Raw eigenvectors or standardized eigenvectors for principal component regression? Thanks to @amoeba, I learned that standardized eigenvectors are sometimes calculated, i.e., eigenvectors are divided by the square root of their eigenvalues.
Now, when I want to do principal component regression (PCR), I first calculate the components that I subsequently use in regression.  There are two procedures that can be used:  


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*I calculate my components once as raw components, that is I do not standardize the eigenvectors.

*I calculate my components as components based on standardized eigenvectors.
Now, I do PCR once with the components in situation 1, and once with the components in situation 2.
My questions: 


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*What are the implications (a) for the coefficients, (b) for the interpretation in the two cases? 

*What are the main differences?

*I know that the variance of the components will differ, but how does that change the behaviour in regression, or does it at all?

*Is there any recommendation to whether I should use the one but not the other?

 A: I'm reviewing this and found a nice paper Parameter Estimation in Factor Analysis: Maximum Likelihood versus Principal Component which makes me think that the eigenvectors are indeed standardized, if only for convenient interpretation. They're operating on a standardized nxp data matrix, $\mathbf{Z}$, where the variance matrix has ones on the diagonal and thus there  are p units of variance (the trace) "to distribute".
For eigenpairs $(\lambda_k, \mathbf{e_k}), k = 1, .., p$, the sum of the eigenvectors is still $p$ and the proportion of the variance is $\lambda_k / p$. To have the nice interpretation of distributing these p units of variance, you'd clearly have to be working with unit eigenvectors (though the paper doesn't specifically say it).
A: It might have something to do with whether your predictor variables are standardized or not (centered they must be!). If they are in original form, some eigenvector will be extremely large so standardizing them is a good idea (Why?). I don't see the point in standardizing them if your PCA used standardized (commensurating) variables already.
