Economic Models - Convex Optimization

216 Athanasios Athanasenas
The estimation results are as follows 12 :
W i = 0.2241+ 0.0977Y i−1 + 0.2256Y i−2 + 0.199Y i−3 + 0.4864W i−1
(0.125) (0.0733) (0.072) (0.073) (0.075)
p value 0.075 0.183 0.002 0.0072 0.000
Hansen 0.075 0.181 0.041 0.7130 0.045
+ 0.0123W i−2 + 0.068W i−3 − 0.03852u i−1 + W ˆv i
(0.87) (0.29) (0.0216)
p value 0.519 0.242 0.076
Hansen 0.062 0.181 0.075 (for all coefficients 2.268)
¯R 2 = 0.541, s = 0.009, DWd = 1.97,
F (7,189) = 34.0(p value = 0.0), CN = 635.4, Revised CN = 3.42.
(18)
Presumably, we are on the right way, since the estimated adjustment coefficient
is almost the same as the one computed by the ML method as in
the previous case. This is the (2, 1) element of matrix A as seen in Eq. (9).
Note that this coefficient is significant for α ≥ 0.08, which means that there
is a rather weak causality effect from Y to W in the long run.
To test the null H 0 : a 1 = a 2 = a 3 = 0, where the coefficients
a j (j = 1, 2, 3) as seen in Eq. (17), we compute the relevant F-statistic,
i.e., F (3,189) = 8.42 (p-value = 0.0). This implies that indeed there is a
causality effect in the short run, from Y to W.
Considering the complete ECVAR, i.e., Eqs. (16) and (18), Eq. (10a) is
used to compute the series {u i }, together with some trivial identities, I form
the following system:
Y i = ˆβ 0 +
W i = ˆb 0 +
3∑
ˆα j Y i−j +
j=1
3∑
â j Y i−j +
j=1
3∑
ˆβ j W i−j −â Y u i−1 + Y ˆv i
j=1
3∑
ˆb j W i−j −â W u i−1 + W ˆv i
j=1
12 According to Hansen statistics, all coefficients seem to be stable for α = 0.05.
The Spearman’s correlation coefficient (r s ) regarding the term u i−1 , is: r s = −0.075,
t =−1.051, p = 0.294, which means that we do not have to bother about heteroscedasticity
problems. Also the revised CN indicates that no multicollinearity problems exist, in
order to affect the reliability of the estimation results.