07350015%2E2014%2E906350
Journal of Business & Economic Statistics
ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
On the Estimation of Integrated Volatility With
Jumps and Microstructure Noise
Bing-Yi Jing, Zhi Liu & Xin-Bing Kong
To cite this article: Bing-Yi Jing, Zhi Liu & Xin-Bing Kong (2014) On the Estimation of Integrated
Volatility With Jumps and Microstructure Noise, Journal of Business & Economic Statistics, 32:3,
457-467, DOI: 10.1080/07350015.2014.906350
To link to this article: http://dx.doi.org/10.1080/07350015.2014.906350
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On the Estimation of Integrated Volatility With
Jumps and Microstructure Noise
Bing-Yi JING
Hong Kong University of Science and Technology, Hong Kong ([email protected])
Zhi LIU
University of Macau, Macau, China ([email protected])
Xin-Bing KONG
Soochow University, Suzhou, China ([email protected])
In this article, we propose a nonparametric procedure to estimate the integrated volatility of an Itˆo semimartingale in the presence of jumps and microstructure noise. The estimator is based on a combination
of the preaveraging method and threshold technique, which serves to remove microstructure noise and
jumps, respectively. The estimator is shown to work for both finite and infinite activity jumps. Furthermore,
asymptotic properties of the proposed estimator, such as consistency and a central limit theorem, are established. Simulations results are given to evaluate the performance of the proposed method in comparison
with other alternative methods.
KEY WORDS: Central limit theorem; High frequency data; Quadratic variation; Semimartingale.
1.
INTRODUCTION
With the availability of high-frequency data, there has been a
rapidly growing interest in the estimation of integrated volatility. For continuous Itˆo process, a commonly used estimator is
the realized volatility (also called realized quadratic variation in
some literature), see, Andersen et al. (2003). The estimation of
integrated volatility becomes tricky when the underlying price
process contains jumps. Two well-behaved estimators are the
multiple-power estimator and the realized threshold quadratic
variation, respectively. The former was proposed by BarndorffNielsen and Shephard (2006) and Barndorff-Nielsen and Shephard (2004), while the latter was proposed by Mancini (2009)
and further developed in Jacod (2008). An interesting comparison of the two different approaches was given in Veraart
(2011).
However, it is widely accepted that the observed prices are
contaminated by microstructure noise, for example, the bid-ask
spreads. Thus, the discretely observed process Yti is
Yti = Xti + ǫti ,
i = 0, 1, . . . , ⌊t/n ⌋,
Xtc = X0 +
Xtd
=
t
0
t
0
|x|≤1
bs ds +
t
σs dWs ,
0
x(μ − ν)(ds, dx) +
t
0
xμ(ds, dx),
|x|>1
(3)
where b and σ are locally bounded optional processes, μ is the
jump measure, with ν its predictable compensator. All these are
defined on a stochastic basis (, F, Ft , P ). An example in the
literature using the process with jumps is by Wu (2008), where,
he uses the L´evy process to model financial security returns.
Under this setting, the quadratic variation of X becomes
[X, X]t = [Xc , Xc ]t + [Xd , Xd ]t
t
(Xs )2 ,
σs2 ds +
=
0
(1)
where Xt represents the latent price process, and ǫti is the microstructure noise at time ti . For simplicity, assume that the observation times in [0, t] are equally spaced, namely, ti =: in
(0 ≤ i ≤ ⌊t/n ⌋) with n → 0 as n → ∞. Our interest lies in
the inference on certain characteristics of the latent process X,
using the contaminated observations Yti ’s.
In this article, we assume that the latent price process X is an
Itˆo semimartingale of the form
Xt = Xtc + Xtd ,
where Xc and Xd are the continuous and discontinuous (or
jump) components, respectively,
0≤s≤t
where Xs = Xs − Xs− is the jump size of X at time s. In this
article, our main interest lies in the estimation of the integrated
t
volatility of the continuous part, that is, [Xc , Xc ]t = 0 σs2 ds,
in the presence of jumps and microstructure noise. Incidentally,
estimation of [Xd , Xd ]t can be done similarly.
(2)
457
© 2014 American Statistical Association
Journal of Business & Economic Statistics
July 2014, Vol. 32, No. 3
DOI: 10.1080/07350015.2014.906350
Color versions of one or more of the figures in the article can be
found online at www.tandfonline.com/r/jbes.
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458
Journal of Business & Economic Statistics, July 2014
For a generic semimartingale X, define (see A¨ıt-Sahalia and
Jacod 2009)
r
|s X| < ∞ ,
(4)
I = r ≥ 0;
0≤s≤t
where s X = Xs − Xs− is the jump size at time s. This is an
interval of the form [β, ∞] or (β, ∞]. We say that Xd is of
finite (or infinite) activity in (0, t], that is Xd has a.s. finitely
(or infinitely) many jumps if 0 ∈ I (or 0 ∈ I ); Xd is of finite
variation in (0, t] if 1 ∈ I .
When the latent price process Xt contains no jumps, that is,
Xd ≡ 0 in (2), the influence of the microstructure noise on the
estimation of the integrated volatility for high-frequency data
has been well documented in the literature. For instance, Zhang
Mykland, and A¨ıt-Sahalia (2005) and Bandi and Russell (2006)
found that microstructure noise, if left untreated, can result in
inconsistent estimators of the integrated volatility. There are
several main approaches to overcome this difficulty, including
(I) subsampling method (Zhang, Mykland, and A¨ıt-Sahalia
2005),
(II) the realized kernel method (Barndorff-Nielsen et al. 2008),
(III) the preaveraging method (Jacod et al. 2009 and Podolskij
and Vetter 2009b),
(IV) the quasi-maximum likelihood method (QMLE) (Xiu
2010).
In this article, we are interested in estimating integrated
volatilities when the latent process Xt contains jumps, that is,
Xd ≡ 0 in (2). Here, given the observations Yti , one needs to
take care of the noise ǫti as well as the jumps Xtdi . The estimation of integrated volatility was considered by Fan and Wang
(2007), and Podolskij and Vetter (2009b) using two fundamentally different techniques, which differ in the order of treating
jumps and noise. Rosenbaum (2009) also considered the integrated volatility estimation under round-off error using wavelet
approach.
To be precise, Fan and Wang (2007) first applied wavelet
methods to try to get rid of the jumps, and then dealt with
the microstructure noise by the multiscale technique of Zhang
(2006), while Podolskij and Vetter (2009a) first applied “preaveraging” to reduce the impact of the microstructure noise,
and then handled the jumps by the multiple-power estimator.
Both papers also showed that their estimators are consistent
for the estimation of the integrated volatility when the latent
price process Xt contains finite activity jumps, which were also
supported by their simulation results.
A natural question is: how will the two methods by Fan and
Wang (2007) and Podolskij and Vetter (2009a) perform if we
increase the jump intensities of the jumps? To answer this, we
conducted some simple simulation studies. In the first simulation, the observations Yti are generated from model (1), where
Xt and ǫt are taken to be
Xt = Wt +
Nt
Ji ,
i=1
iid
ǫi ∼ N (0, 0.012 ),
where Ji ∼ N (0, 0.52 ) and Nt ∼Poisson(λt) and Jt and
Nt are independent. We choose the jump intensity λ =
1, 2, . . . , 10, 20, 50, and n = 23, 400. Now Xt still contains finite activity jumps, but the jumps occur more frequent as λ increases. We then applied the multipower method and the wavelet
method to this model, and the results are presented in Table 1
and Figure 1 in Section 4 (one can ignore column 1 now, which
contains the results for a different method proposed later in the
article). Note that [Xc , Xc ]t=1 = 1. Clearly,
• Podolskij and Vetter’s method works very well when jumps
are rare (e.g., λ = 1), and gradually deteriorates as jumps
become more frequent;
• Fan and Wang’s method is extremely accurate regardless
of the arrival rate λ. This is remarkable and perhaps it may
not come to a complete surprise to us since the wavelet
step can effectively remove all the big jumps in the above
jump-diffusion model, which might explain why it should
perform so well in this case.
The remarkable performance of Fan and Wang’s method in
the jump-diffusion model with noise begs the question: how will
it work for some other jump models with noise? So, in our second
Table 1. Comparisons for Model 1 (jump-diffusion model with noise). Here n = 23, 400
λ
Preaveraging-threshold
(PT)
(bias, s.e., MSE)
Podolskij and Vetter’s
bi-power (PV2)
(bias, s.e., MSE)
Podolskij and Vetter’s
triple-power (PV3)
(bias, s.e., MSE)
Fan and Wang
(FW)
(bias, s.e., MSE)
0
1
2
3
4
5
6
7
8
9
10
20
50
(−0.007, 0.081, 0.007)
(−0.007, 0.081, 0.007)
(−0.003, 0.081, 0.007)
(−0.001, 0.081, 0.007)
(0.004, 0.079, 0.006)
(0.010, 0.081, 0.007)
(0.013, 0.081, 0.007)
(0.019, 0.082, 0.007)
(0.023, 0.082, 0.007)
(0.027, 0.081, 0.007)
(0.027, 0.081, 0.007)
(0.040, 0.082, 0.010)
(0.069, 0.084, 0.012)
(−0.013, 0.091, 0.008)
(0.052, 0.131, 0.020)
(0.109, 0.164, 0.039)
(0.164, 0.186, 0.062)
(0.234, 0.223, 0.104)
(0.299, 0.249, 0.151)
(0.361, 0.275, 0.206)
(0.446, 0.311, 0.295)
(0.502, 0.337, 0.369)
(0.573, 0.370, 0.465)
(0.624, 0.392, 0.510)
(1.438, 0.678, 2.521)
(4.579, 1.521, 23.28)
(−0.020, 0.095, 0.009)
(0.024, 0.114, 0.014)
(0.062, 0.128, 0.020)
(0.098, 0.140, 0.029)
(0.142, 0.153, 0.044)
(0.186, 0.168, 0.063)
(0.226, 0.185, 0.086)
(0.287, 0.205, 0.122)
(0.316, 0.215, 0.146)
(0.363, 0.234, 0.187)
(0.386, 0.252, 0.212)
(0.925, 0.409, 1.023)
(3.033, 0.978, 10.16)
(−0.007, 0.081, 0.007)
(0.008, 0.084, 0.007)
(0.009, 0.083, 0.007)
(0.005, 0.081, 0.007)
(0.006, 0.083, 0.007)
(0.010, 0.083, 0.007)
(0.014, 0.082, 0.007)
(0.020, 0.084, 0.007)
(0.019, 0.084, 0.007)
(0.034, 0.084, 0.008)
(0.034, 0.083, 0.009)
(0.055, 0.082, 0.010)
(0.113, 0.089, 0.021)
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
459
MSE
Preaveraging−Threshold
Bipower
Triplepower
Wavelet
0.4
0.3
0.2
0.1
1
2
3
4
5
6
Jump Intensity (λ)
7
8
9
10
Figure 1. MSE plots for Model 1 (jump-diffusion model with noise, the curve of preaveraging-threshold estimator hides behind the wavelet
estimator curve).
simulation, we generated the observations Yti from model (1),
where Xt and ǫt are taken to be
Xt = Wt + Jt ,
iid
ǫi ∼ N (0, 0.012 ),
where Jt follows a symmetric β-stable process, where β ranges
from 0.3 to 1.5 (all have infinite activity jumps). As β gets
larger, jumps occur more frequently. Again, [Xc , Xc ]t=1 = 1.
