Economic Models - Convex Optimization

A Novel Method of Estimation Under Co-Integration 39
To show the details, underline the first stage of this procedure; we
present some estimation results with different values of q.
For q = 3
Most of the p-values in the 5th column of the table, which is analogous
to Table 2, are equal to zero.
Obviously, this is not the proper lag-length.
For q = 4
All p-values are greater than 0.1, as shown in Table 2. r s =−0.193,
p = 0.05652. F(2, 94) = 4.2524.
This lag-length is acceptable.
For q = 5
All p-values are greater than 0.1, as shown in Table 2. r s =
−0.2094,p= 0.0395.F(2, 92) = 4.2278.
This lag-length is questionable, since for α>0.04, we face the problem
of heteroscedasticity.
q = 6
All p-values are greater than 0.1, as we have seen in Table 2. r s =
−0.166,p= 0.1038.F(2, 90) = 5.937.
This lag-length is quite acceptable.
q = 7
All p-values are greater than 0.1, as shown in Table 2. r s =−0.1928,p=
0.0608.F(2, 88) = 4.768.
This lag-length is also acceptable, but it is inferior when compared to
the previous one.
From these analytical results, it can be easily verified that the proper
lag-length is 6 (q = 6). It may also be worthy to mention that according
to the normality test (Jarque-Bera), we should accept the null. The value
of F = 5.937 is less than the corresponding critical values as shown in
Table 3, for 100 observations and α = (0.01, 0.05). Hence, we accept
Eq. (26) and increase the value of k by one. The estimation results, after
properly selecting the value of q(q = 5), are:
2 m i = 0.0047 − 0.894892 m i−1 + 0.00033 t i
(0.0046) (0.21784) (0.000108)
+
5∑
ˆβ j+3 2 m i−j +û i . (27)
j=1
Regarding the residuals, all p-values in the 5th column of the table,
which is analogous to Table 2, are greater than 0.1. Further, we have:
r s =−0.1556,p= 0.127. Jarque-Bera = 0.8(p = 0.67).F(2, 91) = 8.44.