Economic Models - Convex Optimization

A Novel Method of Estimation Under Co-Integration 37
Another easy and practical way to trace heteroscedasticity, is to select
the explanatory variable, which yields the smallest p-value for the corresponding
Spearman’s correlation coefficient (r s ). Note that in models like
the one we have seen in Eq. (24), such a variable is the trend, resulting
in this case to: r s = 0.3739,t = 2.394,p = 0.0219. This means that
for a significance level α>0.022, heteroscedasticity is present. Hence,
we have an initially NI series {x i }, which has to be weighed somehow in
order to be transformed to a difference stationary series (DSS). In some
cases, this transformation can be achieved if the initial series is expressed
in real terms, or as percentage of growth. In usual applications, the (natural)
logs [i.e., ln(x i )] are considered, given that {x i /x i } > {ln(x i )} >
{x i /x i−1 }.
7. Case Study 3: How to Obtain Reliable Results Regarding
the Order of Integration?
It is a common practice to apply DF/ADF for determining the order of integration
for a given series. However, the analytical step-by-step application
of this test (Enders, 1995, p. 257; Holden and Perman, 1994, pp. 64–65),
leaves the impression that the cure is worse than the disease itself. On the
other hand, by simply adopting the results produced by some commercial
computer programs, which is the usual practice in many applications,
we may reach some faulty conclusions. It should be noted at this point that
according to the results obtained by a well-known commercial package, the
series {x i }, as shown in Table 1, should be considered as I(2), for α = 0.05,
which is not the case.
Here, we present a comparatively simple procedure to determine the
order of integration of a DSS, by inspecting a model analogous to Eq. (24).
Assuming that {z i } is DSS, we proceed as follows:
• Start with k = 1 and estimate the regression:
k z i = β 1 + β 2 k−1 z i−1 + β 3 t i +
q∑
β j+3 k z i−j + u i . (25)
Be sure that the noises in Eq. (25) are white, by applying the relevant
tests as described above. This way, we set the lag-length q. Note that if
k = 1, its value is omitted from Eq. (25) and that 0 z i−1 = z i−1 .
j=1