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I have run a PCA on some returns to get a set of factors. All is good except that the first PC seems to be the short of the market, it has a correlation of -0.9 with the S&P500 but all the loadings are negative (except one which is very small and basically +0).

I am confused as to why the PC is negative and all the loadings are negative, I assume it is ok to switch BOTH signs also?

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  • $\begingroup$ What is your look-back window, and return frequency? Are you using daily rets, monthly rets, etc..? Also, do you have code available yo share? $\endgroup$
    – ma83
    Jan 12 at 13:05
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    $\begingroup$ Depending on the applied method, eigenvalue decompositions can be defined up to a switch of sign. $\endgroup$ Jan 12 at 14:56
  • $\begingroup$ Look back window is 3 years, monthly returns, as for code it’s just basically written in Python where I standardise the returns and then use scikitlearn PCA(10).fit() on the standardised values. $\endgroup$ Jan 12 at 15:35
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    $\begingroup$ eigenvectors are direction agnostic, multiplying by -1 does not impact results. The only reason they might be negative in the first place is as a result of the iterative process that derives them. $\endgroup$
    – Attack68
    Jan 12 at 16:33
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Yes, you can flip the negatives, that cancel out each other side. All of your stocks are negatively correlated to inverse beta, which is the same as saying that stocks tend to be positively correlated to each other, and thus the market.

As Kermitfrog says, different methodologies to calculate the eigenvalues/eigenvectors can produce results with identical magnitudes but the opposite sign. After all, standardizing your data has already removed any positive bias here!

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  • $\begingroup$ You’ve got to stop answering all my questions for me today! :) thank you again, that does make sense when looking at the calculation for the eigen vectors and values! $\endgroup$ Jan 12 at 18:22

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