Estimation of VECM via ML and OLS

Question

Take a vector error correction (VECM) model:

$$\;\;\;\Delta y_t=\Pi y_{t-1}+\Gamma_1\Delta y_{t-1}+...+\Gamma_{p-1}\Delta y_{t-(p-1)}+\varepsilon_t$$

where $\Pi=\alpha \beta'$ and each row of $\beta'$ (or, equivalently, each column of $\beta$) is a cointegrating vector.

Questions:

1. When VECM is estimated by maximum likelihood (ML), is an estimate of the $\beta'$ matrix taken as given (e.g. it could be obtained as a by-product of the Johansen procedure)?
   Or is $\Pi$ estimated simultaneously with all the other parameters in the model, subject to restrictions on $\Pi$ due to the cointegration rank (which needs to be obtained in advance, e.g. via the Johansen procedure)?
2. When VECM is estimated by ordinary least squared (OLS), is an estimate of the $\beta'$ matrix taken as given?

Here is a related question (see Edit of guess 1 and questions 4, 5).

Answer 1

Both procedures are sequential:

1. First obtain an estimate of the $\beta$ parameter (by Johansen ML or Engle-Granger two-step OLS, see details below)
2. Then, conditional on the estimated $\hat\beta$, obtain the $\alpha$ and $\gamma$ parameters with standard (multi-equation) OLS. Note indeed that for a given $\beta$ (whichever the rank of $\beta$), it reduces to a standar (multi-equation) linear regression, for which ML=OLS.

The only difference is in the estimation of the $\beta$:

1. with Johansen ML, you use a reduced rank regression, which allows to estimate multiple cointegrating relationships (rank >=1)
2. with Engle-Granger two-step OLS, you use OLS on a single long-term relationship, so can only estimate 1 cointegrating relationship (rank =1)

Hope this answers your question?