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I have a panel data with identifiers(a,b,c,....z) and different times(t=1,2,3,....100) I have 6 different variables (A,B,C,D,E,F) for every identifier-time observation. I attempt to use those 6 variables for Principal Component Analysis to come up with one variable called "Bias". So here are my questions:

  1. I want to see how "Bias" is different across identifiers cross-sectionally, which means I have to have "Bias" for every identifier-time observation. Do I have to do PCA using all the data, or do I have to do PCA for each identifier?

  2. How do I "normalize" the variables before PCA? In other words, for my case, should I normalize it across different identifiers at each given time, or do I have to normalize it using the mean and standard deviation of full observation?

Thanks in advance.

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  • $\begingroup$ Do the identifiers identify the firms? Sometimes you say "firm", sometimes "identifier", and I wonder whether it's the same thing. $\endgroup$
    – Christian Hennig
    Commented May 3 at 9:34
  • $\begingroup$ Yes, thank you for your point. I edited the question. $\endgroup$
    – Ken
    Commented May 3 at 18:22

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The issue here is how the effect of these decisions is compared to what exactly you are interested in. If you use different PCAs for different identifiers, the values of "Bias" will not mean the same thing across identifiers, which means you can't compare these values across identifiers. So I'd say use the same PCA for all identifiers together.

The normalisation issue can be more subtle. It is certainly possible to make the same case here and say that you should normalise over all observations as this means that all resulting values on Bias are comparable. However, if variances vary between time points and you want to have a comparison that is unaffected by general changes over time, you could normalise at each time point. The difference is that if you normalise over all observations, time points with larger variation will have more influence on your comparison, whereas if you normalise separately for different time points, this influence will be taken out. Whether you want one thing or the other depends on background and research question, and I can't tell from the given information. What you need to ask yourself is whether you think that a time point with larger variation indeed holds more information that you want to use, in which case you'd normalise over all time points. Otherwise by time-wise normalisation you'd unify the impact of the time points (I'd think how this works and what it means is slightly more difficult to explain to other people, in case this plays a role).

I should also say that whether what I say is appropriate depends on what exactly you do with the "Bias" later; in many cases the effect of normalisation is what I state above but there are situations where this isn't the case (in some of those it won't make a difference).

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