a state space approach to estimating integrated...
TRANSCRIPT
A State Space Approach to Estimating IntegratedVariance and Microstructure Noise∗
Daisuke Nagakura † and Toshiaki Watanabe ‡
October, 2008
Abstract
In this paper, we consider estimation of integrated variance (IV) as well as mi-crostructure noise by using a state space method. Our method is based on the resultobtained in Meddahi (2003) and Barndorff-Nielsen and Shephard (2002), namely,when the underlying true log-price process follows a certain class of continuous-timestochastic volatility (SV) models, the IV follows an ARMA(1, 1) process. Assumingthat an observed log-price is the sum of the true log-price and an i.i.d. microstruc-ture noise, we show that the realized variance (RV) calculated from the observedlog-prices, which we call “noise-contaminated RV (NCRV)”, follows an ARMA(1,2) process. We represent the NCRC by a state space form showing that the modelparameters are identifiable. We estimates the IV and microstructure noises by ap-plying the Kalman filtering to the state space form. The proposed method is appliedto yen/dollar exchange rate data. We find that the magnitude of a microstructurenoise is, in average, about 23% ∼ 32% of the NCRV.
Key Words: Realized Variance; Integrated Variance; Microstructure Noise; StateSpace
∗This research is partially supported by Grant-in-Aid for Scientific Research18203901 from the Japanese Ministry of Education, Science, Sports, Culture andTechnology. Views expressed in this paper are those of authors and do not neces-sarily reflect the official views of the Bank of Japan
†Institute for Monetary and Economic Studies, Bank of Japan, E-mail:[email protected]
‡Institute of Economic Research, Hitotsubashi University, E-mail: [email protected]
1
1 Introduction
The variance of financial asset returns are known to change over time. More specif-ically, the variance, or volatility, the square root of the variance, tends to be large(small) following successive large (small) variances in previous periods. This phe-nomenon is known as “volatility clustering.” The model, known as the “stochasticvolatility (SV) model”, in which the volatility is changing over time, have beenused for modeling this kind of volatility dynamics of financial assets. Based on theSV model with estimated model parameters, researchers can estimate the changingvariances. See, for example, Ghysels, Harvey and Renault (1996) for the reviews ofsome of the older papers on SV models and Shephard (2005) for a list of selectedpapers in the SV literature.
Recently, a new class of estimators for the changing variances has been developedby Barndorff-Nielsen and Shephard (2001, 2002), Andersen, Bollerslev, Diebold andEbens (2001) and Andersen, Bollerslev, Diebold and Labys (2001). The estimator iscalled the “realized variance (RV)”. The RV employs very high frequency financialdata such as minute-by-minute return data or entire records of quote or transactionprice data. Under certain assumptions, the RV is a consistent estimator for the“integrated variance (IV)”. The formal definition of IV will be given in Section 2.Intuitively, it is a measure of a variability of financial assets over a period specifiedby researchers (for example, a day). The RV is a model-free estimator in the sensethat we do not have to specify the volatility dynamics.
One of the key assumptions needed for the consistency of the RV is that there areno measurement errors in observed log-prices. The measurement error is particularlycalled “microstructure noise” and is due to, for example, discreteness of prices, bid-ask bounce and irregular trading. When this assumption is violated, the RV is nolonger a consistent estimator of the IV. It can be shown that under the existence ofmicrostructure noises, the RV diverges as the sampling frequency tends to be high.Several alternative estimators of the RV, that are consistent even under the existenceof microstructure noises, have been proposed by Zhou (1996), Zhang, Mykland andAlt-Sahalia (2005), Hansen and Lunde (2006) and Bandi and Russell (2006). Seealso Bandi and Russell (2008) which consider a mean-squared-error optimal samplingtheory for reducing the effect of microstructure noise.
In this paper, we propose a new approach for estimating the IV under the ex-istence of microstructure noises. Our method is based on the result obtained inMeddahi (2003) and Barndorff-Nielsen and Shephard (2002), namely, when the un-derlying true log-price process follows a certain class of continuous-time SV models,the IV follows an ARMA(1, 1) process. Assuming that an observed log-price isthe sum of the true log-price and an i.i.d. microstructure noise, we show that theRV calculated from the observed log-prices, which we call “noise-contaminated RV(NCRV)”, follows an ARMA(1, 2) process. Then, we represent the NCRC by a statespace form showing that the model parameters are identifiable. We estimates theIV as well as microstructure noises by applying the Kalman filtering to the statespace form. The proposed method is applied to yen/dollar spot exchange rate data.We find that the magnitude of a microstructure noise is, in average, about 23% ∼32% of the NCRV for each day depending on the sampling frequency.
The rest of the paper is organized as follows. In the next section, we introduce
2
a class of SV models employed in this paper. We also define formally the RV, IV,and microstructure noise in this section. In Section 3, we explain our approach forestimating IV and microstructure noises in details. Section 4 provides an empiricalanalysis applying our method to yen/dollar spot exchange rate. The last sectionprovides summary and concluding remarks.
2 Continuous Time SV model and RV
2.1 One factor SR-SARV model
Let p(t) be the log of (efficient) spot price at time t. We suppose that p(t) follows theclass of continuous-time SV model considered in Meddahi (2003), which is termedas “one factor square-root stochastic autoregressive variance (SR-SARV) model.”In this class of models, p(t) follows the diffusion process defined as
dp(t) = σ(t)dW (t), σ2(t) = σ20 + ω0P (f(t)), (1)
where f(t) is a state-variable process and the function P (·) is defined so that
E[P (f(t))] = 0, var[P (f(t))] = 1,E[P (f(t+ h))|f(s), p(s), s ≤ t] = exp(−λ0h)P (f(t)), ∀h > 0.
