Economic Models - Convex Optimization

24 Alexis Lazaridis
A straightforward approach to obtain matrices A and C, provided that
some rank conditions of are satisfied, is to solve the eigenproblem (Johnston
and Di Nardo, 1997, pp. 287–296).
= VV −1 (2)
where is the diagonal matrix of eigenvalues and V is the matrix of the
corresponding eigenvectors. Note that since is not symmetric, V ̸= V −1 .
Eq. (2) holds if is (n × n) and all its eigenvalues are real, which implies
that the eigenvectors are real too. In cases that is defined on E n × E m ,
with m>n, Eq. (2) is not applicable. If these necessary conditions hold,
then from Eq. (2) follows that matrix A = V and C = V −1 . Further, the
co-integrating vectors are normalized so that — usually — the first element
to be equal to 1. In other words, if c i 1 (̸=0), denotes the first element1 of
the ith row of matrix C, i.e., 2 c ′ i. and a j the jth column of matrix A, then
the normalization process can be summarized as c ′ i. /ci 1 and a i × c i 1 , for
i = 1,...,n.
It is known that according to the maximum likelihood (ML) method,
matrix C can be obtained by solving the constrained problem:
{min det(I − C ˆ k0 ˆ −1
00 ˆ 0k C ′ ) | C ˆ kk C ′ = I} (3)
where matrices ˆ 00 , ˆ 0k , ˆ kk and ˆ k0 are residual co-variance matrices of
specific regressions (Harris, 1995, p. 78; Johansen Juselius, 1990). These
matrices appear in the last term of the concentrated likelihood of 3 Eq. (5)
together with C, where is replaced by AC.
Note that the minimization of Eq. (3) is with respect to the elements
of matrix C. It is re-called that Eq. (3) is minimized by solving a general
eigenvalue problem, i.e., finding the eigenvalues from:
|λ ˆ kk − ˆ k0 ˆ −1
00 ˆ 0k |=0. (4)
Applying Cholesky’s factorization on the positive definite symmetric
matrix ˆ −1
kk , such that ˆ −1
kk = LL′ ⇒ ˆ kk = (L ′ ) −1 L −1 , where L is lower
1 In some computer programs, ci i is taken to be the ith element of the ith row.
2 Note that dot is necessary to distinguish the jth row of C, i.e., c i. ′ , from the transposed of
the jth column of this matrix (i.e., c i ′).
3 Equation (5) is presented below.