Economic Models - Convex Optimization

38 Alexis Lazaridis
Table 3.
Critical values for the DF F-test.
Sample size α = 0.01 α = 0.05 α = 0.10
25 10.61 7.24 5.91
50 9.31 6.73 5.61
100 8.73 6.49 5.47
250 8.43 6.34 5.39
500 8.34 6.30 5.36
>500 8.27 6.25 5.34
• Proceed to test the hypothesis:
H 0 : β 2 = β 3 = 0. (26)
To compare the computed F-statistic, we have to consider, in this case,
the Dickey-Fuller F-distribution. The critical values (Dickey and Fuller,
1981) are reproduced in Table 3, to facilitate the presentation.
Note that in order to reject Eq. (26), F-statistic should be much greater
than the corresponding critical values, as shown in Table 3. If we reject
Eq. (26), this means that the series {z i } is I(k − 1). If the null hypothesis
is not rejected, then increase k by 1(k = k + 1) and repeat the same
procedure, i.e., starting from estimating Eq. (25), testing at each stage so
that the model disturbances are white noises.
• In case of known structural break(s) in the series, a dummy d i should be
added in Eq. (25), such that:
{ 0 if i ≤ r
d i =
w i if i>r
where w i = 1(∀ i) for the shift in mean, or w i = t i for the shift in
trend. It is re-called that r denotes the date of the break or shift. With
this specification, the results obtained are alike the ones that Perron et al.
(1992a; b) procedure yields.
We applied the proposed technique to find the order of integration of the
variable m i as shown in Eq. (21), which is the log of nominal M1 (money
supply). Harris (1995, p. 38) after applying DF/ADF test reports that “...
and the interest rate as I(1) and prices as (I2). The nominal money supply
might be either, given its path over time....”. Since the latter variable is m i ,
it is clear that, with this particular test, it was not possible to get a precise
answer as to whether {m i } is either I(1) or I(2).