(2)
Note that E[σ2(s)] = σ20 and var[σ2(s)] = ω2
0. This specification of σ2(t) in-cludes many known models such as constant elasticity of volatility (CEV) pro-cesses, GARCH diffusion models (Nelson, 1990), eigenfunction stochastic volatility(ESV) models (Meddahi, 2001) and positive Ornstein-Uhlenbeck Levy-driven mod-els (Barndorff-Nielsen and Shephard, 2001).
2.2 Integrated and realized variance
The integrated variance (IV) is defined as
IVt ≡∫ t
t−1
σ2(s)ds, t = 1, 2, ..., (3)
where the unit of t is determined depending on the objective of research. For ex-ample, if the researcher is interested in changes in the variances of daily returns, tis interpreted as a day.
Under certain conditions, IV can be consistently estimated by the estimatorknown as the realized variance (RV), which is defined as
RV(m)t ≡
m∑i=1
r(m)2
t−1+ im
, (4)
where r(m)t ≡ p(t) − p(t − 1
m) =
∫ tt− 1
mσ(s)dW (s) and m is a positive integer. Here,
and hereafter, the notation “(m)” implies that its value depends on the samplingfrequency m. For example, if t denotes a day and we take observations every fiveminutes, then m = 288 and r
(288)t denotes a five-minute return. It is well known
that, as m→∞, RV(m)t
p−→ IVt.
3
Meddahi (2003, Proposition 3.1) shows that if the true process of p(t) belongs tothe one-factor SR-SARV model introduced in the previous section, then IVt followsan ARMA(1, 1) process:
IVt = cIV + φIVt−1 + ηt + θIV ηt−1, (5)
where ηt is a white noise with var(ηt) = σ2η and cov(ηt, d
(m)s ) = 0, d
(m)t ≡ RV
(m)t −IVt,
for all t and s. The ARMA(1, 1) model parameters, cIV , φ, θIV and σ2η are expressed,
in terms of the one-factor SR-SARV model parameters, σ20, ω2
0 and λ0, as
φ = exp(−λ0), cIV = (1− φ)σ20,
θIV =1−
√1− 4ρ2
2ρ, ρ =
−φ+ corr[IVt, IVt−1]
1 + φ2 − 2φcorr[IVt, IVt−1],
σ2η =
(1 + φ2)var[IVt]− 2φcov[IVt, IVt−1]
1 + θ2IV
,
(6)
where var(IVt), cov(IVt, IVt−1) and corr(IVt, IVt−1) are given as
var[IVt] =2ω2
0
(log φ)2(φ− log φ− 1), cov[IVt, IVt−1] =
ω20
(log φ)2(1− φ)2,
corr[IVt, IVt−1] =(1− φ)2
2(φ− log φ− 1).
(7)
Also, σ2(m)d ≡ var[d
(m)t ] is given as
σ2(m)d =
2σ40
m+
4ω20m
(log φ)2
(φ
1m − 1− log φ
1m
), ∀m ≥ 1. (8)
See Meddahi (2003) for the above results. Meddahi (2003, Proposition 3.2) shows
that RV(m)t = IVt + d
(m)t also follows an ARMA(1, 1) process. Note that the four
ARMA(1, 1) model parameters, cIV , φ, θIV and σ2η, are completely determined as
functions of the three one-factor SR-SARV model parameters, σ20, ω0 and λ0.
2.3 Microstructure noise
Now suppose that the observed log-price p∗(t) is contaminated by a measurementerror or a microstructure noise so that
p∗(t) = p(t) + ε(t). (9)
We assume the following properties on the microstructure noise ε(t).
Assumption 1
(a) ε(t) ∼ i.i.d.(0, σ2ε) with E[ε4(t)] <∞.
(b) ε(t) is independent from p(t) for all s and t.
4
We do not assume any distribution for ε(t). The observed return is defined as
r∗(m)t ≡ p∗(t)− p∗
(t− 1
m
)= r
(m)t + e
(m)t , (10)
where e(m)t = ε(t)− ε(t− 1
m). It is easy to show that E
[e(m)t
]= 0,
var[e(m)t
]= 2σ2
ε and cov[e(m)t , e
(m)
t− im
]=
{−σ2
ε , i = 1,0 i ≥ 2.
(11)
Note that var[e(m)t ] and cov[e
(m)t , e
(m)
t− 1m
] do not depend on m. We define the “noise-
contaminated RV (NCRV)”, denoted by RV∗(m)t , as RV
∗(m)t ≡
m∑i=1
r∗(m)2
t−1+ im
. Write
RV∗(m)t =
m∑i=1
(r(m)
t−1+ im
+ e(m)
t−1+ im
)2
,
= RV(m)t + u
(m)t .
(12)
where
u(m)t ≡ 2
m∑i=1
r(m)
t−1+ im
e(m)
t−1+ im
+m∑i=1
e(m)2
t−1+ im
. (13)
Note that, unlike RV(m)t and RV
∗(m)t , u
(m)t is not necessarily positive since the first
term of u(m)t can be negative. We call u
(m)t “microstructure noise in NCRV” where
we need to distinguish it from ε(t). Otherwise, where there is no ambiguity, we
simply call u(m)t microstructure noise. This microstructure noise will be estimated
in the later section.In the Appendix, we show that
E[u
(m)t
]= 2mσ2
ε , and
cov[u
(m)t , u
(m)s
]=
8σ2
εσ20 + 2(2m− 1)ω2
ε + 4mσ4ε t = s,
ω2ε t = s± 1,
0 otherwise,
(14)
where ω2ε = var[ε2(t)] = E[ε4(t)]− σ4
ε . Thus, u(m)t has the autocovariance structure
of a MA(1) process. Suppose that the MA(1) process is represented as
u(m)t = c(m)
u + ξ(m)t + θ(m)
u ξ(m)t−1 , ξ
(m)t−1 ∼ WN(0, σ
2(m)ξ ). (15)
The mean and autocovariances of u(m)t in terms of c
(m)u , θ
(m)u and σ
2(m)ξ are
E[u
(m)t
]= c
(m)u , and
cov[u
(m)t , u
(m)s
]=
(1 + θ
2(m)u )σ
2(m)ξ t = s,
θ(m)u σ
2(m)ξ t = s± 1,
0 otherwise.