The results containing bias, standard error (s.e.) and mean square
error (MSE) are given in the following table.
Performance of Fan and Wang’s method for different β’s
β
bias
s.e.
MSE
0.3
0.083
0.125
0.027
0.5
0.145
0.148
0.043
0.8
0.274
0.181
0.108
1.0
0.406
0.207
0.208
1.5
0.924
0.259
0.918
Clearly, Fan and Wang’s method works well for small β’s
since the MSEs are under 21% for β ≤ 1. However, as β > 1, the
performance starts to get bad; for example, the MSE is 91.8% for
β = 1.5%. The reason is: the bigger β is, the more small jumps
there are, the more difficult for the wavelet method to separate
the small jumps from the microstructure noise. (Incidentally, the
results for Podolskij and Vetter’s method are even worse in this
case and hence not included here.)
The less than ideal numerical performances of Fan and
Wang’s and Podolskij and Vetter’s methods for the case β > 1
prompt us to search for some alternative methods. Recall that
the observations are
Yti = Xtci + Xtdi + ǫti .
(5)
with 0 < t1 < t2 < · · · < tn ≤ t. Our procedure goes in two
steps:
• Step 1 (Preaveraging). We use preaveraging to reduce the
effect of the microstructure noise ǫti .
• Step 2 (Threshold). We truncate the smoothed data at some
appropriate threshold to remove the jump contribution from
the decomposition of [X, X]t .
We will refer to the method as the preaveraging-threshold (PT)
method, and the resulting estimator as PT estimator.
Simulation studies given later in the article indicate that the PT
method turns out to perform better than the two previously mentioned methods, regardless of finite or infinite activity jumps. In
fact various recent papers have shown that essentially all statistical methods based on increments of an Itˆo semimartingale in
the no-noise case can be translated to the noisy framework by
replacing raw increments with preaveraged statistics, which can
be interpreted as some kind of generalized increments. Given
that intuition, it is perhaps not too surprising that a truncated
version of the estimator from Jacod et al. (2009) works in the
jump case as well, having the same asymptotic variance as their
estimator, since the same holds true for realized volatility (RV)
and truncated RV as shown in Jacod (2008). The present article is further proof that the preaveraged data can be effectively treated as data with no microstructure noise. Recently,
Bajgrowicz, Scaillet, and Treccani (2013) show that, using the
existing jump detection methods, the jumps usually are erroneously identified due to the multiple testing issue and the jump
effect on the market friction has been overstated. They proposed a thresholding testing statistics, however, the test is based
on the microstructure-noise-free data. Christensen, Oomen, and
Podolski (2014) consider the microstructure noise robust jump
test using the Bipower estimator. Since main objective of Bajgrowicz, Scaillet, and Treccani (2013) and Christensen, Oomen,
and Podolski (2014) is the estimation of integrated volatility, the
results of current article provide a possibility to improve the efficiency of their approaches.
The rest of the article is organized as follows. In Section 2,
we introduce the proposed PT estimator. Main results are given
in Section 3. Section 4 is devoted to simulations. We conclude
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460
Journal of Business & Economic Statistics, July 2014
the article in Section 5. The technical proofs are postponed to
the Appendix.
2.
METHODOLOGY
As we described before, the process X is observed with an
error (microstructure noise in practice), namely, at the time
point ti , we observe Xti + ǫi rather than Xti , where the ǫi
are “errors” which are conditionally on the process X, independent. It is convenient to define a “canonical” process ǫt in
R [0,∞) for all t, although we only use its value at some discrete
points. Mathematically, we first have a transition probability
Qt (ω, dx) from (, F) to R. Based on it, we construct a new
probability space (R [0,∞) , B, σ (ǫs : s ∈ [0, t)), Q), where B is
the product Borel σ -field and Q = ⊗t≥0 Qt . Then, the observation is measurable with respect to the filtered probability space
(1)
((1) , F(1) , F(1)
t , P ), which is given by
⎧ (1)
(1)
⎪
⎨ = × R[0, ∞), F = F ⊗ B,
(1)
F(1)
t = Ft ⊗ σ (ǫs : s ∈ [0, t)), P (dω, dx)
⎪
⎩
= P (dω)Q(ω, dx).
Hence, any variable or process in either or R [0,∞) can be
considered in usual way as a variable or a process on . For
more detailed discussion about the new probability space, see,
Jacod, Podolskij, and Vetter (2010).
2.1.
Preaveraging
We briefly describe the idea of the preaveraging. For details,
see Jacod et al. (2009). Define the jth increment by
nj Y := Ytj − Ytj −1 ,
for j = 1, 2, . . . , n.
Together they form a sequence (n1 Y, . . . , nn Y ). Choose an
integer kn such that 1 ≤ kn ≤ ⌊t/n ⌋ and then we formulate
⌊t/n ⌋ − kn + 1 overlapping blocks, the ith being
Bi = ni Y, . . . , ni+kn −1 Y , for 1 ≤ i ≤ ⌊t/n ⌋ − kn + 1.
Within this ith block, we take a weighted average of the increments:
ni,kn Y (g) =
k
n −1
j =1
where the weight function g is chosen such that
• it is continuous, piecewise C 1 with a piecewise Lipschitz
derivative g ′ ,
1
• g(s) = 0 when s ∈ (0, 1), and 0 g 2 (s)ds > 0.
One common choice satisfying the above conditions is g(x) =
x ∧ (1 − x).
Now let us investigate what effect the preaveraging has. In
view of (5), denoting gjn = g(j/kn ), we have
=
k
n −1
:=
Ai1
gjn ni+j Xc
j =1
+
Ai2
+
+
Ai3 ,
(We shall treat the jump component Ai2 in the next subsection.)
Clearly, by choosing kn → ∞ appropriately, we can control the
effect of microstructure noise Ai3 relative to Ai1 . In particular,
1/2
−(1/2+ǫ)
(i) if kn n → ∞, for example, kn = ⌊c n
⌋ for some
ǫ > 0, then the effect of the microstructure noise can be
ignored;
1/2
−1/2
(ii) if kn n = c > 0, for example, kn = ⌊c n ⌋, then Ai1
i
and A3 will be of comparable size.
In either case, the influence of microstructure noise has been
eliminated or substantially reduced.
−1/2
In the following discussion, we will assume kn = ⌊c n ⌋
unless otherwise stated (the only exception is Theorem 2, where
−(1/2+η)
⌋).
we take kn = ⌊c n
2.2.
Threshold Quadratic Variation
The above preaveraging procedure reduces the influence of
the microstructure noise, its effect on the jumps is still very
limited. The remaining task is to get rid of the effect of the
jumps.
After preaveraging, the smoothed increments from the diffu1/4
sion and microstructure noise, Ai1 and Ai3 , are both of size n .
i
However, the smoothed increment from the jump, A2 , may still
1/4
be larger than n . Following the idea of Mancini (2009) or
Jacod et al. (2009), we can propose the following threshold
estimator of [Xc , Xc ]t :
U (Y, g)nt =
⌊t/n ⌋−kn
i=1
n
2
i,kn Y (g) 1{|ni,kn Y (g)|≤un } .
k
n −1
j =1
gjn ni+j Xd
+
k
n −1
gjn ni+j ǫ
j =1
(6)
(7)
Where, un satisfying
1
2
un −̟
→ 0, un −̟
→ ∞, for some 0 ≤ ̟1 < ̟2 <
n
n
g(j/kn )ni+j Y,
for 1 ≤ i ≤ ⌊t/n ⌋ − kn + 1,
ni,kn Y (g)
Some simple variance calculations show that
Ai1 = Op ( kn n ), Ai3 = Op (1/ kn ).
1
.
4
(8)
Such choices of un enable those (smoothed) increments larger
than un to be gradually excluded as n → ∞, and essentially only
those increments due to continuous part and noise are included,
hence we can calculate the integrated volatility after removing
the effect of noise.
3.
MAIN RESULTS
In this section, we study the asymptotic behavior of U (Y, g)nt ,
such as consistency and central limit theorem. Before stating our
main theorems, we need some assumption on the microstructure
noise.
Assumption 1. ν(ω, dt, dx) = dtFt (dx) with Ft (x) =
Ft′ (dx) + Ft′′ (dx), where Ft′ (dx) and Ft′′ (dx) are two mutually singular measures and satisfy, for three constants 0 ≤ β2 ≤
β1 ≤ β < 2, that
Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
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β−β1
f)
• Ft′ (dx) = β(1+|x|
(αt+ 1{0 1/(8 − 2β), we have
P
ˆ n := c3 (U1 (Y, g)nt − c4
ˆ n κˆ 2 − c5 (κˆ 2 )2 ) −
→
Q
t
σs4 ds,
0
where
c3 =
1
,
2
3c g¯ 2 (2)
¯ g¯′ (2),
c4 = 2g(2)
c5 =
2
g¯′ (2)
t.
c2
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462
Journal of Business & Economic Statistics, July 2014
Hence, we easily obtain the asymptotic behavior of the studentized statistics.
ˆ n +212 ˆ κˆ 2
Ŵtn = g¯ 24(2) (c22 Q
c
Corollary 1. Let
+
under the same assumptions as Theorem 3, we have
t
−1/4 ˆ
n − 0 σs2 ds S
n
n
−
→ N (0, 1),
Ŵt
• Model 1—A jump-diffusion process with noise:
2 2
11 (κˆc3) ),
Xt = Wt +
Nt
Ji ,
i=1
where Ji ∼ N (0, 0.52 ), Nt ∼Poisson(λt) and Ji and Nt are
independent. We choose λ = 0, 1, 2, . . . , 10, 20, 50.
• Model 2—A diffusion with infinite activity jumps and
noise:
where N (0, 1) is a standard normal r.v. independent of F.
Xt = Wt + Jt ,
3.2.
We compare the variance of our estimator with that of the
bi-power estimator proposed in Podolskij and Vetter (2009b) in
the special case when X is continuous. From Theorem 3, the
ˆ 1 is
asymptotic variance of
t
c
E|μ2 (σs , κ/c)|ds.
g¯ 2 (2) 0
On the other hand, by Theorem 3 of Podolskij and Vetter
(2009b), the variance of the bi-power type estimator is
t
c
1 + 2m21 − 3m41 2
E|μ2 (σs , κ/c)|ds
g¯ (2) 0
t
c
≈ 1.06 2
E|μ2 (σs , κ/c)|ds,
g¯ (2) 0
4.
SIMULATION STUDY
We now evaluate the performances of our proposed estimator
with other methods by Monte Carlo simulations.
