Economic Models - Convex Optimization

A Novel Method of Estimation Under Co-Integration 25
triangular, Eq. (4) takes the conventional form 4 :
|λI − L ′ ˆ k0 ˆ −1
00 ˆ 0k L|=0. (4a)
After obtaining matrix V = V ′ of the eigenvectors, we can easily
compute the (unormalized) C from C = V ′ L ′ . Note also that C ˆ kk C ′ =
V ′ L ′ (L ′ ) −1 L −1 LV = I. Finally, matrix A is obtained from A = ˆ 0k C ′ .
Hence, the initial problem reduces to finding one of the eigen values and
eigenvectors of a symmetric matrix, which are real. And this is the main
advantage of the ML method.According to this procedure, a constant and/or
trend can be included in the co-integrating vectors, but it is not clear of how
to accommodate additional deterministic factors, such as one or more dummies.
Besides, nothing is said about the variance of the (disequilibrium)
errors, which correspond to the finally adopted co-integration vector.
We propose in the first part of this chapter, a comparatively simple
method, to directly obtain co-integrating vectors, which may include, apart
from the standard deterministic components, any number of dummies, in
cases that this is necessary. Additionally, as we will see in what follows, the
errors corresponding to the proper co-integration vector have the property
of minimum variance.
2. Some Estimation Remarks
It was mentioned that after estimation, matrix can be computed from
Eq. (1a). It is obvious that each equation of the VAR can be estimated by
OLS. At this stage, one should be aware to comply with the basic assumptions
regarding the error terms, in the sense that proper tests must ensure
that the noises are white. This way, the order p of the model is defined
imposing at the same time restrictions — if necessary — on some elements
of A’s and vectors µ and z to be zero.
It can be easily shown that Eq. (1) may be expressed in the form of an
equivalent ECVAR, i.e.,
p−1
∑
x i = δ + µt i + Q j x i−j + x i−p + zd i + w i (5)
j=1
4 It should be noted that it is also necessary to pre-multiply the elements of Eq. (4) by L ′
and post-multiply them by L.