Testing for Jumps and Estimating their Degree of Activity...
Transcript of Testing for Jumps and Estimating their Degree of Activity...
Testing for Jumps and Estimating their Degree ofActivity in High Frequency Financial Data
Yacine A��t-Sahalia Jean JacodPrinceton University Universit�e de Paris VI
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1 INTRODUCTION
1. Introduction
� Di�erent types of jumps
{ Large jumps, which are rather infrequent, are easy to pick out.
{ But visual inspection of most time series in �nance does not provide
clear evidence for either the presence or the absence of smaller,
more frequent, jumps.
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1 INTRODUCTION
� Models with and without jumps do have quite di�erent properties,both mathematical and �nancial:
{ Model calibration
{ Volatility estimation
{ Market (in)completeness
{ Option pricing and hedging
{ Risk management
{ Portfolio choice
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1 INTRODUCTION
� Detecting jumps: other approaches
{ Tests: A��t-Sahalia (2002), Carr and Wu (2003), BNS (2004), ABD
(2004), Huang and Tauchen (2006), Lee and Mykland (2005),
Jiang and Oomen (2006)
{ Separating jumps from volatility: Mancini (2001, 2004), A��t-Sahalia
(2004), A��t-Sahalia and Jacod (2005), Woerner (2006)
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1 INTRODUCTION
� This paper: we propose a very simple family of test statistics for jumpswhich converge as �n ! 0 :
{ To 1 if there are jumps
{ To 2 if there are no jumps.
� We provide a distribution theory (hence a test) for the null hypothesiswhere no jumps are present, but also one for the null where jumps are
present.
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1 INTRODUCTION
� This works as soon as the process X is an Ito semimartingale
{ This is a much weaker condition than what is usually assumed
(compound Poisson processes, or jump-di�usions)
{ The limit depends neither on the law of the process nor on the
coe�cients of the (possibly very complicated) SDE
{ So the test does not require any estimation of these coe�cients.
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2 THE SETUP
2. The Setup
� The structural assumption is that X is an Ito semimartingale on some
�ltered space (;F ; (Ft)t�0;P):
Xt = X0 +At +Mt
= X0 +ACt +A
Jt +M
Ct +M
Jt
where At is a �nite variation and predictable mean component and
Mt is a local martingale
� Each decomposable into a continuous and a pure jump part.
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2 THE SETUP
� The drift, volatility and jump measure are themselves possibly sto-chastic and can possibly jump.
� We assume that the continuous part of X is never degenerate, i.e., we
haveR t0 j�sjds > 0 a.s. for all t > 0.
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3 THE TESTING PROBLEM
3. The Testing Problem
� X is discretely observed at times i�n for all i = 0; 1; � � � ; n with
n�n = T:
� When the jump measure is �nite, there is a positive probability thatthe path X(!) has no jump on [0; T ], although the model itself may
allow for jumps: this is the peso problem.
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3.1 Various Measures of the Variability of X 3 THE TESTING PROBLEM
3.1. Various Measures of the Variability of X
� Here are processes which measure some kind of variability of X and
depend on the whole (unobserved) path of X:
A(p)t =Z t0j�sjpds; B(p)t =
Xs�t
j�Xsjp
where p > 0 and �Xs = Xs �Xs� are the jumps of X.
� The quadratic variation of the process is [X;X] = A(2) +B(2).
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3.1 Various Measures of the Variability of X 3 THE TESTING PROBLEM
� The problem boils down to deciding whether whether B(p)T > 0 for
our particular observed path with any given p:
� Let the observed discrete increments of X (not necessarily due to
jumps) be
�niX = Xi�n �X(i�1)�nand for p > 0 de�ne the estimator
bB(p;�n)t = [t=�n]Xi=1
j�niXjp
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3.1 Various Measures of the Variability of X 3 THE TESTING PROBLEM
� For r > 0; let
mr = E(jU jr) = ��1=22r=2 ��r + 1
2
�denote the rth absolute moment of a variable U � N(0; 1).
