Palgrave Handbook of Econometrics: Applied Econometrics

398 Structural Time Series Models

1960 1980 2000

–2

0

2

4

US GDP growth rates 100 Δ yt

–1.0 –0.5 0.0 0.5 1.0

–317.0

–316.5

–316.0

Profile likelihood for correlation parameter

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

Sample spectrum and parametric spectra

MNZ unrestricted

r = 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

–0.75

–0.50

–0.25

0.00

0.25

0.50

FD cross-validatory estimate of r

Figure 9.1 Quarterly US real growth, 1947:2–2006:4. Sample spectrum and parametric
spectral fit of trend-cycle model with correlated components

of the signal extraction filters

∑

jwψ,jL

jy
tthat yield the cycle estimates in the two
cases.
The real-time estimates support the view that most of the variation in GDP is
permanent, i.e., it should be ascribed to changes in the trend component, whereas
little variance is attributed to the transitory component. In fact, the amplitude of
ψ ̃t|tis small and the interval estimates ofψtin real time are never significantly
different from zero. When we analyze the smoothed estimates the picture changes
quite radically: the cycle estimates are much more variable and there is a dramatic
reduction in the estimation error variance, so that the contribution of the transitory
component to the variation in GDP is no longer negligible. The real-time estimates
provide a gross underestimation of the cyclical component and are heavily revised
as the future missing observations become available. As a matter of fact, the final
estimates depend heavily on future observations, as can be seen from the pattern
of the weights in the last panel of Figure 9.2. That this behavior is typical of the
MNZ model whenˆris high and negative is documented in Proietti (2006a).
The real-time estimates of the trend and cyclical components are coincident
with the Beveridge and Nelson (1981; henceforth, BN) components defined for
the ARIMA(2,1,2) reduced form. The BN decomposition defines the trend com-
ponent at timetas the value of the eventual forecast function at that time, or,
equivalently, the value that the series would take if it were on its long-run path
(see also Brewer, 1979). For an ARIMA(p,1,q) process, this argument defines the