In the following we briefly present two more elaborate, but under specific noise assumptions consistent procedures for estimating IV.
13.4.1 Averaging and Subsampling
The subsampling approach originally suggested byZhang et al.(2005) builds on the idea of averaging over various RVs constructed by sampling sparsely over high-frequency subsamples. To this end the intraday observations are allocated to K subsamples. Using a regular allocation, 5 min returns can for example be sampled at the time points 9:30, 9:35, 9:40, : : :; and at the time points 9:31, 9:36, 9:41, : : : and so forth. Averaging over the subsample RVs yields the so- called average RV estimator:.1=K/PK
kD1RV.k;mk/withmkdenoting the sampling frequency used in the RV computation for subsample k. Usually, mk is equal across all subsamples. The average RV estimator is still biased, but the bias now depends on the average size of the subsamples rather than on the total number of observations.RV constructed from all observations,RV.al l/ can be used for bias correction yielding the estimator:
TTSRV.m;m1;:::;mK;K/D 1 K
XK kD1
RV.k;mk/mN
mRV.al l/; (13.4)
wheremN D.1=K/PK
kD1mk. As the estimator (13.4) consists of a component based on sparsely sampled data and one based on the full grid of price observations, the estimator is also called two time scales estimator.
Under the independent noise assumption, the estimator is consistent. Further- more, under equidistant observations and under regular allocation to the grids, the asymptotic distribution is given by:
m1=6.TTSRVq .m;m1;:::;mK;K/IV/
8
c2.!2/2Cc43IQ
!d N.0; 1/ :
forK D cm2=3. The optimal value ofK, i.e. minimizing the expected asymptotic variance, can be obtained by estimatingcoptD
12!2=IQ1=3
based on data prior to the day under consideration (seeZhang et al. 2005).
A generalization of TTSRV was introduced byA¨ıt-Sahalia et al.(2010) andZhang (2006), which is consistent and asymptotically unbiased also under time-dependent noise. To account for serial correlation in the noise, the RVs are based on overlapping J-period intraday returns. Using these so-called average-lag-J RVs the estimator becomes:
TTSRV.m;K;J /adj Ds 1 K
mKX
iD0
.p.iCK/=mpi=m/2
mN.K/
N m.J /
1 J
mJX
lD0
.p.lCJ /=mpl=m/2
!
(13.5)
withmN.K/ D .mK C1/=K,mN.J / D .mJ C 1/=J,1 J < K < m and the small sample adjustment factors D
1 Nm.K/=mN.J /1
. Note thatK and J now basically denote the slow and fast time scales, respectively. The asymptotic distribution is given by:
m1=6
TTSRV.m;K;J /adj IV q1
c22Cc43IQ
Nd .0; 1/
with2 D16
!22
C32P1
lD1.E.u0;ul//2andd denotes that when multiplied by a suitable factor, then the convergence is in distribution.
Obviously, TTSRVadj converges to IV at rate m1=6, which is below the rate of m1=4, established as optimal in the fully parametric case inA¨ıt-Sahalia et al.(2005).
As a consequence, A¨ıt-Sahalia et al. (2010) introduced the multiple time scale estimator, MTSRV, which is based on the weighted average of average-lag-J RVs computed over different multiple scales. It is computationally more complex, but for suitably selected weights it attains the optimal convergence ratem1=4.
13.4.2 Kernel-Based Estimators
Given the similarity to the problem of estimating the long-run variance of a stationary time series in the presence of autocorrelation, it is not surprising that kernel-based methods have been developed for the estimation of IV. Such an approach was first adopted inZhou(1996) and generalized inHansen and Lunde (2006), who propose to estimate IV by:
KRV.m;H /Z&HLDRV.m/C2 XH hD1
m mhh
withh D Pm
iD1ri.m/riCh.m/. As the bias correction factorm=.mh/increases the variance of the estimator, Hansen and Lunde (2006) replaced it by the Bartlett kernel. Nevertheless, all three estimators are inconsistent.
Recently,Barndorff-Nielsen et al.(2008) proposed a class of consistent kernel based estimators, realized kernels. The flat-top realized kernel:
KRV.m;H /F T DRV.m/C XH hD1
k h1
H .hCh/ ;
wherek.x/forx2Œ0; 1is a deterministic weight function. Ifk .0/D1,k .1/D0 andH D cm2=3 the estimator is asymptotically mixed normal and converges at
ratem1=6. The constantcis a function of the kernel and the integrated quarticity, and is chosen such that the asymptotic variance of the estimator is minimized. Note that for the flat-top Bartlett kernel, where k.x/ D 1 x, and the cubic kernel, kD13x2C2x3, KRV.m;H /FT has the same asymptotic distribution as the TTSRV and the MTSRV estimators, respectively.
Furthermore, ifH D cm1=2,k0.0/ D 0 andk0.1/ D 0 (called smooth kernel functions), the convergence rate becomesm1=4 and the asymptotic distribution is given by:
m1=4
KRV.m;H /FT IV q
4ckıIQC8ckı0!2IVC c43kı00!4
!d N.0; 1/
withkıDR1
0 k.x/2dx,k0ıDR1
0 k0.x/2dxandkı00DR1
0 k00.x/2dx.
For practical applications,Barndorff-Nielsen et al.(2009) consider the non-flat- top realized kernels, which are robust to serial dependent noise and to dependence between noise and efficient price. The estimator is defined as:
KRV.m;H /NFT DRV.m/C XH hD1
k h
H .hCh/ : (13.6)
However, the above mentioned advantages of this estimator come at the cost of a lower convergence rate, i.e.m1=5, and a small asymptotic bias:
m1=5
KRV.m;H /NFT IV ds
!MN
c2ˇˇk00.0/ˇˇ!2; 4ckıIQ
;
where ds denotes stable convergence and MN a mixed normal distribution.
Barndorff-Nielsen et al. (2009) recommend the use of the Parzen kernel as it is smooth and always produces non-negative estimates. The kernel is given by:
k .x/D 8<
:
16x2C6x3for0x < 1=2 2.1x/3 for1=2x 1 0 forx > 1
: (13.7)
For non-flat-top realized kernels, the bandwidthHcan be optimally selected as:
HDc4=5m3=5; cD k00.0/2 kı
!1=5
and 2D p!2 IQ:
For the Parzen kernelc D 3:5134. Obviously, the optimal value ofH is larger if the variance of the microstructure noise is large in comparison to the integrated
quarticity. The estimation of this signal-to-noise ratio2is discussed inBarndorff- Nielsen et al.(2008,2009), see also Sect.13.8.
Realized kernels are subject to the so-called end effects, caused by the missing sample size adjustment of the autocovariance terms. This can be accounted for by using local averages of returns in the beginning and the end of the sample. However, Barndorff-Nielsen et al.(2009) argue that for actively traded assets these effects can be ignored in practice.
Further refinements of the realized kernels in the spirit of the subsampling approach adopted in the TTSRV and MTSRV estimators are considered inBarndorff- Nielsen et al. (2010) by using averaged covariance terms in the realized kernel estimators.