High-Frequency Covariance Estimates With Noisy
and Asynchronous Financial Data
Yacine AÏT-SAHALIA, Jianqing FAN, and Dacheng XIU
This article proposes a consistent and efﬁcient estimator of the high-frequency covariance (quadratic covariation) of two arbitrary assets,
observed asynchronously with market microstructure noise. This estimator is built on the marriage of the quasi–maximum likelihood
estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not inﬂuenced by the Epps effect.
Moreover, the estimation procedure is free of tuning parameters or bandwidths and is readily implementable. Monte Carlo simulations
show the advantage of this estimator by comparing it with a variety of estimators with speciﬁc synchronization methods. The empirical
studies of six foreign exchange future contracts illustrate the time-varying correlations of the currencies during the 2008 global ﬁnancial
crisis, demonstrating the similarities and differences in their roles as key currencies in the global market.
KEY WORDS:
Covariance; Generalized synchronization method; Market microstructure noise; Quasi-Maximum Likelihood Estimator;
Refresh Time.
1. INTRODUCTION
The covariation between asset returns plays a crucial role in
modern ﬁnance. For instance, the covariance matrix and its in-
verse are the key statistics in portfolio optimization and risk
management. Many recent ﬁnancial innovations involve com-
plex derivatives, like exotic options written on the minimum,
maximum or difference of two assets, or some structured ﬁ-
nancial products, such as CDOs. All of these innovations are
built upon, or in order to exploit, the correlation structure of
two or more assets. As technological developments make high
frequency data commonly available, much effort has been put
into developing statistical inference methodologies for continu-
ous time models with intra-day data, enabling us to capture the
daily variation of some interesting statistics that were otherwise
unobservable from daily or weekly data.
Realized variance estimation is an example of such statis-
tics. Unfortunately, unlike those low frequency time series that
are homogeneously spaced, tick-by-tick transactions of differ-
ent assets usually occur randomly and asynchronously; in ad-
dition, with high frequency data comes market microstructure
noise. These factors make it difﬁcult to employ a Realized
Covariance (RC) estimator directly. Popular estimators in the
univariate variance case include Two Scales Realized Volatil-
ity (TSRV) of
Zhang, Mykland, and Aït-Sahalia
(
2005
), the
ﬁrst consistent estimator for integrated volatility in the presence
of noise, Multi-Scale Realized Volatility (MSRV), a modiﬁca-
tion of TSRV which achieves the best possible rate of conver-
gence proposed by
Zhang
(
2006
), Realized Kernels (RK) by
Barndorff-Nielsen et al.
(
2008a
) and the Pre-Averaging (PA)
approach by
Jacod et al.
(
2009
), both of which contain sets of
nonparametric estimators that can also achieve the best con-
vergence rate. In contrast with these nonparametric estimators,