Spectral Analysis of Quadratic Variation in the Presence
of Market Microstructure Noise
Fangfang Wang∗
April 12, 2014
Abstract
We analyze the ex-post variation of equity prices in the frequency domain. A realized
periodogram-based estimator is proposed, which consistently estimates the quadratic variation of the log equilibrium price process. For prices which are contaminated by market
microstructure noise, the proposed estimator behaves like a filter: it removes the noise by
filtering out high frequency periodograms. In other words, the proposed estimator converts
high frequency data into low frequency periodograms. We show, through a simulation study
and an application to the General Electric transaction prices, that the proposed estimator
is insensitive to the choice of sampling frequency and it is competitive with other existing
volatility measures.
Keywords: Jump diffusion, quadratic variation, periodogram, discrete Fourier transform,
spectral density, market microstructure noise
∗
Department of Information and Decision Sciences, University of Illinois at Chicago, Chicago, IL 606077124. Tel.: +1 (312) 996-2679. Email: [email protected]
1
Introduction
Suppose that X(t) is the log equilibrium price of an equity at time t and it follows a continuoustime semi-martingale. Denote by [X, X](T ) its quadratic variation (henceforth denoted by
QV ) over a fixed interval of time [0, T ]. It follows from probability theory that a sum of squared
increments over fine intervals consistently estimates [X, X](T ). To be specific, supposing that
. P
X is discretely sampled at Πn = {0 = t0 < t1 < . . . < tn = T }, RVnX = nj=1(X(tj )−X(tj−1 ))2
converges to [X, X](T ) in probability as the mesh of the partition Πn goes to zero, and RVnX is
known as realized variance in financial econometrics. See for instance (Andersen and Bollerslev
1998), (Andersen, Bollerslev, Diebold, and Labys 2001), (Meddahi 2002), (Barndorff-Nielsen
and Shephard 2002), (Aı̈t-Sahalia and Mykland 2004), (Barndorff-Nielsen, Graversen, Jacod,
and Shephard 2006), (Kanatani 2004), (Mykland and Zhang 2006), among many others.
It has been widely recognized that ultra high frequency prices are contaminated by market microstructure(MS) noise. In other words, when the mesh of Πn is small enough (less
than 1 minute), the observed prices are not true prices. Let Yj be noisy observation of
X(t) at tj for j = 0, . . . , n. The realized variance computed from the noisy prices, RVnY =
Pn
2
j=1 (Yj − Yj−1 ) , can not identify the true price variation – see for instance (Aı̈t-Sahalia,
Mykland, and Zhang 2005), (Hansen and Lunde 2006), (Bandi and Russell 2006), (BarndorffNielsen, Hansen, Lunde, and Shephard 2008), among others. Because RVnY can be decomposed
as a sum of periodograms that are attributable to different Fourier frequencies and RVnY contains both signal X and noise, one can think of extracting signal from RVnY by filtering out
periodograms due to noise.
In this paper, we propose a realized periodogram-based estimator of QV which is the weighted
average of periodograms over non-zero Fourier frequencies. This is motivated by the fact that
the estimation of QV is analogue to estimating spectral density of a stationary process at
zero frequency. The proposed estimator transfers high frequency data into low frequency
periodograms. The estimator is determined by three factors: sample size, a so-called cut-off
frequency which controls the number of periodograms to be included, and spectral window. We
will study the periodogram-based estimator with and without the MS contamination, and derive the optimal cut-off frequency which minimizes the effect of the noise on the periodogrambased estimator.
The proposed estimator generalizes the Fourier estimator of (Malliavin and Mancino 2009),
P
Pn −2π√−1ktj−1 /T
2(X)
2
(X(tj ) −
which is defined as σ̂N,n = (2N + 1)−1 N
k=−N |ck | with ck =
j=1 e
X(tj−1 )). The proposed estimator features a weighting structure and it does not include
2
zero frequency periodogram. Assigning different weight (or a spectral window) to a different
periodogram can potentially improve the finite sample performance. We will show, through
a simulation study, that the weight does matter. Periodogram at zero-frequency is excluded
so that the estimator is invariant to the mean shift in the data. In other words, one does not
need to demean raw data before computing the estimator.
The new estimator has a natural connection with the auto-covariance-based estimators of QV
in the time domain, for instance, the kernel-based estimator of (Zhou 1996) and (Hansen and
Lunde 2006), the realized kernel of (Barndorff-Nielsen, Hansen, Lunde, and Shephard 2008).
Therefore this work fills the gap between time-domain and frequency-domain analysis of QV,
and it enriches the literature of the ex-post measure of price variation. Moreover, compared
with the auto-covariance-based estimators, the proposed estimator is easy to implement and
the fast Fourier transform makes it computationally efficient.
The rest of paper is organized as follows. In section 2, we lay out the model and introduce the
periodogram-based estimator. Its properties are derived in Section 3 and Section 4. The first
focuses on the true price process, while the latter deals with the noisy observations. Section
5 discusses the choice of the cut-off frequency in a finite sample. In Section 6, we perform a
simulation experiment to evaluate the accuracy of the proposed estimator, and an empirical
exercise is presented in Section 7. Section 8 concludes the paper. All the proofs are collected
in an Appendix.
2
A realized periodogram-based estimator of QV
Suppose that the log equilibrium price X(t) is defined on some filtered probability space
(Ω, F , (Ft)t≥0 , P ) and has the dynamics
X(t) = X0 +
Z
0
t
µ(s)ds +
Z
t
σ(s)dB(s) +
Z
[0,t]×R
0
xJ˜X (ds × dx),
(1)
where B is a standard Brownian motion, µ(t) and σ 2 (t) are adapted processes satisfying
RT
RT
E[ 0 µ2 (t)dt] < ∞ and E[ 0 σ 4 (t)dt] < ∞. The volatility σ(t) is almost surely positive. The
R
jump measure JX is a Poisson random measure with Lévy measure ν satisfying |x|≤1 xν(dx) <
R
∞, |x|>1 x2 ν(dx) < ∞, and J˜X is the compensated jump measure.
The MS noise ǫj = Yj − X(tj ) satisfies the following assumption
Assumption 2.1. (1) {ǫj } is white noise with variance σǫ2 , and E(ǫ4j ) = ησǫ4 < ∞. (2) {ǫj }
3
is uncorrelated with X at all lags and leads.
Let ∆Xj = X(tj ) − X(tj−1) and ∆Yj = Yj − Yj−1 . If {∆Xj } is covariance stationary, the
spectral density of ∆Yj is given by
f∆Y (ω) = (2πn)−1 E[X, X](T ) + σǫ2 (1 − cos ω)/π,
(2)
for ω ∈ (−π, π]. At ω = 0, 2πnf∆Y (0) is the expected value of QV, E[X, X](T ). The noise
affects spectral density at non-zero frequencies. The spectral density increases as ω moves
towards π or −π. This is consistent with the empirical feature of tick-by-tick data. Figure 1
plots the power spectrum of log-returns for General Electric transaction prices on Feb 1, 2007.1
The periodogram is small around zero frequency and tends to increase with the frequency. So
the problem of estimating QV in the presence of noise would be analogue to the estimation
of f∆Y (0), i.e., using periodograms close to zero frequency.
Periodogram: [0, π]
−7
x 10
3.5
3
Power
2.5
2
1.5
1
0.5
0
0.5
1
1.5
2
2.5
3
Figure 1: Power Spectrum for GE transaction prices on Feb 1, 2007
Define the discrete Fourier transform of an arbitrary sequence {Zj }nj=1 at frequency ω ∈
(−π, π]:
n
X
Z
Jn (ω) =
e−iωj Zj ,
(3)
j=1
where i =
√
−1 is the imaginary unit. The sample periodogram of {Zj }nj=1 at ω is then
InZ (ω) = n−1 |JnZ (ω)|2 .
1
See Section 7 for details.
4
(4)
We propose a realized periodogram-based estimator of QV:
Fb ∆Y (N, n) = n
X
W (k)In∆Y (ωk ),
(5)
1≤k≤N
where ωk = 2πk/n is the Fourier frequency, and the weight W (k) satisfies:
PN
Assumption 2.2. W (k) ≥ 0 for k ∈ Z.
k=1 W (k) = 1, W (−k) = W (k) for 1 ≤ k ≤ N
PN
2
and 0 otherwise. limN →∞ k=1 W (k) = 0.
P
−1
∆Y
2
As a result Fb∆Y (N, n) =
1≤|k|≤N 2 W (k)|Jn (ωk )| . We exclude periodogram at zero
frequency, in that periodograms are invariant to location shift at the non-zero Fourier frequencies.2 Thus we don’t need to demean raw data before computing the estimator.
The proposed estimator is determined by two types of frequencies: the sampling frequency
T /n and the cut-off Fourier frequency 2πN/n (for irregularly sampled data, T /n measures the
average distance between successive observations). The first determines how often the data
should be sampled, while the latter controls the number of Fourier frequencies to be included
P
in estimation. Note that RVnY = 1≤k≤n |Jn∆Y (ωk )|2 due to the Parseval’s theorem. When n
is even and W (k) = 1/N, we have Fb∆Y (n/2, n) = RVnY . So the periodogram-based estimator
includes realized variance as a special case.
Remark 2.1. The periodogram-based estimator has the following time-domain representation
Fb ∆Y (N, n) =
n−1
X
W ∗ (h)γY (h),
(6)
h=−(n−1)
Pn−|h|
P
where γY (h) = j=1
∆Yj ∆Yj+|h| and W ∗ (h) = N
k=1 W (k) cos(ωk h) is lag window. The right
hand side of (6) is similar in spirit to the Realized Kernel of (Barndorff-Nielsen, Hansen,
Lunde, and Shephard 2008) and (Barndorff-Nielsen, Hansen, Lunde, and Shephard 2011).