(16)
Later, we utilize these two different expressions of the moments of u(m)t to derive the
relationships among parameters.
5
3 State Space Approach
Our approach is in the same spirit of the state space method used in Barndorff-Nielsen and Shephard (2002), where they consider the situation without microstruc-ture noise. The existence of microstructure noises requires additional efforts forchecking identification of state space model parameters. In this section, we describethe identification problem in estimating the IV from the NCRC by a state spacemethod.
3.1 State space form of NCRV
Since RV(m)t = IVt + d
(m)t , we have
RV∗(m)t = IVt + d
(m)t + u
(m)t . (17)
From (5), (15) and (17), we have the following state space form of RV∗(m)t :
(Observation equation)
RV∗(m)t =
[1 1 0 0
] IVtu
(m)t
αtβt
+ d(m)t , (18)
(State equation)IVtu
(m)t
αtβt
=
cIVc(m)u
00
+
φ 0 θIV 0
0 0 0 θ(m)u
0 0 0 00 0 0 0
IVt−1
u(m)t−1
αt−1
βt−1
+
1 00 11 00 1
[ ηtξt], (19)
where d(m)t
ηtξt
∼ 0
00
, σ
2(m)d 0 00 σ2
η 0
0 0 σ2(m)ξ
. (20)
Given the values of φ, cIV , θIV , σ2η, c
(m)u , θ
(m)u , σ
2(m)ξ and σ
2(m)d , we can estimate IVt
and u(m)t by applying the Kalman filtering to the state space form. One problem of
the state space form is how to estimate those parameters. One may simply think thatwe can estimate them from the state space form by, e.g., quasi-maximum likelihoodestimation under Gaussian noise assumption. This is; however, not always possible.It is known that in general the parameters of state space form are not necessarilyidentified (see, for example, Hamilton, 1994, p.388). More precisely, there may beinfinitely many combinations of state space model parameters that give the sameautocovariance structure. We have to confirm whether the model parameters areuniquely identified before parameter estimation. We consider this problem in thenext section. In fact, we show that the above parameters in the state space formcannot be uniquely identified.
6
3.2 Identification of model parameters
Since RV∗(m)t is the sum of three components: IVt (an ARMA(1, 1) process), d
(m)t (a
white noise process) and u(m)t (an MA(1) process), RV
∗(m)t itself follows an ARMA(1,
2) process (see Granger and Morris, 1976) so that it is expressed as
(1− φL)RV∗(m)t = c(m) + (1 + δ
(m)1 L+ δ
(m)2 L2)τt, τt ∼ WN(0, σ2(m)
τ ). (21)
Note that the AR coefficient φ is the same as that of IVt in (5). The ARMA modelof a state space representation is commonly referred to as a reduced form or ARMAreduced form. Note that the parameters of any ARMA reduced form are alwaysidentified (and hence can be estimated). From (5), (17) and (15), we have
(1− φL)RV∗(m)t = (1− φL)IVt + (1− φL)d
(m)t + (1− φL)u
(m)t
= (1− φ)σ20 + ηt + θIV ηt−1 + d
(m)t − φd(m)
t−1 + ξt+(1− φ)c
(m)u + (θ
(m)u − φ)ξt−1 − φθ(m)
u ξt−2.
(22)
These two expressions on the right-hand sides in (21) and (22) are of the same modeland hence their mean and autocovariances must be identical. The autocovariancesof the MA parts in (21) are given as γ
(m)0 = (1 + δ
(m)21 + δ
(m)22 )σ
2(m)τ , γ
(m)1 = (δ
(m)1 +
δ(m)1 δ
(m)2 )σ
2(m)τ , γ
(m)2 = δ
(m)2 σ
2(m)τ , and γj = 0 for j ≥ 3. It is shown in the
Appendix that the autocovariance functions of the MA parts in (22) are
γ(m)0 = (1 + θ2
IV )σ2η + (1 + φ2)σ
2(m)d + [1 + (θ(m)
u − φ)2 + φ2θ(m)2u ]σ
2(m)ξ , (23a)
γ(m)1 = θIV σ
2η − φσ
2(m)d + (θ(m)
u − φ− φθ(m)2u + φ2θ(m)
u )σ2(m)ξ (23b)
γ(m)2 = −φθ(m)
u σ2(m)ξ , (23c)
and γj = 0 for j ≥ 3. By equating the means of the MA parts in (21) and (22),we obtain
c(m) = (1− φ)(σ2
0 + c(m)u
). (23d)
Given ARMA(1, 2) model parameters in (21), we can obtain c(m), φ, and γ(m)j ,
j = 0, 1, 2. Then, unknown parameters in the equations in (23a)∼(23d) are θIV , σ20,
σ2η, c
(m)u , θ
(m)u , σ
2(m)ξ and σ
2(m)d . Observe that the number of unknown parameters
are more than the number of equations. Hence, we cannot uniquely identify theseparameters from these equations. In other word, for a given ARMA(1, 2) model,
there are infinitely many combinations of parameters θIV , σ20, σ2
η, c(m)u , θ
(m)u , σ
2(m)ξ
and σ2(m)d that give the same autocovariance structure as the ARMA(1, 2) model.
