424 Structural Time Series Models
The sources (ii) and (iii) typically arise because the individual components are
unobserved and are dependent through time. The availability of additional time
series observations helps to improve the estimation of an unobserved component.
Multivariate methods are more reliable as they use repeated measures of the same
underlying latent variable and this increases the precision of the estimates. It is
important to measure the uncertainty that surrounds the real time, or concurrent,
estimates,Var(ψt|Ft,) ̃, which are conditional on the information set available to
economic agents and policy makers at the time of making the assessment of the
state of the economy, as opposed to the historical, or final, estimates. Comparing
Var(ψt|Ft,θ ̃)with the final estimation error variance,Var(ψt|Fn,) ̃,n→∞, gives
a clue about the magnitude of the revision of the estimates as new observations
become available.
In the absence of structural breaks, statistical revisions are sound and a fact of
life (i.e., a natural consequence of optimal signal extraction). There is, however,
great concern about revisions, especially for policy purposes; Orphanides and van
Norden (2002) propose temporal consistency as a yardstick for assessing the reliabil-
ity of output gap estimates; temporal consistency occurs when real-time (filtered)
estimates do not differ significantly from the final (smoothed) estimates.
Finally, an additional source of uncertainty is data revision, which concerns
yt. Timely economic data are only provisional and are revised subsequently with
the accrual of more complete information. Data revision is particularly relevant
for national accounts aggregates, which require integrating statistical informa-
tion from different sources and balancing it so as to produce internally consistent
estimates.
9.5 Appendix A: Linear filters
A linear filter applied to a univariate seriesytis a weighted linear combination of
its consecutive values. A time invariant filter can be represented as:
w(L)=∑jwjLj, (9.29)with wjrepresenting the filter weights. The above filter is symmetric if wj=w−j,
in which case we can write w(L)=w 0 +
∑
jwj(L
j+L−j).
Applying w(L)toytyields w(L)ytand has two consequences: the amplitude of
the original fluctuations will be compressed or enhanced and the displacement
over time of the original fluctuations will be altered. These effects can be fully
understood in the frequency domain by considering the frequency response func-
tion (FRF) associated with the filter, which is defined as the Fourier transform of
(9.29): w(e−ıω)=
∑
jwje−ıωj =w
R(ω)+ıwI(ω), where wR(ω)=∑
jwjcosωj
and wI(ω)=
∑
jwjsinωj. The last equality stresses that, in general, the FRF is a
complex quantity, withwR(ω)and wI(ω)representing its real and complex part,
respectively. The polar representation of the FRF, w(e−ıω)=G(ω)e−ıPh(ω), is written
in terms of two crucial quantities, thegain,G(ω)=|w(e−ıω)|=
√
wR(ω)^2 +wI(ω)^2 ,