4.1 Simulation Design
and
iid
ǫi ∼ N (0, 0.012 ),
where Jt is a trimmed symmetric β-stable process with β =
0.5. The algorithm described in Cont and Tankov (2004)
is used to simulate a β−stable process. By “trimmed” it
is meant that 2% of the largest (absolute) values will be
discarded. The reason for trimming will be explained in
the next subsection. Another common technique is to use
“tempered” stable process, which is similar in spirit to
“trimming.”
• Model 3—The stochastic volatility (SV) model with jumps
and noise:
Asymptotic Relative Efficiency Comparison
where mp = E|N (0, 1)|p . Thus, our estimator is asymptotically
about 6% more efficient than the bi-power estimator in the case
of no jumps. In the presence of jumps, our simulations show that
finite sample efficiency can be much greater than the asymptotic
one.
iid
ǫi ∼ N (0, 0.012 ),
dXt = μdt + σt dWt + δdJt ,
and
σt = exp(β0 + β1 τt ),
where dτt = ατt dt + dBt with Corr(dWt , dBt ) = ρ
and Jt follows a tempered symmetric β-stable process as in the last model. We let μ = 0.03, β0 =
0.3125, β1 = 0.125, α = −0.025, ρ = −0.3, δ = 0.1,
and ǫi ∼iid N (0, 0.0012 ). All the coefficients were selected
from Podolskij and Vetter (2009a) to allow a comparison.
It is easy to see that [Xc , Xc ]t = t for Models 1 and 2 while, it
becomes a random variable for Model 3.
We consider four sample sizes: n = 1170, 4680, 7800, and
23,400 (which correspond to sampling every 20 sec, 5 sec, 3
sec, and 1 sec in a trading day). The experiments are repeated
5000 times for each case. The bias and standard error (s.e.) and
MSE are computed for each sample size.
We will compare the performances of our PT estimator with
those by Podolskij and Vetter (2009a,b) and Fan and Wang
(2007). Here are some specifications.
We generate observations Yti from the following model
Yti = Xti + ǫti ,
where ǫi ’s are independent of Xti ’s. The latent process Xt is
generated from the following commonly used models.
• For Fan and Wang’s method, when applying the wavelet
to locate the jumps, we use Daubechies s8 wavelet to calculate the empirical wavelet
√ coefficient (Wang 1995) and
the universal threshold d 2 log n/n, where d is the median
absolute deviation of empirical wavelet coefficient, divided
Table 2. Comparisons for Model 2 (diffusion model with infinite activity jumps and noise)
n
1170
4680
7800
23400
Preaveraging-threshold
(PT)
(bias, s.e., MSE)
Podolskij and Vetter’s
bi-power (PV2)
(bias, s.e., MSE)
Podolskij and Vetter’s
triple-power (PV3)
(bias, s.e., MSE)
Fan and Wang
(FW)
(bias, s.e., MSE)
(0.058, 0.178, 0.035)
(0.034, 0.126, 0.017)
(0.024, 0.109, 0.013)
(0.013, 0.082, 0.007)
(0.274, 0.546, 0.373)
(0.236, 0.394, 0.210)
(0.223, 0.398, 0.194)
(0.173, 0.283, 0.110)
(0.136, 0.357, 0.146)
(0.127, 0.245, 0.076)
(0.118, 0.215, 0.060)
(0.090, 0.148, 0.030)
(0.141, 0.183, 0.053)
(0.116, 0.152, 0.037)
(0.108, 0.138, 0.030)
(0.079, 0.103, 0.017)
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
463
Table 3. Comparisons for Model 3 (SV model with infinite activity jumps and noise)
n
Preaveraging-threshold
(PT)
(bias, s.e., MSE)
Podolskij and Vetter’s
bi-power (PV2)
(bias, s.e., MSE)
Podolskij and Vetter’s
triple-power (PV3)
(bias, s.e., MSE)
Fan and Wang
(FW)
(bias, s.e., MSE)
(0.070, 0.181, 0.038)
(0.034, 0.126, 0.017)
(0.027, 0.111, 0.013)
(0.016, 0.081, 0.007)
(0.311, 0.541, 0.390)
(0.247, 0.397, 0.219)
(0.224, 0.341, 0.166)
(0.187, 0.292, 0.120)
(0.164, 0.364, 0.159)
(0.133, 0.240, 0.076)
(0.121, 0.214, 0.060)
(0.100, 0.161, 0.036)
(0.167, 0.198, 0.067)
(0.141, 0.176, 0.051)
(0.112, 0.152, 0.036)
(0.087, 0.121, 0.022)
1170
4680
7800
23400
MSE
Preaveraging−Threshold
Bipower
Triplepower
Wavelet
0.3
0.2
0.1
20
5
3
1
Sampling time intervel (Seconds)
Figure 2. MSE plots for Model 2 (L´evy model with noise).
MSE
Preaveraging−Threshold
Bipower
Triplepower
Wavelet
0.3
0.2
0.1
20
5
3
Sampling time intervel (Seconds)
Figure 3. MSE plots for Model 3 (SV model with noise).
1
Journal of Business & Economic Statistics, July 2014
QQ Plot of Sample Data versus Standard Normal
5
350
4
300
Quantiles of Input Sample
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464
250
200
150
100
3
2
1
0
−1
−2
50
0
−4
−3
−2
0
2
4
6
−4
−4
−2
0
2
Standard Normal Quantiles
4
Figure 4. The histogram and Q-Q plot of the statistic in Corollary 1.
by 0.6745. When applying the two time scale step, we take
K = 50.
• For Podolskij and Vetter’s method, the weight function
g(x) = x ∧ (1 − x) will be used in the preaveraging step.
• For our PT method, the same weight function g(x) = x ∧
(1 − x) is chosen in the preaveraging step as in Podolskij
and Vetter (2009b), and the threshold level is chosen to be
0.23
n .
4.2.
Simulation Results
The results are presented in Tables 1–3 and Figures 1–3. The
following short-hands are used:
PT = preaveraging-threshold;
FW = Fan and Wang;
PV2 = Podolskij and Vetter’s bi-power;
PV3 = Podolskij and Vetter’s triple-power.
We make the following observations.
1. For all models considered, the performances of all three
methods improve as the sample size increases.
2. In all the cases, the PT method performs the best (in terms
of smaller biases and s.e.’s and MSEs), followed by FW
method, which in turn outperforms PV2 and PV3 methods.
3. In Model 1 (the jump-diffusion model with noise), the PT and
FW methods both work well for all λ’s considered, and both
outperform the PV2 and PV3 methods across the range of λ.
Furthermore, the performances of the PT and FW methods
are not affected by the intensity λ. However, the PV2 and
PV3 methods work well for small λ’s but deteriorate very
rapidly as λ gets bigger.
4. In Models 2 and 3 (with infinite activity jumps and noise),
the PT method consistently outperforms the others. We have
also done simulations when Jt is an untrimmed symmetric
β-stable process, (not shown here to save space). It turns
out that the PT method still performs the best, followed by
the FW method, while PV2 and PV3 perform rather unsatisfactorily. One reason is that the untrimmed β-stable process
can have some rare but huge jumps, which most severely
affect the performance of PV2 and PV3. In reality though,
huge jumps are unrealistic for high frequency tick-by-tick
data so we trimmed 2% increments from largest increments.
“Tempering” is often used for this purpose as well.
5. The FW and PT methods perform equally well in Model 1.
However, in Models 2–3, the FW method does not perform
as well as the PT method. Recall that the FW method tries
to remove jumps first and then deals with noise, while the
PT method does the reverse. So for infinite activity jumps, it
seems better to use PT method.
6. Figure 4 displays the Q-Q plot of the PT estimator for Model
2, where the variance was estimated as in Proposition 1. The
plot is close to linear, indicating asymptotic normality of the
estimator.
5.
CONCLUSION
In this article, we proposed a PT estimator of integrated
volatility in the simultaneous presence of microstructure noise
and jumps. The method is based on two steps, namely, the preaveraging step to reduce microstructure noise, and the threshold
step to remove jumps. The new estimator can handle very general jump processes of finite or infinite activity. The consistency
and asymptotic normality of the proposed estimator have been
established. Simulation studies show excellent performances of
the proposed estimators, in comparison with some alternative
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
methods in the literature. Some possible extensions of the current work include the study of covariation (matrix) estimation,
an important quantity in econometrics, under simultaneous presence of noise and infinity activity jumps.
APPENDIX: THE PROOFS OF THE THEOREMS
By a standard localization procedure, described in details in Jacod
(2012), we can replace the local boundedness hypotheses in assumptions by a boundedness assumption, and also assume that the process
X itself, and thus the jump process Xt , are bounded as well. That is,
we may assume the following:
Assumption A.1. For some
max{|bt |, |σt |, |Xt |, κt (r)} ≤ C.
constant
C > 0,
we
have
We define the continuous part of X by X′ and discontinuous martingale part by X′′ , that is,
t
t
bs′ ds +
σs dWs , X′′ (t) = Xt − X′ (t),
X′ (t) = X0 +
0
When |ni,kn Y ′ (g)| < un /2, from similar considerations, we deduce
that
n
η ≤ K n Y ′ (g)p n X′′ (g)r /ur , if n Y (g) > un ,
i,kn
i,kn
i,kn
n
i
n
η ≤ K n X′′ (g) ∧ un p + 1{p>1} n Y ′ (g)p−1
i,kn
i,kn
i
× ni,kn X′′ (g) ∧ un , if ni,kn Y (g) ≤ un ,
(A.1)
where r is any positive real number. In view of boundedness assumption
of the parameters and repeated use of the H¨older’s and Burkholder’s
inequalities, we have
2
E ni,kn X′′ (g)
(i+kn )n
1 s − in
x 2 Ft (dx)ds
g2
=
kn
n
in
R
≤ K(kn n ),
(A.2)
n ′ l
l/2
for l > 0,
(A.3)
E i,kn Y (g) ≤ Kl (kn n )
n
2
≤ Ks kn n un2−s , for β < s < 2.
E i,kn X′′ (g) ∧ un
0
where bt′ = bt + |x|>1 xFt (dx). Write Yt′ = Xt′ + ǫt , and hence Y =
′
′′
Y + X . Throughout the proof, K denotes a generic constant while
Kα might depend on some parameter α. We also define i,kn W (g) and
i,kn ǫ(g) similarly to i,kn X(g).
We prove a general theorem which is valid for any p > 0, of which
Theorem 1 is a special case. Define
V (Y ′ , g, p)nt =
U (Y, g, p)nt =
⌊t/n ⌋−kn
i=1
⌊t/n ⌋−kn
i=1
where un satisfies (8).
n ′ p
Y (g) ,
i,kn
and
n
Y (g))p 1{|n Y (g)|≤un } ,
i,kn
i,kn
Lemma A.1. Under Assumptions 2, 3, A.1 (with r = 2p), we have
p/2
t
1 ′
1−p/4
n P
2
2
¯
¯
n
U (Y, g, p)t −
→ mp
du,
s + g (2)κ
cg(2)σ
c
0
if either p ≤ 2 or p > 2 with ̟1 >
p−2
.