In particular, the non-negative Realized Kernel of (Barndorff-Nielsen, Hansen, Lunde, and
Shephard 2011) is defined as
K(Y ) =
H
X
h=−H
2
Λ(
h
)γY (h),
H
(7)
This is common practice in estimating spectral density at zero-frequency, see for instance (Grenander and
Rosenblatt 2008), (Brockwell and Davis 2009).
5
where the kernel weight Λ(·) has the property of Λ(0) = 1, Λ(1) = 0, and Λ′ (0) = Λ′ (1) = 0.
P
Moreover, considering γY (h) = 1≤k≤n In∆Y (ωk )eihωk , we can rewrite K(Y ) as
X
K(Y ) =
Λ∗ (k)|Jn∆Y (ωk )|2 ,
(8)
1≤k≤n
P
P
h
ihωk
and nk=1 Λ∗ (k) = 1. The two estimators, Fb∆Y (N, n)
where Λ∗ (k) = n−1 H
h=−H Λ( H )e
and K(Y ), are linked with each other. Yet, they work differently: Fb∆Y (N, n) eliminates the
noise by removing high frequency periodograms — in the frequency domain, while the realized
kernel estimator eliminates the noise by removing the autocovariances at higher lags — in the
time domain.
3
Frequency-domain representation of QV
We first study the periodogram-based estimator in the absence of the MS noise, namely,
P
Fb∆X (N, n) = 1≤k≤N W (k)|Jn∆X (ωk )|2 .
Suppose that there is a continuous record on the sample path of X over [0, T ]. We define a
continuous-time counterpart of Fb∆X (N, n):
X
FbdX (N) = T 2
W (k)|F (dX)(k)|2,
1≤k≤N
RT
where F (dX)(k) = T −1 0 e−2πikt/T dX(t) is the continuous-time Fourier transform of X at
k ∈ Z. Proportion 3.1 below shows that the drift has a negligible effect on FbdX (N).
Rt
Proposition 3.1. Let X c (t) = X(t) − 0 µ(s)ds. Under Assumption 2.2, we have
lim
N →∞
X
1≤k≤N
W (k) |F (dX)(k)|2 − |F (dX c)(k)|2 = 0 in L2 .
Moreover,
Proposition 3.2. Let X c (t) = X(t) −
Assumption 2.2,
c
plimN →∞ FbdX (N) =
Rt
Z
0
c
µ(s)ds, and define FbdX (N) accordingly. Under
T
σ(t)2 dt +
0
X
t≤T
6
(X(t) − X(t−))2 ,
(9)
where ‘plim’ means convergence in probability.
Therefore Fb dX (N) is consistent for [X, X](T ).
Consider the discretely sampled process {X(tj ), j = 0, 1, . . . , n}. The next theorem derives
the consistency of Fb∆X (N, n).
Theorem 3.1. Suppose that the conditions in Proposition 3.2 hold. Let ρ(n) = maxj=1,2,...,n |tj −
T j/n| ∨ |tj−1 − T j/n|. Assume that ρ(n) = O(n−1 ) and N = o(n). Then
b∆X
plimN,n→∞ F
(N, n) =
Z
T
σ(t)2 dt +
0
X
t≤T
(X(t) − X(t−))2 .
(10)
Remark 3.1. As a corollary to Propositions 3.1 and 3.2, we have
F (dX) ⋆B F (dX) =
1
F ([dX, dX]),
T
(11)
RT
where F (d[X, X])(k) = T −1 0 e−2πikt/T d[X, X](t), and ⋆B is the Bohr convolution product.3
The convergence of convolution product in (11) is held in probability. Equality (11) extends
Theorem 2.1 of (Malliavin and Mancino 2009) to jump diffusion processes. More generally,
consider two jump diffusion processes X1 (t) and X2 (t) satisfying (1). Denote by [X1 , X2 ](t)
the quadratic covariation between X1 (t) and X2 (t). Note that [X1 , X2 ](t) = 2−1 {[X1 +X2 , X1 +
X2 ](t) − [X1 , X1 ](t) − [X2 , X2 ](t)}. We have
T −1 F (d[X1 , X2 ])
=2−1 {F (dX1 ) ⋆B F (dX2 ) + F (dX2 ) ⋆B F (dX1 )}.
If the true price process were observable, the peridogram-based estimator consistently estimates QV. For sufficiently large n, Fb∆X (N, n) provides a spectral decomposition of [X, X](T ).
Since the drift µ(t) does not influence the asymptotic behavior of FbdX (N) and periodogram
is invariant to location shift at the non-zero Fourier frequencies, we will assume µ(t) = 0 in
the rest of the paper.
3
For f and g: Z 7→ C, the Bohr convolution product between
f and g, denoted by f ⋆B g, defines a map
PN
from Z to C such that (f ⋆B g)(k) = limN →∞ (2N + 1)−1 s=−N f (s)g(k − s) provided that limN →∞ (2N +
P
1)−1 N
s=−N f (s)g(k − s) exists.
7
4
Estimate QV using noisy observations
In this section, we will study the impact of market MS noise on the periodogram-based
estimator with a focus on the finite sample properties.
Denote by In∆X,∆ǫ (ω) the cross periodogram between {∆Xj } and {∆ǫj }:
In∆X,∆ǫ (ω) = n−1 Jn∆X (ω)Jn∆ǫ (−ω).
For any ω ∈ (−π, π], In∆Y (ω) = In∆X (ω) + In∆ǫ (ω) + In∆X,∆ǫ (ω) + In∆X,∆ǫ (−ω). Consequently,
we can rewrite Fb∆Y (N, n) as the sum of three components:
Fb∆Y (N, n) = Fb ∆X (N, n) + Fb ∆ǫ (N, n) + F̂ ∆X,∆ǫ (N, n),
(12)
PN
P
∆X,∆ǫ
∆ǫ
2
∆X,∆ǫ
(ωk ) +
(N, n) =
where Fb∆ǫ (N, n) = N
k=1 W (k)(nIn
k=1 W (k)|Jn (ωk )| , and F̂
nIn∆X,∆ǫ (−ωk )). The two additional terms are due to the presence of the noise. We measure
the accuracy of Fb∆Y (N, n) as an estimator of QV by its mean squared error: E(Fb∆Y (N, n) −
[X, X](T ))2 . To understand the noise-induced bias, we derive the mean and variance of
Fb∆X (N, n), Fb∆ǫ (N, n), and Fb∆X,∆ǫ (N, n) as functions of N and n.
−1 b ∆ǫ
We consider Fb ∆ǫ (N, n) first. If Fb∆ǫ (N, n) is divided by the sample size n,
n F (N, n)
estimates the spectral density of the MS noise at 0 frequency and hence E n−1 Fb∆ǫ (N, n)
P
2 −1
−1 b ∆ǫ
= o(1) and ( k W (k) ) V ar n F (N, n) = o(1) if N → ∞ and N/n → 0 as n → ∞.4
We show below a refined result for Fb ∆ǫ (N, n).
Proposition 4.1. Suppose that N → ∞, n
assumptions 2.1, 2.2 hold. Then
P
k
W (k)2 → ∞, and N 2 /n → 0 as n → ∞, and
V ar(|Jn∆ǫ (ωs )|2 )
=σǫ4 2(η + 1) + 2(η − 1)nωs2 + O(N 4 /n2 ) ,
Cov(|Jn∆ǫ (ωs )|2 , |Jn∆ǫ (ωl )|2 )
4
=σǫ
4
2(η + 1) − 2ωs2 − 2ωl2 + O(N 4 /n3 ) ,
(13)
0 < s ≤ N,
(14)
0 ≤ s 6= l ≤ N.
See for instance (Bartlett 1977), (Priestley 1981), (Dahlhaus 1985), (Brillinger 2001), (Bloomfield 2004),
(Brockwell and Davis 2009).
8
As a result,
E F̂ ∆ǫ (N, n) =2σǫ2 [1 + 2−1 (n − 1)
V ar F̂ ∆ǫ (N, n) =σǫ4 [2(η + 1) + 4n
X
W (k)ωk2 ] + O(N 4 /n3 ),
1≤k≤N
X
W (k)2 ωk2 ] + o(N 4 /n2 ).
1≤k≤N
We turn to the cross periodogram next which characterizes the interaction between noise and
the true price. Assume that
Assumption 4.1. The volatility σ 2 (u) is stationary, and Πn is an equally spaced partition of
[0, T ].
P
Proposition 4.2. Suppose that N → ∞, n k W (k)2 → ∞, and N 2 /n → 0 as n → ∞, and
assumptions 2.1, 2.2 and 4.1 hold. We have E F̂ ∆X,∆ǫ(N, n) = 0, and
V ar F̂ ∆X,∆ǫ (N, n)
X
2
W (k)2 (1 + nωk2 /2) + o(N 4 /n2 ),
=4σǫ2 σX
k
2
where σX
= E[X, X](T ).
P
Last but not least, we look at F̂ ∆X (N, n). Because In∆X (ωs ) = n−1 |h|<n γX (h)e−iωs h where
Pn−|h|
∆X
2
2
4
4
2
γX (h) =
j=1 ∆Xj ∆Xj+|h| , we have E|Jn (ωs )| = σX . Let E∆Xj = ξn σX /n . Note
that ξn is O(1) due to the Burkholder-Davis-Gundy Inequality. Then V ar(|Jn∆X (ωs )|2 ) =
4
4
(n−1 (ξn − 3) + 1)σX
for 0 < ωs < π, and Cov(|Jn∆X (ωs )|2 , |Jn∆X (ωl )|2 ) = n−1 (ξn − 3)σX
for
ωs 6= ωl . Therefore
2
Proposition 4.3. Under assumption 4.1, E Fb∆X (N, n) = σX
and V ar Fb ∆X (N, n) = n−1 (ξn −
P
4
2 4
2
3)σX
+ N
k=1 W (k) σX , where σX = E[X, X](T ).
