Abridged Bry-Boschan Procedure
Takeshi Otsu
Abstract
This paper examines what kind of filtering would be useful for
the Bry-Boschan (BB) procedure to replicate the official reference dates,
using Index of Industrial Production (IIP) of Japan. We use the official
seasonal-adjustment series, filtering-based equivalents, and business-cycle
components extracted by bandpassing. The main findings are as follows.
First, the business cycle components, extracted by bandpass filters, areuseful for the BB procedure to produce dates of the turns close to thereference dates. Particularly, the Butterworth-based bandpass filtering
works well. Second, the seasonally adjusted series, whether adjusted by
X12-ARIMA (official data) or by a bandpass filter, cannot identify turns in
1960s, or overidentify those in 2000s. Finally, we find a series of themoving averages and outlier adjustment in the BB procedure are notuseful to replicate the official reference dates when we use the tangentbased Butterworth filter. Then, we obtain an abridged edition of the BB
procedure.

1 Introduction
Determination of turning points is essential for empirical analyses ofthe business cycles. It governs basic characteristics of the cycles such as
durations, amplitudes, and synchronization among regions and industries.
It could also influence subsequent analyses of economic modelling and― ８５ ―

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plicy implications, since it gives some guidance or even criteria toevaluate fitness of economic models and effects of economic policies. It is
common that governments organize an official committee to determine
the reference dates of the business cycles. It judges the reference dates
based on not only all the relevant economic variables but also otherunquantifiable, qualitative economy-related information. The reference
dates are publicly available on the internet for the United Sates and Japan.
Judgment of dating by the committee involves arbitrariness becauseinterpretation of economic condition is subjective to some extent anddating rules vary from time to time. Then, it is desirable to reduce such
arbitrariness for objective and systematic business-cycle analyses. Further,clarity of the decision processes would be helpful for revision of thedating rules later on. Such an attempt is made by Bry and Boschan (1971).
It codified dating rules and considerations that are predicated on Burns
and Mitchell (1946), that is, the NBER (National Bureau of Economic
Research) method at that time, and developed a program-based procedureof turning points determination, which is called the Bry-Boschan (BB)procedure hereafter. Nowadays, official agencies do not depend solely on
the BB procedure. But it is still useful for academic research to codify adating rule. For example, Watson (1994) used the BB procedure to study
the business cycles in the U.S. Its modified version is used in Pagan(1997), Harding and Pagan (2001), Harding and Pagan (2002), and Harding

and Pagan (2006). The algorithm is now publicly available and evenembedded in some softwares of economic analyses.
The BB procedure is developed based on the dating rule heuristicallyfound with the U.S. data in the first half of the 20th century. It involves
various kinds of moving averages, so that it could identify peaks and― ８６ ―

Abridged Bry-Boschan Procedure

troughs as close as possible to those selected by the staff at NBER. It
also sets some constraints on durations of the business cycles or phases.
Therefore, when we apply the BB procedure to recent economic data,
other country’s data, or data at regional or industrial levels, we mightneed some adjustments to the procedure. This requires researchers totackle two issues: find appropriate duration constraints and choose asuitable moving-average method. The former is a matter of economic
analysis, that is, the business-cycle analysis itself, and the latter a matterof statistical analysis. However, it is hard to identify what kinds ofmoving averages, or filtering methods in general, should be used. There
would be several statistical criteria, and different criteria lead to different
moving averages. Even a chosen criterion does not designate a specificmoving average. In fact, the choice of the moving averages in the BB
procedure is arbitrary. Therefore, it would be desirable to reduce thisarbitrariness or ambiguity and to fix a moving-average method suitable forthe business-cycle analysis. Then, the duration information of the businesscycle would be suffice to adjust the procedure to a particular analysis.
One way to reduce such arbitrariness is to allow specified businesscycle durations to determine all the weights of moving averages. This canbe done with bandpass filtering. The bandpass filters attempt to extract
certain cyclical components. Thus, once we decide a range of the businesscycle, we could extract particular components that should include useful
information to determine turning dates. The remaining problem is what
filters should be used. To choose a filter among those proposed in theliterature, we could use a criterion in line with Bry and Boschan (1971):
how well the official reference dates are replicated. This criterion is
widely used in the literature (e.g. Hamilton, 1989; Canova, 1994, 1998, 1999).
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With these considerations, this paper examines which filtering method
would be useful for the BB procedure to replicate the official reference
dates of Japan, using Index of Industrial Production (IIP) of Japan. We use
the official seasonal-adjustment series, filtering-based equivalents, andbusiness-cycle components extracted by bandpass filters. We seek a filtering method involving no parameter estimation, so that all researchers needis to determine the periodicities of the cycles appropriate for their research.
To my knowledge, there is not much research in the literature to examine
an alternative choice of the moving averages or the duration rules for the
BB procedure, whereas it is listed as ‘SUGGESTIONS FOR FURTHER
DEVELOPMENT’ in Bry and Boschan (1971, p. 56). We attempt to fillthis gap.
The main findings as follows. First, the business cycle components,
extracted by bandpass filters, are useful for the BB procedure to
produce dates of the turns close to the reference dates. Particularly, theButterworth-based bandpass works well. Second, the seasonally adjusted
series, whether adjusted by XI2-ARIMA (official data) or by a bandpass
filter, cannot identify turns in 1960s, or overidentify those in 2000s.
Finally, we find that the moving averages and outlier adjustment in the
BB procedure are ineffective when we use the tangent-based Butterworth
filter. Thus, we could abridge the BB procedure.
The remaining parts of the paper are organized as follows. In the next
section, we briefly review the BB procedure. Section 3 summarizes thefiltering methods used in the paper. Empirical results are presented in
Section 4. Final discussion and remaining issues are given in Section 5.

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Abridged Bry-Boschan Procedure

2 Bry-Boshcan Procedure
The BB procedure is summarized in Table 1. Watson (1994) foundsome discrepancies between the original description by Bry and Boschan(1971) and the Fortran program they coded. The description here ismodified to be consistent with the Fortran codes. The procedure presumes
to use seasonally adjusted series. In Step I, outliers, if any, are replacedby the values of the Spencer curve. Here, outliers are defined as values
whose ratios to (or differences in absolute value from, depending on data) the 15point Spencer curve are larger than 3.5 standard deviations, the thresholdvalue chosen arbitrarily. This Spencer curve is computed as the 15-monthsymmetric moving average with particular weights (see Kendall and Stuart,
1966, p. 458).

