Regress Prices or Returns?

#1
Hey there,

I've just started a new job and have been asked to take a look at the historical correlation of the spot gold price with the spot copper price. I've gathered all the data and started to run a regression but am stuck on one particular point.

Back in school, I remember being told not to regress prices against each other; but to regress the natural logarithm of their returns (i.e. ln[price today / price yesterday]) instead -- this way you test whether prices move together.

My issue is, if I regress the return on gold against the return on copper my r-squared value is next to nil (which is a perfectly plausible conclusion); however when I regress the gold price against the copper price, my r-squared is very high (>0.65).

There are a couple of things I'm struggling with here:
1) if the underlying prices are correlated (i.e. one price predicts the other), then how is it viable that the returns are not also correlated (at least nominally)
2) these analysis lead me in totally different directions. The first leads me to believe that copper prices are independent of gold prices; while the second seems to indicate that they are very strongly related. What is the right conclusion here?

I've been struggling with this one all night; so any help would be very much appreciated.

The one thing I have been able to identify, is that by looking at returns instead of prices; I am able to transform a lognormally distributed dataset (clearly prices can't fall below 0), into a normal data set (returns appear to be normally distributed). I'm not sure if this is relevant, but I remember from school that residuals in an OLS regression should be Gaussian. Not sure if this also means that the X and Y variables also need to be normally distributed though...

Thanks again guys; really appreciate the time.

John
 
#2
This is an issue of asking two different questions and thinking they are the same. The question of ln(price today/price yesterday) is a question about a differential (as in calculus rather than financial), while the question of the two prices is one of a base level. In physics, these two questions are similar to asking, are the gold and copper vehicles travelling at the same speed? and are they at the same location. You note that you can have one without the other.

So, it seems that your answer is, yes you can predict the long term trends in the spot price of gold by the spot price of copper, but their daily fluctuations are not predictable via that means.