By virtue of Propositions 4.1, 4.2 and 4.3, we have the following theorem which presents the
main result.
Theorem 4.1. Suppose that N = o(n1/2 ), limn,N →∞ n
2.2 and 4.1 hold. We have
P
k
W (k)2 = ∞, and Assumptions 2.1,
lim E(Fb∆Y (N, n) − [X, X](T )) = 2E(ǫ2j ),
N,n→∞
9
(15)
and
lim E(Fb∆Y (N, n) − [X, X](T ))2 = 2E(ǫ4j ) + 6(E(ǫ2j ))2 .
N,n→∞
(16)
Remark 4.1. In the presence of the MS noise, the MSE does not vanish when both n and
N go to infinity. This is due to end-point effects, and it can be eliminated asymptotically by
jittering, namely, average the first m and last m observations (see for instance (Jacod, Li,
Mykland, Podolskij, and Vetter 2009) and (Barndorff-Nielsen, Hansen, Lunde, and Shephard
2011)). The revised estimator is asymptotically unbiased when m → ∞, and it is consistent for
QV.5 Yet, jittering will complicate the computation of the mean and variance of Fb∆X (N, n),
Fb∆ǫ (N, n), and Fb∆X,∆ǫ (N, n) in finite samples. So we will not pursue it in this paper.
Remark 4.2. The asymptotic bias of Fb ∆Y (N, n), 2E(ǫ2j ), does not hinge on the spectral
window. This contradicts Theorem 2 of (Mancino and Sanfelici 2008), which states that the
Fourier estimator, in the presence of the MS noise, is asymptotically unbiased. This is due to
an error in the proof of Theorem 2 of (Mancino and Sanfelici 2008): j ′ should be 2 on line
27 and line 28.
The discussion so far assumes that the MS noise is uncorrelated with the true price process
at all frequencies. This assumption is reasonable when intraday returns are sampled every 15
ticks or so (see for instance (Hansen and Lunde 2006)). We will next consider endogenous
noise, namely, noise correlated with the true price. As in (Hansen and Lunde 2006) and
(Barndorff-Nielsen, Hansen, Lunde, and Shephard 2008), we assume that the noise ǫj has the
following decomposition
Assumption 4.2. ǫj = α∆Xj + ǫ̃j where ǫ̃j is white noise with variance σǫ̃2 and E(ǫ̃4j ) =
ησǫ̃4 < ∞, and {ǫ̃j } is uncorrelated with X at all lags and leads.
The next theorem shows that the asymptotic bias (15) and the asymptotic MSE (16) are
invariant of the interaction between noise and the true price process.
Theorem 4.2. Theorem 4.1 still holds if Assumption 2.1 is replaced with Assumption 4.2.
5
The choice of cut-off frequency in a finite sample
A practical issue arises: how to choose cut-off frequency N for a given data. In this section,
we discuss optimal cut-off frequency by virtue of MSE minimization. Because N should be
5
This is currently studied by (Wang 2014).
10
less than n/2 to avoid aliasing, we define the optimal cut-off frequency as follows
Nopt = arg
min
N ≤n/2,N ∈Z
E(Fb ∆Y (N, n) − [X, X](T ))2 .
(17)
Theorem 5.1 below provides an approximation to choose the optimal cut-off frequency Nopt .
Theorem 5.1. Suppose that N = o(n1/2 ), limn,N →∞ n
2.2 and 4.1 hold. To leading order,
Nopt =arg
+
min
{n
0<N ≤n/2,N ∈Z
X
1≤k≤N
X
P
k
W (k)2 = ∞, and Assumptions 4.2,
W (k)ωk2
(18)
1≤k≤N
W (k) ρ2 + 2ρ + (1 + ρ)nωk2 },
2
2
2
where ωk = 2πk/n, ρ = (2σǫ2 )−1 σX
, and σǫ2 = E(ǫ2j ), σX
=E
RT
0
σ(t)2 dt + T
R
R
x2 ν(dx).
In the absence of the noise, i.e., σǫ2 = 0, we have Nopt = [n/2] the integer part of n/2. This is
consistent with Theorem 3.1.
As an application of Theorem 5.1, we discuss two special cases when one can find Nopt in
closed-form.
Define the rectangular window:
W (k) = 1/N for |k| = 1, . . . , N and 0 otherwise.
(19)
P
∆Y
2
It satisfies Assumption 2.2. We have F̂ ∆Y (N, n) = N −1 N
k=1 |Jn (ωk )| . This is the Fourier
estimator of (Malliavin and Mancino 2002). By virtue of Theorem 5.1, the optimal cut-off
frequency is Nopt = min([N∗ ], [n/2]), where
p
N∗ = − b + (−b3 − d + d(d + 2b3 ))1/3
p
+ (−b3 − d − d(d + 2b3 ))1/3 ,
(20)
and b = 12−1 (5 + 2ρ), d = − 3nρ(ρ+2)
. See also (Wang 2013).
16π 2
Next we consider a triangular window which is defined in the following corollary.
Corollary 5.1. Let H(k) = (N − |k|)/N 2 for |k| = 0, 1, . . . , N − 1. The proposed estimator,
11
with W (1) = H(0) and W (k) = 2H(|k| − 1) for |k| = 2, . . . , N and 0 otherwise, is given by
Fb ∆Y (N, n) =nN −1 In∆Y (ω1 )
X
+ 2n
(N − |k|)N −2 In∆Y (ωk+1).
(21)
1≤k≤N −1
The optimal cut-off frequency is then Nopt = min([N∗ ], [n/2]), where N∗ has the form of (20)
with b = 2(6+ρ)
, and d = − nρ(ρ+2)
.
15
2π 2
The following is a variant of the triangular window defined in Corollary 5.1.
Corollary 5.2. With W (k) = 2(N + 1 − |k|)(N(N + 1))−1 for |k| = 1, . . . , N and 0 otherwise,
P
we have F̂ ∆Y (N, n) = 2(N(N + 1))−1 1≤k≤N (N + 1 − |k|)|Jn∆Y (ωk )|2 . The optimal cut-off
frequency is Nopt = min([N∗ ], [n/2]), where N∗ approximately has the form of (20) with b =
19+4ρ
, and d = − nρ(ρ+2)
.
30
2π 2
Theorem 5.1 is practically important. It provides a simple approximation to compute the
optimal cut-off frequency for a given data set. It should be noted that the results hold
when the volatility σ 2 (t) is stationary and the prices are sampled at regular intervals (i.e.,
Assumption 4.1). Though the stationarity of σ 2 (t) is required, it does not rule out conditional heteroskedasticity in the volatility process. Stationary volatility processes have been
widely discussed in literature. Examples include, but are not limited to, the constant elasticity of variance (CEV) process, the non-Gaussian Ornstein-Uhlenbeck (OU) process, and
the sum or superpositions of independent CEV or OU processes. See also the eigenfunction stochastic volatility(SV) models of (Andersen, Bollerslev, and Meddahi 2011). It has
been shown that these SV models are empirically reasonable. See (Barndorff-Nielsen and
Shephard 2001), (Barndorff-Nielsen and Shephard 2002), (Chernov, Gallant, Ghysels, and
Tauchen 2003), (Meddahi and Renault 2004), (Andersen, Bollerslev, and Meddahi 2011),
(Todorov and Tauchen 2012), among others.
Theorem 5.1 applies to intraday returns that are sampled at evenly spaced intervals, i.e.,
calendar time sampling. Calendar time sampling is common for exchange rates data, for
instance, the data from Olsen & Associates. When dealing with trade and quote data, it is
more natural to sample prices in tick time (number of trades between observations). We will
consider irregularly spaced data in a simulation study, and assess numerically how sampling
frequency affects the optimal cut-off frequency given in equation (18).
12
6
Simulation study
In this section a simulation experiment is performed to evaluate the accuracy of Fb∆Y (N, n)
as an estimator of QV in finite samples, especially at N = Nopt .
Two data generating processes are considered. The first is a GARCH diffusion process
dX(t) = σ(t)dB1 (t),
√
dσ(t)2 = θ(̟ − σ(t)2 )dt + 2λθσ(t)2 dB2 (t),
(22)
where B1 (t) and B2 (t) are Brownian motions, with the leverage correlation corr(dB1 , dB2 ) =
ϕ. We use the parameter setting in (Andersen, Bollerslev, and Lange 1999) and (Huang and
Tauchen 2005): θ = 0.0350, ̟ = 0.6365, λ = 0.2962, ϕ = −0.62. The second model is a
stochastic volatility model with jumps (SVJ)
dX(t) = σ(t)dB1 (t) + J(t)dN(t),
dσ(t)2 = θ(̟ − σ(t)2 )dt + λσ(t)dB2 (t),
(23)
where N(t) is a compound Poisson process with intensity λJ . We use values reported in
(Barndorff-Nielsen and Shephard 2004): θ = 0.01, ̟ = 0.5, λ = 0.05, λJ = 2 and the jump
size is Gaussian with mean 0 and variance 0.64̟, and corr(dB1 , dB2 ) = −0.62.
The true log price process is generated via the Euler discretization: take one day as a reference
measure, and simulate 6.5 hours of trading with dt = 1/23,400, i.e., a total of 23,400 secondby-second returns per day. We simulate 500 daily replications, with X(0) = log(100) and
σ(0)2 = ̟. The noise ǫt is specified as Gaussian with mean 0 and variance σǫ2 . Two variance
levels are considered: σǫ2 = 0.000142 and 0.00142.
