416 Structural Time Series Models
(g) GenerateT 11 (i),T 10 (i)= 1 −T 11 (i)andT 00 (i),T 01 (i)= 1 −T 00 (i)from the posterior:T 11 (i)|S(i−^1 )∼B(
a 1 +N( 11 i−^1 ),b 1 +N 10 (i−^1 ))
,T 00 (i)|S(i−^1 )∼B(
a 0 +N( 00 i−^1 ),b 0 +N 01 (i−^1 ))
,whereB(a,b)is the Beta distribution,N(iji−^1 )is the number of transitions from
St(i−^1 )=itoS(ti+− 11 )=j, andai,bi,i=0, 1, are the parameters of the Beta prior
distributions (set equal toa 1 =b 1 =b 0 =1,a 0 =5). Notice that the transition
probabilities are conditionally independent ofαand the other elements of,
givenS.Figure 9.8 summarizes aspects of the posterior distribution of the cycle, the indi-
catorSt, and some important parameters using a sample ofM=2,000 draws from
the GS scheme outlined above with a burn-in of 2,000 iterations. Interestingly, the
output gap interval estimates are more widely dispersed than in the original speci-
fication with no Markov-switching in the disturbance variances. This is so since the
GM specification has a further source of variation and uncertainty, related to the
state of Markov chainSt, which in turn drives the changes in the volatility regime.
As a result, the Bayesian interval estimates cannot be compared with the classical
ones reported in the bottom left panel of Figure 9.6, since those were derived under
the assumption thatStwas deterministic and known, and they make no allowance
for parameter uncertainty. The estimated posterior probabilities of being in a high
volatility regime confirm the general finding that the main stylized fact is a rela-
tively sharp change point taking place in the first quarter of 1984, although there
remains some uncertainty around that date. The nonparametric estimates of the
posterior distribution of the transition probabilitiesT 11 andT 00 are displayed in the
last panel of the figure. The posterior distributions of the variance parameters for
the trend and cycle disturbances strongly confirm the Great Moderation hypoth-
esis, and quantify it further, as both the permanent and transitory disturbances
underwent a significant volatility reduction. The posterior means do not differ
from the ML estimates reported in Table 9.2: E(ση^2 a|y)=0.60, E(ση^2 b|y)=0.14 and
E(σκ^2 a|y)=0.51, E(σκ^2 b|y)=0.09. As far as the inflation equation is concerned, the
overall conclusion is unchanged: the output gap is a significant source of variation
(the value−θψ 1 =0 is estimated to be the 2.6 percentile of the posterior distribu-
tion of−θψ 1 , which measures the change effect, but it drives inflation only in the
short run, as the null of long-run neutrality is accepted (the 95% credible set for
θψ( 1 )is the interval (−0.01, 0.05)).
9.3.3 Multivariate extensions
The output gap is related to the deviations of the unemployment rate,ut, from
its “natural rate" or NAIRU via Okun’s law. Okun (1962) defined natural unem-
ployment as that level of unemployment occurring when output is equal to its