# Estimation of VECM via ML and OLS

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).

• The accepted answer is correct; $\beta$ consists of the $r$ first eigenvectors of the solution to an eigenvalue problem, where $r$ is the rank. Once you have your $\beta$, you can just do the multiplication $\hat\beta' y_{t-1}$ (call it $\tilde y_{t-1}$) so that in your model you have $\alpha\tilde{y}_{t-1}$. $\alpha$ is full rank so there is no problem estimating it from this, so now what you have is simply multivariate regression. – hejseb Mar 19 '15 at 10:14

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)

• If an estimate of $\beta$ is obtained first and then used to estimate $\alpha$ and the $\Gamma$'s, it seems to me OLS can deal with cointegrating ranks $r>1$ (where $\beta$ has r columns) just as well as with rank $r=1$ (where $\beta$ has 1 column). Why wouldn't it? I assume the same Johansen procedure is used as the first step in both ML and OLS cases. – Richard Hardy Dec 1 '14 at 6:13
• Apparently, what I was interested in was a hybrid procedure where the first step is estimation of $\beta$ using Johansen procedure (as opposed to the first step of Engle-Granger two-step OLS) and the second step is OLS estimation of each equation in the VECM when $\beta$ is taken as given. I think this is how it works in vars package in R, for example. Anyhow, your answer clarifies everything I wanted to know. Thanks! – Richard Hardy Dec 3 '14 at 15:07