabilities of various tests with the 5% level of signicance.

Let us consider rejection probabilities under the null, no-jump case. In many cases, our tests

have quite precise sizes even with 5-minute data, although our tests over-reject under the null

if the volatility process moves too sharply and/or the volatility process stays around zero level

for a long time (Table 3, 4). In those cases, we could reduce the degree of those size distortions

by changing averaging windows for instantaneous volatility. Our tests assume some continuity

of volatility process within the averaging window. If the volatility process moves too sharply,

then our assumptions may not be fullled in a nite sample even though the underlying process

is continuous. To achieve more robust size properties in nite samples, we will consider data-

adaptive rules for averaging window.

Next, we consider the power of the test. Our tests have better power than other tests after

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controlling for size distortions. In our simulations, we consider one jump and check whether tests

detect the jump or not. In that set-up, the LM tests show a large size distortion even with 10-

second data, so their power is signicantly reduced by the size adjustment. Although our tests

sometimes over-reject the null, our tests have the best or comparable power in many cases if we

adjust the critical values to remove size distortions. In particular, in many cases our tests have

a non-trivial power even with jump the order of which is log (n) =n; where n is the sample size,

whereas the BNS and AJ tests have trivial power in that case. The LM tests also have a non-

trivial power but their sizes are not as reliable as ours and their averaging window requirement

is more restrictive than ours.

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ing integrated volatility with noisy high-frequency data, Journal of the American Statistical

Association, 100(472), 13941411.

A Normal and 2 distributions for small and large values

It is well known that for the standard distribution function (x)

limx!1

(1 (x))p

2x exp

x2=2

= 1

or equivalently

limx!1

(log(2 (x) 1)) p2x exp x2=2 =2 = 1we can dene M(")as the smallest value so that for all

x > M(") (28)

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(1 ") p2x exp x2=2 (1 (x)) (1 + ") : (29)

and

2 (1 + ")2 (log(2 (x) 1))p

2x exp

x2=2 2 (1 ")2 (30)

Lemma 6 So let us now choose an arbitrary" >0, and let`,n, wi be the integers dened in the

main section of the paper. If

` 2log n

and

K! 1;

then

pn= P

wi `M(")2 =K

=o

n1

:

:

Proof. Sincewi is distributed according to a 2distribution with` degrees of freedom, we have

pn= 1

(`=2)

Z `M(")2=(K)0

x`=21 exp(x=2) dx:

Sinceexp(x=2) 1, we have withC=

M(")2

Kpn 1

(`=2)

1

`=2C`=2``=2;

and therefore

logpn log(`=2) log(`=2) + `2

log C+ `=2log `:

The well known formula of Stirling implies that for` ! 1 (withm =`=2 1)

log(`=2) n

m log(m) m + logp

2mo

! 0:

Therefore

logpn f(`=2)log ` m (log(m)g + (m + `2

log C) + O(log `)

One can easily see that f(`=2) log ` m (log(m)g = (`=2) log(`=m)+ O(log m). So the terms linearin ` dominate the right hand side of the inequality, Moreover, as M(") is xed and K! 1,

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26/34


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and

Phn

n= (")n

oi> 1 ";

too. Hence it is sucient to show that for all " >0 the dierence between converges to zero. For

showing this, let us rst observe that

min(2(ik)=n

2i=n) s

2i

(s2i1+ s2i2+ ::s

2i`)=`

=2i=ns

2i

(2(i1)=ns2i1+

2(i2)=ns

2i2+ ::

2(i`)s

2i`)=`

max(2(ik)=n2i=n

):

For analyzing the dierence of the left and right side of the above inequality and one, it is sucient

to consider

supk`

log( 2(ik)=n2i=n

)

:Now observe that log(2i=n) log(2(ik)=n) =

Ri=n

(ik)=n Ctdt + DtdV(2)t . Fori <

(")R

i=n

(ik)=n Ctdt

kM=n:Moreover, we have due to Lemma 7

P

"(Z i=n(ik)=n

DtdV(2)t

>2pM`r

log n

n

)# 1

n2

and hence

P

"( sup

i(");k`

Z i=n(ik)=n

DtdV(2)t

> 2pM`r

log n

n

)# `

n! 0:

Hence we can conclude that

P"(supk` log(2(ik)=n

2i=n

)> 4p

M`rlog n

n )#! 0:Since

sup s2i

(s2i1+ s2i2+ ::s

2i`)=`

=O(log n);

we can conclude that the dierence between

sup s2i

(s2i1+ s2i2+ ::s

2i`)=`

and

sup

2i=ns2i

(2(i1)=ns2i1+

2(i2)=ns

2i2+ ::

2(i`)s

2i`)=`

converges to zero.

