Palgrave Handbook of Econometrics: Applied Econometrics

Tommaso Proietti 413

τ=(τ 1 ,...,τn), and:

θ 1 =

⎛

⎝θ− 01 +^1

στη^2 (i−^1 )

∑

t

χtχ′t

⎞

⎠

− 1

,

mφ 1 =φ 1

⎛

⎝θ− 01 mφ 0 +^1

στη^2 (i−^1 )

∑

t

χtτt

⎞

⎠.

(e) Generateσp^2 ε(i)from the full conditional IG distribution:

σp^2 ε|ε(pi−^1 )∼IG

⎛

⎝vε+n

2

,

δε+

∑

t(ε

(i− 1 )

t )

2

2

⎞

⎠.

Hereεpis the stack of the inflation equation measurement disturbances, and

we assume the priorσp^2 ε∼IG(vε/2,δε/ 2 ).

(f) Generateστη^2 (i)from the full conditional IG distribution:

στη^2 |η(τi−^1 )∼IG

⎛

⎝vτ+n

2

,

δτ+

∑

tη

(i− 1 )^2

τt

2

⎞

⎠,

whereητis the stack of the inflation equation core level disturbances, and

we assume the priorσητ^2 ∼IG(vτ/2,δτ/ 2 ).

The above GS scheme defines a homogeneous Markov chain such that the tran-
sition kernel is formed by the full conditional distributions and the invariant
distribution is the unavailable target density.
The IG prior for the variance parameter is centered around the ML estimate and
is not very informative(vη=vκ=vε=vτ=4, andn=426); for the AR param-
eters and the loadings impose a standard normal prior. The number of samples is
M=2,000 after a burn-in sample of size 1,000. Figure 9.7 displays the posterior
means and the 95% interval estimates of the output gap (first panel), along with

a nonparametric estimate of the posterior density of the variance parametersση^2

andσκ^2 (top right panel); the modes are not far from the ML estimates. The bottom

left panel shows theMdraws(φ 1 (i),φ 2 (i))from the posterior of the AR parameter dis-
tribution. The triangle delimits the stationary region of the parameter space; the
posterior means are 1.48 forφ 1 and−0.57 forφ 2. Finally, the last panel shows the
posterior distribution of the change effect,−θψ 1 , and the level effectθψ( 1 ). The
95% confidence interval for the latter is (−0.01, 0.05), which confirms that the
output gap has only transitory effects on inflation. The posterior mean ofψtdoes
not differ from the point estimates arising from the classical analysis. However,
the classical confidence intervals in Figure 9.6 are constructed by replacingwith
the ML estimates and thus do not take into account parameter uncertainty (see
also section 9.4.2). It cannot be maintained that the classical estimates are more
reliable.