3.3 Estimation
We reduce the number of unknown parameters as follows. Equations (6) and (7)
imply that θIV , σ2η and σ2
d are functions of φ, σ20, ω2
0. Below we show that c(m)u , θ
(m)u
and σ2(m)ξ can also be expressed as functions of σ2
0, ω2ε and σ2
ε .1 Substituting these
1They also depend on m as the notation implies.
7
functions into (23a)∼(23d), we can reduce the number of unknown parameters. Toderive the relationships among those parameters, we utilized the expressions givenin (14) and (16).
In views of (14) and (16), we obtain the following equations:
c(m)u = 2mσ2
ε , (24a)
(1 + θ2(m)u )σ
2(m)ξ = 8σ2
0σ2ε + 2(2m− 1)ω2
ε + 4mσ4ε , (24b)
θ(m)u σ
2(m)ξ = ω2
ε . (24c)
Assuming the MA parameter satisfies the invertibility condition, i.e., |θ(m)u | < 1, we
can solve the equations (24a) ∼ (24c) for c(m)u , θ
(m)u and σ
2(m)ξ as
c(m)u = 2mσ2
ε , σ2(m)ξ =
ω2ε
θ(m)u
and θ(m)u = A−
√A2 − 1, (25)
where A = 4σ20σ
2ε
ω2ε
+ 2m − 1 + 2mσ4ε
ω2ε. The detail of the calculation is given in the
Appendix. Note that 0 < θ(m)u < 1 since A > 1.
From (6), (7) and (25), we see that cIV , θIV , σ2η, c
(m)u , θ
(m)u , σ
2(m)ξ and σ
2(m)d can
be expressed as functions of φ, σ20, ω2
0, σ2ε and ω2
ε . To emphasize this, we may writethem as
cIV = cIV (φ, σ20), θIV = θIV (φ), σ2
η = σ2η(φ, ω
20), c
(m)u = c
(m)u (σ2
ε),
θ(m)u = θ
(m)u (σ2
0, σ2ε), σ
2(m)d = σ
2(m)d (φ, σ2
0, ω20) and σ
2(m)ξ = σ
2(m)ξ (σ2
0, σ2ε , ω
2ε).
(26)
Note that θIV is a function of only φ, hence can be assumed to be known (sinceφ is identified from the reduced form). Substituting the expressions in (26) intothe equations in (23a)∼(23d), eventually, we have four equations for the four un-known parameters σ2
0, ω20, ω2
ε and σ2ε . Hence, the order condition for identification is
satisfied. However, this result only does not necessarily mean that for a given auto-covariance structure, we can uniquely identify σ2
0, ω20, ω2
ε and σ2ε . For the uniqueness
of the identification, we must check if the rank condition is also satisfied. To show theuniqueness of the identification, we explicitly derive the solutions for the parametersin terms of c, φ, γ
(m)j j = 0, ..., 2.
In the Appendix, we show that, given c, φ, γ(m)j j = 0, ..., 2 and (26), the equa-
tions in (23a)∼(23d) can be uniquely2 solved for σ20, ω2
0, σ2ε and ω2
ε as
ω2ε = −γ
(m)2
φ, ω2
0 =(log φ)2[φγ
(m)0 + (1 + φ2)γ
(m)1 + 1+φ4
φγ
(m)2 ]
(1− φ)3(1 + φ), (27a)
σ2ε =
√c2
2m2(1− φ)2− (2m− 1)γ
(m)2
2mφ− γ
(m)0 − 2ω2
0D − 2γ(m)2
4m(1 + φ2), (27b)
2More precisely, under the condition σ2ε > 0.
8
and
σ20 =
c
1− φ− 2mσ2
ε , (27c)
where D = B + (1 + φ2)C,
B ≡ φ2 − 1− (1 + φ2) log φ
(log φ)2and C ≡
2m(φ
1m − 1− log φ
1m
)(log φ)2 . (27d)
The equations in (27a) ∼ (27d) show that the four parameters, σ20, ω2
0, σ2ε and ω2
ε
can be uniquely identified from the ARMA(1, 2) reduced form of the state spaceform in (18) and (19). It should be emphasized that this does not imply that wecan directly estimate the state space model parameters in (18) and (19), but impliesthat we can estimate the above four parameters directly by replacing the statespace model parameters with the functions of the four parameters. We estimatethese four parameters by the quasi-maximum likelihood estimation of ARMA(1, 2)model, where innovations are (conveniently) assumed to be Gaussian.
Here, we summarize how to calculate the likelihood.
(Summary on how to calculate the likelihood)
(1) For a given m, calculate RV∗(m)t .
(2) Given φ, σ20, ω2
0, σ2ε and ω2
ε , calculate cIV , θIV , σ2η, c
(m)u , θ
(m)u , σ
2(m)ξ and σ
2(m)d
according to (6), (7) and (25).
(3) With cIV , θIV , σ2η, c
(m)u , θ
(m)u , σ
2(m)ξ and σ
2(m)d obtained in Step (2), compute
γ(m)j , j = 1, ...3 and c(m) in (23a)∼(23d).
(4) Applying a result in Meddahi (2002), (representations of MA(2) parameters interms of the autocovariance functions) calculate corresponding ARMA(1, 2)
parameters, c(m), δ(m)1 , δ
(m)2 and σ
2(m)ξ from γ
(m)j , j = 1, ...3 obtained in Step
(3).