It now remains to show that the dierencesri;n (i1)=nsi;n27


28/34

remain small. Now observe thatri;n (i1)=nsi;n =Z i=n(i1)=n

tdt + tdWt (i1)=ndWt

=

Z i=n(i1)=n

tdt

+

Z i=n(i1)=n

t (i1)=n

dWt

max jtj 1n+ Z i=n(i1)=n u (i1)=n dWu :

For the analysis of Z i=n(i1)=n

u (i1)=n

dWu

we will apply Lemma 7. Since u is a diusion process, where drift and diusion coecients were

assumed to be bounded, we can conclude that for all >0 there exists a Mso that

Phnfor alli and (i 1)=n u i=n u (i1)=n Mju (i 1)=nj1=2oi! 1:Hence

P

"(Z i=n(i1)=n

u (i1)=n

2du 2Mn2+

)#! 1:

To apply Lemma 7, however, we need to guarantee an uniform bound on the integralRi=n(i1)=n

u (i1)

This can easily be achieved by using a stopping time.

We stop the process at time S, where

i=n S (i 1)=n;

if for the rst time Z S(i1)=n

u (i1)=n

2du= 2Mn2+;

otherwise we set

S= 1:

Obviously the denition ofMguarantees that

P(S= 1) 1 "

Hence if we dene

u =

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B.1 Tables and Figures

Following tables summarize rejection probabilities under 5% size.

n 4LN 2LN LIN Ratio ADJ QV BIP SQRT 4LN 2LN

72 3.04 3.74 11.14 8.26 6.42 3.24 3.70 48.64 17.72 48.64

288 4.82 4.56 8.40 6.98 6.18 3.84 3.68 43.38 28.52 66.40

1440 4.76 4.90 5.84 5.30 5.16 4.28 4.30 28.76 39.44 85.96

2880 5.06 5.64 6.02 5.56 5.30 5.10 4.74 22.82 45.60 92.62

8640 5.12 5.18 4.92 4.64 4.58 5.06 4.68 16.46 52.90 96.54


72 33.24 25.44 32.74 28.50 26.36 6.78 9.98 68.32 49.60 68.32

288 59.32 52.58 48.90 46.96 46.06 20.46 36.26 78.62 72.92 87.60

1440 78.80 75.92 65.28 64.56 64.44 34.76 68.26 86.66 88.94 97.74

2880 84.38 81.54 71.26 70.94 70.84 39.62 76.96 88.94 92.32 98.84

8640 90.88 88.78 78.36 78.14 78.08 43.32 85.62 93.12 96.12 99.78


72 22.22 16.44 24.82 20.92 18.58 5.36 7.70 62.32 39.28 62.32

288 25.84 19.14 18.62 16.72 15.68 8.22 12.68 60.38 49.24 76.54

1440 26.80 20.52 12.60 11.64 11.36 9.14 13.76 49.96 57.78 90.68

2880 27.26 21.76 11.30 10.66 10.40 9.52 14.38 46.98 62.76 94.80

8640 27.74 21.94 7.74 7.44 7.32 9.02 13.40 41.70 67.12 97.58

Model1-1 : Pure Diffusion W/O JUMP

LP BNS AJ LM

Model1-2 : Pure Diffusion W 20% JUMP

LP BNS AJ LM

Model1-3 : Pure Diffusion W LN(2N)/N JUMP

LP BNS AJ LM

Table 1: Simulated rejection probability of Model 1 : Pure Diusion


72 3.42 3.76 10.58 8.22 6.38 2.92 3.34 47.24 18.12 47.24

288 4.58 4.02 7.72 6.56 5.84 3.62 3.80 42.98 28.44 65.58

1440 4.30 4.96 6.00 5.52 5.22 4.30 4.50 27.86 38.76 86.52

2880 4.72 4.70 5.86 5.46 5.28 4.14 4.30 23.40 45.06 93.20

8640 5.14 5.58 5.26 5.08 4.98 4.96 5.06 16.10 54.20 96.82


72 5.82 5.30 12.26 9.70 8.12 2.66 3.52 49.62 21.24 49.62

288 13.30 9.98 12.72 11.08 10.40 4.40 7.62 50.60 37.44 70.30

1440 32.14 26.40 18.64 17.84 17.52 11.08 19.92 52.08 59.58 91.36

2880 42.44 36.76 22.52 21.78 21.60 15.68 28.50 58.36 69.46 96.06

8640 59.94 54.92 32.36 31.96 31.84 23.24 44.96 68.44 83.22 98.98


72 4.12 4.16 10.84 8.48 6.82 2.72 3.24 48.02 19.22 48.02

288 4.90 4.28 8.20 6.76 6.02 3.66 3.98 43.50 29.04 65.94

1440 4.64 5.14 6.10 5.58 5.32 4.38 4.66 28.34 39.20 86.60

2880 5.00 4.92 5.98 5.52 5.34 4.18 4.42 23.82 45.48 93.22

8640 5.42 5.82 5.42 5.26 5.12 5.04 5.20 16.70 54.54 96.90

Mode 2-1 : log SV-Diffusion W/O JUMP

LP BNS AJ LM

Mode 2-2 : log SV-Diffusion W 20% JUMP

LP BNS AJ LM

Mode 2-3 : log SV-Diffusion W LN(2N)/N JUMP

LP BNS AJ LM

Table 2: Simulated rejection probability of Model 2 : log SV model

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Jump/MeanVol Meaning 4LN 2LN LIN Ratio ADJ QV BIP SQRT 4LN 2LN

