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.)
Best 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
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.