What is the difference between ECM and VECM?

Question

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.)

Richard Hardy · Accepted Answer · 2021-12-23T11:21:33.567

0

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.

edited Dec 23 '21 at 11:21

answered Dec 23 '21 at 06:44

Richard Hardy

67,272

Now I understand it. Thank you so much! – guest Dec 23 '21 at 06:53
@guest, you are welcome! – Richard Hardy Dec 23 '21 at 06:53
Regarding to your edited answer, why $y_{t-1}$ should be $z_{t-1}$? I refer to this page 454, eq(12.21). – guest Dec 23 '21 at 06:57
@guest, I think I had misunderstood you. If $\phi$ is a coefficient matrix and $y_{t-1}$ is a vector, then it is OK. – Richard Hardy Dec 23 '21 at 11:25

What is the difference between ECM and VECM?

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