4(p−β)
Proof. Since Xt′ is continuous, by Theorem 3.3 of Jacod, Podolskij,
and Vetter (2010), we have
p/2
t
1 ′
P
2
2
cg(2)σ
¯
¯
g
(2)κ
n1−p/4 V (Y ′ , g, p)nt −
→ mp
+
s
du.
c
0
(A.4)
The last inequality follows from (6.25) of Jacod (2012) applied
√ to the
process X′′ , with the sampling interval kn n and αn = un / kn n ,
which goes to ∞ by (8). Let l = r = 1, we deduce from the above
inequalities that, when p ≤ 2,
1/4
n
n1−p/4 E ηin ≤ Kn
+ unp(1−s/2) + 1{p>1} un1−s/2 .
un
When p > 2, we further use the inequality (|x| ∧ un )p ≤ up−2
n (|x| ∧
un )2 to get
1/4
n
1−s/2
.
n1−p/4 E ηin ≤ Kn
+ n1/2−p/4 up−s
+
u
n
n
un
By assumption of un , we have n1/4 /un → 0. Now let p < 2 or p ≥ 2
p−2
and ̟1 ≥ 4(p−β)
, we complete the proof.
Proof of Theorem 1. Letting p = 2, we obtain the required result of
Theorem 1 from Lemma A.1.
Proof of Theorem 2. We redefine U (Y, g)nt involving kn as
U (Y, g, kn )t =
⌊t/n ⌋−kn
i=1
Therefore, it suffices to show that
n1−p/4 (U (Y, g, p)nt − V (Y ′ , g, p)nt ) → 0, in probability.
⌊t/n ⌋−kn n
Rewrite the left-hand side above as n1−p/4 i=1
ηi , where
n ′ p
n
p
n
η = Y (g) 1{|n Y (g)|≤un } − Y (g)
i
i,kn
where
−̟1′
u′n n
for some 0 ≤ ̟1′ < ̟2′ <
i,kn
i,kn
The following two elementary inequalities will be used in the proof:
when p ≤ 1,
||x + y|p − |x|p | ≤ K(|y|p + |x|p−1 |y|),
when p > 1.
We consider two disjoint cases, |ni,kn Y ′ (g)| ≥ un /2 and |ni,kn Y ′ (g)| <
un /2.
When |ni,kn Y ′ (g)| ≥ un /2, if we choose an appropriate constant K
l p
(e.g., 2 (2 + 1) with l > 0), we have
n
η ≤ K n Y ′ (g)p+l /ul .
i
i,kn
n
n
Y (g))2 1{|n
i,kn
i,k
−̟2′
→ 0, u′n n
1
4
n
Y (g))|≤u′n } ,
→ ∞,
− η2 , and
V (Y, g, kn )t =
p
p
= ni,kn Y ′ (g) + ni,kn X′′ (g) 1{|ni,kn Y (g)|≤un } − ni,kn Y ′ (g) .
||x + y|p − |x|p | ≤ |y|p ,
465
⌊t/n ⌋−kn
i=1
n
Y (g))2 .
i,kn
By assumption, kn = cn−(1/2+η) . Theorem 2 follows if we can show
that
t
1
P
V (Y ′ , g, kn )t −
→
σs2 ds,
(A.5)
c1 kn
0
and
1
P
(U (Y, g, kn )t − V (Y ′ , g, kn )t ) −
→ 0.
c1 kn
(A.6)
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466
Journal of Business & Economic Statistics, July 2014
(A.6) follows from the same procedure as in the proof of Theorem 1.
Next, we prove (A.5). Define
γi =
γi′ =
1 n ′ 2
i,kn Y (g) − |σ(i−1)/n i,kn W (g) + i,kn ǫ(g)|2 ,
kn
1
(|σ(i−1)/n i,kn W (g) + i,kn ǫ(g)|2
kn
− |σ(i−1)/n i,kn W (g)|2 ),
i=1
⌊t/n ⌋−kn
i=1
⌊t/n ⌋−kn
i=1
P
γi −
→ 0,
(A.7)
P
γi′ −
→ 0,
(A.8)
i=1
i,j =1
E[(γi′′ − E[γi′′ |F(i−1)/n ])(γj′′ − E[γj′′ |F(j −1)/n ))]
Proof of Theorem 3. When X is a continuous process, by Theorem
4.1 of Jacod, Podolskij, and Vetter (2010),
t
σs2 ds
n−1/4 n1/2 V (Y ′ , g, p)nt ) −
0
1/2
S
P
E[γi′′ |F(i−1)/n ] −
→
(A.9)
c
¯
g(2)
t
0
|μ(σs , κ/c)|dWs′ ,
with the same W ′ and μ defined in Theorem 3. So to prove the theorem,
it is enough to show that
t
σs2 ds.
P
(A.10)
n3/4−p/4 (U (Y, g, p)nt − V (Y ′ , g, p)nt ) −
→ 0.
0
Observe that (since the drift does not affect the estimator of integrated
volatility, so here we assume it is zero for simplicity):
(i+kn −1)n
K
gn2 (s)E(σs − σ(i−1)n )2 ds
E|γi | ≤
kn
(i−1)n
(i+kn −1)n
gn2 (s)E(σs + σ(i−1)n )2 ds
(i−1)n
×
2
k
n −1
j
j +1
+
−g
g
κ2
k
k
n
n
j =1
by Cauchy-Schwarz
inequality and Itˆo’s isometry, where gn (s) =
n
⌋/k
.
Since
g has a piecewise Lipschitz derivative, Itˆo’s
g ⌊ s−(i−1)
n
n
isometry and boundedness of σs together imply
E|γi | ≤ Kn kn n + 1/kn ,
thus, (A.7) follows. Similarly, (A.8) follows from
kn −1
2
j
j +1
K
−g
E|γi′ | ≤
g
κ2
kn j =1
kn
kn
(i+kn −1)n
gn2 (s)E(σ(i−1)n )2 ds
(i−1)n
×
2
k
n −1
j
j +1
+
−g
g
κ2
k
k
n
n
j =1
≤
−
→
P
γi′′ − E[γi′′ |F(i−1)/n ] −
→ 0,
⌊t/n ⌋−kn
=
by Burkholder inequality. Hence, we have shown (A.9).
Finally, (A.10) follows from a standard procedure of Riemann integrability (see Barndorff-Nielsen et al. 2006). Hence the proof is finished.
Therefore, it suffices to prove the following four convergences
i=1
⌊t/n ⌋−kn
≤ Kkn n2n → 0,
1
γi′′ = (|σ(i−1)/n i,kn W (g)|2 ).
kn
⌊t/n ⌋−kn
whenever |i − j | > kn . Hence, we have
⎛
2 ⎞
n ⌋−kn
⌊t/
(γi′′ − E[γi′′ |F(i−1)/n ]) ⎠
E ⎝
K
n + 1/kn2 .
kn
To prove (A.9), since γi′′ is F(i+kn −1)/M -measurable,
E[(γi′′ − E(γi′′ |F(i−1)/n ))(γj′′ − E(γj′′ |F(j −1)/n ))] = 0
The case p = 2 is the exact result of Theorem 3.
⌊t/n ⌋−kn n
ηi . Note that the
The left-hand side reduces to n3/4−p/4 i=1
jumps of Xt will be a finite variation process when β < 1, so we have
the following decomposition,
t
t
′
′
b1s ds +
σs dWs , X1′′ (t) = Xt − X1′ (t),
X1 (t) = X0 +
0
′
b1s
0
where,
= bs − |x|≤1 xFs (dx) and X1′′ = s≤t Xs . Representation of Y changes to Y = X1′′ + Y ′ and Y ′ is the combination of X1′
and the microstructure noise.
Similarly to (A.2)–(A.4), we have the following estimates,
E ni,kn X1′′ (g) ≤ K(kn n ),
l
for l > 0,
E ni,kn Y ′ (g) ≤ Kl (kn n )l/2 ,
n
′′
1−s
E i,kn X1 (g) ∧ un ≤ Kkn n un ,
for β < s < 1,
√
where we have used (6.26) of Jacod (2012) with αn = un / kn n and
r and l as above. We first consider the case of p > 1. For any 0 <
r < 1 and s ∈ (β, 1), using the H¨older’s inequality and the inequality
(|x| ∧ un )m for 0 < m ≤ p, we have
(|x| ∧ un )p ≤ up−m
n
n3/4−p/4 E ηin
⎡
p+l
p
+ 2r
4
n4
⎢ n
≤ Kn3/4−p/4 ⎣ l +
un
urn
p−1
4
+1{p>1} n
+ n1/2 up−s
n
⎤
1/2 1−s ⎥
n un ⎦
1−p
+̟1 (p−s)
≤ Kn nl(1/4−̟2 )−1/4 + nr(1/2−̟2 )−1/4 + n 4
+ 1{p>1} n̟1 (1−s)
:= Kn (cn1 + cn2 + cn3 + 1{p>1} cn4 ).
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
Now we consider the cases of p > 1 and p ≤ 1 separately. Let
s → β. When p > 1,
c1 → l(1/4 − ̟2 ) − 1/4,
c2 → r(1/2 − ̟2 ) − 1/4,
c3 → 1/4 − p/4 + ̟1 (p − β),
c4 → ̟1 (1 − β).
For some large enough l and r, we have c1 > 0 and c2 > 0. When
p−1
, we have c3 > 0. We also have c4 > 0 as ̟1 > 0 and
̟1 > 4(p−β)
β < 1. Let p = 2, we obtain the result of Theorem 3.
When p ≤ 1, using H¨older’s inequality, we have
n3/4−p/4 E |ηin | ≤ Kn nl(1/4−̟2 )−1/4
2 )−1/4
+ r(1/2−̟
+ np(1/4+̟1 (1−s))−1/4
n
c′
:= Kn cn1 + cn2 + n3 .
Clearly, cj > 0, j = 1, 2 and c3′ > 0. The proof follows by combining
the above results and letting p = 2.
Proof of Proposition 1 and Corollary 1. Proposition 1 is an immediate consequence of Lemma A.1 by simply taking p = 4. Because
Theorem 3 holds with stable convergence, the proof of Corollary 1
follows from Theorem 3 and Proposition 1.
ACKNOWLEDGMENTS
The authors thank the Editor and the Associate Editor
for their very extensive and constructive suggestions which
helped to improve this article considerably. Jing’s research
is partially supported by Hong Kong RGC HKUST6019/10P,
HKUST6019/12P, and HKUST6022/13P. Kong’s research is
supported in part by National NSFC No.11201080 and in part
by Humanity and Social Science Youth Foundation of Chinese
Ministry of Education No. 12YJC910003. Liu want to thank the
financial support from The Science and Technology Development Fund of Macau (No.078/2012/A3 and No.078/2013/A3).
[Received February 2012. Revised February 2014.]
