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Reading about PCR I've found many people claiming that PCR is not so good since it doesn't allow you to evaluate the final loading over each base factors. I can't really understand where this problem arises since in my understanding of PCR, we first perform PCA over the set of factors obaining PCs (linear combo of independent variables) and then perform OLS using the first K PCs as new factors.

But at the end of this process, we'll have a loadings $\beta$ for each PC and since every PC contains loadings for base factors, can't we just consider the complex $\beta_i P_{ij}$ as a loading for each base factor?

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Yes, that seems to be right. The main problem in PCR is to decide how many PCs to select

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