# Principal Component Analysis with time series and index construction

I am doing a pca analysis to construct a financial stress index from different variables which I expect they will move together in a period of "financial stress". As I have read in different papers I will take the coefficients of the first PCA (if enough explanatory power) divide them by the first eigenvalue and take this as the weights of the different variables.

My input variables are time series like the VIX Index, CDS spread,... which all seems to be instationary. Now my questions are:

1. Should I do a first differencing on all the variables in order to have stationary data?
2. Should then from this differenced data do the z-score (value - mean)/std in order to have them in the same units?

Or should I do the PCA directly on the instationary Time series data? Or directly on the z-score without differencing them?

In all the paper I have found no one explained how to deal with instationary time series ...

• Since you are working with time series data you may want to look into something more like independent components analysis. – Matt Barstead Feb 5 '18 at 16:57
• What do you mean? I am going tonuse pca since a few papers gave excellent results and we want to apply this method, so my question is not if it's good or not but how to proceed in this analysis with timeseries. – PieroBerna Feb 5 '18 at 17:02
• I will have a look at it for sure, but are you able to help me with my questions? Thanks! – PieroBerna Feb 5 '18 at 17:28
• Check here for further details. – bastian.abaleiv Feb 5 '18 at 17:59
• @MattBarstead ICA is not specifically designed for time series data, nor is PCA improper for it. – Firebug Nov 1 '18 at 22:05

• @PieroBerna the index design is up to you. If you are interested in risk measures, risk (volatility) is measured as the temporal change as you expose. The VIX series appears to be non-stationary, so differencing is needed to obtain a weakly stationary time series (mean value function $\mu_{t}$ is constant and does not depend on time $t$, and the autocovariance function will only depend through the lag difference $|s-t|$ – bastian.abaleiv Feb 7 '18 at 17:27