� We have the following convergences in probability, locally uniform in
t: 8>>>>>>>>><>>>>>>>>>:
p > 2; all X ) bB(p;�n)t P�! B(p)t
p = 2; all X ) bB(p;�n)t P�! A(2)t +B(2)t
p < 2; all X ) �1�p=2nmp
bB(p;�n)t P�! A(p)t
all p; X continuous ) �1�p=2nmp
bB(p;�n)t P�! A(p)t:
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3.2 The Basic Idea 3 THE TESTING PROBLEM
3.2. The Basic Idea
�
8>><>>:p > 2; all X ) bB(p;�n)t P�! B(p)t
all p; X continuous ) �1�p=2nmp
bB(p;�n)t P�! A(p)t:
� We see that when p > 2 the limit B(p)t of bB(p;�n)t does not dependon �n, and B(p)t > 0 is strictly positive if X has jumps between 0
and t.
� On the other hand when X is continuous on [0; t]; then the limit
is B(p)t = 0 but, after a normalization which does depend on �n,bB(p;�n)t converges again to a limit A(p)t not depending on �n.13
3.2 The Basic Idea 3 THE TESTING PROBLEM
� These considerations lead us to compare bB(p;�n) on two di�erent�n�scales.
� Speci�cally, for an integer k, consider:
bS(p; k;�n)t = bB(p; k�n)tbB(p;�n)t :
� Theorem: For any t > 0 the variables bS(p; k;�n)t converge in proba-bility to (
1 if X jumps
kp=2�1 if X is continuous
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4 TESTING FOR JUMPS
4. Testing for Jumps
� The previous theorem provides the �rst step towards constructing a
test for the presence or absence of jumps.
� But to construct actual tests, we need: rates of convergence andasymptotic variances.
� That are applicable under both nulls of jumps and no jumps.
� Consistent estimators of the variances.
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4.1 CLT for Standardized Statistics 4 TESTING FOR JUMPS
4.1. CLT for Standardized Statistics
Theorem:
1. Let p > 3. The variables ( bV jn )�1=2 � bS(p; k;�n)t � 1� converge stablyin law, in restriction to the set
jt to a variable which, conditionally
on F , is centered with variance 1, and which is N(0; 1) if in additionthe processes � and X have no common jumps.
2. If X is continuous, then for p � 2
( bV cn)�1=2 � bS(p; k;�n)t � kp=2�1�! N(0; 1)
stably in law, conditionally on F .16
4.2 Practical Considerations 4 TESTING FOR JUMPS
4.2. Practical Considerations
� Since we must have p > 3, a rather natural choice seems to be p = 4.
� We see that the variances are increasing with k, so it is probably wiseto take k = 2 (although when k > 2 we have to separate the two
points 1 and k, which are further apart than 1 and 2).
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5 SIMULATION RESULTS
5. Simulation Results
� We calibrate the values to be realistic for a liquid stock trading on theNYSE.
� We use an observation length of T = 1 day, consisting of 6:5 hours oftrading, that is 23; 400 seconds.
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5 SIMULATION RESULTS
Simulations: Null of No Jumps, k = 2 and 3
No Jumps: Distribution of the Statistic
-4 -2 -1 0 1 2 4
0.1
0.2
0.3
0.4
No Jumps: k = 2, Standardized
-4 -2 -1 0 1 2 4
0.1
0.2
0.3
0.4
No Jumps: k = 3, Standardized
1.9 2 2.1
2
4
6
8
No Jumps: k = 2, Non−Standardized
2.8 3 3.2 3.4
1
2
3
4
No Jumps: k = 3, Non−Standardized
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5 SIMULATION RESULTS
Simulations: Poisson Jumps
Poisson Jumps: Distribution of the Statistic
-4 -2 0 2 4
0.1
0.2
0.3
0.4
Poisson: 1 Jump per Day, Standardized
-4 -2 0 2 4
0.1
0.2
0.3
0.4
Poisson: 10 Jumps per Day, Standardized
0.95 1 1.05
5
10
15
20
25
30
35Poisson: 1 Jump per Day, Non−Standardized
0.95 1 1.05
5
10
15
20
25
30Poisson: 10 Jumps per Day, Non−Standardized
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5 SIMULATION RESULTS
Simulations: Cauchy Jumps
Cauchy Jumps: Distribution of the Statistic
-4 -2 0 2 4
0.1
0.2
0.3
0.4
Cauchy Jumps θ = 10, Standardized
-4 -2 0 2 4
0.1
0.2
0.3
0.4
Cauchy Jumps θ = 50, Standardized
0.9 0.95 1 1.05 1.1
10
20
30
40
50
Cauchy Jumps θ = 10, Non−Standardized
0.975 1 1.025
50
100
150
200
Cauchy Jumps θ = 50, Non−Standardized
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5 SIMULATION RESULTS
Simulations: Tiny or No Jumps
Tiny Jumps or No Jumps: Distribution of the Statistic
0.8 1 1.2 1.5 1.8 2 2.2
2
4
6
8
10
12
Poisson Jumps: 1 Jump per Day
Non−Standardized
0.8 1 1.2 1.5 1.8 2 2.2
1
2
3
4
5
6
Cauchy Jumps: θ = 1
Non−Standardized
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6 REAL DATA ANALYSIS
6. Real Data Analysis
� In real data, observations of the process X are blurred by market
microstructure noise, which messes things up at very high frequency.