(5) Calculate the Gaussian ARMA(1, 2) log-likelihood for RV ∗t with φ, c(m) δ
(m)1 ,
δ(m)2 and σ
2(m)ξ .
We can obtain the estimates of σ20, ω2
0, σ2ε and ω2
ε by maximizing the abovelog-likelihood with respect to the four parameters. The method provides consistentestimators for the four parameters. Using the proposed methodology, we conductan empirical analysis for exchange rate data in the next section.
4 Empirical Analysis
4.1 Data description
The yen/dollar spot exchange rate series we use are the mid-quote prices observedevery 1 minute, which are obtained from Olsen and Associates. The full sample
9
covers the period from 1 January 2000 to December 31 2006. Figure 1 plots thedaily returns calculated from the data over the period.
We interpolate missing prices by the previous tick method, i.e., interpolating theprevious prices. Also, following Andersen, Bollerslev, Diebold and Labys (2001), weremove the data of inactive trading days. Whenever we did so, we always cut from21:01 GMT on one night to 21:00 the next evening. For details on the motivationof this definition of “day”, see Andersen, Bollerslev, Diebold and Labys (2001),Andersen and Bollerslev (1998) and Bollerslev and Domowitz (1993) . We cut thedata according to the following criterions, which is similar to the criterions used inBeine et al. (2007):
(1) the days that miss more than 500 price observations,
(2) the days where, in total, there are more than 1000 minutes of zero returns
(3) the days where the price did not change for more than 35 minutes.
By these criterions, we can remove all weekend data. However, the days such as U.S.holidays that Andersen, Bollerslev, Diebold and Labys (2001) and Beine et al. (2007)removed are not necessarily cut by these criterions. This is because even when theU.S. market is closed, the transactions are made in other markets. Eventually, weare left with 1809 complete days, or 1809× 1440 = 2604960 price observations fromwhich we calculate the 1-min and 5-min returns.
With these returns, we calculate the series of daily NCRVs. We call the NCRVs1-min NCRV (m = 1440) or 5-min NCRV (m = 288) depending on the intervals ofthe returns. Table 1 reports the descriptive statistics of 1-min and 5-min NCRVs.Figure 2 plots these two NCRVs. The mean of 1-min NCRV is higher than that of5-min NCRV. This can be due to microstructure noises since the mean of NCRVincreases as the sampling frequency tends to be high, or m → ∞, as shown in(23d) and (24a). The first order autocorrelation of NCRVs are somewhat lowerthan usually expected for variances of financial time series (0.4794 for 1-min NCRVand 0.4177 for 5-min RV).
4.2 Estimation of parameters, IV and microstructure noise
For these two series of NCRVs, we estimate parameters, φ, σ20, ω2
0, σ2ε and ω2
ε , asdescribed in the previous section. Note that, in general, the values of 1-min and 5min NCRVs are different and thus the estimates from these two NCRVs are differ-ent. Table 2 shows the estimates of the above parameters and the values of statespace form parameters in (18) and (19) computed from the estimates. The notation“(m)” implies that those values depend on the value of m. Notice that even thoughwe used the series of NCRVs with different m’s for the estimations, the estimatedvalues of parameters that do not depend on m are very similar. For example, theestimated value of σ2
0, the unconditional mean of the IVt, is 0.3581 for m = 1440 and0.3781 for m = 288. The estimated value of ω2, the unconditional variance of IVt, is0.0301 for for m = 1440 and 0.0279 for m = 288. The estimates of the first order au-tocorrelation of IV are significantly higher than those of NCRVs. This suggests thatthe observed low autocorrelations of the NCRVs are due to microstructure noises.One interesting observation is that although we do not assume any distribution for
10
the microstructure noise ε(t), the kurtosis of ε(t), calculated from the estimatedvariances of σ2
ε = var[ε(t)] and ω2ε = var[ε(t)2], are close to 3 in the case of 5-min
NCRV.We display the estimates of the IV by Kalman smoothing in Figure 3(a) for
5-min NCRV, 3(b) for 1-min NCRV and 3(c) for the both. Note that both are theestimate of the same IV series. These two series of IV estimates are very similar asseen in Figure 3(c). Figure 4 plots the smoothed estimates of microstructure noise
u(m)t ’s. The results are natural. We find that 1-min NCRV has a larger bias than
5-min NCRV. The mean of the microstructure noises for 1-min NCRV is higher thanthat for 5-min NCRV. The small biases of 5-min NCRV is consistent with the useof 5-min NCRV in the previous literature.
Last, we calculate the ratio of microstructure noises to the NCRVs, i.e., R(m) =
u(m)t /RV
∗(m)t for each m, t = 1, ..., 1809, where u
(m)t is the smoothed estimates of
u(m)t . The maximum and minimum values are, respectively, 0.6970 and −4.2179
for R(288)(5 min NCRV) and 0.7832 and −0.7201 for R(1440)(1 min NCRV). Wealso calculate the average magnitude of microstructure noise ut(m)’s as the mean ofR(m) and |R(m)|. The values of the means of R(m) and |R(m)| are, respectively,−0.0458 and 0.2341 for m = 288, and 0.3073 and 0.3273 for m = 1440.
