Thank you Patrick! This makes a lot of sense. If I understand you
correctly you are talking about time-varying "instantaneous" correlation
(more precisely covariance) matrix and that a good GARCH can capture the
variations.
Thanks,
Vadim
-----Original Message-----
From: Patrick Burns [mailto:patrick@burns-stat.com]
Sent: Tuesday, August 31, 2004 4:22 PM
To: Vadim Ogranovich; r-sig-finance
Subject: Re: [R-sig-finance] correlation between two stock
market indices
Yes, we are talking about cross-correlation. (Before we get
in an even deeper muddle: for those who can't relate to
cross-correlation, ignore it and just think of correlation.)
I can't think of a very good reference at the moment -- maybe
someone else has ideas.
My statement mainly rests on the following assertion:
Multivariate GARCH is a reasonably good model for the
variance matrix of the returns of assets.
This is most true of daily data. Lower frequency data smooth
out some of the garchiness; things get complicated with intraday data.
More specifically the assertion should be that there exists
some multivariate GARCH model which is reasonably good.
There will be many GARCH models which are not good. One
particular model that almost surely will not be at the head
of the class is a constant correlation model. These were
created because of the ease of estimation rather than from
any empirical or theoretical motivation.
If you think of CAPM with the market being modeled as GARCH,
then assets will be more highly correlated with each other
when the market is in a high volatility period than when it
is in a low volatility period.
Assuming you can believe that correlations change over time
(with some form of continuity), then it shouldn't be too much
of a leap to believe that the time horizon of interest will
influence your estimation procedure.
If you knew that GARCH were the correct model, then it would
be optimal (in the estimation sense) to use GARCH for all
time horizons. But as the time horizon gets longer, all of
the estimates approach the unconditional correlation. So for
long time horizons there is not much sense in going through
the work of fitting a multivariate GARCH model when you will
just end up with the sample correlation anyway. The more
steps you predict ahead, the more model risk you take. GARCH
is not exactly correct, so there is definitely model risk to be had.
For short time horizons, the model doesn't have to be so
perfect in order to outperform the sample correlation.
Assuming that you don't have multivariate GARCH available to
you, there are some half-way measures for getting at
predictions for short time horizons.
A practical option is to use exponential smoothing.
Patrick Burns
Burns Statistics
patrick@burns-stat.com
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and "A Guide for the Unwilling S User")
Vadim Ogranovich wrote:
*) How correlation should be estimated depends on the use
will be put. If the time horizon of interest is long -- on
of two months or longer, then an ordinary sample correlation should
suffice.
If the time horizon is short -- a day or a week, then a
To my embarrassment I do not understand this (we are talking
cross-correlation, aren't we?). Is there a paper I could consult to
close this gap in my education?
Thank you,
Vadim