428 Structural Time Series Models
9.7.3 Smoothing
Smoothing deals with the estimation of the components and the disturbances
based on the full sample of observations. In the Gaussian case thefixed interval
smootherprovides the minimum mean square estimator ofαtusingYn,α ̃t|n=
E(αt|Yn), along with its covariance matrixPt|n=E[(αt−α ̃t|n)(αt−α ̃t|n)′|Yn]. The
computations can be carried out efficiently using the following backwards recur-
sive formulae, given by Bryson and Ho (1969) and de Jong (1989), starting att=n,
with initial valuesrn=0,Rn= 0 andNn=0:
rt− 1 =Lt′rt+Zt′Ft∗−^1 vt, Rt− 1 =L
′
tRt+Z′
tF∗− 1
t Vt,t=n−1,...,1.Nt− 1 =L′tNtLt+Z′tF∗−t^1 Zt,
α ̃t|n=α ̃∗t|t− 1 −At|t− 1 S−n^1 sn+Pt∗|t− 1 (rt− 1 −Rt− 1 S−n^1 sn),
Pt|n=P∗t|t− 1 −P∗t|t− 1 Nt− 1 P∗t|t− 1 +(At|t− 1 +P∗t|t− 1 Rt− 1 )S−n^1 (At|t− 1 +P∗t|t− 1 Rt− 1 )′,
(9.35)
whereLt=Tt+ 1 −KtZ′t. A preliminary forward KF pass is required to store the
quantitiesα ̃t∗|t− 1 ,At|t− 1 ,P∗t|t− 1 ,v∗t,Vt,F∗tandKt.
The smoothed estimates of the disturbances are given byHtη ̃t=E(Htηt|Yn)=
HtH′t(rt− 1 −Rt− 1 S−n^1 sn), andGt ̃t=E(Gtt|Yn)=GtG′t
[
F∗−t^1 (vt−VtS−n^1 sn)+
K′t(rt−RtS−n^1 sn)]
.9.7.4 The simulation smoother
The simulation smoother is an algorithm which draws samples from the condi-
tional distribution of the states and the disturbances given the observations and the
hyperparameters. Carlin, Polson and Stoffer (1992) proposed a single-move state
sampler, by which the states are sampled one at a time. This proves to be ineffi-
cient in the presence of highly autocorrelated state components. Gamerman (1998)
proposed a single move disturbance sampler, which is more efficient since the dis-
turbances driving the components are much less persistent and autocorrelated over
time. Along with reparameterization, an effective strategy is blocking, through the
adoption of a multi-move sampler as in Carter and Kohn (1994) and Früwirth-
Schnatter (1994), who focus on sampling the states. Again, a more efficient
multi-move sampler can be constructed by focusing on the disturbances, rather
than the states. This is the idea underlying the simulation smoother proposed by
de Jong and Shephard (1996).
Letςt=C[′t,η′t]′denote a sub-set of the disturbances of the series, withC
being a selection matrix. The structure of the state-space model model is such
that the states are a (possibly singular) linear transformation of the disturbances
and thatGttcan be recovered fromHtηtvia the measurement equation, which
implies that the distribution of(′,η′)′|Ynis singular. Hence, to achieve efficiency
and to avoid degeneracies, we need to focus on a suitably selected sub-set of the
disturbances. The simulation smoother hinges on the following factorization of