Step II starts with the 12-month moving average (MA12, hereafter) of
the outlier-free series. The MA12 is chosen on the ground that the Spencer curve contains too many minor fluctuations. Any date with the highest
value among the 6 preceding and the 6 following months is tentatively
regarded as the date of a peak. Similarly, any date with the lowest among
the 6 preceding and the 6 following months is considered the date of atentative trough. These peaks and troughs are checked for alternation. For
contiguous peaks or troughs, the highest value is chosen for a peak, andthe lowest for a trough. If the values are same, we set an earlier date for apeak, and a later date for a trough, respectively.
In Step III, the Spencer curve of the outlier-free series is used to ensure
peaks and troughs within ±6 months, because its turns are heuristicallycloser to those of the original series than those of MA12. If there are ties
within ±6 data points on the Spencer curve, an earlier date is chosen for a― ８９ ―

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peak, and a later date for a trough. After alternation check as in Step II,
the durations of a peak to peak or a trough to trough (a full cycle) are
enforced to be at least 15-month period. If the durations are too short, thelower of two peaks or the higher of two troughs are eliminated. If thevalues are same, we set an earlier date for a peak, and a later date for atrough, respectively. Alternation check is conducted if any modification.
In Step IV, a further refinement is conducted with a short-term moving average, which is called MCD (Months for Cyclical Dominance) curve.
The MCD is obtained as follows. First, we compute the Spencer curve of
the original series, taking it as the trend-cycle component. The difference
between the original series and the trend-cycle component gives theirregular component. Next, we take the ratio of the average change in
the irregular component to that in the trend-cycle component. The change
is computed either by the rate of change or by the difference of eachcomponent over various time spans. The MCD is the minimum number ofmonths that gives the ratio less than 1. That is, the MCD is the shortest
period of months that it takes for the change in the trend-cycle component
to dominate that in the irregular component. The BB procedure confines
the MCD between 3 to 6 months. Then, a short-term moving average iscomputed over the span of MCD, and used to ensure peaks and troughs
within ±6 months as in Step III. Alternation is checked as in Step II if
modified.
In the final step (‘V’), a series of tests are conducted to determine
final turns. First, the original series is used to ensure peaks and troughs
within ±4 months or ±4 MCD, whichever is longer (denoted by ‘V.l’). The
second test (‘V.2’) is alternation check as in Step II. Third (‘V.3’), any
turns closer than 6 months from the ends are removed. In the fourth test
― ９０ ―

Abridged Bry-Boschan Procedure(‘V.4’), if the first or the last peak (or trough) takes a smaller (or greater)

value than any value between it and the end of the original series, it isremoved. In the program used by Watson (1994), the first and the last
turns are only compared with the initial and the last data points,
respectively, not with all the values between them. Although this could
make a nontrivial difference, it does not change the results of the paper.
Here, we follow Watson (1994).
The fifth test (‘V.5’) is to check if the duration of a full cycle is at least
15-month length, as in Step III. The final test (‘V.6’) is to check whether aphase (peak to trough or trough to peak) duration is at least 5 months. If it isless than 5 months, the two turning points are eliminated. If the violation
is found at the last turning point, only the last point is removed. In later
experiments, we implement the procedure with various smoothed series in
Step II instead of the 12-month moving average. Also, in some cases, weskip Step III, Step IV, and Step V.1 and V.2 to see their effects.

3 Extracting Business Cycle Components
If it is possible to extract business cycle components, we could use
them in the BB procedure. The cyclical components, if properly extracted,
would incorporate all the turns to be identified as peaks or troughs. Therefore, the BB procedure, specifically the moving average computation,
would be simplified, or even some processes are omitted. In the literature,
there are various methods to extract and measure cyclical information.
Canova (2007) gives a concise description of methods frequently used inmacroeconomic analyses. In this section, we review properties of threemethods used later: Christiano-Fitzgerald filter, Hamming filter, andButterworth filters. To begin with, we consider the following orthogonal― ９１ ―

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decomposition of the observed series xt :
(1)

%t *&
t )%̃
t

[!b #
where &
t is a signal whose frequencies belong to the interval &!a ]%[a #!], while %̃t has the complementary frequencies.
b ]'$[!!#
Suppose that we wish to extract the signal &t . The Wiener-Kolmogorov
theory of signal extraction, as expounded by Whittle (1983, Chapter 3 and 6),
indicates &t can be writtenas:&t *B (L )%t
B (L ) *$#(2)
Bj L j # L %t "%t !k
k

(3)

j *!#

In polar form, we !
have
#1, for "$[!b #!a ] %[a #b]B (e !i $) *"0, otherwise(4)

where 0 (a (b (!. In the business-cycle literature, the values of a andb are often set to the frequencies that correspond to 1.5 and 8 years,
respectively. Theoretically, we need an infinite number of observations,%t’s, to compute &t . In practice, the filtering methods approximate &t by
ˆ t with a finite filter. Now, we briefly review three filtering methods&
mentioned above.

3.1

Christiano-Fitzgerald Filter
Christiano and Fitzgerald (2003) seeked an optimal linear approxi-

mation with finite sample observations. They solved a minimization
problem based on the mean square error (MSE) criterion in the frequency
domain: minimization of a weighted sum of differences between the ideal― ９２ ―

Abridged Bry-Boschan Procedure

bandpass-filter’s weights and their approximates, using a spectral densityof observations as a weight. They derived optimal filter weights, assuming
a difference-stationary process of observed data with a trend or a drift
removed if any.
In their empirical investigations, they examined the effects of the timevarying weights, the asymmetry, and the assumption on the stochastic
process. They compared variance ratios and correlations between thecomponents extracted by the CF filters and the theoretical components
based on the data generating process of observations. To evaluate thesecond moments of the theoretical components, they used the Riemann
sum in the frequency domain. They found that the time-varying weights
and the asymmetry of the filter contribute to a better approximation,
pointing out that the time-varying feature is relatively more important.
Further, they claimed that the time-varying weights should not introduce
severe nonstationarity in the filter approximation because the varianceratios do not vary much through the time. The correlation between thefiltered-out components and the theoretical ones at different leads and lags
symmetrically diminishes as the leads and lags go far away, which might
indicate that the degree of asymmetry is not great. Finally, a CF filter
derived under the Random-Walk data generating process, the so-called
Random Walk filter, gives a good approximation to the optimal filteringthat explicitly uses the estimated coefficients of an optimal moving
average process determined empirically. Therefore, they claimed that we
could use the Random Walk filter without inspecting the data generatingprocess even if the random walk assumption is false. In this paper, wedenote it by CF (RW) henceforth.
Details of the CF filter are given in Christiano and Fitzgerald (2003)― ９３ ―