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ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20
On the Estimation of Integrated Volatility With
Jumps and Microstructure Noise
Bing-Yi Jing, Zhi Liu & Xin-Bing Kong
To cite this article: Bing-Yi Jing, Zhi Liu & Xin-Bing Kong (2014) On the Estimation of Integrated
Volatility With Jumps and Microstructure Noise, Journal of Business & Economic Statistics, 32:3,
457-467, DOI: 10.1080/07350015.2014.906350
To link to this article: http://dx.doi.org/10.1080/07350015.2014.906350
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On the Estimation of Integrated Volatility With
Jumps and Microstructure Noise
Bing-Yi JING
Hong Kong University of Science and Technology, Hong Kong ([email protected])
Zhi LIU
University of Macau, Macau, China ([email protected])
Xin-Bing KONG
Soochow University, Suzhou, China ([email protected])
In this article, we propose a nonparametric procedure to estimate the integrated volatility of an Itˆo semimartingale in the presence of jumps and microstructure noise. The estimator is based on a combination
of the preaveraging method and threshold technique, which serves to remove microstructure noise and
jumps, respectively. The estimator is shown to work for both finite and infinite activity jumps. Furthermore,
asymptotic properties of the proposed estimator, such as consistency and a central limit theorem, are established. Simulations results are given to evaluate the performance of the proposed method in comparison
with other alternative methods.
KEY WORDS: Central limit theorem; High frequency data; Quadratic variation; Semimartingale.
1.
INTRODUCTION
With the availability of high-frequency data, there has been a
rapidly growing interest in the estimation of integrated volatility. For continuous Itˆo process, a commonly used estimator is
the realized volatility (also called realized quadratic variation in
some literature), see, Andersen et al. (2003). The estimation of
integrated volatility becomes tricky when the underlying price
process contains jumps. Two well-behaved estimators are the
multiple-power estimator and the realized threshold quadratic
variation, respectively. The former was proposed by BarndorffNielsen and Shephard (2006) and Barndorff-Nielsen and Shephard (2004), while the latter was proposed by Mancini (2009)
and further developed in Jacod (2008). An interesting comparison of the two different approaches was given in Veraart
(2011).
However, it is widely accepted that the observed prices are
contaminated by microstructure noise, for example, the bid-ask
spreads. Thus, the discretely observed process Yti is
Yti = Xti + ǫti ,
i = 0, 1, . . . , ⌊t/n ⌋,
Xtc = X0 +
Xtd
=
t
0
t
0
|x|≤1
bs ds +
t
σs dWs ,
0
x(μ − ν)(ds, dx) +
t
0
xμ(ds, dx),
|x|>1
(3)
where b and σ are locally bounded optional processes, μ is the
jump measure, with ν its predictable compensator. All these are
defined on a stochastic basis (, F, Ft , P ). An example in the
literature using the process with jumps is by Wu (2008), where,
he uses the L´evy process to model financial security returns.
Under this setting, the quadratic variation of X becomes
[X, X]t = [Xc , Xc ]t + [Xd , Xd ]t
t
(Xs )2 ,
σs2 ds +
=
0
(1)
where Xt represents the latent price process, and ǫti is the microstructure noise at time ti . For simplicity, assume that the observation times in [0, t] are equally spaced, namely, ti =: in
(0 ≤ i ≤ ⌊t/n ⌋) with n → 0 as n → ∞. Our interest lies in
the inference on certain characteristics of the latent process X,
using the contaminated observations Yti ’s.
In this article, we assume that the latent price process X is an
Itˆo semimartingale of the form
Xt = Xtc + Xtd ,
where Xc and Xd are the continuous and discontinuous (or
jump) components, respectively,
0≤s≤t
where Xs = Xs − Xs− is the jump size of X at time s. In this
article, our main interest lies in the estimation of the integrated
t
volatility of the continuous part, that is, [Xc , Xc ]t = 0 σs2 ds,
in the presence of jumps and microstructure noise. Incidentally,
estimation of [Xd , Xd ]t can be done similarly.
(2)
457
© 2014 American Statistical Association
Journal of Business & Economic Statistics
July 2014, Vol. 32, No. 3
DOI: 10.1080/07350015.2014.906350
Color versions of one or more of the figures in the article can be
found online at www.tandfonline.com/r/jbes.
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458
Journal of Business & Economic Statistics, July 2014
For a generic semimartingale X, define (see A¨ıt-Sahalia and
Jacod 2009)
r
|s X| < ∞ ,
(4)
I = r ≥ 0;
0≤s≤t
where s X = Xs − Xs− is the jump size at time s. This is an
interval of the form [β, ∞] or (β, ∞]. We say that Xd is of
finite (or infinite) activity in (0, t], that is Xd has a.s. finitely
(or infinitely) many jumps if 0 ∈ I (or 0 ∈ I ); Xd is of finite
variation in (0, t] if 1 ∈ I .
When the latent price process Xt contains no jumps, that is,
Xd ≡ 0 in (2), the influence of the microstructure noise on the
estimation of the integrated volatility for high-frequency data
has been well documented in the literature. For instance, Zhang
Mykland, and A¨ıt-Sahalia (2005) and Bandi and Russell (2006)
found that microstructure noise, if left untreated, can result in
inconsistent estimators of the integrated volatility. There are
several main approaches to overcome this difficulty, including
(I) subsampling method (Zhang, Mykland, and A¨ıt-Sahalia
2005),
(II) the realized kernel method (Barndorff-Nielsen et al. 2008),
(III) the preaveraging method (Jacod et al. 2009 and Podolskij
and Vetter 2009b),
(IV) the quasi-maximum likelihood method (QMLE) (Xiu
2010).
In this article, we are interested in estimating integrated
volatilities when the latent process Xt contains jumps, that is,
Xd ≡ 0 in (2). Here, given the observations Yti , one needs to
take care of the noise ǫti as well as the jumps Xtdi . The estimation of integrated volatility was considered by Fan and Wang
(2007), and Podolskij and Vetter (2009b) using two fundamentally different techniques, which differ in the order of treating
jumps and noise. Rosenbaum (2009) also considered the integrated volatility estimation under round-off error using wavelet
approach.
To be precise, Fan and Wang (2007) first applied wavelet
methods to try to get rid of the jumps, and then dealt with
the microstructure noise by the multiscale technique of Zhang
(2006), while Podolskij and Vetter (2009a) first applied “preaveraging” to reduce the impact of the microstructure noise,
and then handled the jumps by the multiple-power estimator.
Both papers also showed that their estimators are consistent
for the estimation of the integrated volatility when the latent
price process Xt contains finite activity jumps, which were also
supported by their simulation results.
A natural question is: how will the two methods by Fan and
Wang (2007) and Podolskij and Vetter (2009a) perform if we
increase the jump intensities of the jumps? To answer this, we
conducted some simple simulation studies. In the first simulation, the observations Yti are generated from model (1), where
Xt and ǫt are taken to be
Xt = Wt +
Nt
Ji ,
i=1
iid
ǫi ∼ N (0, 0.012 ),
where Ji ∼ N (0, 0.52 ) and Nt ∼Poisson(λt) and Jt and
Nt are independent. We choose the jump intensity λ =
1, 2, . . . , 10, 20, 50, and n = 23, 400. Now Xt still contains finite activity jumps, but the jumps occur more frequent as λ increases. We then applied the multipower method and the wavelet
method to this model, and the results are presented in Table 1
and Figure 1 in Section 4 (one can ignore column 1 now, which
contains the results for a different method proposed later in the
article). Note that [Xc , Xc ]t=1 = 1. Clearly,
• Podolskij and Vetter’s method works very well when jumps
are rare (e.g., λ = 1), and gradually deteriorates as jumps
become more frequent;
• Fan and Wang’s method is extremely accurate regardless
of the arrival rate λ. This is remarkable and perhaps it may
not come to a complete surprise to us since the wavelet
step can effectively remove all the big jumps in the above
jump-diffusion model, which might explain why it should
perform so well in this case.
The remarkable performance of Fan and Wang’s method in
the jump-diffusion model with noise begs the question: how will
it work for some other jump models with noise? So, in our second
Table 1. Comparisons for Model 1 (jump-diffusion model with noise). Here n = 23, 400
λ
Preaveraging-threshold
(PT)
(bias, s.e., MSE)
Podolskij and Vetter’s
bi-power (PV2)
(bias, s.e., MSE)
Podolskij and Vetter’s
triple-power (PV3)
(bias, s.e., MSE)
Fan and Wang
(FW)
(bias, s.e., MSE)
0
1
2
3
4
5
6
7
8
9
10
20
50
(−0.007, 0.081, 0.007)
(−0.007, 0.081, 0.007)
(−0.003, 0.081, 0.007)
(−0.001, 0.081, 0.007)
(0.004, 0.079, 0.006)
(0.010, 0.081, 0.007)
(0.013, 0.081, 0.007)
(0.019, 0.082, 0.007)
(0.023, 0.082, 0.007)
(0.027, 0.081, 0.007)
(0.027, 0.081, 0.007)
(0.040, 0.082, 0.010)
(0.069, 0.084, 0.012)
(−0.013, 0.091, 0.008)
(0.052, 0.131, 0.020)
(0.109, 0.164, 0.039)
(0.164, 0.186, 0.062)
(0.234, 0.223, 0.104)
(0.299, 0.249, 0.151)
(0.361, 0.275, 0.206)
(0.446, 0.311, 0.295)
(0.502, 0.337, 0.369)
(0.573, 0.370, 0.465)
(0.624, 0.392, 0.510)
(1.438, 0.678, 2.521)
(4.579, 1.521, 23.28)
(−0.020, 0.095, 0.009)
(0.024, 0.114, 0.014)
(0.062, 0.128, 0.020)
(0.098, 0.140, 0.029)
(0.142, 0.153, 0.044)
(0.186, 0.168, 0.063)
(0.226, 0.185, 0.086)
(0.287, 0.205, 0.122)
(0.316, 0.215, 0.146)
(0.363, 0.234, 0.187)
(0.386, 0.252, 0.212)
(0.925, 0.409, 1.023)
(3.033, 0.978, 10.16)
(−0.007, 0.081, 0.007)
(0.008, 0.084, 0.007)
(0.009, 0.083, 0.007)
(0.005, 0.081, 0.007)
(0.006, 0.083, 0.007)
(0.010, 0.083, 0.007)
(0.014, 0.082, 0.007)
(0.020, 0.084, 0.007)
(0.019, 0.084, 0.007)
(0.034, 0.084, 0.008)
(0.034, 0.083, 0.009)
(0.055, 0.082, 0.010)
(0.113, 0.089, 0.021)
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
459
MSE
Preaveraging−Threshold
Bipower
Triplepower
Wavelet
0.4
0.3
0.2
0.1
1
2
3
4
5
6
Jump Intensity (λ)
7
8
9
10
Figure 1. MSE plots for Model 1 (jump-diffusion model with noise, the curve of preaveraging-threshold estimator hides behind the wavelet
estimator curve).
simulation, we generated the observations Yti from model (1),
where Xt and ǫt are taken to be
Xt = Wt + Jt ,
iid
ǫi ∼ N (0, 0.012 ),
where Jt follows a symmetric β-stable process, where β ranges
from 0.3 to 1.5 (all have infinite activity jumps). As β gets
larger, jumps occur more frequently. Again, [Xc , Xc ]t=1 = 1.
