# What is the difference between ECM and VECM?

From what I understood,
ECM is for two variables and apply OLS to estimate EC term and
VECM is for multi-variables (vector form) and apply VAR to estimate EC term.

But as I read other papers, I think I may have misunderstood.
Is VECM just an expansion of ECM?
However, in equation form, how could it be justified?

Suppose that $$y_t =\alpha +\beta x_t +e_t$$.
ECM: $$\Delta y_t = \alpha+\sum_{j=1}^{k} \phi{_j} \Delta y_{t-j} +\sum_{j=1}^{k} \pi_j \Delta x_{t-j} +\delta z_{t-1} +v_t$$ where $$z_{t-1}=y_t-a-b x_t$$
Suppose that $$y_t =\alpha +\beta y_t +e_t$$ (vector form).
VECM: $$\Delta y_t = \alpha+\phi y_{t-1} +\sum_{j=1}^{p-1} \pi_j \Delta y_{t-j} +v_t$$ for VAR(p).
VECM's EC term $$y_{t-1}$$ (vector) is just an expansion of ECM's EC term $$z_{t-1}$$? (I don't think so.)

## 1 Answer

ECM consists of a single equation for a single dependent variable (that is cointegrated with another variable), while VECM consists of multiple equations for multiple dependent variables (that are cointegrated). You could take one equation from a VECM, analyze it separately, and you could call that an ECM.

Regarding

ECM is for two variables and apply OLS to estimate EC term and VECM is for multi-variables (vector form) and apply VAR to estimate EC term

VAR is a model, not an estimation technique. A VECM has an equivalent restricted-VAR representation. A VECM can be estimated using equation-by-equation OLS with the error correction term(s) taken as given. The coefficients of the error correction terms are estimated beforehand using OLS or maximum likelihood.

• Now I understand it. Thank you so much! Commented Dec 23, 2021 at 6:53
• @guest, you are welcome! Commented Dec 23, 2021 at 6:53
• Regarding to your edited answer, why $y_{t-1}$ should be $z_{t-1}$? I refer to this page 454, eq(12.21). Commented Dec 23, 2021 at 6:57
• @guest, I think I had misunderstood you. If $\phi$ is a coefficient matrix and $y_{t-1}$ is a vector, then it is OK. Commented Dec 23, 2021 at 11:25