Denote by NoptR and NoptT the optimal cut-off frequencies for the rectangular window and
the triangular window respectively, which are calculated from (20) and Corollary 5.1. Figure 2
displays NoptR and NoptT as functions of sampling interval ranging from 1 second to 3 minutes
for σǫ2 = 0.000142, and from 20 seconds to 9 minutes for σǫ2 = 0.00142, where X(t) follows
GARCH diffusion (22). The dashed curve corresponds to the Nyquist frequency n/2. When
the sampling interval is smaller than 30 seconds for σǫ2 = 0.000142 and 3 minutes for σǫ2 =
0.00142, NoptR and NoptT are much smaller than the Nyquist frequency. This observation
shows that the effect of the MS noise is pronounced at ultra high frequencies. The plots for
SVJ model (23) are similar and hence are omitted.
13
Rectangular
Trianglar
Nyquist frequency
Rectangular
Trianglar
Nyquist frequency
550
500
9000
450
8000
400
7000
350
opt
10000
6000
N
Nopt
11000
5000
300
250
4000
200
3000
150
2000
100
1000
50
0.5
1
1.5
Minutes
2
2.5
1
2
3
(a) σǫ2 =0.000142
4
5
Minutes
6
7
8
(b) σǫ2 =0.00142
Figure 2: Nopt versus sampling interval, GARCH diffusion (22)
We then compute the periodogram-based estimators with rectangular and triangular windows.
For comparison purposes, we also use the Realized Kernel to estimate daily QV. As suggested
in (Barndorff-Nielsen, Hansen, Lunde, and Shephard 2011), we compute the Realized Kernel
with the Parzen weight

2
3

 1 − 6x + 6x
Λ(x) =
2(1 − x)3


0
if 0 ≤ x ≤ 1/2,
if 1/2 ≤ x ≤ 1,
if x > 1.
2
The preferred bandwidth is H ∗ = 3.51ξ 4/5 n3/5 with ξ 2 = √R 1 σǫ
0
σ(u)4 du
(24)
. Prior to computing the
realized kernel, we average the first two prices and the last two prices to remove end-effects as
suggested in (Barndorff-Nielsen, Hansen, Lunde, and Shephard 2011). Since the MS effect is
pronounced at ultra high frequencies, we focus on prices at frequencies of 1 second, 5 seconds,
10 seconds, 20 seconds and 30 seconds. Meanwhile, we also consider returns that are sampled
at randomly spaced time points. Noisy prices are picked up at times {t̃j } where t̃j − t̃j−1
is Exponential distributed with mean duration τ . We consider the following values of τ : 5
seconds, 10 seconds, 20 seconds and 30 seconds.
Tables 1 and 2 report mean squared errors for regularly-sampled and irregularly-sampled
prices, respectively. For the periodogram-based estimators, we report not only the mean
squared error at the optimal cut-off frequency — see the number in the parentheses, but
also the minimum mean squared error (min MSE) with the associated cut-off frequency —
14
see the number in the parentheses. Evidently, the approximation in equation (18) offers a
good representation of the optimal cutting frequency, and it is not sensitive to the sampling
frequency. Moreover, the periodogram-based estimators are competitive with the Realized
Kernel. When the sample path does not feature jumps, the triangular window is preferred over
the rectangular window. When jumps are present, the rectangular window achieves a smaller
mean squared error for σǫ2 = 0.000142. The simulation results also suggest using returns
sampled at the highest possible frequency to construct the periodogram-based estimators.
7
Real data analysis
In this section, we consider an application on the transaction prices of General Electric on Feb
1, 2007. The intraday data are extracted from the Wharton Research Data Services, TAQ
database. We focus on the prices from the New York Stock Exchange. Transaction prices
before 9:30 am and after 4:00 pm were removed. We deleted entries with zero prices. For
multiple transactions with the same time stamp, we used the median price. We also filtered
out prices that are more than one spread away from the bid and ask quotes. There are 8,052
observations left in the sample, so the prices are recorded every 3 seconds on average. The
price series are plotted in Figure 3.
36.2
36.15
36.1
36.05
36
35.95
35.9
35.85
35.8
930
1000
1030
1100
1130
1200
1230 13001330 1400 1430 1500
1530
Figure 3: GE transaction prices on Feb 1, 2007
As suggested by the simulation study, we will use all the available data to construct the
periodogram-based estimator. To compute the optimal cut-off frequency, we need to estimate
2
σX
and σǫ2 . Denote by RVsparse the subsampled RV based on 20-minute returns, which is
15
the average of 1,200 RV’s by shifting the time of the first observation in 1 second increment.
Define
q
X
2
−1
(2n(j) )−1 RV (j) ,
σ̂e = q
j=1
where RV (1) , RV (2) , . . . , RV (q) are q distinct realized variances based on every qth trade by
varying the starting point, and n(j) is the number of non-zero returns in the calculation of
2
RV (j) . We use RVsparse and σ̂e2 as estimators of σX
and σǫ2 respectively, with q = 50 (see
(Barndorff-Nielsen, Hansen, Lunde, and Shephard 2009) for details). We obtained σ̂e2 =
2
0.0065% and σ̂X
= 1.98%. The optimal cut-off frequencies are 169 for the rectangular window
and 247 for the triangular window. The periodogram-based estimators at the optimal cutoff frequency are 1.66% and 1.71% for the rectangular window and the triangular window
respectively.
Figure 4 presents volatility signature plot for the realized variance, periodogram-based estimators using rectangular window and triangular window. We consider both tick time sampling
and calendar time sampling. The horizontal line is the subsampled 20-minute RV. The signature plots for the periodogram-based estimators are almost level, especially for the triangular
window, which shows that the periodogram-based estimators are robust to market microstructure noise.
0.065
0.07
RV
Rec
Tri
0.06
RV
Rec
Tri
0.06
0.055
0.05
0.05
0.045
0.04
0.04
0.035
0.03
0.03
0.025
0.02
0.02
0.015
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Ticks
0.01
(a) Tick time sampling
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Seconds
(b) Calendar time sampling
Figure 4: Volatility Signature Plot
16
8
Conclusion
In this paper, we proposed a new frequency-domain estimator of quadratic variation. It is
based on the weighted average of periodograms at non-zero Fourier frequencies. It is nonparametric in nature. The performance of the estimator depends on both sample size n and cut-off
frequency N. We provide a simple approximation to calculate the optimal cut-off frequency for
a given data set. This work fills the gap between the time-domain and the frequency-domain
analysis of the ex-post measure of price variation. The appeal of the proposed estimator is
its flexibility and simplicity. It is easy to implement and is computationally efficient as well.
We show, through a simulation study, that the proposed estimators with rectangular window
and triangular window are competitive with the realized kernel of (Barndorff-Nielsen, Hansen,
Lunde, and Shephard 2011), and they are insensitive to the choice of the sampling frequency.
9
9.1
Technical Appendices
Proof of Proposition 3.1
Let A(t) =
Rt
0
µ(s)ds. Note that F (dX)(k) = F (dA)(k) + F (dX c )(k). We obtained
|F(dX)(k)|2
=|F(dA)(k)|2 + |F(dX c )(k)|2
+ F(dA)(k)F(dX c )(−k) + F(dA)(−k)F(dX c )(k).
It suffices to show that
lim
N →∞
X
W (k)|F (dA)(k)|2 = 0
in L2 .
(9.1)
1≤k≤N
RT
RT
P
Note that 1≤k≤N W (k)|F (dA)(k)|2 = T −2 0 b(s)µ(s)ds where b(s) = 0 DN ((s−t)/T )µ(t)dt,
P
and DN (v) = 1≤k≤N W (k)e2πikv . Because
E
≤E
Z
Z
0
T
T
b(s)µ(s)ds
2
b(s) dsE
0
Z
2
T
0
17
µ(s)2 ds
≤T
2
Z
1
Z
|DN (u)| du E
2
T
2
µ(s) ds
0
0
2
,
R1
and 0 DN (v)e−2πikv dv = W (k) for 1 ≤ k ≤ N and 0 otherwise, it follows from the Parseval’s
R1
P
theorem that 0 |DN (v)|2 dv = 1≤k≤N W (k)2 and hence (9.1) holds.
9.2
Proof of Proposition 3.2
Note that |F (dX c)(k)|2 = F (dX c)(k)F (dX c )(−k). It follows from the Ito’s lemma that
T 2 |F (dX c )(k)|2
Z T
Z
=[X, X](T ) +
Ik (h−)dI−k (h) +
0
T
I−k (h−)dIk (h),
0
Rh
RT
RT
where Ik (h) = 0 e−2πikt/T dX c (t). Let R(k) = 0 Ik (h−)dI−k (h) + 0 I−k (h−)dIk (h). Next
we will show that
X
lim
W (k)R(k) = 0 in L2 .
(9.2)
N →∞
1≤k≤N
R1
It suffices to prove that (E|AN |2 )2 ≤ C 0 |DN (v)|4 dv for some constant C > 0, where AN =
R T R h−
P
DN ((h − t)/T )dX c (t)dX c (h) and DN (v) = 1≤k≤N W (k)e2πikv .
0
0
Rt
R h−
Let Z(t) = 0 σ(s)dB(s) and G(h) = 0 DN ((h − t)/T )dX c (t). Then
AN =
Note that
Z
|
Z
T
0
T
G(h)dZ(h) +
{z
} |0
Z
T1
∞
G(h)xJ˜X (dh × dx) .
−∞
{z
}
T2
Z T
|G(h)|2 σ(h)2 dh
E|T1 |2 =E
s 0
s
Z T
Z
4
≤ E
|G(h)| dh E
E|T2 |2 =E
=E
Z
0
T
0
Z
0
T
Z
∞
−∞
(9.4)
T
σ(h)4 dh,
0
|G(h)|2 x2 dhν(dx)
|G(h)|2 dh
18
(9.3)
Z
∞
−∞
x2 ν(dx).
