I understand that Principal Component Analysis (PCA) can be applied for cross-sectional as well as for time-series data. Nevertheless, I am trying to figure out if there is anything wrong with applying it to covariances instead of raw data?
In particular, PCA tries to partition the number of explanatory variables in order to maximize the overall explained variance. Applying this to covariance time-series sounds like partitioning pairs of covariances so that the explained variance is maximized. Which by definition of covariance means that the focus is on the variance of covariances.
Does anyone see any mathematical issue beneath this procedure?