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.