Economic Models - Convex Optimization

Credit and Income 209
errors computed from u i = c ′ k. ˜x i must be a stationary series. Considering
now the kth column of matrix A, that is a k , and given that u i−1 = c ′ k. x i−1,
then the ECVAR in Eq. (4) can be written as:
p−1
∑
x i = Q j x i−j + a k u i−1 + w i . (5)
j=1
This is the conventional form of an ECVAR, when matrix ˜ instead
of is considered. In any case, the maximum lag in Eq. (5) is p − 1. It
should be noted that this specification of an ECM is fully justified from the
theoretical point of view. However, if we want to include an intercept in
Eq. (5), i.e., in the short-run model, it may be assumed that the intercept in
the co-integration vector is cancelled by the intercept in the short-run model,
leaving only an intercept in Eq. (5). Thus, when a vector of constants is to
be included in Eq. (4), which, in this case, takes the form:
p−1
∑
x i = δ + Q j x i−j + ˜˜x i−1 + w i (6)
j=1
then Eq. (5), matrix ˜ and the augmented vector ˜x will become:
p−1
∑
x i = δ + Q j x i−j + a k u i−1 + w i
j=1
(6a)
˜ = [.µ]
(6b)
[ ]
xi−1
˜x i−1 = . (6c)
Lt i−1
According to Eq. (6a), the co-integrating vector c ′ k.
, used to compute
u i = c ′ k. ˜x i, includes only the coefficient of the time-trend variable. The
inclusion of a trend in the model can be justified, if we examine the estimation
results of a long-run relationship considering Y and W. In such a
case, we immediately realize that a trend should be present.
After adopting the VAR as seen in Eq. (1), the next step is to determine
the value of p from Table 1 (Holden and Perman, p. 108).
We see from Table 1 that for α ≤ 0.05, p = 4, which means that we
have to estimate a VAR(4), with only one deterministic terms (i.e., trend).
After estimation, we found that matrix ˜ (hat is omitted for simplicity),