404 Structural Time Series Models
1960 1980 200089US GDP trend
Series
Trend – ARIMA(2,1,2)1960 1980 2000–0.0250.025US GDP cycleRW ARIMA(1,1,0)
ARIMA(2,1,2) Leser – HPHP1995 2000 20050.000.02US GDP cycle (1995:1–2006:4)
RW ARIMA(1,1,0)
ARIMA(2,1,2) HP1960 1980 20000.00010.00020.0003
Estimation error variance
RW
ARIMA(1,1,0)
ARIMA(2,1,2)1960 1980 2000–0.0250.0000.025Bandpass component and BK cycle
Bandpass component BK1960 1980 20000.050.00US GDP bandpass and highpass cycleBandpass
HighpassFigure 9.4 Model-based filtering. Estimates of the lowpass component (using the
ARIMA(2,1,2) model) and of the highpass and bandpass components in US GDP, and their
comparison with the Leser–HP and BK cycles
The estimates for the three models are obtained as the conditional mean ofψt
given the observations by applying the Kalman filter and smoother to the rep-
resentation (9.17); the algorithm also provides their estimation error variance. It
must be stressed that the Leser–HP filter is optimal for a restricted IMA(2,2) process
and thus it does not yield the minimum mean square estimator of the cycle, nor
its standard error. In general, also looking at the middle panel, which displays the
estimated cycles for the last 12 years, the model-based estimates are almost indis-
tinguishable, and are quite close to the Leser–HP cycle estimates in the middle
of the sample. Large differences with the latter arise at the beginning, where the
lowpass component had greater amplitude, and at the end of the sample period.
The particular model that is chosen matters little as far as the point estimates of
ψtare concerned. Nevertheless, it is relevant for the assessment of the accuracy of
the estimates, as can be argued from the right middle panel of the figure, which
shows the estimation error variance, Var(ψt|Yn), for the three models of US GDP.
It is also evident that the standard errors obtained for the Leser–HP filter would
underestimate the uncertainty of the estimates.
We conclude this first illustration by estimating the deviation cycle as a
bandpass component, assuming that the true model is the ARIMA(2,1,2) and using
the cut-off frequenciesωc 1 = 2 π/32,ωc 2 = 2 π/6, and the valuesm=2,r=0; as
a consequence, the componentψtin (9.20) selects all the fluctuations in a range