80 How much Structure in Empirical Models?
0.60.8
0.960.98–2–2–10ωDistance, all shocksβ
0.60.8
0.960.98–1060–1055–1050ωLikelihood, all shocksβ 0.60.8
0.960.98–10–8–6–4–42x 10^4ωPosterior, all shocksβ0.20.40.60.8
1.6
1.8
2
2.20hDistance, all shocksφ 0.20.40.60.8
1.6
1.8
2
2.2–1200–1150–1100–1050hLikelihood, all shocksφ 0.20.40.60.8
1.6
1.8
2
2.2–1200–1150–1100–1050hPosterior, all shocksφFigure 2.3 Impulse responses: determinate versus indeterminate equilibrium
the coefficients of the aggregate decision rules, but always in combination with
other parameters. The coefficients of the restricted VAR solution are inverted to
compute impulse responses and their distance from the “truth” is then squared
and summed. One would guess that it is just by chance that such a complex set
of operations will allow the mapping fromωto the objective function to be well
behaved.
The standard answer to the problems shown in Figures 2.1 and 2.2 is to fix param-
eters with difficult identification features (after all, it does not matter what value
we select) and estimate the remaining ones. While this approach is common, there
is no guarantee that it will give meaningful answers to the questions of interest.
In fact, while such a mixed calibration-estimation approach will be successful, at
least in population, if the parameters that are treated as fixed are set at their true
value, setting them at values which are only slightly different from the true ones
may lead estimation astray. Intuitively this happens because, for example, setting
βto the wrong value implies adjustments in parameters which enter jointly withβ
in the coefficients of the aggregate decision rules and this may move the minimum
of the function in a somewhat unpredictable way. Canova and Sala (2005) show,
in the context of a simple RBC example, that these shifts may be significant and
may drive inference the wrong way.