ABSTRACT
We derive the asymptotic distribution for the LU decomposition, that is, the Cholesky decomposition, of realized covariance matrix. Distributional properties are combined with an existing generalized heterogeneous autoregressive (GHAR) method for forecasting realized covariance matrix, which will be referred to as a generalized HARQ (GHARQ) method. An out-of-sample forecast comparison of a real data set shows that the proposed GHARQ method outperforms other existing methods in terms of optimizing the variances of portfolios.
Acknowledgments
The authors are very thankful for the valuable comments of a referee which improve the article considerably. This research is supported by a grant from the National Research Foundation of Korea (2016R1A2B4008780)
Disclosure statement
No potential conflict of interest was reported by the authors.
Appendix – Asymptotic normality of ![](//:0)
and consistent estimators of variances of elements of ![](//:0)
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The analysis of this Appendix provides materials for the GHARQ method proposed for two-asset portfolio in Section 2 to be extended to a general multiple-asset setup. Let us consider a portfolio of q assets, . Let
be given. Let a realized covariance matrix
and the corresponding intraday log return
be given. Let
. Recall that the LU decomposition of
is
in which
is upper triangular. The elements of
are computed by the following algorithm:
for
Forecasts of is obtained by fitting q-variate version of (4) for which all the elements other than
can be computed from (7)–(9). Estimators of the other elements are obtained from the asymptotic normality of
which is, as
,
where is the asymptotic variance matrix of
which is given in Barndorff-Nielsen and Shephard (Citation2004). Then,
is the estimated value of the corresponding element of
. We need estimators of
and
. Obvious estimator of
is
. Let
for
, the elements of
. Differentiating (7)–(9), we get,
for
As discussed in Barndorff-Nielsen and Shephard (Citation2004), a consistent estimator of the element of
corresponding to the asymptotic covariance of
is
for . Since
is consistent, so is the proposed estimator of
.