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and its properties are discussed in Iacobucci and Noullez (2005). As
argued in Otsu (2015), the cyclical components extracted by CF (RW)
might be distorted in magnitude and timing. The gain function, defined as
the modulus of the frequency response function, shows large ripples over
the target ranges, indicating a large distortion in estimating the cyclicalcomponents. It also shows leakage effects over higher frequencies of morethan 8 periods per cycle. Further, phase shifts are indicated by values of
its phase function indicate, defined as arctangent of the ratio of thereal-valued coefficient of the imaginary part of the frequency response
function to the real part value.

3.2

Hamming-Windowed Filter
Iacobucci and Noullez (2005) claimed that the Hamming-windowed

filter be a good candidate for extracting frequency-defined components.
The proposed filter has a flatter response over the passband than otherfilters in the literature, such as the HP filter (Hodrick and Prescott, 1997), theBK filter (Baxter and King, 1999), and the CF (RW) filter. This means that it
has almost no leakage and compression, and eliminates high-frequency
components better than these three filters. The filtering is implemented in
the frequency domain. The procedure is described as follows. First, we
subtract, if necessary, the least-square regression line to detrend theobservation series to make it suitable for the Fourier transform. Second,
we implement the Fourier transform of the detrended series, Third, weconvolve the ideal response with a spectral window to find the windowed
filter response in the frequency domain. The window is the so-called
Tukey-Hamming window (Priestly, 1981, pp. 433-442).

― ９４ ―

Abridged Bry-Boschan Procedure

3.3

Butterworth Filters
Pollock (2000) have proposed the tangent-based Butterworth filters in

the two-sided expression, which are called rational square-wave filters.
The one-sided Butterworth filters are widely used in electrical engineering,
and well documented in standard text books, such as Oppenheim andSchafer (1999) and Proakis and Manolakis (2007). The two-sided version
guarantees phase neutrality or no phase shift. It has finite coefficients, andits frequency response is maximally flat over the pass band: the first(2n !1) derivatives of the frequency response are zero at zero frequency
for the nth-order filter. The filter could stationarize an integrated processof order up to 2n. The order of the filter can be determined so that theedge frequencies of the pass band and/or the stop band are aligned tosome designated frequencies. Further, Gomez (2001) pointed out that thetwo-sided Butterworth filters could be interpreted as a class of statisticalmodels called UCARIMA (the unobserved components autoregressive-integrated
moving average) in Harvey (1989, p. 74). Since the Butterworth filters are notso often used in the literature, we present relevant equations to look atthem a little bit more closely.
The lowpass filter is expressed as
BFTL #(1 "L )n (1 "L !1)n
(1 "L )n (1 "L !1)n "!(1 !L )n (1 !L !1)n(5)

where L d "t #"t !d , and L !d "t #"t "d . Similarly, the highpass filter isexpressed as
BFTH #!(1 !L )n (1 !L !1)n
(1 "L )n (1 "L !1)n "!(1 !L )n (1 !L !1)n(6)

Note BETL "BFTH #1 ，which is the complementary condition discussed
by Pollock (2000, p. 321). Here, ! is the so-called smoothing parameter.
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We observe that the Butterworth highpass filter in eq.(6) can handle
nonstationary components integrated of order 2n or less. Let %c the cutoff
point at which the gain is equal to 0.5. It is shown
!2n
2)$
"(#
tan (%c )(7)

To see this, we replace the L by e !i % in eq.(5) to obtain the frequency
response function in polar form as$L (e !i %; "'
n) (
(

1
(1 'e !i %))2n1 '"(i (1 !e !i %))

(8)

1
2n
2)$
1 '"#
tan (%)(9)

5. We also
Here, it is easy to see that eq.(7) holds when $L (e !i %) (0&observe in eq.(9) that the first (2n !1) derivatives of $L (e !i %) are zero at%(0; thus, this filter is maximally flat. Note that the gain is the modulusof the frequency response function, and indicates to what degree the filter
passes the amplitude of a component at each frequency. The Butterworth
filters considered here are symmetric and their frequency response
functions are non-negative. Therefore, the gain is equivalent to thefrequency response. Then, we can use eq.(9) to specify %c so that the gainat the edge of the pass band is close to one and that of the stop band%p ], and the stop band [%s '#],close to zero. Let the pass band (0'where %p is smaller than %s . As in Gomez (2001, p. 372), we consider thefollowing conditions for some small positive values of !
1 and !
2,1 !!
&1
$L (e !i %; "'
n) %
1 (%

%p ]
for %"[0'(10)0 &%$L (e !i %; "'
#]
(!
n) %
for %"[%s '2(11)― ９６ ―

Abridged Bry-Boschan ProcedureThat is, we can control leakage and compression effects with precision
specified by the values of !
1 and !
2 . These conditions can be written as
follows:

!
"2n
2)
1
tan (#p $
1$
%
2)
1 !!
tan (#c $
1
!
"2n
2)
1
tan (#s $
1$
%
(#
2)
$
!
tan c
2

(12)

(13)

Then, we can solve for the cutoff frequency (#c ) and the filter’s order (n),given #p , #s , !
1 and !
2 . The closer to zeros both !
2 and !
2 , the smaller
the leakage and the compression effects. If n turns out not an integer, thenearest integer is selected.
The Butterworth filters could be based on the sine function. Insteadof eq.(5) and eq.(6), the lowpass and the highpass filters can be written as
follows, respectively.
BFSL %

1
1 $"(1 !L )n (1 !L !1)n(14)

BFSH %"(1 !L )n (1 !L !1)n
1 $"(1 !L )n (1 !L !1)n

(15)

where
!2n
2)#
2sin (#c $
"%"