5 Summary and Concluding Remarks
In this paper, we proposed a new approach, using a state space form of the realizedvariance, to estimating the integrated variance. Our method is based on the resultin Meddahi (2001), which shows that when the prices follows a certain class ofstochastic volatility models, the integrated variance follows a ARMA(1, 1) model.We showed that under the existence of microstructure noises, the observed realizedvariance follows a ARMA(1 ,2) model. We represented the ARMA(1, 2) model by astate space form and established the uniqueness of the identification of state spaceform parameters. The proposed method is applied to yen/dollar exchange rate data,where we find that the RV calculated with 5-min returns are less biased than with1-min returns. The two series of IV estimates obtained by the proposed methodwith 1-min returns and 5-min returns are very similar. The method was also usedfor estimating microstructure noises.
In the estimation, we constructed the likelihood using only either 1-min or 5-min RV. It is more desirable to use both RVs for estimating common parameters.Investigating the optimal way for combining RVs of different scales is a subject offuture research. It is also interesting to apply our method to stock return data andexamine the effects of microstructure noises.
11
Appendix
Hereafter, we suppress “(m)” in the notations r(m)t , u
(m)t and e
(m)t , for simplicity.
Derivation of (14)
First, we derive E[ut]. Since rt and et are independent by Assumption 1 andvar(et) = 2σ2
ε , we have
E[ut] = 2m∑i=1
E[rt−1+ i
met−1+ i
m
]+
m∑i=1
E[e2t−1+ i
m
]=
m∑i=1
E[rt−1+ i
m
]E[et−1+ i
m
]+mvar(et)
= 2mσ2ε .
(28)
The following results are used for deriving var[ut] in (34) and cov[ut, ut−1] in (35)below: For any s and t, we have
cov [rses, rtet] = E [rsesrtet]− E [rses]E [rtet]= E [eset]E [rsrt]− E [rs]E [es]E [rt]E [et] .
(29)
Thus, when t = s we have
cov [rses, rtet] = 2σ2εE [r2
t ]
= 2σ2εE[(
∫ tt−1/m
σ(s)dW (s))2]
= 2σ2εE[∫ tt−1/m
σ2(s)ds]
= 2σ2εE[∫ t
t− 1mσ2(s)ds
]=
2σ2εσ
20
m.
(30a)
The third equality comes from the Ito isometry. When t 6= s, we have
cov [rses, rtet] = E [eset]E [rs] [rt]− E [rs]E [es]E [rt]E [et] = 0. (30b)
Next, we derive cov [e2t , e2s]. When t = s, we have
cov[e2t , e2t ] = var[e2t ]
= E[e4t ]− (E[e2t ])2
= E[ε4t − 4ε3
t εt− 1m
+ 6ε2t ε
2t− 1
m
− 4εtε3t− 1
m
+ ε4t− 1
m
]− 4σ4
ε
= 2E[ε4t ] + 2σ4
ε .
(31)
When t = s± 1m
, we have
cov[e2s, e
2s− 1
m
]= cov
[e2s+ 1
m
, e2s
]= cov
[ε2s+ 1
m
− 2εs+ 1mεs + ε2
s, ε2s − 2εsεs− 1
m+ ε2
s− 1m
]= var[ε2
s]= ω2
ε .
(32)
12
When t = s± im
for i ≥ 2, we have cov[et, es] = 0. We have cov[rtet, e2s] = 0 for any
t and s since
cov[rtet, e2s] = E[rtete
2s]− E[rtet]E[e2s]
= E[rt]E[ete2s]− E[rt]E[et]E[e2s]
= 0.(33)
From (30) ∼ (33) we have.
var [ut] = var
[2m∑i=1
rt−1+ imet−1+ i
m+
m∑i=1
e2t−1+ i
m
]= 4var
[m∑i=1
rt−1+ imet−1+ i
m
]+ var
[m∑i=1
e2t−1+ i
m
]+4cov
[m∑i=1
rt−1+ imet−1+ i
m,m∑i=1
e2t−1+ i
m
]= 4cov
[m∑i=1
rt−1+ imet−1+ i
m,m∑i=1
rt−1+ imet−1+ i
m
]+ cov
[m∑i=1
e2t−1+ i
m
,m∑i=1
e2t−1+ i
m
]+4cov
[m∑i=1
rt−1+ imet−1+ i
m,m∑i=1
e2t−1+ i
m
]= 4
m∑i=1
m∑j=1
cov[rt−1+ i
met−1+ i
m, rt−1+ j
met−1+ j
m
]+
m∑i=1
m∑j=1
cov[e2t−1+ i
m
, e2t−1+ j
m
]+4
m∑i=1
m∑j=1
cov[rt−1+ i
met−1+ i
m, e2t−1+ i
m
]= 8σ2
εσ20 + 2m(E[ε4
t ] + σ4ε) + 2(m− 1)(E[ε4
t ]− σ4ε)
= 8σ2εσ
20 + 2(2m− 1)ω2
ε + 4mσ4ε ,
(34)
and
cov[ut, ut+1] = cov
[2m∑i=1
rt−1+ imet−1+ i
m+
m∑i=1
e2t−1+ i
m
, 2m∑i=1
rt+ imet+ i
m+
m∑i=1
e2t+ i
m
]= 4cov
[m∑i=1
rt−1+ imet−1+ i
m,m∑i=1
rt+ imet+ i
m
]+2cov
[m∑i=1
rt−1+ imet−1+ i
m,m∑i=1
e2t+ i
m
]+2cov
[m∑i=1
rt+ imet+ i
m,m∑i=1
e2t−1+ i
m
]+ cov
[m∑i=1
e2t−1+ i
m
,m∑i=1
e2t+ i
m
]= cov
[e2t , e
2t+ 1
m
]= ω2
ε .
(35)
It is easy to check cov[ut, ut±i] = 0 for i ≥ 2, and hence we have (14).
