Economic Models - Convex Optimization

Credit and Income 207
Since the variables Y i , W i are used in this study, it should be re-called that
the parameters of a long-run relationship represent constant elasticities.
It should be re-called that if two series are not stationary but they have
common trends so that there exist a linear combination of these series that
is stationary, which means that they are integrated in a similar way and for
this reason, they are called co-integrated, in the sense that one or the other
of these variables will tend to adjust so as to restore a long-run equilibrium.
Also, if there is a linear trend in the data, as in the case of Y i and W i as
shown in Fig. 1, then we specify a model including a time-trend variable,
allowing thus the non-stationary relationships to drift.
4.1. The VAR Model and The Equivalent ECVAR
We start this section by formulating and estimating an unrestricted VAR of
the form:
p∑
x i = δ + µt i + A j x i−j + w i (1)
j=1
where x ∈ E n (here E denotes the Euclidean space), δ is the vector of
constant terms, t i is a time-trend variable, µ is the vector of corresponding
coefficients, and w is the noise vector. The matrices of coefficients A j , are
defined on E n × E n . It is noted that variable t i is included for the reason
mentioned above. It can be easily shown (Johansen, 1995), that (1) may be
expressed in the form of an equivalent ECVAR, that is:
p−1
∑
x i = δ + µt i + Q j x i−j + x i−1 + w i (2)
j=1
where Q j = ( ∑ j
k=1 A k − I).
After estimation, we may compute matrix from:
⎛ ⎞
p∑
= ⎝ A j
⎠ − I
j=1
(2a)
or, directly from the corresponding ECVAR as given in Eq. (2). It is re-called
that this type of VAR models like the one presented above, is recommended
by Sims (1980a; b), as a way to estimate the dynamic relationships among
jointly endogenous variables. It should be noted that for the case under