(16)

These are the so-called sine-based Butterworth filters. When n is equal
to two, eq.(15) is the HP cyclical filter, derived in King and Rebelo (1993,
p. 224). Thus, as pointed out by Gomez (2001, p. 336), the sine-based

Butterworth filter with order two (n %2) can be viewed as the HP filter.
As in the case of the tangent-based one, the cutoff point, #c , can be
determined with the following conditions:― ９７ ―

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!
"2n
2)
1
sin ($p '
1"
#
2)
1 !!
sin ($c '
1
!
"2n
2)
1
sin ($s '1"#2)!
sin ($c '
2

(17)

(18)

We observe that the Butterworth highpass filter in eq.(6) or eq.(15)
can handle nonstationary components integrated of order 2n or less. Thus,
the HP filter can stationarize the time series with unit root components up
to the fourth order. Gomez (2001, p. 367) claimed that the BFT would give
better approximations to ideal low-pass filters than the BFS. A simulation
study in Otsu (2007) confirmed it. In the following analysis, we use both
BFT and BFS for completeness.
In the paper, we apply the Butterworth filters to extraction of compo$2 ], where $1 is smaller than $2 . The
nents over a certain band [$1 %bandpass filter is obtained as the difference between two highpass filtersin eq.(6), or two lowpass filters in eq.(5) with different values of ", as in
Baxter and King (1999, p. 578). Suppose a lowpass filter has the pass band[0%$p 1] and the stop band [$1 %#]. Here, $p 1 indicates a frequency atwhich the cycle is longer by some periods than at $1 and corresponds to$p in eq.(12), while $1 corresponds to $s in (13). This lowpass filter has
the cutoff frequency of $c 1 and the order of n1 determined in eq.(12) and(13). Let "1 the corresponding value of ". Similarly, another lowpass$2 ] and the stop band [$p 2%#]. Here, $p 2
filter has the pass band [0%indicates a frequency at which the cycle is shorter by some periods than at$2 . In short, we assume that $p 1 &$1 &$2 &$p 2. $2 corresponds to $pin (12), and $p 2 corresponds to $s in (13). The filter has the cutoff
frequency of $c 2 and the order of n2 . Then, the value of " is "2 . The
bp
"2 %
n1 %
n2 ), can be obtained as
bandpass filter, BFT ("1 %― ９８ ―

Abridged Bry-Boschan Procedure!2 $BFT bp (!1 $n1 $n2 ) "BFTL (!2 $n2 ) !BFTL (!1 $
n1 )

(19)

The corresponding frequency response is expressed as!2 $h (#; !1 $n1 $n2 ) ""L (e !i #; !2 $n2 ) !"L (e !i #; !1 $
n1 )

(20)

bp
!2 $
n1 $
n2 ),
We can obtain the bandpass filter for the sine-type, BFS (!1 $and its frequency response in a similar manner.
Alternatively, we sequentially apply the highpass filter with a lower
cutoff frequency to a series, and then further apply the lowpass filter witha higher cutoff frequency to the filtered series. Although Pedersen (2001,
p. 1096) reported that the sequential filtering has less distorting effects thanuse of the linear combination of the filters, the empirical results in thefollowing sections do not change whether we use the difference method(the linear combination) or the sequential method. Yet another method is toconvert the lowpass filter to the bandpass filter by the frequency transformation, described in a standard textbook (e.g. Proakis and Manolakis, 2007,
p. 733), and explicitly obtain the bandpass filter (see Gomez, 2001, p. 371).

This filter, however, has only one order parameter, implicitly assuming n1
is equal to n2 . But, the values of n1 and n2 are very different in fact (seeOtsu, 2015). Therefore, we would not use the transformation method laterin the paper.
Finally, Harvey and Trimbur (2003, pp. 248-249) derived the generalized Butterworth bandpass filter in the context of unobserved-component
models, taking advantage of the Wiener-Kolmogorov formula. To compute the values of the smoothing parameter and the filter’s order, we needdetermine the locational parameter values of the band and the bandwidth.
Still, a numerical calculation is involved. Here, we use the difference― ９９ ―

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method, because it is easy to control leakage and compression effects at a
specific frequency.
Turning to implementation, we can implement the Butterworth filtering either in the time domain or in the frequency domain. Following
Pollock (2000), Otsu (2007) implemented it in the time domain, and foundthat when the cycle period is longer than seven, the matrix inversion is so
inaccurate that it is impossible to control leakage and compression effects
with a certain precision specified by eq.(12) and eq.(13), or eq.(17) andeq.(18). Further, the filters at the endpoints of data have no symmetry
due to the finite truncation of filters. This implies that the time-domain
implementation introduces phase shifts. Therefore, we do not choose thetime-domain filtering.
Alternatively, we can implement the Butterworth filtering in thefrequency domain. In the frequency-domain filtering, cyclical components
are computed via the inverse discrete Fourier transform, using the Fouriertransformed series with the frequency response function as their weights.
In contrast to the time-domain filtering, the frequency-domain filtering
does not introduce any phase shifts, as the theoretical background of thesymmetrical filters dictates. For the frequency-domain procedures to work
well, it is required that a linear trend be removed and circularity bepreserved in the time series, which we discuss next.

3.4

Detrending Method
To obtain better estimates of cyclical components, it is desirable toremove a linear trend in the raw data. The linear regression line, recommended by Iacobucci and Noullez (2005), is often used for trend removal.
As shown by Chan, Hayya, and Ord (1977) and Nelson and Kang (1981),―１００―Abridged Bry-Boschan Procedure

however, this method can produce spurious periodicity when the true
trend is stochastic. Another widely-used detrending method is the firstdifferencing, which reweighs toward the higher frequencies and can distortthe original periodicity, as pointed out by Baxter and King (1999), Chan,
Hayya, and Ord (1977), and Pedersen (2001).
Otsu (2011) found that the drift-adjusting method employed by
Christiano and Fitzgerald (2003, p. 439) could preserve the shapes of
autocorrelation functions and spectra of the original data better than thelinear-regression-based detrending. Therefore, this detrending method""""
T . Then, we
would create less distortion. Let the raw series zt "t $1"
compute the drift-adjusted series, #t as follows:ˆ#t $zt !(t #s )!