(9.5)
Let Gc (h) =
Then
Rh
0
DN ((h − t)/T )dZ(t), and GJ (h) =
E
Z
T
0
Z
|G(h)| dh ≤ 27 E
T
4
0
c
R h− R ∞
0
−∞
4
|G (h)| dh + E
Z
T
0
DN ((h − t)/T )xJ˜X (dt × dx).
|G (h)| dh .
J
4
It follows from the Burkholder-Davis-Gundy inequality that there exists a constant C1 > 0
such that
Z
E
≤C1 E
Z
=C1
|G (h)| dh
T
c
0
Z
0
T
Z
4
h
|DN ((h − t)/T )|4 σ(t)4 dtdh
0
0
Z T
T
σ(t)4 dt,
|DN (u/T )|4 duE
(9.6)
0
and
E|GJ (h)|4
Z h− Z
≤C1 E(
≤C1 T
Z
0
h
0
∞
−∞
DN ((h − t)/T )2 x2 JX (dt × dx))2
DN ((h − t)/T )4 dt
Z
∞
−∞
2
x2 ν(dx) .
Therefore,
Z
T
|GJ (h)|4 dh
Z
Z T
2
4
≤C1 T
DN (u/T ) du
E
(9.7)
0
0
∞
2
x ν(dx)
−∞
In view of (9.6) and (9.7), we have
2 2
(E|AN | ) ≤ C
Z
for some constant C > 0.
19
0
1
|DN (v)|4 dv
2
.
9.3
Proof of Theorem 3.1
It suffices to prove that
for some C > 0.
X
|Jn (ωk )|2 − T 2 |F (dX)(k)|2 ≤ Ckρ(n),
2
It follows from Lemma 3.1 and Proposition 3.2 that E|F (dX)(k)|4 < ∞.
Pn −iωj
1(tj−1 ,tj ] (t). Then
j=1 e
JnX (ω)
=
Z
|0
+
Z
T
Z
|
T0
0
Z
Let H(t) =
T
H(t)σ(t)dB(t)
H(t)µ(t)dt +
{z
} |0
{z
}
T
(9.8)
(9.9)
T1
∞
−∞
H(t)xJ˜X (dt × dx) .
{z
}
T2
R
2
T
Since |H(t)| ≤ 1, there exists a constant C1 > 0 such that E|T0 |4 ≤ C1 E 0 µ(s)2 ds ,
RT
R∞
E|T1 |4 ≤ C1 E 0 σ(t)4 dt , E|T2 |4 ≤ C1 ( −∞ x2 ν(dx))2 due to the Burkholder-Davis-Gundy
inequality. Hence E|JnX (ω)|4 is bounded uniformly in n.
P
Let M(t) = nj=1 e−2πijk/n − e−2πikt/T 1(tj−1 ,tj ] (t). We have
X
|J (ωk )|2 − T 2 |F (dX)(k)|2
n
2
X
≤ kJn (ωk )k4 + kT F (dX)(k)k4 kTk4 ,
RT
where T = 0 M(t)dX(t). Decompose T as T = T0 + T1 + T2 , where Tk is defined in (9.9)
with H(·) replaced by M(·). Note that |M(t)| ≤ 2πkρ(n)/T . For some constant C2 > 0,
Z
E|T0 | ≤C2 k ρ(n) E
4
4
4
Z
0
T
T
2
µ(s) ds
2
σ(t)2 dt)2 ,
E|T1 | ≤C2 k ρ(n) (E
0
Z ∞
4
4
4
x2 ν(dx))2 ,
E|T2 | ≤C2 k ρ(n) (
4
4
4
,
(9.10)
−∞
where (9.10) is due to equation (35) of (Malliavin and Mancino 2009). Therefore there exists a
constant C3 > 0 which is independent of n such that kTk4 ≤ C3 kρ(n). The proof is complete.
20
9.4
Proof of Equality (11)
By definition,
(F(dX) ⋆B F(dX)) (k)
N
X
1
F(dX)(s)F(dX)(k − s).
N →∞ 2N + 1
= lim
s=−N
Rh
RT
Let Ik (h) = 0 e−2πikt/T dX(t). Because T 2 F (dX)(s1)F (dX)(s2) = 0 e−2πi(s1 +s2 )t/T d[X, X](t)
RT
RT
+ 0 Is1 (h−)dIs2 (h) + 0 Is2 (h−)dIs1 (h) due to Ito’s Lemma, we have
N
X
1
F(dX)(s)F(dX)(k − s)
2N + 1
s=−N
1
= F(d[X, X])(k)
T
Z T
N Z T
X
1
I
(h−)dI
(h)
.
I
(h−)dI
(h)
+
+
s
s
k−s
k−s
(2N + 1)T 2
0
0
s=−N
We need to show that 2N1+1
k ∈ Z. It suffices to prove
PN
s=−N
RT
0
Is (h−)dIk−s (h) +
RT
0
Ik−s (h−)dIs (h) = op (1) for any
Z T
N
X
1
Is (h−)dIk−s (h) = op (1).
2N + 1
0
(9.11)
s=−N
Note that
N Z T
X
1
Is (h−)dIk−s (h)
2N + 1
0
s=−N
Z T Z h−
DN ((h − t)/T )dX(t)dX(h),
=
0
where DN (v) = e−2πikh/T 2N1+1
sition 3.2, we have
0
PN
s=−N
e2πisv . Similar to the proofs of Lemma 3.1 and Propo-
Z T Z h−
2 !2
E DN ((h − t)/T )dX(t)dX(h)
0
0
Z 1
|DN (v)|4 dv,
≤C
0
21
R1
for some constant C > 0. Since
and this completes the proof.
9.5
0
|DN (v)|4 dv ≤
R1
0
|DN (v)|2 dv = (2N + 1)−1 , (9.11) is true
Proof of Proposition 4.1
Equalities (13) and (14) follow from Lemma 9.1. Therefore
V ar F̂ ∆ǫ (N, n)
X
=
W (k)2 V ar|Jn∆ǫ (ωk )|2
k
+
X
W (k)W (k′ )Cov(|Jn∆ǫ (ωk )|2 , |Jn∆ǫ (ωk′ )|2 )
k6=k ′
=σǫ4 [2(η − 1) + 4(η − 3)(n − 1)(1 −
X
W (k) cos ωk )2
k
X
X
+ 4(
W (k) cos ωk )2 + 4n
W (k)2 (1 − cos ωk )(n − (n − 2) cos ωk )]
k
=σǫ4 [2(η
+ 1) + 4n
X
k
W (k)2 ωk2 ]
+ o(N 4 /n2 ).
k
Note that In∆ǫ (ω) = n−1
P
|h|<n
γǫ (h)e−iωh where γǫ (h) =
E F̂ ∆ǫ (N, n)
=2σǫ2 [n − (n − 1)
=2σǫ2 [1 + (n − 1)
X
Pn−|h|
j=1
∆ǫj ∆ǫj+|h| . We have
W (k) cos ωk ]
1≤k≤N
X
W (k)ωk2 /2] + O(N 4 /n3 ).
1≤k≤N
The proof of Proposition 4.1 needs the following lemma.
Lemma 9.1. Consider an MA(1) process Pj = (1 − θB)ej where B is the backshift operator
and et is white noise with variance σe2 and Ee4t = ηe σe4 < ∞. Then for 0 < ωs = 2πs/n < π,
V ar(|JnP (ωs )|2 )
=σe4 (ηe − 3)(1 + θ 4 + (n − 1)(1 + θ 2 − 2θ cos ωs )2 )
+ 4θ 2 σe4 + σe4 ((1 + θ 2 )n − 2θ(n − 1) cos ωs )2 ,
22
and for 0 ≤ ωs 6= ωl ≤ π,
Cov(|JnP (ωs )|2 , |JnP (ωl )|2 )
=σe4 (ηe − 3)(n − 1)(1 + θ2 − 2θ cos ωs )(1 + θ2 − 2θ cos ωl )
+ σe4 (ηe − 3)[1 + θ4 ] + 4θ2 σe4 (1 + cos ωs cos ωl ) .
Proof of Lemma 9.1: Let ω = ωj , λ = ωk . According to the proof of Theorem 10.3.2 of
(Brockwell and Davis 2009),
Cov(I P (ω), I P (λ))
n
X
=n−2 (ηe − 3)σe4
+
n
n
X
−1
s,u=1
+
n
n
X
−1
s,v=1
∗
ψs,t,u,v
e−iω(t−s) e−iλ(v−u)
s,t,u,v=1
γ(u − s)e
γ(v − s)e
−iωs−iλu
−iωs+iλv
!
!
n
−1
n
X
t,v=1
n
−1
n
X
t,u=1
γ(v − t)e
γ(u − t)e
iωt+iλv
iωt−iλu
!
!
∗
where ψs,t,u,v
= ψt−s ψu−s ψv−s + ψ1 ψt−s+1 ψu−s+1 ψv−s+1 , and ψ0 = 1, ψ1 = −θ and ψk = 0 for
k 6= 0, 1; and γ(h) = (1 + θ2 )σe2 if h = 0, −θσe2 if h = ±1 and 0 for the others.
Note that for 0 ≤ ω, λ ≤ π,
n
X
ψt−s ψu−s ψv−s e−iω(t−s) e−iλ(v−u)
s,t,u,v=1
=
n−s
n−s
n−s
n
X
X
X
X
ψu eiλu )
ψv e−iλv )(
ψt e−iωt )(
(
u=1−s
v=1−s
s=1 t=1−s
−iω
=1 + (n − 1)(1 − θe
−iλ 2
)|1 − θe
| ,
and
n
X
ψt−s+1 ψu−s+1 ψv−s+1 e−iω(t−s) e−iλ(v−u)
s,t,u,v=1
= − θ 3 + (n − 1)eiω (1 − θe−iω )|1 − θe−iλ |2 .
