Gunnar Bårdsen and Ragnar Nymoen 857it follows that the main problems of macroeconometrics are model specification
and model evaluation, rather than finding the best estimator under the assumption
that the model is identical to the DGP.
The local DGP is changing with the evolution of the real world economy–through
technical progress, changing patterns of family composition and behavior, and
political reform. Sometimes society evolves gradually and econometric models are
then usually able to adapt to the underlying real-life changes, i.e., without any
noticeable loss in “usefulness.” Often, however, society evolves so quickly that
estimated economic relationships break down and cease to be of any aid in under-
standing the current macroeconomy and in forecasting its development even over
the first couple of years. In this case we speak of a changing local approximation in
the form of a regime shift in the generating process, and a structural break in the
econometric model. Since the complexity of the true macroeconomic mechanism,
and the regime shifts also contained in the mechanism, lead us to conclude that
any model will at best be a local approximation to the DGP, judging the quality of,
and choosing between, the approximations becomes central.
In the rest of this section we present our approach to finding a local approxima-
tion useful for policy.^4
17.2.1 Linearization
Consider a very simple example of an economic model in the form of the
differential equation:
dy
dt
=f(y,x), x=x(t), (17.1)
in which a constant inputx= ̄xinducesy(t)to approach asymptotically a con-
stant statey ̄ast→∞. Clearlyx ̄and ̄ysatisfyf(y ̄, ̄x)=0. For example, standard
DSGE models usually take this form, with the models having deterministic steady-
state values. The usual procedure then is to expand the differential (or difference)
equation about this steady-state solution (see, e.g., Campbell, 1994; Uhlig, 1999).
Employing this procedure yields:
f(y,x)=f(y ̄,x ̄)+∂f(y ̄,x ̄)
∂y
(y−y ̄)+∂f(y ̄, ̄x)
∂x
(x−x ̄)+R, (17.2)where:
R=1
2!(
∂^2 f(ξ,η)
∂x^2(x−x ̄)^2 + 2∂^2 f(ξ,η)
∂x∂y
(x− ̄x)(y− ̄y)+∂^2 f(ξ,η)
∂y^2(y−y ̄)^2)
,and(ξ,η)is a point such thatξlies betweenyandy ̄whileηlies betweenxandx ̄.
Sincey ̄andx ̄are the steady-state values foryandxrespectively, then the expression
forf(y,x)takes the simplified form:
f(y,x)=a(y−y ̄)+δ(x−x ̄)+R, (17.3)wherea=∂f(y ̄,x ̄)/∂yandδ=∂f(y ̄,x ̄)/∂xare constants.