(21)

where s is any integer and
z !z1
ˆ$ T
!
T !1

(22)

Note that the first and the last points are the same values:#1 $#T $Tz1 !zT #s (z1 !zT )
T !1

(23)

In Christiano and Fitzgerald (2003, p. 439), s is set to !1 . Although Otsu
(2011) suggested some elaboration on the choice of s, it does not affectthe results of our subsequent analyses in the paper. Thus, we also set s to
!1 .
It should be noted that the drift-adjusting procedure in eq.(21) would
make the data suitable for filtering in the frequency domain. Since thediscrete Fourier transform assumes circularity of data, the discrepancy
in values at both ends of the time series could seriously distort the―１０１―

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frequency-domain filtering. The eq.(23) implies that this adjustment
procedure avoids such a distortionary effect.
A final remark here is that the removed trend is put back before
the BB procedure is implemented. In the business cycle literature, it isimportant to distinguish a classical cycle and a growth one, as pointed out
by Pagan (1997). The classical cycle consists of peaks and troughs in thelevels of aggregate economic activities, often represented by the grossnational product (GDP). The classical cycle is studied by Burns and
Mitchell (1946), one of the influential seminal works, which found thatbusiness cycles range from 18 months (1.5 years) to 96 months (8 years) for
the United States.
On the other hand, the growth cycle exists in the detrended series, on
which the real business cycle literature focuses. The two types of thecycles show different dates of the peaks and the troughs. When a series
has a cyclical component around a deterministic upward trend, typicalas in economic data, detrending would make the peaks earlier, while
delaying the troughs (see Bry and Boschan, 1971, p. 11). For this reason, thedating based on the growth cycle generically tends to deviate from that on
the classical cycle. Then, Canova (1994) and Canova (1999) judged theestimated dates matched the official dates as long as deviations were
within two or three quarters. The results in Otsu (2013) also show that theestimated dates of peaks based on the detrended series tend to markearlier and those of troughs later than the official dates. To mitigate thisdeviation, the linear trend component is incorporated into input of the BB
procedure.

―１０２―Abridged Bry-Boschan Procedure

3.5

Boundary Treatment
In addition to the detrending method, we make use of another device

to reduce variations of the estimates at ends of the series: extension with aboundary treatment. As argued by Percival and Walden (2000, p. 140), it
might be possible to reduce the estimates’ variations at endpoints if wemake use of the so-called reflection boundary treatment to extend theseries to be filtered. We modify the reflection boundary treatment so thatthe series is extended antisymmetrically instead of symmetrically as in theconventional reflecting
rule. Let the extended series fj ,!
#!
if 1 "j "Tjfj $"2! !! if !T #3 "j "0
1
2!j

(24)

That is, the T !2 values, folded antisymmetrically about the initial data
point, are appended to the beginning of the series. We call this extension
rule the antisymmetric reflection, distinguished from the conventionalreflection.
It is possible to append them to the end of the series. The reason toappend the extension at the initial point is that most filters give accurate
and stable estimates over the middle range of the series. When we put theinitial point in the middle part of the extended series, the starting parts of
the original series would have estimates more robust to data revisions or
updates than the ending parts. Since the initial data point indicates thefarthest past in the time series, it does not make sense that the estimate of
the initial point is subject to a large revision when additional observations
are obtained in the future. Otsu (2010) observed that it moderately reduced
compression effects of the Butterworth and the Hamming-windowed filters.
We note that this boundary treatment makes the estimates at endpoints
identically zero when a symmetric filter is applied. We filter the extended―１０３―

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series, fj , and extract the last T values to obtain the targeted components.
3.6

Parameter Values
To implement the Butterworth filtering, we need specify four parame-

ter values, n1 , n2 , "1 , and "2 in eq.(19). We obtain these values from eqs.
(7), (12), and (13) for target frequency bands, that is, values of $p and $s
and for given values of !
1 and !
2 . We set both !
1 and !
2 to 0.01.
The precision parameter values command orders of the Butterworth
filters (see Gomez, 2001). But higher orders lead to numerical instability as
noted in Pollock (2000, pp. 324-325). In fact, we could not implement the
Butterworth filtering in the frequency domain when we set the transition
bands to the range of 55 to 56 cycles per period with a precision valuesof less or equal to 1%. If we allow a little bit wider transition bands, we
can avoid this instability in computation.
Otsu (2015) examined width of transition bands to have numericalstability when we extract the business cycle components from the Index ofIndustrial Production of Japan. It is shown that the transition bands withfrequencies of 12 to 18 months per cycle and 96 to 132 months per cycle
give rise to the dates of peaks and troughs consistent with the officiallypublished reference dates of the business cycles. The corresponding ordersof the tangent-based Butterworth filter are 11 and 14 for each transition
band under the precision value of 0.01, which are low enough to secure
numerical stability.
In the paper, we attempt to extract cyclical components with periods
per cycle of 1.5 years to 8 years. In terms of a period per cycle (p), a
frequency ($) is expressed as 2#. Therefore, using the notation in thep2#]%$2 ], is [2#previous section, the target band, [$1 %96 18 in months. Following―１０４―Abridged Bry-Boschan Procedure
2#
Otsu (2015), we set $p 1 to 132
and $p 2 to 2#
12 . In this case, the transition
2# 2#
2#
]
[2#%
]
%
$p 1 to $p in eq.(12)bands are [13296 and 18 12 , respectively. Setting

and $1 to $s in eq.(13), we find n1 and $c . "1 is obtained from eq.(7).
Similarly, we find n2 and "1 by setting $2 to $p in eq.(12) and $p 2 to $s
in eq.(13), together with eq.(7). In a similar way, we compute the parameter values of the sine-based Butterworth filter from eq.(16), eq.(17) andeq.(18).
Two remarks are in order. As is always the case, the sine-based filter
commands a higher order than the tangent-based on under the same
precision values of !
1 and !
2 . In addition, as already mentioned, the wellknown HP filter is viewed as the sine-based Butterworth filter with anorder of two. This implies that the HP filter either does not preserve theprecision or requires a very wide transition band. In the literature, it ispointed out that it might mislead researchers to false empirical results(Harvey and Jaeger, 1993), or it could generate spurious business cycle

dynamics (Cogley and Nason, 1995). In the paper, we use the HP filter forcompleteness.

