Principal Component Analysis (PCA) before Vector Autoregressive model (VAR) The PCA is intended to convert covariates to linearly uncorrelated. On the opposite VAR model captures the linear interdependencies among covariates. Will then the PCA transformation improve VAR model or there is risk of decreasing its predictive power?
 A: $$
pca: X=ST
$$
$$
VAR: S_t=\phi S_{t-1}+\epsilon_t=\hat{S_t}+\epsilon_t
$$
$$
Forecasting: \hat{X}_t=\hat{S}_tT=\phi S_{t-1}T=\phi X_{t-1} T^{-1}T=\phi X_{t-1}
$$
Couple of thoughts:
1) Performing PCA on your dependent variables won't change the predictive power, as shown above
2) if you do PCA on some set of exogenous variables (not shown above), it may give you better insight on whether all variables included in the set of exogenous variables provide predictive power, which is something you might consider doing for model selection... But it would have no affect on the predictive power on the model (it'd be equivalent to transforming your betas), so there's no reason to do it if your purpose is to strengthen your model.
A: Pure PCA will have no effect. If you 


*

*run PCA, 

*estimate a VAR on the principal components (PCs) and 

*convert the estimated coefficients on the PCs to the coefficients on the original variables, 


you will get the same coefficient values as from simply running a VAR model on the original data.
Even if you skip 3. and use the model from 2. for forecasting, you will still get the same forecasts as from the simple VAR on the original data.
The intuition is that PCs span the same space as the original variables; thus a projection on the PCs will have the same residuals and the same proportion of explained variance as a projection on the original variables.
A: It depends on how you use PCA in your analysis. If you are using it to remove variables/variance from the dataset/model, then the PCA-derived model will not retain the SAME predictive power as a fully dimensional model. That being said, different is not always worse. In fact, by squeezing more of your data's variance into fewer variables (i.e. forming principle components from the data) and removing some less informative PCs (ones capturing minuscule variance), there's a real possibility that the PCA-derived model's predictions become more accurate due to the fact that you're less likely to overfit (VARs tend to run into overfitting issues often due to model parameter count exploding...). If you keep all principle components after PCA, then the predictive power should remain constant across methods. The real tricky part can be making sense of the model in its principle component form. Evidently, there are ways to back-project pca-derived model coefficients to their full dimensional form (http://scot-dev.github.io/scot-doc/vartransform.html#covbivar1) so long as you retain all PCs after PCA (which seems utterly useless in most cases). I'm not sure you you would backproject when you remove PCs, but would love to find an answer to that question if anyone has any pointers.