Moreover,
n
X
s,u=1
γ(u − s)e−iωs−iλu
23
(9.12)

2 −iω + e−iλ )

 θσe (e
=
(n + nθ 2 − 2(n − 1)θ)σe2


(n + nθ 2 + 2(n − 1)θ)σe2
and
n
X
s,v=1
=
(
for 0 < ω, λ < π,
for ω = λ = 0,
for ω = λ = π,
γ(v − s)e−iωs+iλv
(9.13)
θσe2 (e−iω + eiλ )
σe2 ((1 + θ 2 )n − 2θ(n − 1) cos λ)
for 0 ≤ ω =
6 λ ≤ π,
for 0 ≤ ω = λ ≤ π.
As a result, for 0 ≤ ω 6= λ ≤ π,
Cov(nI P (ω), nI P (λ))
=σe4 (ξn − 3)(1 + θ 4 + (n − 1)(1 + θ 2 − 2θ cos ω)(1 + θ 2 − 2θ cos λ))
+ θ 2 σe4 [4 + 4 cos ω cos λ] ,
and for 0 < ω < π
V ar(nI P (ω))
=σe4 (ξn − 3)(1 + θ 4 + (n − 1)(1 + θ 2 − 2θ cos ω)2 )
+ 4θ 2 σe4 + σe4 ((1 + θ 2 )n − 2θ(n − 1) cos ω)2 .
9.6
Proof of Proposition 4.2
By the definition of F̂ ∆X,∆ǫ (N, n), E F̂ ∆X,∆ǫ (N, n) = 0.
P
∆X
∆ǫ
Let F̂1X,ǫ (N, n) = N
k=1 W (k)Jn (ωk )Jn (−ωk ). We have
2
V ar F̂1X,ǫ (N, n) = E F̂1X,ǫ (N, n)
X
=
W (k)2 E|Jn∆X (ωk )|2 E|Jn∆ǫ (ωk )|2
k
+
X
k6=k′
W (k)W (k ′ )E Jn∆X (ωk )Jn∆X (−ωk′ ) E Jn∆ǫ (−ωk )Jn∆ǫ (ωk′ ) .
Note that E(Jn∆ǫ (ωs )Jn∆ǫ (ωk )) =
Pn
−iωs j−iωk l
γ(j
j,l=1 e
24
− l). In view of (9.12) and (9.13), we
have
E(Jn∆ǫ (ωs )Jn∆ǫ (ωk ))
=σǫ2 (e−iωs + e−iωk ),
for 0 ≤ ωs , ωk ≤ π,
E(Jn∆ǫ (ωs )Jn∆ǫ (−ωk ))
(
σǫ2 (e−iωs + eiωk )
=
2σǫ2 (n − (n − 1) cos(ωs ))
for 0 ≤ ωs =
6 ωk ≤ π,
for 0 ≤ ωs = ωk ≤ π.
P
Note also that E(Jn∆X (ωs )Jn∆X (ωk )) = nj=1 e−iωs+k j E∆Xj2 = 0 for ωs + ωk 6= 0 (mod 2π),
2
. As a result,
otherwise, EJn∆X (ωs )Jn∆X (ωk ) = σX
X
2
W (k)2 (n − (n − 1) cos(ωk )).
V ar F̂1X,ǫ (N, n) = 2σǫ2 σX
k
Let F̂2ǫ,X (N, n) =
∆ǫ
∆X
W
(k)
J
(ω
)J
(−ω
)
. We have for N < n/2,
k
k
n
n
k=1
PN
Cov F̂1X,ǫ (N, n), F̂2ǫ,X (N, n)


n
X
X
=
e−iωk+k′ j E∆Xj2  σǫ2 (eiωk + eiωk′ )
W (k)W (k′ ) 
j=1
k,k ′
=0.
2
It follows that V ar(F̂ ∆X,∆ǫ (N, n)) = 4σǫ2 σX
nωk2 /2) + o(N 4 /n2 ).
9.7
P
2
2 2
k W (k) (n−(n−1) cos ωk ) = 4σǫ σX
P
k
W (k)2 (1+
Proof of Theorem 4.1
Let V = [X, X](T ). Because Cov F̂ ∆X,∆ǫ (N, n), Fb∆ǫ (N, n) = 0 and Cov F̂ ∆X,∆ǫ (N, n), Fb∆X (N, n)
= 0, we have
E(Fb∆Y (N, n) − V )2
2
= E Fb∆ǫ (N, n) + V ar Fb ∆X (N, n)
+ V ar Fb∆ǫ (N, n) + V ar F̂ ∆X,∆ǫ (N, n).
25
Note that
2
X
E Fb∆ǫ (N, n) =4σǫ4 + 4σǫ4 n
W (k)ωk2 + O(N 4 /n2 ),
k
b ∆X
V ar F
(N, n) =n
−1
4
(ξn − 3)σX
+
V ar F̂
(N, n)
2
=4σǫ2 σX
X
4
W (k)2 σX
,
k
X
V ar Fb ∆ǫ (N, n) =σǫ4 [2(η + 1) + 4n
∆X,∆ǫ
X
W (k)2 ωk2 ] + o(N 4 /n2 ),
k
W (k) (1 + nωk2 /2) + o(N 4 /n2 ).
2
k
As a result,
E(Fb∆Y (N, n) − V )2
4
=(2η + 6)σǫ4 + n−1 (ξn − 3)σX
+ 4σǫ4 n
+
X
2
W (k)
k
4
σX
+
4σǫ4 nωk2
+
X
k
2
4σǫ2 σX
(1
(9.14)
W (k)ωk2
+ nωk2 /2) + O(N 4 /n2 ).
Therefore with N 2 = o(n), E(Fb∆Y (N, n) − V )2 = (2η + 6)σǫ4 + o(1).
9.8
Proof of Theorem 4.2
Let Zj = (1 − (1 + α)−1 αB)∆Xj where B is the backshift operator. We have
Fb∆Y (N, n) =(1 + α)2 FbZ (N, n) + Fb∆ǫ̃ (N, n)
(9.15)
+ (1 + α)F̂ Z,∆ǫ̃(N, n),
P
where F̂ Z,∆ǫ̃(N, n) = 1≤k≤N W (k)(JnZ (ωk )Jn∆ǫ̃ (−ωk )+JnZ (−ωk )Jn∆ǫ̃ (ωk ). Because Proposition
4.1 applies also to ǫ̃, we obtain the following
E F̂ ∆ǫ̃ (N, n) =2σǫ̃2 [1 + (n − 1)
V ar F̂
∆ǫ̃
(N, n)
=σǫ̃4 [2(η
X
+ 1) + 4n
k
X
k
26
W (k)ωk2 /2] + O(N 4 /n3 ),
W (k)2 ωk2 ] + o(N 4 /n2 ).
Moreover,
2
E F̂ Z (N, n) =σX
(1 − θ)2 + O(N 2 /n2 ),
X
4
W (k)2 (1 − θ)4 + 4θ(1 − θ)2 n−1
V ar F̂ Z (N, n) =σX
(9.16)
(9.17)
k
4
+ n−1 (ξn − 3)σX
(1 − θ)4 + o(N 2 /n2 ),
and
E F̂ Z,∆ǫ̃ (N, n) =0,
2
V ar(F̂ Z,∆ǫ̃ (N, n)) =4(1 − θ)2 σǫ̃2 σX
+
X
2 2 −1
σǫ̃ n
16θσX
W (k)2 (1 + nωk2 /2)
(9.18)
k
+ o(N 2 /n2 ),
where θ = (1 + α)−1 α. (They will be verified later.)
2
Because the variance of ǫj is σǫ2 = α2 σX
/n + σǫ̃2 , we have
2
E Fb∆Y (N, n) − σX
2
=(1 + α)2 E FbZ (N, n) + E Fb∆ǫ̃ (N, n) − σX
=2E(ǫ2j ) + o(N 2 /n).
Z,∆ǫ̃
∆ǫ̃
Z,∆ǫ̃
Z
b
b
Note that Cov F̂
(N, n), F (N, n) = 0, Cov F̂
(N, n), F (N, n) = 0 and Eǫ4j =
2 −1
4 −2
ησǫ̃4 + 6α2 σǫ̃2 σX
n + α4 ξn σX
n . Consequently,
E(Fb∆Y (N, n) − [X, X](T ))2
2
= E Fb∆ǫ̃ (N, n) + V ar Fb ∆ǫ̃ (N, n) + (1 + α)4 V ar FbZ (N, n)
+ (1 + α)2 V ar F̂ Z,∆ǫ̃ (N, n)
−1
4
2
=2Eǫ4j + 6σǫ4 + (ξn − 3)σX
+ 8α(2 − α)σǫ2 σX
n
X
X
4
2
+ 4σǫ4 n
W (k)ωk2 + (σX
+ 4σǫ2 σX
)
W (k)2
k
+
(4σǫ4
+
2
2σǫ2 σX
)n
X
k
W (k)2 ωk2
+ O(N 4 /n2 ) + o(N 2 /n
k
=2Eǫ4j
+
6σǫ4
+ o(1).
27
X
(9.19)
W (k)2 )
Proof of (9.16): Note that InZ (ω) = n−1
and cos ω = 1 − ω 2 /2 + O(ω 4). We have
P
−iωh
where γZ (h) =
|h|<n γZ (h)e
E F̂ Z (N, n)
2
=σ∆X
[n(1 + θ 2 ) − 2θ(n − 1)
2
=σX
(1
2
2
X
W (k) cos ωk ]
1≤k≤N
2
− θ) + O(N /n ).
