PCA seems to be very popular in dimension reduction applications and for extracting the top PCs which explain the data. One such application in futures is on the term structure to obtain the level, slope and curvature components.

However, since the sign of the PCs can be anything(because +EigenVector and -EigenVector are both valid solutions), how do we infer anything about the slope of the term structure by using the slope PC? The slope could be either positive or negative since the sign of PC is not significant.

How do I handle this sign issue when working with PCA?



1 Answer 1


A PCA explains the variation in data. A slope PC is usually identified by the pattern of the signs of the loadings.

If the loadings of short term contracts have the same sign which is different from the sign of the loading of longer term contracts then such a PC is identified as slope PC.

It means that if this PC goes up or down it affects short term contracts in the opposite way to long tgerm contracts. The sign is irrelevant as this holds for up and down moves.

  • $\begingroup$ I was hoping to use PCA to extract the slope information and implement a contango/backwardation strategy using information about the roll yields inferred by this slope PC. If the sign is arbitrary, I'm left with only the magnitude. Am I missing something? PS - I am trying to replicate the results of this paper: papers.ssrn.com/sol3/papers.cfm?abstract_id=1605211 $\endgroup$
    – Mindstorm
    Commented Jan 15, 2014 at 11:10
  • $\begingroup$ Please have a look at my answer here about PCA on yield curves: quant.stackexchange.com/questions/7202/… $\endgroup$
    – Richi Wa
    Commented Jan 17, 2014 at 13:06
  • $\begingroup$ another comment: yes the magnitude matters and for interpretation the sign strucutre as I try to piont out above. $\endgroup$
    – Richi Wa
    Commented Jan 17, 2014 at 13:07
  • $\begingroup$ The sign of the same loadings can be different across software packages. One thing I have observed across packages is that if the sign of the largest abs(loadings) on the first PC in the range ~0.8-0.9 are negative, then these would be positive when determined using another program. $\endgroup$
    – user6430
    Commented Jan 29, 2014 at 12:56
  • $\begingroup$ You are aware of the fact that you can flip all signs simultaneously? If not then I would advise you to study more details on how PCA works. $\endgroup$
    – Richi Wa
    Commented Jan 29, 2014 at 13:15

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