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First of all, PCA analysis is not something I came across in my economics studies. But, recently, I wanted to make a PCA analysis of American GDP.

I started to read about the fundamentals of PCA and played around with it in R. Now, I have finally produced a result that can make one period ahead forecast.

I thought: "Great - now I could use the PCA analysis and find the variables that explain most of the variation in GDP and sort them out. I want to know which variables have the most influence". For example: Does consumer spending explain more of the variation in GDP than, let's say, imports of goods and services.

But then I recalled: As far as I understand, the PCA analysis takes the original variables and replaces them with latent components. So my thought is more a matter of correlation - not the PCA analysis itself.

Is this problem more suited for a partial least squares regression PLS analysis?

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  • $\begingroup$ If you have a bunch of variables which correlate you can use PCA to replace them with a smaller number of PCs (constructs or latents). You can even choose to rotate the produced PCs. A variable loaded high by a component can be regarded as a natural indicator, representative of that component. $\endgroup$
    – ttnphns
    Nov 25 '21 at 10:24
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    $\begingroup$ As A stands for Analysis, the phrase PCA analysis reads very awkwardly to me. (Call this pedantic by all means.) $\endgroup$
    – Nick Cox
    Nov 25 '21 at 12:19
  • $\begingroup$ I can't see clearly what the question is. $\endgroup$ Nov 26 '21 at 6:25
  • $\begingroup$ What aren't the previous questions about PCA explaining it, and what is the new thing here? $\endgroup$ Nov 26 '21 at 6:31
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If the aim is not just forecasting but trying to get a good handle on which predictors are most important in some sense, in my view PCA is likely to be a distraction and its use to turn out to be a detour. You would be better off applying some flavour of regression directly. That is unlikely to be trivially easy and selection of predictors is a deep and difficult art, but you are more likely to get a good handle on which are the important predictors.

The longer answer on what does important mean and how do you find a good model is covered by every worthwhile regression text.

All that said, I am unclear how PCA leads to predictions in time unless you are using something else as well.

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You could answer your boss's question in focusing on the PCs that best explain the variance of your dataset (as suggested above).

Then, you could decompose those PCs in combinations of your initial variables. The coefficients in front of the variables will tell you how strongly they influence the variability.

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Yes and no. PCA doesn't replace anything with latent variables you might be thinking of factor analysis (FA). PCA does construct principal components (PC) which are linear combinations of all the variables.

If for example you were to find that the first PC explains 90 % of the variability, you could focus on this PC and check which variables mostly contribute to this PC, by checking the loadings/weights/coefficients (which makes sense if your data was normalized). In this case you can find the variables which explain most of the variability... through the PCs.

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