Palgrave Handbook of Econometrics: Applied Econometrics

Fabio Canova 81

0.6

0.8

0.96

0.98

–2

–2

–1

0

ω

Distance, all shocks

β

0.6

0.8

0.96

0.98

–1060

–1055

–1050

ω

Likelihood, all shocks

β 0.6

0.8

0.96

0.98

–10

–8

–6

–4

–4

2

x 10^4

ω

Posterior, all shocks

β

0.20.4

0.60.8

1.6

1.8

2

2.2

0

h

Distance, all shocks

φ 0.20.4

0.60.8

1.6

1.8

2

2.2

–1200

–1150

–1100

–1050

h

Likelihood, all shocks

φ 0.2

0.40.6

0.8

1.6

1.8

2

2.2

–1200

–1150

–1100

–1050

h

Posterior, all shocks

φ

Figure 2.4 Distance function, likelihood and posterior plots

What can one then do to conduct structural estimation? The distance function
we have employed can be obtained by approximating the likelihood function of the
model. Therefore, the resulting estimators can be thought of as quasi-maximum
likelihood (ML) estimators of the structural parameters. However, there is no rea-
son to use such an approximation. Once the decision rules are written in a state
space format, the likelihood function can be easily and efficiently computed with
the Kalman filter. Therefore, identification problems could be reduced if informa-
tion about the covariance matrix of the shocks or the steady-states of the model –
which are not used when normalized impulse response matching is performed – are
brought into the estimation. Figure 2.4, which plots the distance function when
all the shocks are considered and the likelihood function inβandω, andφandh,
indeed suggests that these parameters could be better identified from the likelihood
than from the distance function – the curvature of the latter is much larger than
the curvature of the former. Nevertheless, the problem with ridges remains. Since
the likelihood has all the information that the model delivers, one can conclude
that it is the solution mapping, rather than the objective function mapping, that
induces under- and partial identification problems in this example.
It has become quite common to estimate the parameters of a DSGE model by
Bayesian methods. Bayesian methods attempt to trace out the shape of the posterior