410 Structural Time Series Models
This will be referred to as the GM specification. We shall consider two cases: (i) the
sequenceStis deterministic, taking the value 1 before 1984:1, and 0 thereafter; (ii)
Stis a random process, which we model as a first-order Markov chain with initial
probabilityp(S 0 = 1 )=1, i.e., we know for certain that the process started in a
high variance state, and transition probabilitiesP(St=j|St− 1 =i)=Tij,i=0, 1,
withTij= 1 −Tiiforj =i.
9.3.2.1 ML estimation
The bivariate model and its GM extension under assumption (i) were estimated
by ML in the time domain. The likelihood is evaluated by the Kalman filter (see
Appendix C for details). The parameter estimates and the associated standard errors
are reported in Table 9.2. The estimated trend and cycle disturbance variances are
smaller after 1984:1 (regimeb), as expected, and the likelihood ratio test of the
homogeneity hypothesis,H 0 :ση^2 a=ση^2 b,σκ^2 a=σκ^2 b, clearly leads to a rejection.
The roots of the AR polynomial for the output gap are complex and the loadings
of core inflation on the output gap are significantly different from zero at the 5%
level. The table also reports the Wald test for the null of long-run neutrality of
the output gap,H 0 :θψ( 1 )=0, which is accepted under both specifications, with
p-values equal to 0.16 and 0.19. The evidence is thus that the output gap has only
transitory effects on the level of inflation.
Figure 9.6 displays the point and 95% interval estimates of the output gap and
the core component of inflation for both specifications. It is interesting that the
explicit consideration of the Great Moderation of volatility makes the estimates of
the output gap after the 1984 break more precise. In interpreting this result, we
must stress that the interval estimates make no allowance for parameter uncertainty
and for the uncertainty in dating the transition from the high volatility state to
the low volatility one.
9.3.2.2 Bayesian estimation
Let us focus on the standard bivariate model (9.22) first and denote byythe stack of
the observations(yt,pt)fort=1,...,n,α=(α′ 0 ,...,α′n)′, where the state vector at
timetisαt=(μt,βt,ψt,ψt− 1 ,τt). Also, letμ,ψ,η,κdenote, respectively, the stack
of potential output, the output gap, the disturbancesηt, and the cycle disturbances,
where, e.g.,ψ=(ψ 1 ,...,ψn), and let=[φ 1 ,φ 2 ,ση^2 ,σκ^2 ,σp^2 ε,στη^2 ,θψ 0 ,θψ 1 ]denote
the vector of hyperparameters.^3 Notice that knowledge ofαimplies knowledge of
both the individual state components and the disturbances. Our main interest lies
in aspects of the posterior marginal densities of the states given the observations,
e.g.,f(ψ|y)andf(|y): e.g., E[h(ψ)]=
∫
h(ψ)f(ψ|y)dψ, for some functionh(·)such
ash(ψ)=ψt. The computation of the integral is carried out by stochastic simula-
tion: given a sampleψt(i),i=1,...,M, drawn from the posteriorf(ψ|y),E[h(ψ)]is
approximated byM−^1
∑
ih(
ψt(i)). The required sample is obtained by Monte Carlo
Markov chain methods and, in particular, by a Gibbs sampling (GS) scheme that
we now discuss in detail. This scheme produces correlated random draws from the
joint posterior densityf(α,|y), and thus fromf(ψ|y), by repeatedly sampling an