� Assume that each observation is a�ected by an additive noise, that isinstead of Xi�n we really observe Yi�n = Xi�n + "i, and the "i are
i.i.d. with E("2i ) and E("4i ) �nite.
� We show that, in the presence of noise, the limit of our test statisticsbS(4; k;�n)t becomes as �n ! 0:
bS(4; k;�n)t P�! 1
k
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6 REAL DATA ANALYSIS
Real Data Analysis: 30 DJIA Stocks, All 2005 Trading Days
Empirical Distribution of the Test Statistic: DJIA30 All 2005 Trading Days
0.5 1 1.5 2 2.5 3
100
200
300
∆ = 15 seconds
0.5 1 1.5 2 2.5 3
50
100
150
200
250
∆ = 30 seconds
0.5 1 1.5 2 2.5 3
200
400
600
800∆ = 5 seconds
0.5 1 1.5 2 2.5 3
100
200
300
400
500∆ = 10 seconds
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7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7. Estimating the Degree of Jump Activity
� For modelling purposes one would like to infer the characteristics ofX; that is, its drift, its volatility and its L�evy jump measure, from the
observations.
{ When the time interval �n goes to 0; it is well known that one
can infer consistently the volatility, under very weak assumptions.
{ But such consistent inference is impossible for the drift or the L�evy
measure, if the overall time of observation [0; T ] is kept �xed.
{ In fact, even in the unrealistic case where the whole path of X is
observed over a �xed [0; T ], one can infer neither the drift nor the
L�evy measure.
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7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� One can however hope to be able to characterize the behavior of theL�evy measure near 0:
{ First whether it does not explode near 0, meaning that the number
of jumps is �nite;
{ Second, when the number of jumps is in�nite, we would like to be
able to say something about the concentration of small jumps.
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7.1 De�ning an Index of Jump Activity 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.1. De�ning an Index of Jump Activity
� Recall our de�nition of the process B(p)t corresponding to the semi-martingale X :
B(p)t =Xs�t
j�Xsjp
where �Xs = Xs �Xs� is the size of the jump at time s, if any.
� De�ne
It = fp � 0 : B(p)t <1g:
� Necessarily, the (random) set It is of the form [�t;1) or (�t;1) forsome �t � 2, and 2 2 It always, and t 7! �t is non-decreasing.
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7.1 De�ning an Index of Jump Activity 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� We call �T (!) the jump activity index for the path t 7! Xt(!) at time
T .
� We de�ne this index in analogy with the special case where X is a
L�evy process:
{ Then �T (!) = � does not depend on (!; T ), and it is also the
in�mum of all r � 0 such thatRfjxj�1g jxjrF (dx) < 1, where F
is the L�evy measure
{ This property shows that, for a L�evy process, the jump activity
index coincides with the Blumenthal-Getoor index of the process.
{ In the further special case where X is a stable process, then � is
also the stable index of the process.
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7.1 De�ning an Index of Jump Activity 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� When X is a L�evy process, the index � is only a partial element of the
whole L�evy measure F
� But this is the most informative knowledge one can draw about F
from the observation of the path t 7! Xt for all t � T; T �nite.