4 Empirical Study
4.1

Reference Dates and Data
The reference dates of business cycles in Japan are determined by the

Economic and Social Research Institute (ESRI), affiliated with the Cabinet
Office, Government of Japan. The ESRI organizes the Investigation
Committee for Business Cycle Indicators to inspect historical diffusionindexes compiled from selected series and other relevant information. To
construct a historical diffusion index, the peaks and troughs of eachindividual time series are dated by the Bry-Boschan method. Thus, the―１０５―成城・経済研究

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reference dates correspond to those of peaks and troughs of the classicalcycles, that is, the Burns-and-Mitchell-type cycle based on the level ofaggregate economic activity. Typically, the final determination of thedates is made about two to three years later.
Table 2 shows the reference dates of peaks and troughs identified by
the ESRI. It also contains periods of expansion, contraction, and durationof a complete cycle (trough to trough). There are 14 peak-to-trough phases
identified after World War II. The average period is about 36 months for
expansion, 17 for contraction, and 53 for the complete cycle. We compare
the reference dates with those obtained with various filtering methods.
We use Index of Industrial Production (IIP) of Japan in monthly
basis, retrieved from Nikkei NEEDS CD-ROM (2008). We use seasonallyunadjusted (NEEDS series mnemonic: IIP00P001) and seasonally adjusted(NEEDS series mnemonic: IIP00P001@) series. The base year is 2000. The

sample ranges from January 1955 to January 2008, 637 observations in all.
Although we should examine all series composed of the coincident index,
they are available only after July 1978. Here, we focus on the IIP which
provides the largest sample size among the coincident-index related series
of Japan. Note the sample period does not allow analyses of the peaks andtroughs before January 1955 and after January 2008 in Table 2.

4.2

Comparison with Reference Dates
To assess goodness of a dating procedure, one criterion used in theliterature is how well it replicates the official reference dates of thebusiness cycle. Canova (1994) examined the estimated growth cycle toevaluate performance of 11 different detrending methods to replicate
NBER dating, assuming that the detrending removes a secular component.
―１０６―Abridged Bry-Boschan Procedure

Similar analyses are conducted by Canova (1999) with 12 methods including Hamilton (1989)’s procedure. They found that the Hodrick-Prescott(HP) filter proposed by Hodrick and Prescott (1997) and a frequency

domain filter as an approximation to the Butterworth filter (see Canova,
1998, p. 483) would be the most reliable tools to reproduce the NBER

dates. Otsu (2013) also conducted a comparative analysis among bandpass
filters such as the Christiano-Fitzgerald (CF) filter (Christiano and Fitzgerald,2003), the Hamming-windowed filter (Iacobucci and Noullez, 2005) and the

Butterworth filters (e.g. Gomez, 2001; Pollock, 2000), using Japanese real
GDP data. It shows that the Butterworth filters give the business-cycle
dates closest to the official reference dates.
In the paper, we compute dates of peaks and troughs of IIP series
for various filtering methods. The dating rule follows that of the BB
procedure. We examine how well estimated dates replicate the official
reference dates of the business cycles. We use official seasonal-adjustment
series, seasonal adjusted by the Hamming-windowed filter and the Butterworth filters, and the series bandpassed by the three filtering methods.
Since Bry and Boschan (1971) presumed to use seasonally adjusted
series, a seasonal adjustment would be required for dating procedures. The
official seasonal adjustment is implemented with the X12-ARIMA method.
To see effects of the adjustment methods, we also use the Hammingwindowed filter, Christiano-Fitzgerald filter and the Butterworth filters toseasonally adjust the series and use them in the BB procedure. Althoughconventional seasonal adjustment attempts to remove seasonal frequencies
only, these filters attenuate all the frequencies higher than seasonal ones.
As shown in Otsu (2009), however, the adjusted data have a power
spectrum in frequency domain that is identical to that of the X12-ARIMA―１０７―

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adjusted data. Therefore, these filters can be used for seasonal adjustment.
As for the CF filter, since the algorithm provided by Christiano andFitzgerald (2003) allows a one-sided highpass filter, seasonally adjusted
series are obtained by subtracting the cyclical components shorter than orequal to 12-month cycle from the original series. Figure 1 shows the seasonally adjusted series via these three methods. They are very similar tothe official seasonal-adjustment data. The difference is in that the official
data show more wriggles and no drop at the end of the sample.
Table 3 shows the results based on official seasonal-adjustment data.
In the second column, the data is processed through all steps in the BB
procedure. We find the BB procedure is not able to identify the peaks
from 1960s to 1970 and 1977. Although there is no peak in 1981 in thereference dates, it identifies October 1981 as a peak. If we use the outlierfree Spencer curve instead of the 12-month moving average in Step II, theBB procedure picks up two dates close to the 1960s’ peaks: March 1962
and December 1964 in the third column. In addition, it identifies April
1995 and January 2003 that are not in the reference dates. Further, we
drop the Spencer filtering in the fourth column. Then, Step I, Step II.1,
and Step III are ineffective. In Step IV, the 3-month moving average
is used. It is shown that more peaks are picked up: January 2005 andDecember 2006. We observe similar results for troughs in Table 4. The
identified dates in 1990s and 2000s are rather different from the reference
dates. In sum, without the 12-month moving average and/or the Spencer
filtering, the turns in 1960s are detected, and the turns in 1990s and 2000s
are overidentified.
To see effects of steps in the BB procedure, some steps or their
combinations are skipped to identify the dates. Table 5 and Table 6 show
―１０８―

