Tommaso Proietti 405of periodicity that goes from one and a half years (6 quarters) to eight years (32
quarters). The gain of the filter is displayed in Figure 9.3. The estimates ofψtare
compared to the BK cycle in the bottom left panel of Figure 9.4 and to the corre-
sponding highpass estimates (ψt+ (^) t). With respect to the BK cycle, the estimates
are available also in real time.
The conclusion is that model-based filtering improves the quality of the esti-
mated lowpass component, providing estimates at the boundary of the sample
period that are automatically adapted to the series under investigation, and enables
the investigator to assess the reliability of the estimates (conditional on a particular
reduced form).
The second application deals with assessing the uncertainty in estimating the
business cycle chronology. According to the classical definition, the business cycle
is a recurrent sequence of expansions and contractions in the aggregate level of
economic activity (see Burns and Mitchell, 1946, p. 3). Dating the business cycle
amounts to establishing a set of reference dates that mark the phases or states of
the economy. Usually two phases, recessions and expansions, are considered, that
are delimited by peaks and troughs in economic activity. Dating is carried out by an
algorithm, such as that due to Bry and Boschan (1971), or that proposed by Artis,
Marcellino and Proietti (2004), which aims at estimating the location of turning
points, enforcing the alternation of peaks and troughs and minimum duration ties
for the phases and the full cycle. Downturns and upturns have to be persistent
to qualify as cycle phases; thus, they need to fulfill minimum duration con-
straints, such as at least two quarters for each phase; moreover, to separate it from
seasonality, a complete sequence, recession-expansion or expansion-recession,
i.e., a full cycle, has to last longer than one year. Depth restrictions, motivated
by the fact that only major fluctuations qualify for the phases, should also be
enforced.
An integral part of the dating algorithm is prefiltering, which is necessary in
order to isolate fluctuations in the series with period greater than the minimum
cycle duration. For instance, in the quarterly case we need to abstract from all
the fluctuations with periodicity less than five quarters, i.e., from high-frequency
fluctuations that do not satisfy the minimum cycle duration. In lieu of thead
hocand old-fashioned moving averages adopted by Bry and Boschan, one can use
model-based lowpass signal extraction filters.
The advantages are twofold: first, it is possible to select the cut-off frequency
so as to match the minimum cycle duration; e.g., in our caseωc= 2 π/5. Sec-
ond, the uncertainty in dating arising from prefiltering can be assessed by Monte
Carlo simulation, by means of an algorithm known as the simulation smoother (see
de Jong and Shephard, 1995; Durbin and Koopman, 2002; and Appendix C, section
9.7.4). This repeatedly draws simulated samples from the posterior distribution of
the lowpass component with a cut-off frequency corresponding to five quarters,
μ ̃(ti)∼μt|Yn, so that by repeating the draws a sufficient number of times we can
get Monte Carlo estimates of different aspects of the marginal and joint distribu-
tion of the lowpass component, intended here as the level of output devoid of all
fluctuations with a periodicity smaller that five quarters.