� Things are very di�erent when T ! 1, though, since observing Xover [0;1) completely speci�es F .
� However, � captures an essential qualitative feature of F , which is itslevel of activity: when � increases, the (small) jumps tend to become
more and more frequent.
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7.2 The Brownian Motion... 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.2. The Brownian Motion...
� Recall that the semimartingale X is only observed at times i�n, over
[0; T ].
� The problem is made more challenging by the presence in X of a
continuous, or Brownian, martingale part:
{ � characterizes the behavior of Fnear 0:
{ Hence it is natural to expect that the small increments of the
process are going to be the ones that are most informative about
�:
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7.2 The Brownian Motion... 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
{ But that is where the contribution from the continuous martingale
part of the process is inexorably mixed with the contribution from
the small jumps.
{ We need to see through the continuous part of the semimartingale
in order to say something about the number and concentration of
small jumps.
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7.3 Counting Increments 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.3. Counting Increments
� For �xed $ > 0 and � > 0, we consider the functionals
U($;�;�n)t =[t=�n]Xi=1
1fj�ni Xj>��$n g:
� U($;�;�n)t simply counts the number of increments whose magni-tude is greater than ��$n .
� In all cases below, we will set $ < 1=2:
� This way, we are retaining only those increments of X that are notpredominantly made of contributions from the continuous part, whichare Op(�
1=2n ):
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7.3 Counting Increments 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� A more general class of estimators can be constructed from the trun-
cated power variation functionals
Ur($;�;�n)t =[t=�n]Xi=1
j�niXjr1fj�ni Xj>��$n g:
� Here we focus on U = U0.
� While one could imagine looking at other (small) values of r; theredoes not appear to be immediate bene�ts from doing so in the present
problem.
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7.4 Behavior of the L�evy Measure 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.4. Behavior of the L�evy Measure
� Our regularity assumption is that for some � 2 (0; 2) and �0 2 [0; �=2),we have for all (!; t):
Ft = F0t + F
00t + F
000t ;
where F 0t is locally of the ��stable form
F 0t(dx) =1
jxj1+�
a(+)t 1
f0<x�z(+)t g+ a
(�)t 1
f�z(�)t �x<0g
!dx;
for some predictable non-negative processes a(+)t ; a
(�)t ; z
(+)t and z
(�)t .
� Any additional components F 00 and F 000 in the L�evy measure beyondthe most active part F 0 must have jump activity indices (which areat most �0 and �=2; respectively) that are su�ciently apart from theleading jump activity index �:
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7.4 Behavior of the L�evy Measure 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� For example, any process of the following formwill satisfy the assump-tion
dXt = btdt+ �tdWt + �t�dYt + �0t�dY0t
where:
{ � and �0 are cadlag adapted processes
{ Y is ��stable
{ Y 0 is any L�evy process with jump activity index less that �.
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7.5 Estimators of the Jump Activity Index 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.5. Estimators of the Jump Activity Index
� The key property of the functionals U($;�;�n) is
�$�n U($;�;�n)tP�!
�At
�
where �At =1�
R t0
�a(+)s + a
(�)s
�ds:
� This leads us to propose two di�erent estimators, at each stage n.
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7.5 Estimators of the Jump Activity Index 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� For the �rst one, �x 0 < � < �0 and de�ne
b�n(t;$; �; �0) = log(U($;�;�n)t=U($;�0;�n)t)
log(�0=�);
� b�n is constructed from a suitably scaled ratio of two Us evaluated
on the same time scale �n but at two levels of truncation of the
increments, � and �0.
� Based on �$�n U($;�;�n)tP�! �At
��, this will be consistent.
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7.5 Estimators of the Jump Activity Index 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� Our second estimator is
b�0n(t;$; �) = log(U($;�;�n)t=U($;�; 2�n)t)
$ log 2:
� b�0n is constructed from a suitably scaled ratio of two Us evaluated at
the same level of truncation �; but on two time scales, �n and 2�n.
� Based on �$�n U($;�;�n)tP�! �At
��, this will be consistent.
