22 Methodology of Empirical Econometric Modeling
1.4.2.1 From DGP to LDGP
“The prettiest are always further!” she said at last. (Quote from Alice in
Lewis Carroll, 1899)Granted a stochastic basis for individual agent decision taking, such that any eco-
nomic transaction can be described as an event in an event space, which could
have been different for a myriad of reasons, then outcomes are measurable ran-
dom variables with (possibly different) distributions at each point in time. Let
U^1 T=(u 1 ,...,uT)be the complete set of random variables relevant to the econ-
omy under investigation over a time spant=1,...T, defined on the probability
space(,F,P), whereis the sample space,Fthe event space andPthe probability
measure. Denote the vast, complex, and ever-changing joint distribution of
{
ut}conditional on the pre-sample outcomesU 0 and all necessary deterministic terms
Q^1 T=(q 1 ,...,qT)(like constants, seasonal effects, trends, and shifts) by:
DU(
U^1 T|U 0 ,QT^1 ,ξT^1)
, (1.1)whereξT^1 ∈⊆Rkare the parameters of the agents’ decision rules that led to
the outcomes in (1.1). ThenDU(·)is the unknown, and almost certainly unknow-
able, data-generation process of the relevant economy. The theory of reduction
discussed in,inter alia, Hendry (1987), Florens, Mouchart and Rolin (1990) and
Hendry (1995a, Ch. 9) shows that a well-defined sequence of operations leads
to the “local” DGP (LDGP), which is the actual generating process in the space
of the variables under analysis. The resulting LDGP may be complex, non-linear
and non-constant from aggregating, marginalizing (following the relevant data
partition), and sequential factorization (the order of these reductions below is
not a central aspect), so the choice of the set of variables to analyze is crucial
if the LDGP is to be viably “captured” by an empirical modeling exercise. In
turn, that LDGP can be approximated by a “general unrestricted model” (GUM)
based on truncating lag lengths, approximating the functional form (perhaps after
data transformations) and specifying which parameters are to be treated as con-
stant in the exercise. Finally, a further series of reductions, involving mapping to
non-integrated data, conditioning, and simultaneity, lead to a parsimonious rep-
resentation of the salient characteristics of the dataset. Tests of losses from all these
reductions are feasible, as discussed in section 1.4.2.4.
Aggregation. Almost all econometric data are aggregated in some way, implicitly
discarding the disaggregates: although some finance data relate to point individual
transactions, their determinants usually depend on aggregates (such as inflation).
We represent this mapping asU^1 T→VT^1 , where the latter matrix is a mix of the data
to be analyzed and all other variables. The key issue is the impact of that mapping
onξT^1 →φ^1 T, where the latter set may include more or fewer constant parameters
depending on the benefits or costs of aggregation. Aggregates are linear sums, so
their means have variances proportional to population size: ifxi,t∼IN
[
μt,σt^2]