Economic Models - Convex Optimization

214 Athanasios Athanasenas
in relevant works, the stability analysis applied for the system presented
in Eq. (8).
4.5. The ECM Formulation
The lagged values of the disequilibrium errors, that is u i−1 , serve as an
error correction mechanism in a short-run dynamic relationship, where the
additional explanatory variables may appear in lagged first differences. All
variables in this equation, also known as ECM, are stationary so that, from
the econometric point of view, it is a standard single equation model, where
all the classical tests are applicable. It should be noted, that the lag structure
and the details of the ECM, should be in line with the formulation as seen
in Eq. (6a). Hence, I started from this relation considering the errors u i
and estimated the following model, given that the maximum lag length is
p − 1 i.e., 3.
Y i = α 0 +
3∑
α j Y i−j +
j=1
3∑
β j W i−j − a Y u i−1 + Y v i . (15)
j=1
Note that Y v i are the model disturbances. If the adjustment coefficient
a Y is significant, then we may conclude that in the long run, W causes Y.
If a Y = 0, then no such a causality effect exists. In case that all β j are
significant, then there is a causality effect in the short run, from W to
Y. Ifallβ j = 0, then no such causality effect exists. The estimation
results are presented below.
Y i = 0.5132+ 0.2618Y i−1 + 0.161Y i−2 − 0.00432Y i−3 + 0.0273W i−1
(0.125) (0.074) (0.073) (0.0738) (0.075)
p value 0.0001 0.0005 0.0276 0.953 0.719
Hansen 0.0370 0.1380 0.2380 0.234 0.128
+ 0.0487W i−2 − 0.076W i−3 − 0.0879u i−1 + Y ˆv i
(0.075) (0.065) (0.022)
p value 0.519 0.242 0.0001
Hansen 0.065 0.098 0.025 (for all coefficients 2.316) (16)
¯R 2 = 0.167,s= 0.009, DWd = 1.98,F (7,189) = 6.65,
Condition number (CN) = 635.34.
(p value = 0.0)