Peerless


Total Posts: 108 
Joined: Aug 2012 


We have two assets. We compute their respective daily returns and weekly returns over the same period. Now we compute two correlations (Pearson): one based on daily returns and one based on weekly returns.
Is there an approximation to scale daily correlation to weekly correlation? If not, why not?
My google searches have not been fruitful. I would tend to say that there is no reason it should scale but I cannot articulate any sound reasoning. 



tbretagn


Total Posts: 256 
Joined: Oct 2004 


I don't see why there would be any reason those two correlations could be scaled. You can build a counter example intuitively by forcing the weekly returns to be the same, while the daily returns can be completely different.

Et meme si ce n'est pas vrai, il faut croire en l'histoire ancienne 


I would say that in a theoretical sense the correlations should be the same ... in a framework where the volatility is low and constant, correlations are stable, and both assets follow a random walk
in reality you will have clustering of volatility, possible meanreversion, and correlations will drift over periods of months or years.
Even so I would guess that using correlations of nonoverlapping weekly returns is more likely to give sensible results, it is a crude way of reducing the noise due to difference in market close times and so on.
Did you find anything googling "estimation error of realised correlation" ?




Peerless


Total Posts: 108 
Joined: Aug 2012 


Thanks for confirming my intuition. I have came across this link. However, the assumptions (multivariate joint normal distribution and iid) are such that in practice it is not really relevant imho. 



For a continuous time stochastic processes:
* Assuming the instantaneous covariance is stationary...
* Variance scales with sqrt(horizon). *Unless* the process exhibits autocorrelation. Either meanreversion, which means variance scales slower. Or momentum: variance scales faster.
* Covariance scales with the the square root of horizon length. *Unless* past returns on process A have nonzero correlation with future returns on process B. I.e. there's some leadlag dependency.
Since correlation is the just the ratio of covariance to variance, we can say that it should remain invariant across all horizons as long as the above conditions hold.
The latter two conditions imply a violation of market efficiency. That is, if they're broken it implies that we can with some nonzero accuracy predict forward returns on one or both assets from their past returns. That's not to say markets are perfectly efficient, but it's unlikely that there's gross violations of efficiency. Particularly if you're not looking at very short horizons where microstructure effects come into play. Therefore you should expect that correlations at any mediumfrequency or longer horizon should be nearly identical.
(Minor caveat being as Silverside mentioned, if you're using different times, like comparing European close prices against American close prices. That introduces leadlag effects without actually violating market efficiency)
As for stationarity, there's two likely sources of violation: periodic and trend. For the former, a common case might be overnight vs intraday returns. Two stocks may have repeatedly different correlations for their overnight returns. So if you're comparing minutewise returns against daily this affect may come into play. Anything with interday or longer though, it's pretty unlikely there'd be major periodic affects. Maybe weekend or some January effect, but that's usually pretty weak.
As for trend nonstationarity, this is probably most likely due to using derivatives. An example might be looking at the correlation between some specific future and the spot index. Correlation tends to rise as you get closer to expiry, so you'd expect longer horizons, sampled at the same starting point, to exhibit higher correlation. 
Good questions outrank easy answers.
Paul Samuelson 

