Varying orders of integration - VAR/VECM model

I am building a VAR model, and have gotten a thorough set of guidelines through a question I asked a little while ago.

However, I am left with some questions based on the following quote from Step 3 of the linked answer:

"If all tests confirm that your series are I(2) (remember to use the log of the series before running the tests) then take the second difference, but note that your interpretation of the results will change, since now you are working with the difference of the continuously compound returns."

My questions are as follows:

• In what way will the interpretation of my results change as a result of having I(2)?

• What happens if I have variables that have different orders of integration? say two I(1) variables and one that is I(2).

• I would think twice before concluding that an asset price is an I(2) process. As I commented under your earlier question, unit root tests are to be used carefully. – Richard Hardy Feb 1 '16 at 19:50
• @RichardHardy Fair enough, I see the logic behind that, but given the below test stats that is the conclusion I draw. May of course be wrong. ADF results are as follows (critical values left to right 1, 5, 10pct). test-statistic: -2.8563 4.3594, tau2: -3.51 -2.89 -2.58, phi1: 6.70 4.71 3.86 diff 1: test: -2.231 2.49 diff 2: test: -7.6637 29.4035 tau2 and phi1 stay the same for both differences. – youjustreadthis Feb 1 '16 at 20:31

What will change is the interpretation of the coefficients in the model. Suppose that there is no cointegration between your variables and you estimate a bivariate VAR(1) model with the first difference of the log of prices:

$$\Delta Y_t = a_0 + a_1 \Delta Y_{t-1} + a_2 \Delta X_{t-1} + \epsilon_{1,t} \\ \Delta X_t = b_0 + b_1 \Delta Y_{t-1} + b_2 \Delta X_{t-1} + \epsilon_{2,t}$$

In this model, you know that an increase of 1 (100%) in asset $X$ is usually followed by an increase of around $a_2$ in asset $Y$ (in terms of continuously compounded returns).

But what if you run the same model with both series in the second differences? Well, no problem, but you need to remember that you are working with the difference of the continuously compounded returns. Now $a_2$ is measuring something different (and more unusual).

The same reasoning applies to the case where one variable is I(1) and the other I(2). There is no problem in running a VAR with one variable in the first-difference and the other in the second-difference, but the interpretation of the coefficients will be harder. Also, if you want to predict values of $Y$, you need to know if the algorithm you are applying is predicting $Y$, $\Delta Y$ or $\Delta^2 Y$.

It is unusual that your returns are not stationary, but it may happen with some commodity prices. If you are working with monthly data, make sure there is no seasonality in it. Here is a suggestion for you:

• Run the model in first differences;
• Check if residuals are not correlated;
• If they are, try to add more lags;
• If they are still correlated, then try to run the model with the second differences.
• It is "continuously compounded returns". – Richard Hardy Feb 2 '16 at 9:48