vectors can be normalized as βc = β(c0β)-1. An important special case of
this normalization is obtained by letting c = [I, 0]0, so that β = [I, -B]0,
where B is a r × (k - r) parameter matrix. Such a normalization is used by
Ahn and Reinsel (1990) and Phillips (1991, 1995).
The ML estimator is obtained by maximizing the log-likelihood function
(2) subject to the normalization restrictions (cf. Pesaran et al. 1999). In
practice this approach may become computationally burdensome and for a
small number of time periods, problems with the convergence of the Gauss-
Newton algorithm may occur. In the following section a simple two-step
approach is suggested to obtain an estimator that is asymptotically equiva-
lent to the ML estimator.
3 The two-step estimator
Since the ML procedure is computationally burdensome it is preferable to
employ a simple two-step1 estimation procedure that has the same large sam-
ple properties as the ML procedure. Engle and Yoo (1991) have shown that
the information matrix of the Gaussian likelihood is asymptotically block
diagonal with respect to the “short-run parameters” (αi, Σi) and the ma-
trix of cointegration vectors β . Therefore, the matrix β can be estimated
conditional on some consistent initial estimator of αi and Σi (i = 1, . . . , N).
To motivate the two-step estimator, consider the transformed VECM
model
γi0∆yit = γi0αiβ0yi,t-1 + γi0εit
zit = β0yi,t-1 + vit , (3)
where zit = (γi0αi)-1γi0∆yit, vit = (γi0αi)-1γi0εit and γi is a k × r matrix with
rk(γi0αi) = r. From
Σv = E(vitvi0t) = (γi0αi)-1γi0Σiγi (α0iγi)-1 (4)
it follows that Σv - (α0iΣ-1αi)-1 is positive semi-definite and, therefore, the
optimal choice of the transformation is γi0 = α0iΣi-1 . The resulting estimator
is asymptotically equivalent to the Gaussian ML estimator (cf. Reinsel 1993,
p. 170).
1This estimator is called a “three-step estimator” by Engle and Yoo (1991). Here we
follow Reinsel (1993, p. 170f) and refer to it as the two-step estimation procedure.