I am investigating how to appropriately combine PCA with VAR modeling. I am using PCA to reduce the number of vars I fit to a VAR model, and am attempting to recover the non-dim. reduced coefficients via projecting dim.-reduced coefficients to their full dimensional state using the PCA coefficients (following the method outlined here: http://scot-dev.github.io/scot-doc/vartransform.html#covbivar1). In the above link, they mention that one of the requirements for recovering full dimensional coefficients is: "A VAR model can have zero cross-correlation despite having causal structure." and argue that this "requirement relates to the fact that in order to reconstruct model A from model B all information about A must be present in B. Thus, information about the causal structure of A must be preserved in B, although y⃗ n=VAR(B,r⃗ ) is uncorrelated. Covariance of a bivariate AR(1) process shows that it is possible to construct models where causal structure cancels cross-correlation." I am entirely confused by this statement. My basic question is--am I required to impose a constraint that makes my residuals have zero covariance (what I believe they're indicating in the last equation on the page)? Can someone try to explain their second requirement to me in a clearer manner so that I can have more confidence in my model-fitting approach?