[R-sig-finance] correlation between two stock market indices

[Patrick Burns]

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 at 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 to 
>>which it will be put.  If the time horizon of interest is 
>>long -- on the order 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 GARCH 
>>model is going to be appropriate. 
>>    
>>
>
>To my embarrassment I do not understand this (we are talking about the
>cross-correlation, aren't we?). Is there a paper I could consult to
>close this gap in my education?
>
>Thank you,
>Vadim
>
>  
>