Palgrave Handbook of Econometrics: Applied Econometrics

412 Structural Time Series Models

1950 1960 1970 1980 1990 2000

–5

0

5

Output gap

1950 1960 1970 1980 1990 2000

0

1

2

3

4

CPI quarterly inflation and trend

1950 1960 1970 1980 1990 2000

–5

0

5

Output gap – GM model

1950 1960 1970 1980 1990 2000

0

1

2

3

4

CPI quarterly inflation and trend – GM model

Figure 9.6 Estimates of the output gap and core inflation using the ML estimates of the
parameters of the bivariate models of output and inflation under two specifications

The generations are repeated until a draw falls inside the stationarity region.

(b) Generateση^2 (i)from the full conditional inverse gamma (IG) distribution:

ση^2 |η(i−^1 )∼IG

⎛

⎝vη+n

2

,

δη+

∑

tη

(i− 1 )^2

t

2

⎞

⎠.

This assumes that the prior distribution isση^2 ∼IG(vη/2,δη/ 2 ).

(c) Generateσκ^2 (i)from the full conditional IG distribution:

σκ^2 |κ(i−^1 )∼IG

⎛

⎝vκ+n

2

,

δκ+

∑

tκ

(i− 1 )^2

t

2

⎞

⎠.

This assumes that the prior distribution isσκ^2 ∼IG(vκ/2,δk/ 2 ).

(d) Generate (θ(ψi 0 ),θψ(i) 1 )′. Assuming the Gaussian prior (θψ 0 ,θψ 1 )′ ∼

N(mθ 0 ,θ 0 ), the full posterior is(θψ 0 ,θψ 1 )′|τ,στη^2 (i−^1 )∼N(mθ 1 ,θ 1 ), where