Economic Models - Convex Optimization

A Novel Method of Estimation Under Co-Integration 29
which in this case, is the first one. Hence, the errors corresponding to this
co-integration vector are obtained from:
û i = C i − 0.9206225I i − 0.06545528W i + 0.1110597. (15)
These errors will be denoted by usvd i . It should be pointed out here
that the smallest norm condition is just an indication as to which row of
matrix C to be considered first.
It may be useful to re-call at this point that the co-integration vector
reported by the authors (p. 111), which is obtained by applying OLS after
omitting the first two observations from the initial sample is:
1 − 0.9109 − 0.0790 + 0.1778.
It should be clarified that this vector has been obtained by applying OLS
to estimate directly the long-run relationship. Repeating this procedure,
we get:
1 − 0.910971 − 0.079761 + 0.178455.
Hence, the errors are computed from:
û i = C i − 0.910971I i − 0.079761W i + 0.178455. (16)
These errors will be denoted by ubook i .
The next task is to compare the above three series of errors, having a
first look at the corresponding graphs, which are shown in Fig. 1.
Although by a first glance the three series seem to be stationary, we shall
go a bit further to see their properties in detail. Before going to the next
section, it is worth to mention here that for the three series, the normality
test using the Jarque-Bera criterion favors the null.
Figure 1. The series {uml i }, {usvd i }, and {ubook i }.