13
Derivation of (23a)∼(23c)
γ0 = cov{ηt + θIV ηt−1 + dt − φdt−1 + ξt + (θu − φ)ξt−1 − φθuξt−2,ηt + θIV ηt−1 + dt − φdt−1 + ξt + (θu − φ)ξt−1 − φθuξt−2}
= σ2η + θ2
IV σ2η + σ2
d + φ2σ2d + σ2
ξ + (θu − φ)2σ2ξ + φ2θ2
uσ2ξ
= (1 + θ2IV )σ2
η + (1 + φ2)σ2d + [1 + (θu − φ)2 + φ2θ2
u]σ2ξ ,
γ1 = cov{ηt + θIV ηt−1 + dt − φdt−1 + ξt + (θu − φ)ξt−1 − φθuξt−2,ηt−1 + θIV ηt−2 + dt−1 − φdt−2 + ξt−1 + (θu − φ)ξt−2 − φθuξt−3}
= θIV σ2η − φσ2
d + (θu − φ− φθ2u + φ2θu)σ
2ξ
γ2 = cov{ηt + θIV ηt−1 + dt − φdt−1 + ξt + (θu − φ)ξt−1 − φθuξt−2,ηt−2 + θIV ηt−3 + dt−2 − φdt−3 + ξt−2 + (θu − φ)ξt−3 − φθuξt−4}
= −φθuσ2ξ .
Derivation of (25)
From (24c), we have σ2ξ = ω2
ε/θu. Substituting this into (24b), we have
(1 + θ2u)
ω2ε
θu= 8σ2
0σ2ε + 2(2m− 1)ω2
ε + 4mσ4ε
⇔ ω2εθ
2u − [8σ2
0σ2ε + 2(2m− 1)ω2
ε + 4mσ4ε ]θu + ω2
ε = 0
⇔ θ2u − 2[4
σ20σ
2ε
ω2ε
+ 2m− 1 + 2mσ4ε
ω2ε]θu + 1 = 0
⇔ θu = A±√A2 − 1,
(36)
where A = 4σ20σε
ω2ε
+2m−1+2mσ4ε
ω2ε. Since A > 1 for m ≥ 1, we have A+
√A2 − 1 > 1.
Assuming that θu satisfies the invertibility condition, we obtain θu in (25).
Derivation of (27a) and (27c)
From (23c) and (24c), we have ω2ε = −γ2
φ. From (6) and (7), we have
σ2η =
2ω20
1 + θ2IV
B and σ2d =
2σ40
m+ 2ω2
0C, (37)
where B and C are given in (27d). From ω2ε = θuσ
2ξ in (24c), we have
(1 + θ2u − 2θuφ+ φ2 + φ2θ2
u)σ2ξ =
(1θu
+ θu − 2φ+ φ2
θu+ φ2θu
)ω2ε
=[(
1θu
+ θu
)(1 + φ2)− 2φ
]ω2ε ,
(38)
and
(θu − φ− φθ2u + φ2θu)σ
2ξ =
(1− φ
θu− φθu + φ2
)ω2ε
=[1 + φ2 −
(1θu
+ θu
)φ]ω2ε .
(39)
Substituting (37), (38) and (39) into (23a) and (23b), we have
γ0 = 2ω20D + 2σ4
0
1 + φ2
m+
[(1
θu+ θu
)(1 + φ2)− 2φ
]ω2ε , (40a)
14
and
γ1 = 2ω20E − 2σ4
0
φ
m−[(
1
θu+ θu
)φ− (1 + φ2)
]ω2ε , (40b)
where D = B + (1 + φ2)C and E = θIV
1+θ2IVB − φC. From (6), we have
θIV1 + θ2
IV
=2ρ(1−
√1− 4ρ2)
4ρ2 + (1−√
1− 4ρ2)2
=2ρ(1−
√1− 4ρ2)(1 +
√1− 4ρ2)
4ρ2(1 +√
1− 4ρ2) + (1−√
1− 4ρ2)2(1 +√
1− 4ρ2)= ρ,
(41)
and
ρB =−2φ(φ− log φ− 1) + (1− φ)2
2(log φ)2(42)
Hence, we have
φγ0 + (1 + φ2)γ1 = 2 [φD + (1 + φ2)E]ω20 + [(1 + φ2)2 − 2φ2]ω2
ε ,= 2 [φ+ (1 + φ2)ρ]Bω2
0 + (1 + φ4)ω2ε ,
=(1− φ)3(1 + φ)
(log φ)2ω2
0 + (1 + φ4)ω2ε ,
(43)
or
ω20 =
(log φ)2[φγ0 + (1 + φ2)γ1 − (1 + φ4)ω2ε ]
(1− φ)3(1 + φ). (44)
Substituting ω2ε = −γ2
φ, we obtain (27a). Next, note that from (25), we have
1
θu+ θu =
1 + θ2u
θu
=1 + (A−
√A2 − 1)2
A−√A2 − 1
=A+√A2 − 1 + (A−
√A2 − 1)2(A+
√A2 − 1)
(A−√A2 − 1)(A+
√A2 − 1)
= 2A.
(45)
From (23d) and (24a), we have
c = (1− φ)(σ2
0 + 2mσ2ε
)or σ2
ε =c− (1− φ)σ2
0
2(1− φ)m. (46)
Substituting σ2ε in (46) into A, we have
2A = 2
[4σ2
0
(c−(1−φ)σ2
0
2(1−φ)m
)1ω2
ε+ 2m− 1 + 2m
(c−(1−φ)σ2
0
2(1−φ)m
)21ω2
ε
]
= 2[
2cσ20
(1−φ)mω2ε− 2σ4
0
mω2ε
+ 2m− 1 +(c2−2(1−φ)cσ2
0+(1−φ)2σ40
2(1−φ)2mω2ε
)]=
4cσ20
(1−φ)mω2ε− 4σ4
0
mω2ε
+ 2(2m− 1) + c2
(1−φ)2mω2ε− 2cσ2
0
(1−φ)mω2ε
+σ40
mω2ε
=2cσ2
0
(1−φ)mω2ε− 3σ4
0
mω2ε
+ 2(2m− 1) + c2
(1−φ)2mω2ε.