The results containing bias, standard error (s.e.) and mean square
error (MSE) are given in the following table.
Performance of Fan and Wang’s method for different β’s
β
bias
s.e.
MSE
0.3
0.083
0.125
0.027
0.5
0.145
0.148
0.043
0.8
0.274
0.181
0.108
1.0
0.406
0.207
0.208
1.5
0.924
0.259
0.918
Clearly, Fan and Wang’s method works well for small β’s
since the MSEs are under 21% for β ≤ 1. However, as β > 1, the
performance starts to get bad; for example, the MSE is 91.8% for
β = 1.5%. The reason is: the bigger β is, the more small jumps
there are, the more difficult for the wavelet method to separate
the small jumps from the microstructure noise. (Incidentally, the
results for Podolskij and Vetter’s method are even worse in this
case and hence not included here.)
The less than ideal numerical performances of Fan and
Wang’s and Podolskij and Vetter’s methods for the case β > 1
prompt us to search for some alternative methods. Recall that
the observations are
Yti = Xtci + Xtdi + ǫti .
(5)
with 0 < t1 < t2 < · · · < tn ≤ t. Our procedure goes in two
steps:
• Step 1 (Preaveraging). We use preaveraging to reduce the
effect of the microstructure noise ǫti .
• Step 2 (Threshold). We truncate the smoothed data at some
appropriate threshold to remove the jump contribution from
the decomposition of [X, X]t .
We will refer to the method as the preaveraging-threshold (PT)
method, and the resulting estimator as PT estimator.
Simulation studies given later in the article indicate that the PT
method turns out to perform better than the two previously mentioned methods, regardless of finite or infinite activity jumps. In
fact various recent papers have shown that essentially all statistical methods based on increments of an Itˆo semimartingale in
the no-noise case can be translated to the noisy framework by
replacing raw increments with preaveraged statistics, which can
be interpreted as some kind of generalized increments. Given
that intuition, it is perhaps not too surprising that a truncated
version of the estimator from Jacod et al. (2009) works in the
jump case as well, having the same asymptotic variance as their
estimator, since the same holds true for realized volatility (RV)
and truncated RV as shown in Jacod (2008). The present article is further proof that the preaveraged data can be effectively treated as data with no microstructure noise. Recently,
Bajgrowicz, Scaillet, and Treccani (2013) show that, using the
existing jump detection methods, the jumps usually are erroneously identified due to the multiple testing issue and the jump
effect on the market friction has been overstated. They proposed a thresholding testing statistics, however, the test is based
on the microstructure-noise-free data. Christensen, Oomen, and
Podolski (2014) consider the microstructure noise robust jump
test using the Bipower estimator. Since main objective of Bajgrowicz, Scaillet, and Treccani (2013) and Christensen, Oomen,
and Podolski (2014) is the estimation of integrated volatility, the
results of current article provide a possibility to improve the efficiency of their approaches.
The rest of the article is organized as follows. In Section 2,
we introduce the proposed PT estimator. Main results are given
in Section 3. Section 4 is devoted to simulations. We conclude
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460
Journal of Business & Economic Statistics, July 2014
the article in Section 5. The technical proofs are postponed to
the Appendix.
2.
METHODOLOGY
As we described before, the process X is observed with an
error (microstructure noise in practice), namely, at the time
point ti , we observe Xti + ǫi rather than Xti , where the ǫi
are “errors” which are conditionally on the process X, independent. It is convenient to define a “canonical” process ǫt in
R [0,∞) for all t, although we only use its value at some discrete
points. Mathematically, we first have a transition probability
Qt (ω, dx) from (, F) to R. Based on it, we construct a new
probability space (R [0,∞) , B, σ (ǫs : s ∈ [0, t)), Q), where B is
the product Borel σ -field and Q = ⊗t≥0 Qt . Then, the observation is measurable with respect to the filtered probability space
(1)
((1) , F(1) , F(1)
t , P ), which is given by
⎧ (1)
(1)
⎪
⎨ = × R[0, ∞), F = F ⊗ B,
(1)
F(1)
t = Ft ⊗ σ (ǫs : s ∈ [0, t)), P (dω, dx)
⎪
⎩
= P (dω)Q(ω, dx).
Hence, any variable or process in either or R [0,∞) can be
considered in usual way as a variable or a process on . For
more detailed discussion about the new probability space, see,
Jacod, Podolskij, and Vetter (2010).
2.1.
Preaveraging
We briefly describe the idea of the preaveraging. For details,
see Jacod et al. (2009). Define the jth increment by
nj Y := Ytj − Ytj −1 ,
for j = 1, 2, . . . , n.
Together they form a sequence (n1 Y, . . . , nn Y ). Choose an
integer kn such that 1 ≤ kn ≤ ⌊t/n ⌋ and then we formulate
⌊t/n ⌋ − kn + 1 overlapping blocks, the ith being
Bi = ni Y, . . . , ni+kn −1 Y , for 1 ≤ i ≤ ⌊t/n ⌋ − kn + 1.
Within this ith block, we take a weighted average of the increments:
ni,kn Y (g) =
k
n −1
j =1
where the weight function g is chosen such that
• it is continuous, piecewise C 1 with a piecewise Lipschitz
derivative g ′ ,
1
• g(s) = 0 when s ∈ (0, 1), and 0 g 2 (s)ds > 0.
One common choice satisfying the above conditions is g(x) =
x ∧ (1 − x).
Now let us investigate what effect the preaveraging has. In
view of (5), denoting gjn = g(j/kn ), we have
=
k
n −1
:=
Ai1
gjn ni+j Xc
j =1
+
Ai2
+
+
Ai3 ,
(We shall treat the jump component Ai2 in the next subsection.)
Clearly, by choosing kn → ∞ appropriately, we can control the
effect of microstructure noise Ai3 relative to Ai1 . In particular,
1/2
−(1/2+ǫ)
(i) if kn n → ∞, for example, kn = ⌊c n
⌋ for some
ǫ > 0, then the effect of the microstructure noise can be
ignored;
1/2
−1/2
(ii) if kn n = c > 0, for example, kn = ⌊c n ⌋, then Ai1
i
and A3 will be of comparable size.
In either case, the influence of microstructure noise has been
eliminated or substantially reduced.
−1/2
In the following discussion, we will assume kn = ⌊c n ⌋
unless otherwise stated (the only exception is Theorem 2, where
−(1/2+η)
⌋).
we take kn = ⌊c n
2.2.
Threshold Quadratic Variation
The above preaveraging procedure reduces the influence of
the microstructure noise, its effect on the jumps is still very
limited. The remaining task is to get rid of the effect of the
jumps.
After preaveraging, the smoothed increments from the diffu1/4
sion and microstructure noise, Ai1 and Ai3 , are both of size n .
i
However, the smoothed increment from the jump, A2 , may still
1/4
be larger than n . Following the idea of Mancini (2009) or
Jacod et al. (2009), we can propose the following threshold
estimator of [Xc , Xc ]t :
U (Y, g)nt =
⌊t/n ⌋−kn
i=1
n
2
i,kn Y (g) 1{|ni,kn Y (g)|≤un } .
k
n −1
j =1
gjn ni+j Xd
+
k
n −1
gjn ni+j ǫ
j =1
(6)
(7)
Where, un satisfying
1
2
un −̟
→ 0, un −̟
→ ∞, for some 0 ≤ ̟1 < ̟2 <
n
n
g(j/kn )ni+j Y,
for 1 ≤ i ≤ ⌊t/n ⌋ − kn + 1,
ni,kn Y (g)
Some simple variance calculations show that
Ai1 = Op ( kn n ), Ai3 = Op (1/ kn ).
1
.
4
(8)
Such choices of un enable those (smoothed) increments larger
than un to be gradually excluded as n → ∞, and essentially only
those increments due to continuous part and noise are included,
hence we can calculate the integrated volatility after removing
the effect of noise.
3.
MAIN RESULTS
In this section, we study the asymptotic behavior of U (Y, g)nt ,
such as consistency and central limit theorem. Before stating our
main theorems, we need some assumption on the microstructure
noise.
Assumption 1. ν(ω, dt, dx) = dtFt (dx) with Ft (x) =
Ft′ (dx) + Ft′′ (dx), where Ft′ (dx) and Ft′′ (dx) are two mutually singular measures and satisfy, for three constants 0 ≤ β2 ≤
β1 ≤ β < 2, that
Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
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β−β1
f)
• Ft′ (dx) = β(1+|x|
(αt+ 1{0 1/(8 − 2β), we have
P
ˆ n := c3 (U1 (Y, g)nt − c4
ˆ n κˆ 2 − c5 (κˆ 2 )2 ) −
→
Q
t
σs4 ds,
0
where
c3 =
1
,
2
3c g¯ 2 (2)
¯ g¯′ (2),
c4 = 2g(2)
c5 =
2
g¯′ (2)
t.
c2
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462
Journal of Business & Economic Statistics, July 2014
Hence, we easily obtain the asymptotic behavior of the studentized statistics.
ˆ n +212 ˆ κˆ 2
Ŵtn = g¯ 24(2) (c22 Q
c
Corollary 1. Let
+
under the same assumptions as Theorem 3, we have
t
−1/4 ˆ
n − 0 σs2 ds S
n
n
−
→ N (0, 1),
Ŵt
• Model 1—A jump-diffusion process with noise:
2 2
11 (κˆc3) ),
Xt = Wt +
Nt
Ji ,
i=1
where Ji ∼ N (0, 0.52 ), Nt ∼Poisson(λt) and Ji and Nt are
independent. We choose λ = 0, 1, 2, . . . , 10, 20, 50.
• Model 2—A diffusion with infinite activity jumps and
noise:
where N (0, 1) is a standard normal r.v. independent of F.
Xt = Wt + Jt ,
3.2.
We compare the variance of our estimator with that of the
bi-power estimator proposed in Podolskij and Vetter (2009b) in
the special case when X is continuous. From Theorem 3, the
ˆ 1 is
asymptotic variance of
t
c
E|μ2 (σs , κ/c)|ds.
g¯ 2 (2) 0
On the other hand, by Theorem 3 of Podolskij and Vetter
(2009b), the variance of the bi-power type estimator is
t
c
1 + 2m21 − 3m41 2
E|μ2 (σs , κ/c)|ds
g¯ (2) 0
t
c
≈ 1.06 2
E|μ2 (σs , κ/c)|ds,
g¯ (2) 0
4.
SIMULATION STUDY
We now evaluate the performances of our proposed estimator
with other methods by Monte Carlo simulations.
