Hi Adam Roble ,

Yes, you can apply a Vector Autoregression (VAR) model to variables that are integrated of order zero (I(0)) or stationary. A VAR model is a multivariate extension of a univariate autoregression model, and it can be used to model the dynamic relationships among multiple stationary time series.

A Vector Error Correction Model (VECM) is also possible to be used on stationary variables with cointegration relationship. Johansen Cointegration test examine if there is a linear combination of non-stationary series that give stationary series. If the result of Johansen cointegration test indicate that there is a linear combination of your variables that is stationary, then it's appropriate to use VECM.

You can find more information about VAR and VECM models in the following books:

• "Introduction to Time Series and Forecasting" by Peter J. Brockwell and Richard A. Davis
• "Time Series Analysis: With Applications in R" by Jonathan D. Cryer and Kung-Sik Chan

These are some of the more general references, more focus on applied econometrics can be found in books like:

• "Econometric Analysis of Cointegrated Variables" by Pedro Santa-Clara,
• "Cointegration, Causality, and Forecasting: A Festschrift in Honour of Clive W.J. Granger" by Richard J. Smith and J. Denis Sargan.

Hope this can help !

Lamzihri

If your two variables are I(0), you can certainly estimate a VAR. Any time series econometrics book will verify this. The I(0) assumption for all variables is a classic VAR assumption (although Nobel prize winner Chris Sims says it is not necessary). For example, see Applied Econometric Time Series by Walter Enders. If all n variables are I(0), the Johansen test will likely indicate n cointegrating vectors, because each variable is stationary around its own mean. There then exists error correction where each variable adjusts to its own mean. But estimating the typical VECM imposes some restrictions not imposed by estimation a corresponding VAR in levels.

Assume a 2-variable model with x1 and x2. Let's focus on the x1 equation. The x1 error correction equation is (assuming one lag in first differences):

x1 - x1(-1) = ex1*x1((-1) - x1bar) + bx1*(x1(-1) - x1(-2)) + bx2*(x2(-1) - x2(-2)) + ux1

where x1bar is x1's mean and ux1 the residual. I omit a constant, so there's no time trend. Now rearrange this to the implied VAR(2):

x1 = -ex1*x1bar + (1+ex1+bx1)*x1(-1) - bx1*x1(-2) + bx2*x2(-1) - bx2*x2(-2) + ux1

This reveals that the VAR(2) implied by the VECM has restrictions compared to a standard VAR(2). Here in the x1 equation, it is obvious that coefficients on x2(-1) and x2(-2) are restricted to being equal to each other. Likewise, you can work out that the coefficients on x1(-1) and x1(-2) are also restricted.

Thus, I would say that if you think the variables are stationary in levels, then estimate a standard VAR, not a VECM.

Hello Adam Roble ,

As your variables are stationary at level I(0), then you should go for VAR model directly after selecting the lag length. VECM is not ok for your dataset.

You can use Johansen cointegration test when all variables are stationary at first difference I(1). When Johansen test result shows cointegration among variables only then you can run VECM model.

And you can also conduct Granger Causality test to find out if the lag values of independent variables affect the endogenous variable or not.

Yes you can do that without any doubt. You need to choose the appropriate lag length before running the model. For lag length, you can run varsoc command in stata which will fetch you the appropriate lag length.