# 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

## 1 Answer

As far as I understand, there is no need to difference the series. In this paper the authors provide a very intuitive explanation of PCA to capture the intra-day variation without taking differences of any type. I know its not the same DGP, but the analysis should be similar.

• I went quickly through the paper but difin't find anything helpful. Nobody who could help? – PieroBerna Feb 5 '18 at 20:37
• Just another question, for indices like VIX should I use the index itself or for example the daily change in % ? – PieroBerna Feb 6 '18 at 16:52
• @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
• Yes this is what would make sense (and what is done in linear regression,...), but I wasnt sure if for PCA we can also work with unstationary time series. The literature is somewhat confusing on this topic. I still didn't find a clear answer. In the construction of other indices like the St. Louises, Kansas City and Cleveland Fed Financial Stress Indexes nothing is said about stationarity/unstationarity of the data (but everything else is very clear). They also used dvariables like an index value (MSCI world) divided by the 200 MA which also is unstationary. I am still confused – PieroBerna Feb 8 '18 at 6:18
• Here a few links link and link – PieroBerna Feb 8 '18 at 12:04