26 Methodology of Empirical Econometric Modeling
and leads from (1.6) to:
∏Tt= 1fxt(
xt|Xtt−− 1 s,X^10 −s,qt,ρ 0). (1.8)
VARs like (1.7) are often formulated forrt, rather thanxt, as occurs in the first stage
of some cointegration analyses.
Contemporaneous conditioning. Conditioning concerns both contemporaneous
variables in models and current-dated instrumental variables (IVs), so letx′t=
(y′t:z′t), where the former are thekvariables to be modeled and the lattern−kare
taken as given. Then forρ 0 =(κ 1 :κ 2 ):
fxt(
xt|Xtt−− 1 s,X^10 −s,qt,ρ 0)
=fyt|zt(
yt|zt,Xtt−−s 1 ,X^10 −s,qt,κ 1)fzt(
zt|Xtt−− 1 s,X^10 −s,qt,κ 2). (1.9)
A viable analysis from the conditional distribution alone in (1.9) requires that
θ=h 1 (κ 1 ); and there will be no loss of information only if(κ 1 ,κ 2 )satisfy a cut
so(κ 1 ,κ 2 )∈K 1 ×K 2 , in which caseztis weakly exogenous forθ. When (1.7)
holds, both conditional and marginal distributions in (1.9) will be Normal, and
the relationships linear. The former leads to VAR-type modeling as noted, whereas
the conditional representation in (1.9) underpins more “structural” approaches
when theztare instruments: we return to conditioning in section 1.4.5 below.
Simultaneity. Finally, at least for the order of reductions considered here, simul-
taneity can allow a more parsimonious representation of the conditional distribu-
tion by modeling in terms ofyt, whereis a non-singular matrix that captures the
current-dated interdependencies. Ifztdoes not enter the conditional distribution,
xtcould be modeled directly relative to lagged information (see, e.g., Demiralp
and Hoover, 2003).
1.4.2.4 Implications
Five important issues are clarified by these reductions from the DGP down to a
specific model of a sub-set of the variables.
Econometric concepts. First, there exists an LDGP as in (1.4) for whatever choices
are made ofxt. When all reductions are without loss, the statistical modelfyt|zt(·)in
(1.9) could also be the LDGP. Although most empirical analyses seem to commence
by specifying what is included, rather than what is eliminated, almost all the central
concepts in econometrics (in italics below) correspond to when reductions (in bold
face) can be achieved without loss of relevant information:
- Aggregationentails no loss of information on marginalizing with respect to
disaggregates when the formulation retainssufficient statisticsforθ. - Data transformationshave no associated reduction, but relate toparameters of
interest,θ, and hence the need for these to beinvariant and identifiable.