If I decide to use PCA to estimate a high-dimensional asset portfolio covariance matrix using reduced dimensions, I can use the following procedure to transform the low-dimensional matrix back to the high-dimensional matrix:
I understand that because we are using PCA on the covariance matrix, we are implicitly using centered data which is fine.
But we then also have the choice of scaling the asset-returns by their corresponding standard deviations before applying PCA (z-scaling), which would result in correlation-based PCA. This may make sense if we want to avoid results being dominated by single highly volatile assets.
If we apply above standardization procedures, will the resulting reconstructed full-dimensional matrix also be a correlation matrix? And if yes, do we not have to transform it back to a covariance matrix by de-standardizing it using following formula?
Because the correlation matrix does not have the correct real-world scaling of the underlying assets? Hence, value at risk for example may very well be under- or over-estimated?