4
Proof of (9.17): It follows from Lemma 9.1 that, with E∆Xj4 = ξn σX
/n2 ,
V ar(nInZ (ωs ))
4
=n−2 σX
(ξn − 3)(1 + θ 4 + (n − 1)(1 + θ 2 − 2θ cos ωs )2 )
4
4
+ n−2 σX
((1 + θ 2 )n − 2θ(n − 1) cos ωs )2 ,
+ 4θ 2 n−2 σX
for 0 < ωs = 2πs/n < π, and
Cov(nInZ (ωs ), nInZ (ωl ))
4
=n−2 σX
(ξn − 3)(n − 1)(1 + θ 2 − 2θ cos ωs )(1 + θ 2 − 2θ cos ωl )
4
4
+ n−2 σX
(ξn − 3)(1 + θ 4 ) + 4θ 2 n−2 σX
(1 + cos ωs cos ωl ) ,
for 0 ≤ ωs 6= ωl ≤ π. Note also that ξn = O(1). We have
V ar F̂ Z (N, n)
X
=
W (s)2 V ar|JnZ (ωs )|2
s
+
X
s6=l
W (s)W (l)Cov(|JnZ (ωs )|2 , |JnZ (ωl )|2 )
4
4
=n−2 σX
(ξn − 3)(1 + θ 4 ) + 4θ 2 n−2 σX
X
2
4
4
W (s)2
+ n−2 σX
(ξn − 3)(n − 1) 1 + θ 2 + (1 + θ 2 )2 σX
4
− 4θ(1 + θ 2 )n−2 σX
(ξn − 3)(n − 1)
4
+ 4θ 2 n−2 σX
[(ξn − 3)(n − 1) + 1]
4
+ 4θ 2 σX
(1 − 2n−1 )
X
"
"
s
X
s
X
W (s)2 (cos ωs )2
s
28
W (s)(cos ωs )
#2
W (s)(cos ωs )
s
#
Pn−|h|
j=1
Zj Zj+|h|,
4
− 4(1 + θ 2 )θ(1 − n−1 )σX
4
=σX
X
+n
s
−1
2
W (s)
4
X
W (s)2 cos ωs
s
(1 − θ) + 4θ(1 − θ)2 n−1
4
(ξn − 3)σX
(1 − θ)4 + o(N 2 /n2 ).
Proof of (9.18): Let F̂1Z,∆ǫ̃ (N, n) =
PN
Z
∆ǫ̃
k=1 W (k)Jn (ωk )Jn (−ωk ).
We have
V ar F̂1Z,∆ǫ̃ (N, n)
X
=
W (k)W (k′ )E JnZ (ωk )JnZ (−ωk′ ) E Jn∆ǫ̃ (−ωk )Jn∆ǫ̃ (ωk′ )
k6=k ′
+
X
W (k)2 E|JnZ (ωk )|2 E|Jn∆ǫ̃ (ωk )|2 .
k
In view of (9.12) and (9.13), we obtain
E(Jn∆ǫ̃ (ωs )Jn∆ǫ̃ (ωk ))
=σǫ̃2 (e−iωs + e−iωk ),
for 0 < ωs , ωk < π,
E(Jn∆ǫ̃ (ωs )Jn∆ǫ̃ (−ωk ))
(
σǫ̃2 (e−iωs + eiωk )
=
2σǫ̃2 (n − (n − 1) cos ωs )
for 0 < ωs 6= ωk < π,
for 0 < ωs = ωk < π,
and
E(JnZ (ωs )JnZ (ωk ))
2
=θσ∆X
(e−iωs + e−iωk ),
for 0 < ωs , ωk < π,
E(JnZ (ωs )JnZ (−ωk ))
(
2 (e−iωs + eiωk )
θσ∆X
=
2 ((1 + θ 2 )n − 2θ(n − 1) cos ω )
σ∆X
s
for 0 < ωs =
6 ωk < π,
for 0 < ωs = ωk < π,
2
2
where σ∆X
= n−1 σX
. Consequently,
V ar F̂1Z,∆ǫ̃ (N, n)
X
2
= 2σǫ̃2 σ∆X
W (k)2 (n − (n − 1) cos ωk )((1 + θ 2 )n − 2θ(n − 1) cos ωk )
|
k
{z
T1
29
}
2
+ θσ∆X
σǫ̃2
X
W (k)W (s)(e−iωs + eiωk )(eiωs + e−iωk ) .
k6=s
|
{z
}
T2
Because (n − (n − 1) cos ωk )((1 + θ2 )n − 2θ(n − 1) cos ωk ) = (1 − θ)2 n + 2θ + (1 − θ)2 n2 ωk2 /2 +
O(nωk2), we have
2
T1 =2σǫ̃2 σX
X
k
W (k)2 ((1 − θ)2 + 2θn−1 + (1 − θ)2 nωk2 /2)
2
+ O(N /n2
X
W (k)2 ).
Note also that (e−iωs + eiωk )(eiωs + e−iωk ) = 2(1 + cos ωk cos ωs − sin ωk sin ωs ). With some
algebra, we have
i
h
X
2 2 −1
1−
W (k)2 + O(N 2 /n3 ).
σǫ̃ n
T2 =4θσX
It follows that
V ar F̂1Z,∆ǫ̃(N, n)
X
2
W (k)2 (1 + nωk2 /2)
=2(1 − θ)2 σǫ̃2 σX
k
+
Let F̂2∆ǫ̃,Z (N, n) =
PN
k=1 W (k)
2 2 −1
4θσX
σǫ̃ n
+ O(N 2 /n2
X
W (k)2 ).
∆ǫ̃
Jn (ωk )JnZ (−ωk ) . Then
V ar F̂2∆ǫ̃,Z (N, n) = V ar F̂1Z,∆ǫ̃(N, n) ,
and
Cov F̂1Z,∆ǫ̃ (N, n), F̂2∆ǫ̃,Z (N, n)
X
2 2
=n−1 θσX
σǫ̃
W (k)W (k′ )(e−iωk + e−iωk′ )(eiωk + eiωk′ ).
k,k ′
Because
P
k,k ′
Consequently,
W (k)W (k ′)(e−iωk + e−iωk′ )(eiωk + eiωk′ ) = 4 + O(N 2 /n2 ), we have
∆ǫ̃,Z
Z,∆ǫ̃
2 2
Cov F̂1 (N, n), F̂2 (N, n) = 4n−1 θσX
σǫ̃ + O(N 2 /n3 ).
V ar F̂1Z,∆ǫ̃ (N, n) + F̂2∆ǫ̃,Z (N, n)
30
2 2 −1
2
=16θσX
σǫ̃ n + 4(1 − θ)2 σǫ̃2 σX
9.9
X
W (k)2 (1 + nωk2 /2) + o(N 2 /n2 ).
k
Proof of Theorem 5.1
It follows from (9.19) that for a fixed n, the optimal cut-off frequency should be such that
minimizes to leading order
4σǫ4 n
X
4
2
+ 4σǫ2 σX
)
W (k)ωk2 + (σX
k
+
(4σǫ4
+
2
)n
2σǫ2 σX
X
W (k)2
k
X
W (k)2 ωk2
k
=4σǫ4
n
X
W (k)ωk2 +
k
X
k
W (k)2 ρ2 + 2ρ + (1 + ρ)nωk2
!
(9.20)
2
where ρ = (2σǫ2 )−1 σX
.
9.10
Proof of Corollary 5.1
Note that
P
1≤k≤N
P
h
−1) P
(N +1)2 (N +2)
1
2
+ 2(N −1)(2N
,
1≤k≤N W (k)k =
6N
N2
3N 3
2(N +1)(N 3 +4N 2 +6N +4)
3
− N 2 . Equation (9.20) becomes
15N 3
2
1≤k≤N W (k) =
W (k)2 k2 =
−
1
N
i
, and
2π 2 2 8π 2 (6 + ρ)
4(ρ2 + 2ρ) 1
N +
N+
.
3n
15n
3
N
2
2
(6+ρ)
N 2 + 8π 15n
N + 4(ρ 3+2ρ) N1 is minimized at N∗ = −b + (−b3 −
For N ∈ (0, ∞), N 7→ 2π
3n
p
p
, and d = − nρ(ρ+2)
, and
d + d(d + 2b3 ))1/3 + (−b3 − d − d(d + 2b3 ))1/3 where b = 2(6+ρ)
15
2π 2
2
2
ρ = σX /(2σǫ ). The proof is complete.
2
9.11
Proof of Corollary 5.2
Because
to
P
k
W (k)k 2 =
(N +1)(N +2)
6
and
P
k
W (k)2 k 2 =
2
15
N +3+
4
N
2(ρ2 + 2ρ) 1
1
4π 2 (N + 1)(N + 2)
+
+
3
N
N +1
n
6
31
−
1
N +1
, (9.20) is equal
8π 2 (1 + ρ)
1
4
+
−
N +3+
15n
N
N +1
2
2
4(ρ2 + 2ρ) 1
2π 2 2π (19 + 4ρ)
N +
N+
.
≈
3n
15n
3
N
2
2
N 2 + 2π (19+4ρ)
N + 4(ρ 3+2ρ) N1 is minimized at N∗ = −b + (−b3 −
For N ∈ (0, ∞), N 7→ 2π
3n
15n
p
p
, and d = − nρ(ρ+2)
. The proof is
d + d(d + 2b3 ))1/3 + (−b3 − d − d(d + 2b3 ))1/3 b = 19+4ρ
30
2π 2
complete.
2
References
Aı̈t-Sahalia, Y., and P. Mykland (2004): “Estimators of diffusions with randomly spaced
discrete observations: A general theory,” The Annals of Statistics, 32(5), 2186–2222.
Aı̈t-Sahalia, Y., P. Mykland, and L. Zhang (2005): “How often to sample a continuoustime process in the presence of market microstructure noise,” Review of Financial Studies,
18(2), 351–416.
Andersen, T., and T. Bollerslev (1998): “Answering the skeptics: Yes, standard volatility models do provide accurate forecasts,” International Economic Review, pp. 885–905.
Andersen, T., T. Bollerslev, F. Diebold, and P. Labys (2001): “The distribution
of exchange rate volatility,” Journal of the American statistical Association, 96, 42–55.
