# VECM with first differences? [closed]

Is it ok to take first differences of data which is non stationary in levels but stationary in first differences (and cointegrated), and input these differenced variables into the VECM? Or does this constitute over differencing given th VECM differences the data as well?

• If your variables are cointegrated then you don't need to difference them. That's the whole point of cointegration. Sep 11, 2019 at 5:05
• Is it ok to difference them again though? as this provides better results in my Impulse Response Functions Sep 11, 2019 at 5:56
• Possible duplicate of Levels or First Differences, VECM or VAR for Ultimate Impulse Response Functions? Sep 11, 2019 at 7:31

## 1 Answer

When the variables are integrated of order one, I(1), and cointegrated, VECM is specified with variables in differences on the left hand side of the equation: $$\Delta y_t=\Pi y_{t-1}+\Gamma_1\Delta y_{t-1}+...+\Gamma_{p-1}\Delta y_{t-(p-1)}+\varepsilon_t.$$ The right-hand-side variables are all differenced except for the error correction term which is a linear combination (or several of them) of $$y_{t-1}$$ in levels.

If you specify a VECM of the form $$\Delta^2 y_t=\Pi\Delta y_{t-1}+\Gamma_1\Delta^2 y_{t-1}+...+\Gamma_{p-1}\Delta^2 y_{t-(p-1)}+u_t$$ instead, you have overdifferencing your series, and that is a problem.

Could you be getting "better" results from the latter model in terms of impulse response function? That is not impossible. Models are but approximations to reality. Assumptions for making these models and the coefficient estimators work may be violated to a smaller or larger degree. Perhaps the behavior of variables you are modelling is indeed better described by the overdifferenced model than the regular/"proper" one.

However, there is also a question of how you know what is "better". Perhaps the IRFs you obtain from the overdifferenced model are more like what you expected beforehand, but that does not mean they are closer to the truth than those from the regular VECM.

In any case, when choosing the overdifferenced model you would need to come up with a justification for that, while choosing the regular VECM is already justified by the theory around cointegrated processes (Granger's representation theorem etc.).