� One could look at a third estimator obtained from two Us evaluated attwo di�erent rates of truncation $ and $0; but there does not appearto be immediate bene�ts from doing so.
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7.6 Asymptotic Distribution of the Estimators 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.6. Asymptotic Distribution of the Estimators
Theorem: Under regularity assumptions, both variables
log(�0=�)�1
U($;�0;�n)t� 1U($;�;�n)t
�1=2 � b�n(t;$; �; �0)� ��
$ log 2�1
U($;�;2�n)t� 1U($;�;�n)t
�1=2 � b�0n(t;$; �)� ��
converge stably in law, in restriction to the set f �At > 0g, to a standardnormal variable N (0; 1) independent of X.
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7.6 Asymptotic Distribution of the Estimators 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
� The quali�er \in restriction to the set f �At > 0g" is essential in thisstatement.
{ On the (random) set f �At > 0g, the jump activity index is �.
{ On the complement set f �At = 0g; anything can happen: on thatset, the number � has no meaning as a jump activity index for X
on [0; T ]:
� These results are model-free, because the drift and the volatility processesare totally unspeci�ed apart from the regularity assumption on the L�evy
measures Ft.
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7.7 Simulation Results 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.7. Simulation Results
� The data generating process is dXt=X0 = �tdWt + dYt
� Y is a pure jump process, ��stable or Compound Poisson (� = 0).
� Stochastic volatility �t = v1=2t
dvt = �(� � vt)dt+ v1=2t dBt + dJt;
� Leverage e�ect: E[dWtdBt] = �dt; � < 0
� With jumps in volatility: J is a compound Poisson process with uniformjumps.
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7.7 Simulation Results 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
Simulations: � = 1:25 and � = 1
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
b β = 1.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
b β = 1
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
200
bβ = 0.75
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
200
250
β = 0.5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
bβ = 0.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
200
400
600
800
1000
b β = 0
Estimator Based on Two Truncation Levels
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7.7 Simulation Results 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
Simulations: � = 0:75 and � = 0:5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
b β = 1.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
b β = 1
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
200
bβ = 0.75
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
200
250
β = 0.5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
bβ = 0.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
200
400
600
800
1000
b β = 0
Estimator Based on Two Truncation Levels
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7.7 Simulation Results 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
Simulations: � = 0:25 and � = 0
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
b β = 1.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
b β = 1
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
200
bβ = 0.75
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
200
250
β = 0.5
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
50
100
150
bβ = 0.25
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2
200
400
600
800
1000
b β = 0
Estimator Based on Two Truncation Levels
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7.8 Empirical Results: Intel & Microsoft 2005 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
7.8. Empirical Results: Intel & Microsoft 2005
INTC�n 2 sec 5 sec 15 sec� 4 5 6 4 5 6 4 5 6
Qtr 1 1:70 1:69 1:69 1:86 1:87 1:76 1:61 1:36 1:46Qtr 2 1:06 1:06 1:05 1:23 1:13 1:09 1:09 1:13 1:14Qtr 3 1:15 1:20 1:40 1:20 1:21 1:18 1:27 1:34 1:45Qtr 4 1:32 1:51 1:59 1:54 1:35 1:42 1:77 1:72 1:42All Year 1:30 1:35 1:40 1:44 1:36 1:32 1:40 1:36 1:32
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7.8 Empirical Results: Intel & Microsoft 2005 7 ESTIMATING THE DEGREE OF JUMP ACTIVITY
MSFT�n 2 sec 5 sec 15 sec� 4 5 6 4 5 6 4 5 6
Qtr 1 1:72 1:92 1:94 1:74 1:86 1:86 1:75 1:89 2:00Qtr 2 1:59 1:60 1:43 1:60 1:48 1:56 1:47 1:17 1:27Qtr 3 1:50 1:60 1:63 1:52 1:54 1:63 1:66 1:81 1:97Qtr 4 1:64 1:79 1:72 1:82 1:66 1:65 1:71 1:37 1:24All Year 1:60 1:71 1:66 1:66 1:62 1:66 1:65 1:54 1:68
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8 CONCLUSIONS
8. Conclusions
� Jumps are prevalent in these data
� Especially if one accounts for small, in�nite activity, jumps.
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