Abridged Bry-Boschan Procedurethe selected results. In these cases, we start with the outlier-free Spencer
curve instead of the 12-month moving average in Step II. Any of thesecannot identify peaks in 1970 and 1977, and troughs in 1971 and 1977.
When Step III through Step IV.2 are skipped, the turns in 1995 are not
identified. This is because a trough in 1995 is tentatively identified inAugust in this case, instead of September in other two cases, being
removed in Step V.6. Similarly, when only Step III is skipped, the lasttrough is tentatively identified in April 2003, while it is in August 2003 incase of skipping Step III and IV. Then, Step V.6 removes the trough ofApril 2003 in the second column. These suggest that steps involving the
Spencer curve and the short-term moving average do not contribute to
replicating the official reference dates.
As already mentioned, the bandpass filters can be used for seasonal
adjustment. In Table 7 and Table 8, all frequencies higher than the
12-month cycle are removed by the Butterworth filters. The sine-based
and the tangent-based give the same results. The full BB procedure (thesecond column) gives a very similar result to that of the official seasonal-

adjustment series. Skipping Step III only, or together with Step IV, does
not change the results (the third column). Further, when all steps from III to
V.2 are omitted, the final turning points are same as the tentative ones
based on the 12-month moving average. Then, the date of June 2007 isadditionally identified as a peak, and September 1986 is identified as atrough, instead of January 1987. But, we do not see much improvement in
replicating the reference dates. In the fifth column, the moving averages
within the BB procedure are not executed. Then, it not only identifies theturns in 1960s, but also overidentifies the turns in 2000s. A similar result
is obtained with the Hamming-windowed filter in Table 9 and Table 10,―１０９―

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and with the CF filter in Table 11 and Table 12.
Turning to results with the bandpassed series, we find the BB-based
dates get closer to the reference dates. As in the literature, we presume
that business cycles range from 18 months to 96 months. Figure 2 shows
the bandpassed series. Table 13 and Table 14 show results with the
Butterworth filters. The results with the sine-based one slightly change
whether the BB procedure uses the moving averages or not: the turns in
1962 are detected without the moving averages. The BB procedure withthe tangent-based filter gives robust results. It does not matter whether themoving averages are used or not. In fact, we could skip all steps from III
to V.2 without any consideration of outlier removal if we use the
Butterworth-filtered series in Step II instead of the 12-month moving
average. The turns in 1977 are not yet identified, and those in 1981 areoveridentified. In experimental exercises, we also examined the case that
2! 2!
2!
]
[2!"
]
"
the transition bands are [14496 and 18 12 , and obtained similar results.

The dates of peaks are identical. Those of troughs are different by one
month: March vis-à-vis April in 1958, October vis-à-vis September in
1965, and December vis-à-vis November in 1982. Table 15 and Table 16
show results for other bandpass filters. With the Hamming-windowed andthe CF filters, the BB procedure identifies too many turns in 2000s andthe beginning of 1980s, while the HP filter misses the first two turns andone turn in 1977. In sum, the BB procedure replicates the official reference dates better with the Butterworth filtering than with other filters.

5 Discussion
This paper examines what kind of filtering would be useful for the BB
procedure to replicate the official reference dates, using Index of Industrial―１１０―Abridged Bry-Boschan Procedure

Production (IIP) of Japan. We use the official seasonal-adjustment series,
filtering-based equivalents, and business-cycle components extracted bybandpassing. The main findings are as follows. First, the business cycle
components, extracted by bandpass filters, is useful for the BB procedure
to produce dates of the turns close to the reference dates. Particularly, theButterworth-based bandpass works well. Second, the seasonally adjusted
series, whether adjusted by X12-ARIMA (official data) or by a bandpass
filter, cannot identify turns in 1960s, or overidentify those in 2000s.
Finally, we can drop the moving averages and outlier adjustment in
the BB procedure when we use the tangent-based Butterworth filter. A
possible abridged version of the Bry-Boschan procedure is described inTable 17. Here, we do not bother what threshold value is appropriate to
identify outliers: it is set to 3.5 in the original procedure. Moreover, wehave to determine neither what kinds of moving averages to use nor
length of their leads and lags. All we need to do is to determine what
duration is appropriate for a minimum cycle or a minimum phase.
A couple of caveats are in order. First, it is desirable to examine
robustness of the results obtained here. Analyses with different data intime and region would be helpful. Second, we might tune dating rules, sothat the BB-based dating gives a closer correspondence to the official
reference dates. For example, it might be necessary to include depths ofthe business cycles into dating rules or taking into consideration otherrules discussed in Webb (1991) and Harding and Pagan (2016, pp. 26-31).
Finally, although we use the official reference dates as a criterion ofevaluation, we may need to reinvestigate whether official turning points
are appropriate. The turns in 1977 are hardly detected in many cases. It isretrospectively recognized that the period of March 1975 to January 1977
―１１１―

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was an upturn period without a good economic sentiment, while that ofJanuary to October 1977 was a ‘slack’ period, but not a serious slump.
Further, the political turbulence during 1970s might affeet the dating
decision as well as economic data. Thus, it is natural that no turns aredetected in these period. These are left for the future research.

Acknowledgements

A part of this work is financially supported by Tokubetsu

Kenkyu Josei (Seiio University) under the title of「景気動向指数の簡便推計に関する研究」．

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81(4), 575-593.
Bry, G. and C. Boschan, 1971, Cyclical Analysis of Time Series: Selected Procedures and Computer Programs, Technical Paper 20, NBER, New York.
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in Business Cycles No. 2, NBER, New York.
Canova, F., 1994, “Detrending and turning points,” European Economic Review 38,
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Canova, F., 1998, “Detrending and business cycle facts,” Journal of Monetary
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Canova, F., 1999, “Does detrending matter for the determination of the reference
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Canova, F., 2007, Methods for Applied Macroeconomic Research, Princeton
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Christiano, L.J. and T.J. Fitzgerald, 2003, “The band pass filter,” International
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Cogley, T. and J.M. Nason, 1995, “Effects of the Hodrick-Prescott filter on trend
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Gomez, V., 2001, “The use of Butterworth filters for trend and cycle estimation ineconomic time series,” Journal of Business and Economic Statistics 19(3), 365373.
Hamilton, J., 1989, “A new approach to the economic analysis of nonstationary
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Harding, D. and A. Pagan, 2001, “Extracting, analysing and using cyclical information,” presented at CEPR/Banca d’ltalia conference on Monitoring the European
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Harding, D. and A. Pagan, 2002, “Dissecting the cycles: a methodological investigation,” Journal of Monetary Economics 49, 365-381.
Harding, D. and A. Pagan, 2006, “Synchronization of cycles,” Journal of Monetary
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Harding, D. and A. Pagan, 2016, The econometric analysis of recurrent events inmacroeconomics and finance, Princeton University Press, Princeton and Oxford.
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Hodrick, R.J. and E.C. Prescott, 1997, “Post-war U.S. business cycles: An empiricalinvestigation,” Journal of Money, Credit, and Banking 29, 1-16. Also in Discussion Paper No. 451, Department of Economics, Carnegie-Mellon University,
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Iacobucci, A. and A. Noullez, 2005, “A frequency selective filter for short-length
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King, R.G. and S.T. Rebelo, 1993, “Low frequency filtering and real businesscycles,” Journal of Economic Dynamics and Control 17, 207-231.
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Whittle, P., 1983, Prediction and regulation by linear least-square methods, 2nd ed.
(revised), Basil Blackwell, Oxford.

