Economic Models - Convex Optimization

208 Athanasios Athanasenas
consideration, vector x in Eqs. (1) and (2) is 2-dimensional, i.e.,
[ ]
Yi
x i = and it is x ∼ I(1).
W i
It is known that the matrix can be decomposed such that = AC,
where the elements of A are the coefficients of adjustment and the rows
of matrix C are the (possible) co-integrating vectors. In some books, A is
denoted by α and B by β ′ , although capital letters are used for matrices.
Besides, β denotes the coefficient vector in a general linear econometric
model. To avoid any confusion, we adopted the notation used here, as well
as in Mouza and Paschaloudis (2007).
Usually, matrices A and C are obtained by applying the ML method
(Harris, 1995; Johansen and Juselius, 1990, p. 78). According to this procedure,
a constant and/or trend can be included in the co-integrating vectors
in the following way.
Considering Eq. (2), we can augment matrix to accommodate, as an
additional column, the vectors δ and µ in the following way.
˜ = [.µδ]. (3)
In this case, ˜ is defined on E n × E m , with m>n. For conformability,
the vector x i−1 in Eq. (2) should be augmented accordingly by using the
linear advance operator L (such that L k y i = y i+k ), i.e.,
⎡
˜x i−1 = ⎣ x ⎤
i−1
Lt i−1
⎦ .
(3a)
1
Hence, Eq. (2) takes the form:
p−1
∑
x i = Q j x i−j + ˜˜x i−1 + w i . (4)
j=1
Note that no constants (vector δ), as well as coefficients of the time trend
(vector µ) are explicitly presented in the ECVAR in Eq. (4), which specifies
precisely a relevant set of n ECM.
It is assumed at this point that matrices C and A refer to matrix ˜ as
seen in Eq. (4). Let us consider now the kth row of matrix C, that is c ′ k. .8 If
this row is assumed to be a co-integration vector, then the (disequilibrium)
8 Note that dot is necessary to distinguish the kth row of C, i.e., c ′ k. , from the transposed of
the kth column of this matrix that is c ′ k .