(47)
15
From (40), (45) and (47), we have
γ0 = 2ω20D−
(1 + φ2)
mσ4
0 +2(1 + φ2)c
(1− φ)mσ2
0 +2(2m−1)(1+φ2)ω2ε +
(1 + φ2)c2
(1− φ)2m−2φω2
ε .
(48)
Multiplying both sides in (48) by m/(1 + φ2) and arranging, we have
σ40 −
2c
1− φσ2
0 −c2
(1− φ)2+m (γ0 − 2ω2
0D + 2φω2ε)
1 + φ2− 2m(2m− 1)ω2
ε = 0. (49)
Solving this quadratic equation for σ20, we have
σ20 =
c
1− φ±
√2c2
(1− φ)2+ 2m(2m− 1)ω2
ε −m (γ0 − 2ω2
0D + 2φω2ε)
(1 + φ2). (50)
From σ2ε > 0 and (46), we must have c
1−φ > σ20. Hence, the sign of the second term
in (50) is negative. From (46), we have
σ2ε =
1
2m
√2c2
(1− φ)2+ 2m(2m− 1)ω2
ε −m (γ0 − 2ω2
0D + 2φω2ε)
(1 + φ2). (51)
From the above results, we obtain (27b) and (27c).
16
References
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18
Table 1: Descriptive statistics of the NCRV
1-min NCRV 5-min NCRV
m 1440 288
Mean 0.5317 0.4039Variance 0.0629 0.0620
S.D. 0.2507 0.2490
SACORR(1) 0.4794 0.4177SACORR(2) 0.3628 0.3292SACORR(3) 0.3261 0.2819SACORR(4) 0.3294 0.2595SACORR(5) 0.3246 0.2577
Note: the table reports the mean, variance and standard deviation of RV calculatedwith different m. SACORR(k) denotes the sample autocorrelation of order k.
19
Table 2: Estimates of parameters
1-min NCRV 5-min NCRV
φ 0.9301 0.8849(0.0516) (0.0329)
σ20 0.3581 0.3781
(0.0895) (0.2155)ω2
0 0.0301 0.0279(0.0084) (0.0324)
σ2ε 6.0915× 10−5 4.5457× 10−5
(5.4324× 10−5) (1.9086× 10−4)ω2ε 5.8662× 10−6 2.9568× 10−5
(9.6479× 10−7) (7.5890× 10−6)L 240.0671 181.187
corr(IVt, IVt−1) 0.9531 0.9225cIV 0.0372 0.0435
θIV 0.2679 0.2677σ2η 0.0025 0.0038
m 1440 288
c(m)u 0.1754 0.0262
θ(m)u 1.7267× 10−4 8.6660× 10−4
σ2(m)ξ 0.0340 0.0341
σ2(m)d 2.199× 10−4 0.0012
var[IVt]
var[IVt] + var[u(m)t ] + var[d
(m)t ]
0.4618 0.4313
var[u(m)t ]
var[IVt] + var[u(m)t ] + var[d
(m)t ]
0.5348 0.5496
σ2η
σ2η + σ2
ξ
0.0690 0.0992
σ2ξ
σ2η
0.0742 0.1101
E[ε4t ]/σ
4ε 1.9239 3.0907
Note: the robust standard errors are inside the parenthesis. L is the log-likelihood.
20
21
Figure 1 : Daily returns of yen/dollar exchange rate
2000.7 2001.7 2002.7 2003.7 2004.7 2005.7 2006.7-3
-2
-1
0
1
2
3
YEAR
RET
UR
N (%
)
Figure 2: 1-min NCRV and 5 min NCRV
2000.7 2001.7 2002.7 2003.7 2004.7 2005.7 2006.70
0.5
1
1.5
2
2.5
3
3.5
YEAR
VAR
IAN
CE
(%)
1 min NCRV5 min NCRV
22
Figure 3: Smoothed IV estimates for 1 min and 5 min NCRVs (a) 5 min
2000.7 2001.7 2002.7 2003.7 2004.7 2005.7 2006.70
0.5
1
1.5
2
2.5
3
3.5
YEAR
VAR
IAN
CE
(%)
Smoothed IV (5 min) 5 min NCRV
(b) 1 min
2000.7 2001.7 2002.7 2003.7 2004.7 2005.7 2006.70
0.5
1
1.5
2
2.5
3
3.5
YEAR
VAR
IAN
CE
(%)
Smoothed IV (1 min)1 min NCRV
23
(c) 1 min and 5 min
2000.7 2001.7 2002.7 2003.7 2004.7 2005.7 2006.70
0.5
1
1.5
2
2.5
3
3.5
YEAR
VAR
IAN
CE
(%)
Smoothed IV (1 min)Smoothed IV (5min)
Figure 4: Estimate of microstructure noise for 1 min and 5 min NCRVs
2000.7 2001.7 2002.7 2003.7 2004.7 2005.7 2006.7
-0.5
0
0.5
1
1.5
2
2.5
YEAR
MIC
RO
STR
UC
TUR
E N
OIS
E(%
)
Microstructure noise (1 min)Microstructure noise (5 min)