Economic Models - Convex Optimization

A Novel Method of Estimation Under Co-Integration 35
The series { ∆xi}, { ∆ 2 x i }, and { ∆
3 x i }
Figure 2. Differences of the NI series {x i }.
Using the series {x i } as shown in Table 1, we can easily verify from the
corresponding graph, that even for n = 5, the series { 5 x i } is not stationary.
In Fig. 2, the series {x i }, { 2 x i }, and { 3 x i } are presented, for a better
understanding of this situation. We will call the variables, which belong to
this category near the integrated (NI) (Banerjee et al., 1993, p. 95) series.
To trace that a given series is NI, we may run the regression:
x i = β 1 + β 2 x i−1 + β 3 t i +
q∑
β j+3 x i−j + u i (24)
which is a re-parametrized AR(q) with constant, where we added a trend
too. As shown in Eq. (17), the value of q is set such that the noises u i to
be white. A comparatively simple way for this verification is to compute
the residuals û i and to consider the corresponding Ljung-Box Q statistics
and particularly their p-values, which should be much greater than
0.1, to say that no autocorrelation (AC) is present. On the other hand, if
we trace heteroscedasticity, this is a strong indication that we have a NI
series. For the data presented in Table 1, we found that q = 2. The corresponding
Q statistics (Column 4) together with p-values are presented
in Table 2.
We see that for all k (column 1), the corresponding p-values (column 5)
are greater than 0.1. But, if we look at the residuals graph (Fig. 3), we can
verify the presence of heteroscedasticity.
j=1