# Stationarity in multivariate time series

I am working with a multivariate time series and using VAR (Vector Autoregression) model for forecasting. My question is What does stationarity actually means in a multivariate framework.

1) I know that if in VAR setup if determinant of inverse of |I-A|matrix has eigen values less than 1 in modulus , the overall VAR system is stable/stationary, but does that mean I can proceed without bothering about differencing the non stationary component present in the multivariate time series

2) How to proceed if one of the component series is non stationary rest are stationary?

3) How to proceed if more than one component time series are non stationary but are " Not Co-integrated"?

Above all are there any other methods to deal with multivariate time series.I am also exploring the machine learning methods

I've got the same problem and I can understand your thoughts very well! After dealing with this subject and reading several books I'm also a little bit confused. But as I understand: if the whole VAR system is stationary it follows that EVERY single component is stationary. So if you test the stationary of the VAR system (by means of the determinant of inverse of |I-A|matrix as described) it will be enough and you can proceed.

Currently I'm working with VAR-models, too. In my cases the VAR system is always stationary because the modulus of the eigenvalues are all less than 1. But when I look at the single time series I would think that some series are not stationary. I think, this is your problem, too...

So I think one has to decide which criterion to use. Either looking at the eigenvalue-condition and proceed if all are less than one in modulus or first have a look at single time series and than put the stationary time series (after differencing / polynomial subtraction if needed) in the VAR analysis.

By the way, if it helps, I found one reference which says that the single components do not necassary have to be stationary but only the vector of time series (the VAR system). This is a german reference [B. Schmitz: Einführung in die Zeitreihenanalyse, p. 191]. But in my opinion this conflicts with the proposition that VAR system stationarity results in single component stationarity...

Hoping for more arguments from others.

• You've put it really nicely. To update on the topic, right now I am trying both approaches on different multivariate time series data sets and see which one performs better, Although this is not an exact way to deal with this. It is the best I could come up with Another doubt : What is the minimum no. of records you need to fit a VAR model on a time series with n variables – NG_21 Dec 24 '13 at 17:01

I think I've figured out the possible solution. It all depends on the nature of eigen values. Lets say we have 3 time series in our system. Correspondingly there are different possibilites for eigen values

1) Case 1 : All the eigen values are less than 1 in modulus => VAR model is stationary and can be built and used for forecasting after other diagnostic checks.

2) Case 2 : All the eigen values are > 1 in modulus => VAR is non-stationary, We have to go for a co integration check. If none of them are co-integrated , then differencing or log transformation is the suggested way

3) Case 3: Eigen Value =1 i.e a unit root => We will have to go the VECM (Vector Error Correction Model) approach

4) Case 4 : Now this is interesting, some of the eigen values are < 1 and rest are > 1, none of them being equal to 1 , => System is exploding i.e one of the series is stationary around a mean/variance, while other one is not. In this case either transforming the series via differencing or log transformation , is the logical way or rather dealing only with the non stationary series with univariate methods gives better forecasts.

I sounds logical to me that, if one of the series is non stationary and other is stationary, Then the stationary one might not be impacting the non stationary series at all. But I don't have any rigorous mathematical proof for that

1) A stationary VAR means that all of its variables are stationary. So I suggest testing each variable individually for stationarity, and thereafter for co-integration if they happen to be non-stationary.

2/3) You should difference the non-stationary components before attempting to use them in a VAR. If there is one non-stationary component, difference it before using it in the VAR, same goes if there are several non-stationary components, or if all are non-stationary, use the differenced series in you model.

You can probably use other methods for analyzing, like machine learning, but that is a field I'm not very familiar with.

• My doubt still remains, if the modulus of Eigen Values of the [I-A] matrix is less than 1 , the overall VAR system is stationary , even though its component series might be non stationary. So Should I go for differencing the series or proceed without it Thanks – NG_21 Dec 11 '13 at 4:49