0.000 No Jump 33.04 21.40 10.60 8.64 8.56 4.20 5.46 72.92 66.40 84.30

0.022 LN(2N)/N 45.64 34.08 17.90 15.78 15.60 4.84 7.08 78.14 72.70 87.56

0.107 63.70 53.90 40.92 38.26 38.14 9.84 17.54 86.44 83.42 91.22

0.192 71.38 63.24 51.54 49.28 49.18 13.52 24.26 88.88 86.04 94.06

0.277 76.02 69.10 57.94 56.16 56.02 15.78 29.72 90.94 88.42 94.82

0.362 78.16 72.14 62.18 60.48 60.40 16.92 31.76 91.64 89.64 95.16

0.447 20% 80.36 73.80 64.60 63.08 63.04 18.66 34.96 92.96 90.74 95.72

Size Distortion 28.04 1 6.40 5.60 3.64 3.56 -0.80 0.46 67.92 6 1.40 7 9.30

Size Adjusted Power

Jump/MeanVol

0.000 No Jump 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00

0.005 LN(2N)/N 17.60 17.68 12.30 12.14 12.04 5.64 6.62 10.22 11.30 8.26

0.093 35.66 37.50 35.32 34.62 34.58 10.64 17.08 18.52 22.02 11.92

0.182 43.34 46.84 45.94 45.64 45.62 14.32 23.80 20.96 24.64 14.76

0.270 47.98 52.70 52.34 52.52 52.46 16.58 29.26 23.02 27.02 15.52

0.359 50.12 55.74 56.58 56.84 56.84 17.72 31.30 23.72 28.24 15.86

0.447 20% 52.32 57.40 59.00 59.44 59.48 19.46 34.50 25.04 29.34 16.42

LP BNS AJ LM

Table 5: Simulated rejection probability of Model 3-1 (1 Factor CIR SV model) with 5minfrequency

Jump/MeanVol Meaning 4LN 2LN LIN Ratio ADJ QV BIP SQRT 4LN 2LN

0.000 No Jump 19.94 9.12 10.48 8.74 8.56 4.54 5.54 65.28 57.40 78.50

0.022 LN(2N)/N 33.32 21.58 16.62 13.96 13.80 5.44 7.16 73.60 67.20 83.22

0.107 58.20 47.60 39.22 36.68 36.58 10.50 18.76 84.02 80.34 90.36

0.192 66.96 58.80 50.24 48.08 48.04 13.08 25.62 88.14 85.10 92.68

0.277 72.10 63.94 55.48 53.66 53.54 16.04 29.64 89.34 86.56 93.72

0.362 74.82 67.16 60.60 58.54 58.52 17.58 33.54 90.18 87.68 94.26

0.447 20% 77.40 71.36 63.86 62.14 62.10 17.90 34.94 91.38 89.44 95.04

Size Distortion 14.94 4.12 5.48 3.74 3.56 -0.46 0.54 60.28 5 2.40 7 3.50

Size Adjusted Power

Jump/MeanVol

0.000 No Jump 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00

0.005 LN(2N)/N 18.38 17.46 11.14 10.22 10.24 5.90 6.62 13.32 14.80 9.72

0.093 43.26 43.48 33.74 32.94 33.02 10.96 18.22 23.74 27.94 16.86

0.182 52.02 54.68 44.76 44.34 44.48 13.54 25.08 27.86 32.70 19.18

0.270 57.16 59.82 50.00 49.92 49.98 16.50 29.10 29.06 34.16 20.22

0.359 59.88 63.04 55.12 54.80 54.96 18.04 33.00 29.90 35.28 20.76

0.447 20% 62.46 67.24 58.38 58.40 58.54 18.36 34.40 31.10 37.04 21.54

LP BNS AJ LM

Table 6: Simulated rejection probability of Model 3-2 (2 Factor CIR SV model) with 5min

frequency

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Figure 1: A time path of pure diusion model under the null

Figure 2: A time path of 1 factor CIR SV model under the null

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Documents

Taesuk Lee (2011) Rate-optimal Test for Jumps in Diffusion Processes.pdf