Palgrave Handbook of Econometrics: Applied Econometrics

414 Structural Time Series Models

1950 1960 1970 1980 1990 2000

–5

0

5

1.0

0.5

0.0

–0.5

Output gap

0.0 0.2 0.4 0.6 0.8 1.0

Trend and cycle dist. variances

ση^2 σκ^2

–2 –1 0 1 2

Autoregressive parameters

φ 1

φ 2

–0.1 0.0 0.1 0.2 0.3 0.4 0.5

Inflation output gap loadings

−θψ 1 θψ 0 + θψ 1

Figure 9.7 Bayesian estimation of the standard bivariate output gap model. Point and 95%
interval estimates of the output gap; posterior densities of variance and loadings parameters;
draws from the posterior of the AR parameters

For the GM model, the parameter setis such that the trend and cycle distur-

bance variances are replaced by the variances in the two regimes,ση^2 a,ση^2 b,σκ^2 a,σκ^2 b,
and under the Markov switching specification (ii), according to whichStis a
first-order Markov chain, includes the transition probabilitiesT 11 ,T 00.
The steps of the GS algorithm need to be amended. An additional step is necessary
to draw a sample from the distribution ofS=(S 0 ,...,Sn)conditional onαand.
Notice that this distribution depends on these random vectors only viaη,κ, and
the elements of,ση^2 a,ση^2 b,σκ^2 a,σκ^2 b,T 11 ,T 00. Sampling from the full posterior of
the indicator variableSis achieved by the following algorithm (Carter and Kohn,
1994):

1. Sample S(ni) from the filtered state probability distribution P(Sn|α,,y) =
   P(Sn|η,κ,).


2. Fort=n−1,..., 1, 0, sampleS(ti)from the conditional probability distribution:

P(St|S(t+i) 1 ,η,κ,)=

P(S(t+i) 1 |St,)P(St|ηt,κt,)

∑

StP(S

(i)

t+ 1 |St,)P(St|η

t,κt,)

,

whereηt=(η 0 ,...,ηt)andκt=(κ 0 ,...,κt).