Economic Models - Convex Optimization

212 Athanasios Athanasenas
vector:
[1 − 0.289613 − 0.003621] (10)
producing the errors u i , which are the disequilibrium errors obtained from:
u i = Y i − 0.289613W i − 0.003621t i .
(10a)
It should be noted that the vector specified in Eq. (10) is a co-integration
vector, iff the series {u i } computed from Eq. (10a) is stationary. To find out
that this series is stationary, we run the following regression with a constant
term, since ∑ u i ̸= 0.
q∑
u i = a + b 1 u i−1 + b j+1 u i−j + ε i . (11)
Note that the value of q is set such that the noises ε i to be white. With q = 3,
the estimation results are as follows:
3∑
u i = 0.4275 − 0.07384 u i−1 + ˆb j+1 u i−j +ˆε i
(0.1174) (0.02029) (11a)
j=1
t =−3.64.
It should be noted that in Eq. (11a), there is no problem regarding
autocorrelation and heteroscedasticity. We have to compute the t u -statistic
from MacKinnon (1991, pp. 267–276) critical values (see also Granger and
Newbold, 1974; Hamilton, 1994; Harris, 1995, Table A6, p. 158; Harris,
1995, pp. 54–55). This statistic (t u = ∞ + 1 /T + 2 /T 2 , where T
denotes the sample size) is evaluated from the relevant table, taking into
account Eqs. (10a) and (11a). Note that in the latter equation, there is only
one deterministic term (constant). The value of the t u -statistic for this case
is −3.3676 (α = 0.05). Hence, since −3.64 < −3.3676, we reject the
null that the series {u i } is not stationary. Recall that the null [u i ∼ I(1)] is
rejected in favor of H 1 [u i ∼ I(0)], if t