4.1 Simulation Design
and
iid
ǫi ∼ N (0, 0.012 ),
where Jt is a trimmed symmetric β-stable process with β =
0.5. The algorithm described in Cont and Tankov (2004)
is used to simulate a β−stable process. By “trimmed” it
is meant that 2% of the largest (absolute) values will be
discarded. The reason for trimming will be explained in
the next subsection. Another common technique is to use
“tempered” stable process, which is similar in spirit to
“trimming.”
• Model 3—The stochastic volatility (SV) model with jumps
and noise:
Asymptotic Relative Efficiency Comparison
where mp = E|N (0, 1)|p . Thus, our estimator is asymptotically
about 6% more efficient than the bi-power estimator in the case
of no jumps. In the presence of jumps, our simulations show that
finite sample efficiency can be much greater than the asymptotic
one.
iid
ǫi ∼ N (0, 0.012 ),
dXt = μdt + σt dWt + δdJt ,
and
σt = exp(β0 + β1 τt ),
where dτt = ατt dt + dBt with Corr(dWt , dBt ) = ρ
and Jt follows a tempered symmetric β-stable process as in the last model. We let μ = 0.03, β0 =
0.3125, β1 = 0.125, α = −0.025, ρ = −0.3, δ = 0.1,
and ǫi ∼iid N (0, 0.0012 ). All the coefficients were selected
from Podolskij and Vetter (2009a) to allow a comparison.
It is easy to see that [Xc , Xc ]t = t for Models 1 and 2 while, it
becomes a random variable for Model 3.
We consider four sample sizes: n = 1170, 4680, 7800, and
23,400 (which correspond to sampling every 20 sec, 5 sec, 3
sec, and 1 sec in a trading day). The experiments are repeated
5000 times for each case. The bias and standard error (s.e.) and
MSE are computed for each sample size.
We will compare the performances of our PT estimator with
those by Podolskij and Vetter (2009a,b) and Fan and Wang
(2007). Here are some specifications.
We generate observations Yti from the following model
Yti = Xti + ǫti ,
where ǫi ’s are independent of Xti ’s. The latent process Xt is
generated from the following commonly used models.
• For Fan and Wang’s method, when applying the wavelet
to locate the jumps, we use Daubechies s8 wavelet to calculate the empirical wavelet
√ coefficient (Wang 1995) and
the universal threshold d 2 log n/n, where d is the median
absolute deviation of empirical wavelet coefficient, divided
Table 2. Comparisons for Model 2 (diffusion model with infinite activity jumps and noise)
n
1170
4680
7800
23400
Preaveraging-threshold
(PT)
(bias, s.e., MSE)
Podolskij and Vetter’s
bi-power (PV2)
(bias, s.e., MSE)
Podolskij and Vetter’s
triple-power (PV3)
(bias, s.e., MSE)
Fan and Wang
(FW)
(bias, s.e., MSE)
(0.058, 0.178, 0.035)
(0.034, 0.126, 0.017)
(0.024, 0.109, 0.013)
(0.013, 0.082, 0.007)
(0.274, 0.546, 0.373)
(0.236, 0.394, 0.210)
(0.223, 0.398, 0.194)
(0.173, 0.283, 0.110)
(0.136, 0.357, 0.146)
(0.127, 0.245, 0.076)
(0.118, 0.215, 0.060)
(0.090, 0.148, 0.030)
(0.141, 0.183, 0.053)
(0.116, 0.152, 0.037)
(0.108, 0.138, 0.030)
(0.079, 0.103, 0.017)
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
463
Table 3. Comparisons for Model 3 (SV model with infinite activity jumps and noise)
n
Preaveraging-threshold
(PT)
(bias, s.e., MSE)
Podolskij and Vetter’s
bi-power (PV2)
(bias, s.e., MSE)
Podolskij and Vetter’s
triple-power (PV3)
(bias, s.e., MSE)
Fan and Wang
(FW)
(bias, s.e., MSE)
(0.070, 0.181, 0.038)
(0.034, 0.126, 0.017)
(0.027, 0.111, 0.013)
(0.016, 0.081, 0.007)
(0.311, 0.541, 0.390)
(0.247, 0.397, 0.219)
(0.224, 0.341, 0.166)
(0.187, 0.292, 0.120)
(0.164, 0.364, 0.159)
(0.133, 0.240, 0.076)
(0.121, 0.214, 0.060)
(0.100, 0.161, 0.036)
(0.167, 0.198, 0.067)
(0.141, 0.176, 0.051)
(0.112, 0.152, 0.036)
(0.087, 0.121, 0.022)
1170
4680
7800
23400
MSE
Preaveraging−Threshold
Bipower
Triplepower
Wavelet
0.3
0.2
0.1
20
5
3
1
Sampling time intervel (Seconds)
Figure 2. MSE plots for Model 2 (L´evy model with noise).
MSE
Preaveraging−Threshold
Bipower
Triplepower
Wavelet
0.3
0.2
0.1
20
5
3
Sampling time intervel (Seconds)
Figure 3. MSE plots for Model 3 (SV model with noise).
1
Journal of Business & Economic Statistics, July 2014
QQ Plot of Sample Data versus Standard Normal
5
350
4
300
Quantiles of Input Sample
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464
250
200
150
100
3
2
1
0
−1
−2
50
0
−4
−3
−2
0
2
4
6
−4
−4
−2
0
2
Standard Normal Quantiles
4
Figure 4. The histogram and Q-Q plot of the statistic in Corollary 1.
by 0.6745. When applying the two time scale step, we take
K = 50.
• For Podolskij and Vetter’s method, the weight function
g(x) = x ∧ (1 − x) will be used in the preaveraging step.
• For our PT method, the same weight function g(x) = x ∧
(1 − x) is chosen in the preaveraging step as in Podolskij
and Vetter (2009b), and the threshold level is chosen to be
0.23
n .
4.2.
Simulation Results
The results are presented in Tables 1–3 and Figures 1–3. The
following short-hands are used:
PT = preaveraging-threshold;
FW = Fan and Wang;
PV2 = Podolskij and Vetter’s bi-power;
PV3 = Podolskij and Vetter’s triple-power.
We make the following observations.
1. For all models considered, the performances of all three
methods improve as the sample size increases.
2. In all the cases, the PT method performs the best (in terms
of smaller biases and s.e.’s and MSEs), followed by FW
method, which in turn outperforms PV2 and PV3 methods.
3. In Model 1 (the jump-diffusion model with noise), the PT and
FW methods both work well for all λ’s considered, and both
outperform the PV2 and PV3 methods across the range of λ.
Furthermore, the performances of the PT and FW methods
are not affected by the intensity λ. However, the PV2 and
PV3 methods work well for small λ’s but deteriorate very
rapidly as λ gets bigger.
4. In Models 2 and 3 (with infinite activity jumps and noise),
the PT method consistently outperforms the others. We have
also done simulations when Jt is an untrimmed symmetric
β-stable process, (not shown here to save space). It turns
out that the PT method still performs the best, followed by
the FW method, while PV2 and PV3 perform rather unsatisfactorily. One reason is that the untrimmed β-stable process
can have some rare but huge jumps, which most severely
affect the performance of PV2 and PV3. In reality though,
huge jumps are unrealistic for high frequency tick-by-tick
data so we trimmed 2% increments from largest increments.
“Tempering” is often used for this purpose as well.
5. The FW and PT methods perform equally well in Model 1.
However, in Models 2–3, the FW method does not perform
as well as the PT method. Recall that the FW method tries
to remove jumps first and then deals with noise, while the
PT method does the reverse. So for infinite activity jumps, it
seems better to use PT method.
6. Figure 4 displays the Q-Q plot of the PT estimator for Model
2, where the variance was estimated as in Proposition 1. The
plot is close to linear, indicating asymptotic normality of the
estimator.
5.
CONCLUSION
In this article, we proposed a PT estimator of integrated
volatility in the simultaneous presence of microstructure noise
and jumps. The method is based on two steps, namely, the preaveraging step to reduce microstructure noise, and the threshold
step to remove jumps. The new estimator can handle very general jump processes of finite or infinite activity. The consistency
and asymptotic normality of the proposed estimator have been
established. Simulation studies show excellent performances of
the proposed estimators, in comparison with some alternative
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
methods in the literature. Some possible extensions of the current work include the study of covariation (matrix) estimation,
an important quantity in econometrics, under simultaneous presence of noise and infinity activity jumps.
APPENDIX: THE PROOFS OF THE THEOREMS
By a standard localization procedure, described in details in Jacod
(2012), we can replace the local boundedness hypotheses in assumptions by a boundedness assumption, and also assume that the process
X itself, and thus the jump process Xt , are bounded as well. That is,
we may assume the following:
Assumption A.1. For some
max{|bt |, |σt |, |Xt |, κt (r)} ≤ C.
constant
C > 0,
we
have
We define the continuous part of X by X′ and discontinuous martingale part by X′′ , that is,
t
t
bs′ ds +
σs dWs , X′′ (t) = Xt − X′ (t),
X′ (t) = X0 +
0
When |ni,kn Y ′ (g)| < un /2, from similar considerations, we deduce
that
n
η ≤ K n Y ′ (g)p n X′′ (g)r /ur , if n Y (g) > un ,
i,kn
i,kn
i,kn
n
i
n
η ≤ K n X′′ (g) ∧ un p + 1{p>1} n Y ′ (g)p−1
i,kn
i,kn
i
× ni,kn X′′ (g) ∧ un , if ni,kn Y (g) ≤ un ,
(A.1)
where r is any positive real number. In view of boundedness assumption
of the parameters and repeated use of the H¨older’s and Burkholder’s
inequalities, we have
2
E ni,kn X′′ (g)
(i+kn )n
1 s − in
x 2 Ft (dx)ds
g2
=
kn
n
in
R
≤ K(kn n ),
(A.2)
n ′ l
l/2
for l > 0,
(A.3)
E i,kn Y (g) ≤ Kl (kn n )
n
2
≤ Ks kn n un2−s , for β < s < 2.
E i,kn X′′ (g) ∧ un
0
where bt′ = bt + |x|>1 xFt (dx). Write Yt′ = Xt′ + ǫt , and hence Y =
′
′′
Y + X . Throughout the proof, K denotes a generic constant while
Kα might depend on some parameter α. We also define i,kn W (g) and
i,kn ǫ(g) similarly to i,kn X(g).
We prove a general theorem which is valid for any p > 0, of which
Theorem 1 is a special case. Define
V (Y ′ , g, p)nt =
U (Y, g, p)nt =
⌊t/n ⌋−kn
i=1
⌊t/n ⌋−kn
i=1
where un satisfies (8).
n ′ p
Y (g) ,
i,kn
and
n
Y (g))p 1{|n Y (g)|≤un } ,
i,kn
i,kn
Lemma A.1. Under Assumptions 2, 3, A.1 (with r = 2p), we have
p/2
t
1 ′
1−p/4
n P
2
2
¯
¯
n
U (Y, g, p)t −
→ mp
du,
s + g (2)κ
cg(2)σ
c
0
if either p ≤ 2 or p > 2 with ̟1 >
p−2
.
