Carlo Favero 845wherex ̃tis the output gap,π ̃tis the inflation rate,R ̃tis the short-term interest rate
andg ̃tand ̃ztare two stationary AR(1) processes for government and technology,
respectively.
The first equation is an intertemporal Euler equation obtained from the house-
hold’s optimal choice of consumption and bond holdings. There is no investment
in the model and so output is proportional to consumption up to an exogenous
process that describes fiscal policy. The net effects of these exogenous shifts on
the Euler equation are captured in the processg ̃t. The parameter 0<β<1isthe
household’s discount factor andτ>0 is the inverse of the elasticity of intertem-
poral substitution. The second equation is the forward-looking Phillips curve,
which describes the dynamics of inflation and whereκdetermines the degree of
the short-run trade-off between output and inflation. The third equation is the
monetary policy reaction function. The central bank follows a nominal interest
rate rule by adjusting its instrument to deviations of inflation and output from
their respective target levels. The shock (^) R,tcan be interpreted as an unanticipated
deviation from the policy rule or as policy implementation error. Fiscal policy
is simply described by an autoregressive process. The set of structural shocks is
thus (^) t=
(
(^) R,t, (^) g,t, (^) z,t
)′
, which collects technology, government and monetary
shocks.
To cast the model in the form of:
0
∼
Zt= 1
∼
Zt− 1 +C+!
t+ηt, (16.15)
specify the relevant matrices as follows:
∼
Zt=
⎡
⎢
⎢⎢
⎢
⎢⎢
⎢
⎢⎢
⎢⎢
⎢
⎢
⎣
x ̃t
π ̃t
R ̃t
̃R∗
t
̃gt
̃zt
Et ̃xt+ 1
Etπ ̃t+ 1
⎤
⎥
⎥⎥
⎥
⎥⎥
⎥
⎥⎥
⎥⎥
⎥
⎥
⎦
(^) t=
⎡
⎢⎢
⎣
(^) tR
Gt
(^) tZ
⎤
⎥⎥
⎦ ηt=
[
ηxt=xt−Et− 1 (xt)
ηtπ=πt−Et− 1 (πt)
]
0 =
⎡
⎢⎢
⎢⎢
⎢
⎢⎢
⎢
⎢⎢
⎢⎢
⎣
(^101) τ 0 −( 1 −ρg) −ρτz − 1 −^1 τ
−κ 10 0 κ 00 −β
001 −( 1 −ρR) 0000
−ψ 2 −ψ 1 01 0 000
000 0 1 000
000 0 0 100
100 0 0 000
010 0 0 000
⎤
⎥⎥
⎥⎥
⎥
⎥⎥
⎥
⎥⎥
⎥⎥
⎦