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I have two financial series data, x and x', where x' was formed form x using the orthogonal transformation as explained in Orthogonalized Equity Risk Premia and Systematic Risk Decomposition.

I was just wondering what statistical methods exist to see if x and x' are "similar" to each other. As of now, I do not have precise definition of "similar", but I can think of comparing simple summary statistics such as mean, variance, skewness, kurtosis, and correlation coefficient between x and x'. What statistical methods exist to safely conclude that x and x' are similar to each other?

Thank you very much in advance for your help!

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  • $\begingroup$ Whether the two time series are beams of light, sounds or something else, their comparison can done using spectral analysis. If the spectrum of X is similar to that of X', the two time series are similar. $\endgroup$
    – stans
    Aug 3, 2018 at 19:53
  • $\begingroup$ @stans What if they have wildly different amplitudes like $\sin(x)$ and $20\sin(x)?$ $\endgroup$
    – Dave
    Dec 31, 2021 at 2:32

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