4(p−β)
Proof. Since Xt′ is continuous, by Theorem 3.3 of Jacod, Podolskij,
and Vetter (2010), we have
p/2
t
1 ′
P
2
2
cg(2)σ
¯
¯
g
(2)κ
n1−p/4 V (Y ′ , g, p)nt −
→ mp
+
s
du.
c
0
(A.4)
The last inequality follows from (6.25) of Jacod (2012) applied
√ to the
process X′′ , with the sampling interval kn n and αn = un / kn n ,
which goes to ∞ by (8). Let l = r = 1, we deduce from the above
inequalities that, when p ≤ 2,
1/4
n
n1−p/4 E ηin ≤ Kn
+ unp(1−s/2) + 1{p>1} un1−s/2 .
un
When p > 2, we further use the inequality (|x| ∧ un )p ≤ up−2
n (|x| ∧
un )2 to get
1/4
n
1−s/2
.
n1−p/4 E ηin ≤ Kn
+ n1/2−p/4 up−s
+
u
n
n
un
By assumption of un , we have n1/4 /un → 0. Now let p < 2 or p ≥ 2
p−2
and ̟1 ≥ 4(p−β)
, we complete the proof.
Proof of Theorem 1. Letting p = 2, we obtain the required result of
Theorem 1 from Lemma A.1.
Proof of Theorem 2. We redefine U (Y, g)nt involving kn as
U (Y, g, kn )t =
⌊t/n ⌋−kn
i=1
Therefore, it suffices to show that
n1−p/4 (U (Y, g, p)nt − V (Y ′ , g, p)nt ) → 0, in probability.
⌊t/n ⌋−kn n
Rewrite the left-hand side above as n1−p/4 i=1
ηi , where
n ′ p
n
p
n
η = Y (g) 1{|n Y (g)|≤un } − Y (g)
i
i,kn
where
−̟1′
u′n n
for some 0 ≤ ̟1′ < ̟2′ <
i,kn
i,kn
The following two elementary inequalities will be used in the proof:
when p ≤ 1,
||x + y|p − |x|p | ≤ K(|y|p + |x|p−1 |y|),
when p > 1.
We consider two disjoint cases, |ni,kn Y ′ (g)| ≥ un /2 and |ni,kn Y ′ (g)| <
un /2.
When |ni,kn Y ′ (g)| ≥ un /2, if we choose an appropriate constant K
l p
(e.g., 2 (2 + 1) with l > 0), we have
n
η ≤ K n Y ′ (g)p+l /ul .
i
i,kn
n
n
Y (g))2 1{|n
i,kn
i,k
−̟2′
→ 0, u′n n
1
4
n
Y (g))|≤u′n } ,
→ ∞,
− η2 , and
V (Y, g, kn )t =
p
p
= ni,kn Y ′ (g) + ni,kn X′′ (g) 1{|ni,kn Y (g)|≤un } − ni,kn Y ′ (g) .
||x + y|p − |x|p | ≤ |y|p ,
465
⌊t/n ⌋−kn
i=1
n
Y (g))2 .
i,kn
By assumption, kn = cn−(1/2+η) . Theorem 2 follows if we can show
that
t
1
P
V (Y ′ , g, kn )t −
→
σs2 ds,
(A.5)
c1 kn
0
and
1
P
(U (Y, g, kn )t − V (Y ′ , g, kn )t ) −
→ 0.
c1 kn
(A.6)
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466
Journal of Business & Economic Statistics, July 2014
(A.6) follows from the same procedure as in the proof of Theorem 1.
Next, we prove (A.5). Define
γi =
γi′ =
1 n ′ 2
i,kn Y (g) − |σ(i−1)/n i,kn W (g) + i,kn ǫ(g)|2 ,
kn
1
(|σ(i−1)/n i,kn W (g) + i,kn ǫ(g)|2
kn
− |σ(i−1)/n i,kn W (g)|2 ),
i=1
⌊t/n ⌋−kn
i=1
⌊t/n ⌋−kn
i=1
P
γi −
→ 0,
(A.7)
P
γi′ −
→ 0,
(A.8)
i=1
i,j =1
E[(γi′′ − E[γi′′ |F(i−1)/n ])(γj′′ − E[γj′′ |F(j −1)/n ))]
Proof of Theorem 3. When X is a continuous process, by Theorem
4.1 of Jacod, Podolskij, and Vetter (2010),
t
σs2 ds
n−1/4 n1/2 V (Y ′ , g, p)nt ) −
0
1/2
S
P
E[γi′′ |F(i−1)/n ] −
→
(A.9)
c
¯
g(2)
t
0
|μ(σs , κ/c)|dWs′ ,
with the same W ′ and μ defined in Theorem 3. So to prove the theorem,
it is enough to show that
t
σs2 ds.
P
(A.10)
n3/4−p/4 (U (Y, g, p)nt − V (Y ′ , g, p)nt ) −
→ 0.
0
Observe that (since the drift does not affect the estimator of integrated
volatility, so here we assume it is zero for simplicity):
(i+kn −1)n
K
gn2 (s)E(σs − σ(i−1)n )2 ds
E|γi | ≤
kn
(i−1)n
(i+kn −1)n
gn2 (s)E(σs + σ(i−1)n )2 ds
(i−1)n
×
2
k
n −1
j
j +1
+
−g
g
κ2
k
k
n
n
j =1
by Cauchy-Schwarz
inequality and Itˆo’s isometry, where gn (s) =
n
⌋/k
.
Since
g has a piecewise Lipschitz derivative, Itˆo’s
g ⌊ s−(i−1)
n
n
isometry and boundedness of σs together imply
E|γi | ≤ Kn kn n + 1/kn ,
thus, (A.7) follows. Similarly, (A.8) follows from
kn −1
2
j
j +1
K
−g
E|γi′ | ≤
g
κ2
kn j =1
kn
kn
(i+kn −1)n
gn2 (s)E(σ(i−1)n )2 ds
(i−1)n
×
2
k
n −1
j
j +1
+
−g
g
κ2
k
k
n
n
j =1
≤
−
→
P
γi′′ − E[γi′′ |F(i−1)/n ] −
→ 0,
⌊t/n ⌋−kn
=
by Burkholder inequality. Hence, we have shown (A.9).
Finally, (A.10) follows from a standard procedure of Riemann integrability (see Barndorff-Nielsen et al. 2006). Hence the proof is finished.
Therefore, it suffices to prove the following four convergences
i=1
⌊t/n ⌋−kn
≤ Kkn n2n → 0,
1
γi′′ = (|σ(i−1)/n i,kn W (g)|2 ).
kn
⌊t/n ⌋−kn
whenever |i − j | > kn . Hence, we have
⎛
2 ⎞
n ⌋−kn
⌊t/
(γi′′ − E[γi′′ |F(i−1)/n ]) ⎠
E ⎝
K
n + 1/kn2 .
kn
To prove (A.9), since γi′′ is F(i+kn −1)/M -measurable,
E[(γi′′ − E(γi′′ |F(i−1)/n ))(γj′′ − E(γj′′ |F(j −1)/n ))] = 0
The case p = 2 is the exact result of Theorem 3.
⌊t/n ⌋−kn n
ηi . Note that the
The left-hand side reduces to n3/4−p/4 i=1
jumps of Xt will be a finite variation process when β < 1, so we have
the following decomposition,
t
t
′
′
b1s ds +
σs dWs , X1′′ (t) = Xt − X1′ (t),
X1 (t) = X0 +
0
′
b1s
0
where,
= bs − |x|≤1 xFs (dx) and X1′′ = s≤t Xs . Representation of Y changes to Y = X1′′ + Y ′ and Y ′ is the combination of X1′
and the microstructure noise.
Similarly to (A.2)–(A.4), we have the following estimates,
E ni,kn X1′′ (g) ≤ K(kn n ),
l
for l > 0,
E ni,kn Y ′ (g) ≤ Kl (kn n )l/2 ,
n
′′
1−s
E i,kn X1 (g) ∧ un ≤ Kkn n un ,
for β < s < 1,
√
where we have used (6.26) of Jacod (2012) with αn = un / kn n and
r and l as above. We first consider the case of p > 1. For any 0 <
r < 1 and s ∈ (β, 1), using the H¨older’s inequality and the inequality
(|x| ∧ un )m for 0 < m ≤ p, we have
(|x| ∧ un )p ≤ up−m
n
n3/4−p/4 E ηin
⎡
p+l
p
+ 2r
4
n4
⎢ n
≤ Kn3/4−p/4 ⎣ l +
un
urn
p−1
4
+1{p>1} n
+ n1/2 up−s
n
⎤
1/2 1−s ⎥
n un ⎦
1−p
+̟1 (p−s)
≤ Kn nl(1/4−̟2 )−1/4 + nr(1/2−̟2 )−1/4 + n 4
+ 1{p>1} n̟1 (1−s)
:= Kn (cn1 + cn2 + cn3 + 1{p>1} cn4 ).
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Jing, Liu, and Kong: On the Estimation of Integrated Volatility With Jumps and Microstructure Noise
Now we consider the cases of p > 1 and p ≤ 1 separately. Let
s → β. When p > 1,
c1 → l(1/4 − ̟2 ) − 1/4,
c2 → r(1/2 − ̟2 ) − 1/4,
c3 → 1/4 − p/4 + ̟1 (p − β),
c4 → ̟1 (1 − β).
For some large enough l and r, we have c1 > 0 and c2 > 0. When
p−1
, we have c3 > 0. We also have c4 > 0 as ̟1 > 0 and
̟1 > 4(p−β)
β < 1. Let p = 2, we obtain the result of Theorem 3.
When p ≤ 1, using H¨older’s inequality, we have
n3/4−p/4 E |ηin | ≤ Kn nl(1/4−̟2 )−1/4
2 )−1/4
+ r(1/2−̟
+ np(1/4+̟1 (1−s))−1/4
n
c′
:= Kn cn1 + cn2 + n3 .
Clearly, cj > 0, j = 1, 2 and c3′ > 0. The proof follows by combining
the above results and letting p = 2.
Proof of Proposition 1 and Corollary 1. Proposition 1 is an immediate consequence of Lemma A.1 by simply taking p = 4. Because
Theorem 3 holds with stable convergence, the proof of Corollary 1
follows from Theorem 3 and Proposition 1.
ACKNOWLEDGMENTS
The authors thank the Editor and the Associate Editor
for their very extensive and constructive suggestions which
helped to improve this article considerably. Jing’s research
is partially supported by Hong Kong RGC HKUST6019/10P,
HKUST6019/12P, and HKUST6022/13P. Kong’s research is
supported in part by National NSFC No.11201080 and in part
by Humanity and Social Science Youth Foundation of Chinese
Ministry of Education No. 12YJC910003. Liu want to thank the
financial support from The Science and Technology Development Fund of Macau (No.078/2012/A3 and No.078/2013/A3).
[Received February 2012. Revised February 2014.]
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