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I am new to PCA and am having trouble understanding some parts of the methodology. In multi-factor models you can run regressions like:

Stock Return = A(1)*Size + A(2)*Market Return + ...

So in this case, you have a clear understanding of what is driving the stocks return. Now, when you have a system of say 100 stocks and extract the 3 most important principal components, is there any way of actually understanding where the risk is coming from? For example, if you ran a regression like:

Stock Return = A(1)*PC1 + A(2)*PC2

While that would tell you the effect of each component, how can that actually help an investor manage their portfolio? Because the PC doesn't really tell you what the actual risk factor is right? Or are you supposed to then run something like:

PC1 = A(1)*Size +...+

To determine what underlies each component?

Please help me with an intuitive answer.

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  • $\begingroup$ Since each PC is (by definition) a linear combination of the original variables, by merely expanding each PC out in the second model, you will have written a model that is exactly in the first form you present. Could you explain what you mean by an "actual risk factor"? $\endgroup$ – whuber Oct 18 '17 at 22:36
  • $\begingroup$ I basically want to study the determinants of returns for various stocks in an emerging economy. I am currently deciding between quantile regression, PCA and other factor models. What I mean is if I estimate Returns = A(1)Size, I know that the effect is coming from the size of the firm. In contrast, if I use PCs I don't really know what the underlying determinants are right? So if I estimate Returns = A(1)PC1, I know the effect of the first component on returns but if I am an investor this doesn't really tell me what is driving returns. Does this make sense? $\endgroup$ – Vladmir Putin Oct 18 '17 at 22:49
  • $\begingroup$ Sorry I am just using individual stocks as an example I am actually using data on the stock index. $\endgroup$ – Vladmir Putin Oct 18 '17 at 22:55
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Stock returns is not the good place to start with learning PCA, because it will not produce you interpretable factors. If your objective is to understan PCA then it's better to start with interest rates, yield curves. There you often get first three factors roughly corresponding to the level, slope and the curvature of the yield curves.

With the stock returns, maybe the first factor could be mapped to the market factors, i.e. similar to CAPM. Otherwise, you'll get a bunch of PCs that are hard to assign to any intuitive or usual risks.

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