Andersen, T., T. Bollerslev, and S. Lange (1999): “Forecasting financial market
volatility: Sample frequency vis-à-vis forecast horizon,” Journal of Empirical Finance, 6(5),
457–477.
Andersen, T. G., T. Bollerslev, and N. Meddahi (2011): “Realized volatility forecasting and market microstructure noise,” Journal of Econometrics, 160(1), 220–234.
Bandi, F., and J. Russell (2006): “Separating microstructure noise from volatility,” Journal of Financial Economics, 79(3), 655–692.
Barndorff-Nielsen, O., S. Graversen, J. Jacod, and N. Shephard (2006): “Limit
theorems for bipower variation in financial econometrics,” Econometric Theory, 22(4), 677.
32
Barndorff-Nielsen, O., P. Hansen, A. Lunde, and N. Shephard (2008): “Designing
realized kernels to measure the ex post variation of equity prices in the presence of noise,”
Econometrica, 76(6), 1481–1536.
(2009): “Realized kernels in practice: Trades and quotes,” The Econometrics Journal, 12(3), C1–C32.
Barndorff-Nielsen, O., and N. Shephard (2002): “Econometric analysis of realized
volatility and its use in estimating stochastic volatility models,” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(2), 253–280.
(2004): “Power and bipower variation with stochastic volatility and jumps,” Journal
of Financial Econometrics, 2(1), 1.
Barndorff-Nielsen, O. E., P. R. Hansen, A. Lunde, and N. Shephard (2011):
“Multivariate realised kernels: consistent positive semi-definite estimators of the covariation
of equity prices with noise and non-synchronous trading,” Journal of Econometrics, 162(2),
149–169.
Barndorff-Nielsen, O. E., and N. Shephard (2001): “Non-Gaussian Ornstein–
Uhlenbeck-based models and some of their uses in financial economics,” Journal of the
Royal Statistical Society: Series B (Statistical Methodology), 63(2), 167–241.
Bartlett, M. (1977): An introduction to stochastic processes: with special reference to
methods and applications. Cambridge University Press.
Bloomfield, P. (2004): Fourier analysis of time series: an introduction. Wiley-Interscience.
Brillinger, D. (2001): Time series: data analysis and theory, vol. 36. Society for Industrial
Mathematics.
Brockwell, P., and R. Davis (2009): Time series: theory and methods. Springer Verlag.
Chernov, M., A. R. Gallant, E. Ghysels, and G. Tauchen (2003): “Alternative
models for stock price dynamics,” Journal of Econometrics, 116(1), 225–257.
Dahlhaus, R. (1985): “Asymptotic normality of spectral estimates,” Journal of Multivariate
Analysis, 16(3), 412–431.
Grenander, U., and M. Rosenblatt (2008): Statistical analysis of stationary time series.
Chelsea Pub Co.
33
Hansen, P. R., and A. Lunde (2006): “Realized variance and market microstructure
noise,” Journal of Business & Economic Statistics, 24(2), 127–161.
Huang, X., and G. Tauchen (2005): “The relative contribution of jumps to total price
variance,” Journal of Financial Econometrics, 3(4), 456–499.
Jacod, J., Y. Li, P. A. Mykland, M. Podolskij, and M. Vetter (2009): “Microstructure noise in the continuous case: the pre-averaging approach,” Stochastic Processes and
their Applications, 119(7), 2249–2276.
Kanatani, T. (2004): “Integrated volatility measuring from unevenly sampled observations,”
Economics Bulletin, 3(36), 1–8.
Malliavin, P., and M. Mancino (2002): “Fourier series method for measurement of multivariate volatilities,” Finance and Stochastics, 6(1), 49–61.
(2009): “A Fourier transform method for nonparametric estimation of multivariate
volatility,” The Annals of Statistics, 37(4), 1983–2010.
Mancino, M., and S. Sanfelici (2008): “Robustness of Fourier estimator of integrated
volatility in the presence of microstructure noise,” Computational Statistics & Data Analysis, 52(6), 2966–2989.
Meddahi, N. (2002): “A theoretical comparison between integrated and realized volatility,”
Journal of Applied Econometrics, 17(5), 479–508.
Meddahi, N., and E. Renault (2004): “Temporal aggregation of volatility models,” Journal of Econometrics, 119(2), 355–379.
Mykland, P., and L. Zhang (2006): “ANOVA for diffusions and Ito processes,” The
Annals of Statistics, 34(4), 1931–1963.
Priestley, M. (1981): Spectral analysis and time series. Academic press.
Todorov, V., and G. Tauchen (2012): “The realized Laplace transform of volatility,”
Econometrica, 80(3), 1105–1127.
Wang, F. (2013): “Optimal Design of Fourier Estimator in the Presence of Microstructure
Noise,” Computational Statistics & Data Analysis, forthcoming.
34
(2014): “An Unbiased Measure of Integrated Volatility in the Frequency Domain,”
Working paper, UIC.
Zhou, B. (1996): “High-frequency data and volatility in foreign-exchange rates,” Journal of
Business & Economic Statistics, pp. 45–52.
35
n
1 sec
5 sec
10 sec
20 sec
30 sec
1 sec
5 sec
10 sec
20 sec
30 sec
1 sec
5 sec
10 sec
20 sec
30 sec
1 sec
5 sec
10 sec
20 sec
30 sec
Triangular window
min MSE
MSE at Nopt
Rectangular window
min MSE
MSE at Nopt
σǫ2 = 0.000142, GARCH diffusion
4.25 (1528)
4.82 (2017)
4.45 (1169)
7.69 (860)
8.70 (1091)
7.90 (577)
10.19 (671)
10.95 (828)
10.53 (439)
12.76 (585)
12.76 (585)
13.49 (386)
15.99 (390)
15.99 (390)
14.39 (389)
Realized Kernel
MSE
(22)
4.83 (1343)
8.77 (699)
11.26 (520)
13.58 (384)
15.99 (320)
9.01
18.96
25.07
32.94
38.71
23400
4680
2340
1170
780
10.17
21.80
28.48
38.38
43.71
(22)
12.97 (321)
27.93 (176)
34.33 (134)
44.88 (102)
52.49 (86)
25.00
47.10
62.52
84.21
97.12
23400
4680
2340
1170
780
σǫ2 = 0.000142, SVJ model (23)
11.59 (1960) 14.46 (2799) 11.63 (1105) 13.96 (1843)
17.01 (978) 22.61 (1498) 17.25 (707)
20.34 (945)
21.56 (846) 24.23 (1130) 23.71 (675)
24.46 (699)
26.83 (585)
26.83 (585)
27.29 (577)
29.82 (513)
35.67 (390)
35.67 (390)
31.40 (373)
32.38 (390)
23.71
46.00
60.93
80.08
96.87
23400
4680
2340
1170
780
23400
4680
2340
1170
780
22.65
43.19
57.68
81.86
92.41
σǫ2
(645)
(343)
(260)
(196)
(171)
(778)
(384)
(302)
(249)
(227)
= 0.00142, GARCH diffusion
12.01 (463)
10.38 (408)
24.03 (260)
22.76 (231)
31.17 (201)
30.97 (186)
41.14 (155)
40.73 (142)
47.63 (133)
47.93 (106)
σǫ2 = 0.00142,
23.45 (675)
43.20 (376)
57.91 (290)
83.89 (222)
98.12 (190)
SVJ model (23)
23.14 (543)
25.28 (463)
46.36 (322)
52.21 (251)
62.35 (199)
65.96 (191)
89.48 (163)
98.31 (143)
99.76 (160) 113.88 (121)
Table 1: Mean squared error, evenly sampled data
Note: The mean squared errors have been multiplied by 104 .
36
66.75
163.29
229.08
302.53
352.59
τ
5 sec
10 sec
20 sec
30 sec
5 sec
10 sec
20 sec
30 sec
5 sec
10 sec
20 sec
30 sec
5 sec
10 sec
20 sec
30 sec
Triangular window
min MSE
MSE at Nopt
Rectangular window
min MSE
MSE at Nopt
Realized Kernel
MSE
9.40 (834)
13.11 (666)
18.04 (614)
22.87 (402)
σǫ2 = 0.000142,
10.40 (1095)
14.04 (828)
18.04 (614)
22.87 (402)
GARCH diffusion (22)
9.39 (583)
10.46 (701)
13.33 (452)
13.69 (520)
18.79 (389)
19.01 (392)
21.65 (379)
22.65 (325)
20.92
28.24
38.78
47.37
23.97
34.16
44.01
54.47
(335)
(255)
(198)
(176)
σǫ2 = 0.00142,
25.73 (260)
36.34 (200)
47.67 (155)
59.35 (131)
GARCH diffusion (22)
23.76 (235)
26.72 (176)
36.89 (163)
40.23 (133)
45.76 (152)
52.02 (102)
57.91 (115)
66.49 (85)
50.34
63.27
89.83
113.29
(883)
(858)
(610)
(389)
σǫ2 = 0.000142, SVJ (23)
27.13 (1511)
21.23 (637)
30.45 (1140)
30.29 (424)
43.23 (610)
41.48 (514)
70.64 (389)
66.06 (384)
27.07
31.03
42.76
67.42
(954)
(706)
(523)
(389)
45.84
71.77
104.94
113.59
46.12 (386)
67.69 (323)
85.97 (250)
110.59 (184)
σǫ2 = 0.00142, SVJ (23)
46.32 (373)
56.08 (302)
68.72 (294)
74.85 (192)
88.37 (219)
94.44 (168)
110.63 (187)
115.59 (108)
59.59 (249)
75.12 (194)
107.37 (141)
121.02 (120)
257.97
312.55
409.86
555.99
20.93
28.03
43.23
70.64
Table 2: Mean squared error, unevenly sampled data
Note: The mean squared errors have been multiplied by 104 .
37