A parametric approach to the estimation of cointegration vectors in panel data

vectors can be normalized as β_c = 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” (α_i, Σ_i) 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

γ_i⁰∆yit = γ_i⁰αiβ⁰yi,t-1 + γ_i⁰εit

zit = β⁰yi,t-1 + vit ,                               (3)

where zit = (γ_i⁰αi)^-1γ_i⁰∆yit, vit = (γ_i⁰αi)^-1γ_i⁰εit and γi is a k × r matrix with
rk(γ_i⁰αi) = r. From

Σv = E(vitv_i⁰_t) = (γ_i⁰αi)^-1γ_i⁰Σiγi (α⁰_iγi)^-1                   (4)

it follows that Σ_v - (α⁰_iΣ^-1α_i)^-1 is positive semi-definite and, therefore, the
optimal choice of the transformation is γ_i⁰ = α⁰_iΣ_i^-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.

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