David F. Hendry 25The obvious check on the validity of such a reduction is whether longer lags matter;
and as before, the key criterion is the impact on{ψt}.
Parameter constancy. The parameters in question are those that characterize the
distributionfr(·)in (1.6). Then their constancy entails that the{ψt}depend on
a smaller set of parameters that are constant, at least within regimes. Complete
constancy requiresψt =ψ 0 ∀t, and while unlikely in economics, is often the
assumption made, at least until there is contrary evidence. When there is no loss,
θ=f(ψ 0 ), so all parameters of interest can be recovered from the model.
Linearity. The distribution in (1.6) may correspond to the linear Normal when the
functional form is chosen appropriately to ensure that a homoskedastic process also
results:
frt(
rt|Rtt−− 1 s,R^10 −s,qt,ψ 0)
̃appINk⎡
⎣∑si= 1irt−i+s+ 1 qt,⎤
⎦. (1.7)The LDGP distribution need not be Normal, but that is partly dependent on the
specification ofqt, especially whether breaks in deterministic terms are mod-
eled therein. The constancy of the coefficients of any model also depends on
the functional forms chosen for all the data transformations, and an opera-
tional GUM presumes that
{
rt}
has been transformed appropriately, based on
theoretical and empirical evidence. Checks for various nonlinear alternatives and
homoskedasticity are merited.
1.4.2.3 From the general to the specific
Providing that a viable set of basic parameters is postulated (and below we will
allow for the possibility of many shifts), then a variant of (1.7) can act as the GUM
for a statistical analysis. When the LDGP is nested in the GUM, so none of the
reductions above led to important losses, a well-specified model which embeds the
economic theory and can deliver the parameters of interest should result. When the
LDGP is not nested in the GUM, so the reductions in the previous sub-section entail
losses, it is difficult to establish what properties the final specific model will have,
although a well-specified approximation at least will have been found. Because
wide-sense non-stationarity of economic variables is such an important problem,
and within that class, location shifts are the most pernicious feature, section 1.5
considers the recent approach of impulse saturation (see Hendry, Johansen and
Santos, 2008; Johansen and Nielsen, 2008).
Mapping to a non-integrated representation. Many economic variables appear to be
integrated of at least first order (denotedI(1)), so there is a mappingrt→(rp,t:
β′rt)=xt, where there aren−pcointegrating relations andpunit roots, soxtis
nowI(0). Processes that areI(2) can be handled by mapping to second differences
as well (see, e.g., Johansen, 1995). This reduction toI(0) transformsψ 0 toρ 0 (say)