24 Methodology of Empirical Econometric Modeling
Sequentially factorizing. Next, lettingRt^1 − 1 =(r 1 ,...,rt− 1 ), the retained marginal
density from (1.3) can be sequentially factorized as (see, e.g., Doob, 1953):
DR(
RT^1 |U 0 ,QT^1 ,ω^1 T)
=∏Tt= 1Drt(
rt|R^1 t− 1 ,U 0 ,qt,λt). (1.4)
The right-hand side of (1.4) completes the intrinsic reductions from the DGP to
the LDGP for the set of variables under analysis (generally, the effects of the initial
conditionsU 0 are ignored and assumed to be captured byR 0 ). The sequential
densities in (1.4) create a martingale difference (or innovation) process:
(^) t=rt−E
[
rt|R^1 t− 1 ,U 0 ,qt
]
, (1.5)
whereE[ (^) t|R^1 t− 1 ,U 0 ,qt]= 0 by construction.
Parameters of interest. These are the targets of the modeling exercise, and are
hypothesized – on the basis of prior reasoning, past studies, and institutional
knowledge – to be the features of interest. We denote them byθ∈, and any
later reduction choices must be consistent with obtainingθfrom the final specifi-
cation. To the extent that the economic theory supporting the empirical analysis
is sufficiently comprehensive, the
{
λt
}
in (1.4) should still contain the required
information about the agents’ decision parameters, soθ=h(ω^1 T). The next stage is
to formulate a general model of (1.4) that also retains the necessary information.
1.4.2.2 From LDGP to general unrestricted model
The LDGP in (1.4) can be approximated by a model based on a further series of
reductions, which we now discuss. Indeed, (1.4) is often the postulated basis of
an empirical analysis, as in a vector autoregression, albeit with many additional
assumptions to make the study operational. There are no losses when the LDGP
also satisfies these reductions, and if not, evidence of departures can be ascertained
from appropriate tests discussed in section 1.4.2.4, so that such reductions are then
not undertaken.
Lag truncation. The potentially infinite set of lags in (1.4) can usually be reduced
to a small number, soR^1 t− 1 Rtt−− 1 s=(rt−s...rt− 1 ), where the maximum lag length
becomessperiods, with initial conditionsR^10 −s. Long-memory and fractional inte-
gration processes are considered in, e.g., Granger and Joyeux (1980), Geweke and
Porter-Hudak (1983), Robinson (1995) and Baillie (1996). Lettingfrt(·)denote the
resulting statistical model of the
{
rt
}
, which could coincide with the LDGP when
the reduction is without loss, then the mapping is:
∏T
t= 1
Drt
(
rt|R^1 t− 1 ,U 0 ,qt,λt
)
⇒
∏T
t= 1
frt
(
rt|Rtt−− 1 s,R 01 −s,qt,ψt
)
. (1.6)