# OLS versus ML estimation of VECM

A vector error correction (VECM) model has an equivalent vector autoregression (VAR) representation.

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

(VAR) $\;\;\;\;\;\;\;\; y_t=A_1 y_{t-1}+...+A_p y_{t-p}+\varepsilon_t$

where on one hand

(A) $\;\;\Pi=-(I-A_1-...-A_p) \;$ and $\;\;\Gamma_i=-(A_{i+1}+...+A_p)$

while on the other hand

(B) $\;\; A_1=\Pi+I+\Gamma_1$, $\;A_i=\Gamma_i-\Gamma_{i-1}$ for $i=2,...,p-1$, and $A_p=-\Gamma_{p-1}$.

Compare the following estimation techniques (suppose the cointegration rank is given):

(1) Estimate the VECM representation by maximum likelihood (ML)
(2) Estimate the VECM representation by ordinary least squares (OLS)
(3) Estimate the equivalent VAR representation by OLS with linear restrictions due to (B), then algebraically convert it to the VECM representation.

I am assuming normally distributed disturbances for simplicity.

Questions:

1. Is (1) more efficient than (2)?
2. Will (2) and (3) give exactly the same estimates?
3. Should any of the alternatives, or maybe yet another approach, be generally preferred?

My guesses:

1. The knowledge of the cointegration rank amounts to nonlinear restrictions on the $\Pi$ matrix. This can be fully utilized in (1) but not in (2) as OLS will not handle nonlinear restrictions. Therefore, I guess the knowledge of the cointegration rank will be ignored by OLS and thus (1) should be more efficient than (2).
Edit: $\Pi=\alpha \beta'$ factors the cointegration matrix $\Pi$ into a loading matrix $\alpha$ and a matrix of cointegrating vectors $\beta'$. Perhaps what OLS does is use the matrix of estimated cointegrating vectors $\beta'$ from the Johansen procedure so that only $\alpha$ is remains to be estimated and there is no problem of nonlinear restrictions (actually, there are no restrictions at all in this case). If so, OLS is still not completely efficient because the cointegrating vectors are estimated (via the Johansen procedure) without taking into account all the dynamics implied in the VECM model. But what about ML estimation? Does it rely on Johansen procedure as an initial step, too? If so, then (1) and (2) would be equivalent.
2. I guess that (2) and (3) will give identical estimates as they both use equivalent representations of the same model with the same number of parameters to be estimated by the same estimation technique.
3. (1) should be generally preferred to (2) and (3) due to its efficiency as per my guess in point 1. (2) and (3) must be computationally faster than (1) since the latter will likely use numerical optimization. Also, (2) and (3) will not have convergence problems possibly found in (1).

Questions (continued):

1. Does ML estimator of VECM use the matrix of estimated cointegrating vectors $\beta'$ from the Johansen procedure?
2. Does OLS estimator of VECM use the matrix of estimated cointegrating vectors $\beta'$ from the Johansen procedure?

Edit: questions 4. and 5. are asked here, and should preferably be answered there. But since they are intimately related to the topic of this post, I still keep them visible here.

You are asking a complicated questions, to which there are no clear answers.

1. Is (1) more efficient than (2)? Note actually that the Johansen ML estimator has a strange finite-sample distribution with no finite moments, and hence has a large variance. So it is most likely not more efficient than the "Granger 2-SLS". On the other side, "Granger 2-SLS" has a large bias in the first stage, which contaminates the second stage. A simple correction for that involves adding leads and lags in the first stage, as done by Saikonnen's estimator.

2. Will (2) and (3) give exactly the same estimates? Mmh... I do not think you can use this procedure, as the restrictions due to B obviously have to be imposed with knowledge of B. So these will be the same only if you know in advance B, but then the restricted OLS procedure is of little interest...

3. Should any of the alternatives, or maybe yet another approach, be generally preferred? There is no final answer to that, as in every case in statistics, it depends... There have been quite a lot of theoretical and Monte-Carlo comparisons, of these estimators, check Maddala and Kim (1998) for a discussion, or, one among others, or Gonzalo (1994).

4. and 5. as you asked these questions in a separate post: Estimation of VECM via ML and OLS maybe you could remove from this one?

Refs:

OLS assumptions do not cover the case when one or more predictors are equal to the lagged response. The so-called strict exogeneity assumption requires the predictor to be uncorrelated with the innovation. E.g. if in AR(1) we consider $y_{t-1}$ a predictor of $y_t$, then $y_{t-1}$ is correlated with $e_{t-1}$, $e_{t-2}$, etc. As a result, applying OLS to AR(1) will produce biased estimates that are still consistent.

The relevant links are here, here and Section 3.3 of this paper

• I would disagree. As long as the regressors are "predetermined", their being lagged values of the response is not a problem. Recall that VAR models are normally estimated by OLS (a counterexample to your statement). – Richard Hardy Nov 4 '14 at 21:03
• I think the main reason why AR(p) or VAR(p) are estimated by OLS is because it's straightforward. With a certain degree of abuse the lagged responses can be treated as independent covariates. On the other hand, stretching OLS to estimate MA(q) is not that simple, and that's why it is done by MLE. – James Nov 4 '14 at 21:30
• Actually, AR(p) and VAR(p) satisfy the OLS assumptions. That OLS is also straightforward is a side issue here. – Richard Hardy Nov 4 '14 at 21:55
• Just on a note. MA(q) models cannot be estimated by OLS! You need NLS or to assume a distribution and use MLE since the lagged error is not observed at time t! It is usual to use OLS to estimate VAR(k) models and to assume normality of the errors. That way we get the equivalent to ML estimate. – Plissken Nov 11 '14 at 18:28
• Even under the strict exogeneity assumption OLS is not equivalent to MLE because the estimates of error term variance are not the same. – James Nov 12 '14 at 2:08