Economic Models - Convex Optimization

A Novel Method of Estimation Under Co-Integration 33
Following the procedure described above, we obtain matrix C which is:
⎡
⎤
1 −48.30517 24.60144 37.75685
1 5.491415 −0.6510755 7.423318
C = ⎢
⎥
⎣ 1 −2.055229 −5.466846 0.9061699 ⎦
1 0.0051181894 0.1603713 −0.1244312
⎛ ⎞
66.06966
9.310488
Euclidean norm = ⎜ ⎟
⎝ 5.99429 ⎠
1.020406
The singular values of are:
f 1 = 1.503069, f 2 = 0.7752298, f 3 = 0.1977592
and f 4 = 0.012560162
The fact that one singular value is greater than one, is an indication that
unless a dummy is present — we hardly can trace one row of matrix C,
which may be related to stationary errors û i . As shown in Fig. 1, from the
graphs, from which refer to the errors corresponding to the rows of C, we
see that only the second row produces reasonably results. In this case, the
errors are computed from:
û i = (m − p) i + 5.491415p i − 0.6510755y i + 7.42318R i . (22)
As usual, these errors will be denoted by usvd i .
The minimum variance property can be seen from:
Standard deviation of {uml i } Standard deviation of {usvd i }
0.2671 0.2375
To see that none of the two error series is stationary, as it was indicated
by the first singular value, we present the following results.
u ˆml i = 0.268599 − 0.178771 uml i−1 − 0.214098uml i−1
(0.065634)
t =−2.724
uŝvd i = 0.832329 − 0.193432 usvd i−1 − 0.160135usvd i−1
(0.066421)
t =−2.912
For a long-run relationship with three (independent) variables without
trend and intercept, the critical value t u for α = 0.10 is less than −3.