recent developments in high frequency financial ... - Index of

300 V. Voev
where �kl(t) is the (k, l) element of the �(t) process. With this notation, we will
now interpret the ‘true’ covariance matrix as
� t+1
�t+1 =
t
�(u)du. (18)
Thus the covariance matrix at time t + 1 is the increment of the integrated
covariance matrix of the continuous local martingale from time t to time t + 1.
The realized covariance as defined in expression (1) consistently estimates �t+1 as
given in Eq. (18). Furthermore, Barndorff-Nielsen and Shephard (2004) show that
under a set of regularity conditions the realized covariation matrix follows asymptotically,
as M →∞, the normal law with N × N matrix of means � t+1
t �(u)du.
The asymptotic covariance of
�
�
√ M
� RC
t+1 −
� t+1
t
�(u)du
is �t+1, aN 2 × N 2 array with elements
�� t+1
�
�t+1 = {�kk ′(u)�ll ′(u) + �kl ′(u)�lk ′(u)} du
t
k,k ′ ,l,l ′ =1,...,N
Of course, this matrix is singular due to the equality of the covariances in the
integrated covariance matrix. This can easily be avoided by considering only its
unique lower triangular elements, but for our purposes it will be more convenient
to work with the full matrix. The result above is not useful for inference, since the
matrix �t+1 is not known. Barndorff-Nielsen and Shephard (2004) show that a
consistent, positive semi-definite estimator is given by a random N 2 ×N 2 matrix:
Ht+1 =
M�
j=1
xj,t+1x ′ j,t+1
− 1
2
M−1 �
j=1
�
xj,t+1x ′ j+1,t+1 + xj+1,t+1x ′ �
j,t+1 , (19)
�
where xj,t+1 = vec r j r t+ M
′
t+ j
�
M
and the vec operator stacks the columns of a
matrix into a vector. It holds that MHt+1
p
→ �t+1 with M →∞.
With the knowledge of this matrix, we can combine the asymptotic results for
the realized covariance, with the result in Eq. (7) to compute the estimates for πt,
ρt and νt.
For the equicorrelated matrix F we have that4 �
fij =¯r σ (RC)
ii σ (RC)
jj , where ¯r
is the average value of all pairwise correlations, implied by the realized covariance
matrix, and σ (RC)
is the (i,j) element of the realized covariance matrix. Thus �,
ij
the population equicorrelated matrix, has a typical element φij =¯̺ √ σiiσjj , where
σij is the (i,j) of the true covariance matrix � and ¯̺ is the average correlation
4 In the following exposition, the time index is suppressed for notational convenience.
.