Palgrave Handbook of Econometrics: Applied Econometrics

76 How much Structure in Empirical Models?

that these features may be indistinguishable when one looks just at normalized
impulse responses.
How can one avoid observational equivalence? Clearly, part of the problem
emerges because normalized impulse responses carry no information for the unsta-
ble rootλ 2. However, the variance of the shocks does have this information and,
for example, the likelihood function of the first process will be different from those
of the other two. Hence, adding information could help limit the extent of obser-
vational equivalence problems. In the case where one is not willing to or cannot
use this information and only employs the dynamics in response to normalized
shocks to recover structural parameters, information external to the models needs
to be brought in to disentangle various structural representations (as it is done, for
example, in Boivin and Giannoni, 2006).

2.2.1.2 Example 2: identification problems in a New Keynesian model

Throughout this sub-section we assume that the investigator knows the correct
model and the restrictions needed to identify the shocks. Initially, we assume that
he/she chooses as an objective function the distance between the responses in the
model and in the data. Later on, we examine how identification is affected when
additional information is brought into the estimation process.
We consider a well-known small-scale New Keynesian (NK) model, which has
become the workhorse in academic and policy discussions and constitutes the
building block of larger-scale models currently estimated in the literature. Several
authors, including Ma (2002), Beyer and Farmer (2004), Nason and Smith (2005)
and Canova and Sala (2005), have pointed out that such a structure is liable to
identification problems. Here we discuss where and how these problems emerge.
The log-linearized version of the model consists of the following three equations
for the output gapyt, inflationπtand the nominal ratert:

yt=

h

1 +h

yt− 1 +

1

1 +h

Etyt+ 1 +

1

φ

(it−Etπt+ 1 )+v 1 t (2.10)

πt=

ω

1 +ωβ

πt− 1 +

β

1 +ωβ

Etπt+ 1 +

(φ+ν)( 1 −ζβ)( 1 −ζ)

( 1 +ωβ)ζ

yt+v 2 t (2.11)

it=λrit− 1 +( 1 −λr)(λππt− 1 +λyyt− 1 )+v 3 t, (2.12)

wherehis the degree of habit persistence,φis the relative risk aversion coeffi-
cient,βis the discount factor,ωis the degree of price indexation,ζis the degree
of price stickiness, andνis the inverse elasticity of labor supply, whileλr,λπ,λy
are monetary policy parameters. The first two shocks follow an AR(1) process with
parametersρ 1 ,ρ 2 , whilev 3 tis i.i.d. The variances of the shocks are denoted by

σi^2 ,i=1, 2, 3. For the sake of presentation, we assume that the shocks are con-
temporaneously uncorrelated even though, in theory, some correlation must be
allowed for.
Since the model features three shocks and three endogenous variables, we can
construct several limited information objective functions, obtained by considering