458 Fractional Integration and Cointegration
1.341, and then Box-Jenkins-type procedures identified models within the ARMA
class. This resulted in AR(1) and ARMA(1,1) models foru 1 tand white noise and
ARMA(1,1) models foru 2 t, and we fitted all four combinations. We also fitted
bivariate versions of Bloomfield’s (1973) model, where:
A(s)=diag{
exp(∑
p
j= 1 θ^1 jsj)
, exp(∑
p
j= 1 θ^2 jsj)}
,forp=1, 2, 3. For each model we applied the univariate Whittle procedure of
Velasco and Robinson (2000), using untapered, differenced data and adding back
- We summarize the seven models and the resulting (δˆ,γ)ˆ as follows:
Model 1:u 1 tis AR(1) andu 2 tis white noise (δˆ,γ)ˆ =(1.612, 0.669)
Model 2:u 1 tis AR(1) andu 2 tis ARMA(1,1) (δˆ,γ)ˆ =(1.408, 0.669)
Model 3:u 1 tis ARMA(1,1) andu 2 tis white noise (δˆ,γ)ˆ =(1.612, 0.660)
Model 4:u 1 tis ARMA(1,1) andu 2 tis ARMA(1,1) (δˆ,γ)ˆ =(1.408, 0.660)
Model 5:utis bivariate Bloomfield withp= 1 (δˆ,γ)ˆ =(1.214, 0.710)
Model 6:utis bivariate Bloomfield withp= 2 (δˆ,γ)ˆ =(1.434, 0.701)
Model 7:utis bivariate Bloomfield withp= 3 (δˆ,γ)ˆ =(1.323, 0.547)Theγˆseem very robust to the short memory specification, theδˆrather less so.
We also took the opportunity to examine a further question which, in one form
or another, always arises with applications of fractional models, and perhaps most
acutely when non-stationary data are involved. This is the matter of truncation.
When estimated innovations from a stationary fractional model are computed, the
(infinite) AR representation has to be truncated because the data begins at time “1,”
not at time “−∞.” Now, in our model (10.5)–(10.6) for non-stationary data, the
truncation is actually inherent in the model, so strictly speaking there is no “error”
associated with it. However, the model reflects the time when the data begin, and
if we were to drop the first observation, say, and start the model off at the second,
the degree of filtering applied to all subsequent observations would change, and
it is possible that this could have a marked effect, especially with non-stationary
data. Thus, in Table 10.1 we report computations of our estimatesν(ˆγˆ,δˆ,θ)ˆ = ̄νi
and Wald statistics:
b(γˆ,θ)ˆ{ˆν(γˆ,δˆ,θ)ˆ − 1 }^2 =Wi,for modelsi=1,..., 7, based on the lastn′=n−jobservations, forj=0, 1,..., 10,
in order to explore sensitivity to starting value. Substantial variation is evident
across the largern′, with allν ̄iexceeding 1 and the homogeneity hypothesis being
strongly rejected whenn′=123 across all seven models, but asn′decreases, things
stabilize. Forn′ ≤119 some sensitivity to theu 2 tspecification was found, the
white-noise cases (Models 1 and 3) providing estimates ofνless than 0.9, whereas
for the other models they all exceed 0.9, with the largest values for Model 7. For
n′≤122 the homogeneity hypothesisν=1 is never rejected even at the 10% level.
From certain perspectives, practitioners could consider our empirical analysis
simplistic, as we do not take into account possible alternative features of our