VAR or VECM for a mix of stationary and nonstationary variables?

Question

I have 4 time series. One of them is stationary and rest of them are not. I need to find relation between them. I will use AIC to decide lag length.

1. Should I use VAR or VECM to find relation between them?
2. Will VAR or VECM give me relation in terms of equation which can be used for forecasting?
3. Do I need to perform Johansen's test of cointegration?
4. What good would it do?

Answer 1

So you have three nonstationary series and one stationary series. Let us call them $x_1$, $x_2$, $x_3$, and $x_4$, respectively. Suppose the nonstationarity of $x_1$, $x_2$, $x_3$ is of a unit-root kind (rather than of some other kind); that is, each of $x_1$, $x_2$, $x_3$ is integrated of order one, I(1). You can determine the order of integration using, for example, the augmented Dickey-Fuller test (ADF test).

Test each pair of the nonstationary series ($x_1$ and $x_2$; $x_1$ and $x_3$; $x_2$ and $x_3$) for cointegration using the Johansen or the Engle-Granger test.
Then test all three series ($x_1$, $x_2$, $x_3$) for cointegration using the Johansen test.
Depending on the results of the tests, you may find yourself in one of the following situations:

(A) No cointegration
(B) Two of the variables (say, $x_1$ and $x_2$) are cointegrated while the third variable (say, $x_3$) is not
(C) The three variables ($x_1$, $x_2$, $x_3$) are cointegrated

In general, you want the following:

• Models for cointegrated variables should have an error-correction representation; otherwise the model would be misspecified (cointegration goes hand-in-hand with the error correction representation).
• Models for stationary dependent variables should not have nonstationary explanatory variables (except perhaps for stationary combinations of cointegrated nonstationary variables); otherwise the linear combination of the regressors would diverge from the regressand.
• Models for nonstationary dependent variables should have at least one nonstationary explanatory variable; otherwise the regressand would diverge from the linear combination of the regressors. Mind nonstandard distributions of estimators for the integrated variables.

Based on these principles, you may do the following:

If (A) then first-difference each of the three variables ($x_1$, $x_2$, $x_3$), and use them together with the stationary variable $x_4$ to build a VAR model.

If (B) then build a model where

• $\Delta x_1$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
  
• $\Delta x_2$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
  
• $\Delta x_3$ depends on lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
  
• $x_4$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$.
  

If (C) then build a model where

• $\Delta x_1$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
  
• $\Delta x_2$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
  
• $\Delta x_3$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$;
  
• $x_4$ depends on the error correction term and lags of $\Delta x_1$, $\Delta x_2$, $\Delta x_3$, $x_4$.
  

These are pretty general models with lots of regressors. You may find it beneficial to exclude some variables from some equations or use penalization to avoid overfitting.

_{Relevant additional keywords: I(0), I(1).}

Answer 2

Should I use VAR or VECM to find relation between them?

In practice, it depends on the power of cointegration tests:

• If your variables are cointegrated and you used a VAR model: you could have done better by estimating a VECM model. Your estimations are still consistent (in fact superconsistent), but inefficient.
• If your variables are not cointegrated and you use a VECM: You have used wrong information. The estimations are not consistent.

Will VAR or VECM give me relation in terms of equation which can be used for forecasting?

In practice, You can use both for forecasting. Of course, if the goal is to forecast, there are other criteria to check for models performance. This article introduces the concept.

Do I need to perform Johansen's test of cointegration?

If the goal is forecast, estimate as much possible model as you can and compare their forecast performance. If the goal is to estimate the structure of the model, sure, you should test for cointegration. Of course, in this case you should do sensitivity analysis by estimating other unrestricted VAR models, because the power of statistical tests are limited.