In principal components regression, should I weight the regression the same as the PCA? Or at all?

Question

I am using PCA on foreign exchange return series to find a market "beta". I am using 10 years of daily data with a 2-year half life weighting in the PCA using the package FactoMineR's PCA function. I extract the first principal component return series (so the product of the first eigenvector and the returns matrix) and I want to regress that against each foreign exchange return vector to find the residuals, that is, to find each currency's returns independently of the market beta.

Should I use the same 2 year half-life weighting in the regressions? Will this "double up" the weighting somehow? Conversely if I don't weight the regression, will I implicitly be putting too much weight on PC1 returns that are less "relevant"?

For what it's worth market participants tend psychologically to put a higher weight on recent than long past currency behaviour.

Thanks for the help.

Answer 1

I think your question is interesting, and I was waiting to see an answer, but since I still don't see one. How about I try myself.

Am I following you correctly?

1)You have data on 10 years exchange rates, you weighted the data such that more recent observation are more important (weighted more). 
2)From this data you extracted the PC, which gives you the market factor.
3)And you want to see how each currency compares to the market factor, by regressing each currency on the market factor.

I think it would make more sense to regress weighted data for exchange rates as well (the ones that you used to calculate the market factor). If you are discounting based on the fact that "further back is less important", then you should discount both, since this holds for the market, just like for every individual currency.

I do have concern from economics/finance point of view regarding the market factor of the currency markets though. In context of let's say US stock market, assets generally move together, so sensitivity to the market, which is your coefficient on the market factor, makes perfect sense. But how would this work for currencies? Currencies are zero-sum game, so an average value of all currencies would not change over time; that is your market factor should roughly be a constant. So measuring sensitivity to that doesn't make sense to me. Maybe I am missing something, and your approach makes perfect sense, but I wanted to raise that point.