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Table 1

Summary of Bry-Boschan Procedure

Step

ProcedureI

Outlier-removed series (XO):
The data point of the original series (X) is replaced by that of the Spencer-filtered series(XSP) if its normalized difference in absolute value is larger than or equal to 3.5.

II

Dating with 12-month moving average:
1. Moving average:
Compute 12-month moving average with 6 lags and 5 leads (X12), using XO.
2. Identification of peaks and troughs:
Find the maximum (peak) or the minimum (trough) of X12 values within 6-monthleads and lags.
3. Enforcement of alternation:
Ensure the peaks and the troughs are alternate. If not, choose a peak with a greater
value and a trough with a smaller value. If the values are same, choose an earlier peak
and a later trough.

III

Dating with Spencer filtering:
1. Spencer filtering:
Filtering XO with the Spencer filter to obtain a series named XOSP.
2. Identification of peaks and troughs:
Ensure the peaks and the troughs in Step II within ±6 months, using XOSP. Modify ifnecessary.
3. Enforcement of alternation: Ensure alternation as in Step II.
4. Enforcement of minimum cycle duration:
Check if the duration of a peak-to-peak or trough-to-trough takes at least 15-month
period. If not, choose higher peaks and lower troughs, or if equal, an earlier date for apeak and a later one for a trough.

IV

Dating with short-term moving average:
1. Spencer filtering:
Use the Spencer curve of the original series (X) as the trend-cycle component, andcompute the irregular component by the difference between X and the Spencer curve.
Find the minimum number of months (MCD) over which the average rate of change
in the trend-cycle component exceeds the average change in the irregular component.
If it is less than 3 months, the MCD is set to 3, while set to 6 if more than 6 months.
2. Short-term moving average:
Compute the short-term moving average (MCDX) of the original series (X) with thespan of MCD obtained above. The values at the first and the last dates with missing
values in leads and lags, are to set to the same values as those at the nearest dates.
3. Identification of peaks and troughs:
Ensure the peaks and the troughs as in Step III within ±6 months, using MCDX series.
Modify if necessary.
4. Enforcement of alternation: Check alternation as in Step II.

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Summary of Bry-Boschan Procedure (continued)

Step

ProcedureV

Dating with the original series:
1. Identification of peaks and troughs:
Ensure the peaks and the troughs as in Step IV within ±4 months or MCD, whichever longer, using the original series (X). Modify if necessary.
2. Enforcement of alternation: Ensure alternation as in Step II.
3. Elimination of turns within 6 months at endpoints:
Eliminate peaks and troughs within 6 months of beginning and end of series.
4. Enforcement of the first and last peak (or trough) to be extrema:
Eliminate peaks (or troughs) at both ends of series which are lower (or higher) thanvalues closer to end.
5. Enforcement of the minimum cycle duration:
Check if the peak-to-peak and the trough-to-trough cycles are less than 15 months.
If not, eliminate lower peaks (or higher troughs), or if equal, a later peak and an
earlier trough.
6. Enforcement of the minimum phase duration:
Eliminate phases (peak to trough or trough to peak) whose duration is less than 5months.

Table 2

Reference Dates of Business Cycles in Japan
Dates (month, year)
Peak

June,
January,
June,
December,
October,
July,
November,January,
February,
June,
February,May,
November,February,

Number of Periods (in months)
Trough

1951
1954
1957
1961
1964
1970
1973
1977
1980
1985
1991
1997
2000
2008

October,November,
June,
October,
October,
December,
March,
October,
February,
November,
October,January,
January,
March,

1951
1954
1958
1962
1965
1971
1975
1977
1983
1986
1993
1999
2002
2009

Expansion

Contraction

Duration

―
27
31
42
24
57
23
22
28
28
51
43
22
73

4
10
12
10
12
17
16
9
36
17
32
20
14
13

―
37
43
52
36
74
39
31
64
45
83
63
36
86

Source: Indexes of Business Conditions, Economic and Social Research Institute, Cabinet Office,
Government of Japan, October 7, 2012.

―１１６―

Abridged Bry-Boschan Procedure

Table 3 Comparison with Reference Dates: Peaks, IIP (official S.A.)
Official Ref. Dates

BB proc.

No MA12No MA12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

1957
1961

6
12

1957

5

1957

5

1957

5

1964
1970
1973

10
7
11

1962
1964

3
12

1962
1964

3
12

1977
1980

1
2

1985
1991

1974
1977
1980
1981
1985
1991
1995
1997
2000
2003
2005
2006

1
1
2
10
5
5
4
5
12
1
1
12

1974

1

1974

1

6
2

1980
1981
1985
1991

2
10
5
5

1997
2000

5
11

1997
2000

5
12

1980
1981
1985
1991
1995
1997
2000
2003

2
10
5
5
4
5
12
1

2008

2

* No outliers are adjusted.

Table 4 Comparison with Reference Dates: Troughs, IIP (official S.A.)
Official Ref. Dates

BB proc.

No MA12No MA12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

1958
1962
1965
1971
1975
1977

6
10
10
12
3
10

1958

6

1958
1962
1965

6
12
5

1958
1962
1965

6
12
5

1975

3

1975

3

1980
1982

8
10

1980
1982

8
10

1975
1977
1980
1982

3
7
8
10

1986

8

1986

8

1986

8

1994

1

1998

8

1994
1995
1998

1
9
8

1994
1995
1998

1
9
8

2001

11

2001

11

2001

11

2003
2005
2007

8
7
5

1983
1986
1993

2
11
10

1999

1

2002

1

2009

3

* No outliers are adjusted.

―１１７―

成城・経済研究

第２
１
５号 （２
０
１
７年１月）

Table 5 Comparison with Reference Dates: Peaks, IIP (official S.A.) w/o MA12Official Ref. Dates

Skip BB.III

Skip BB.III-IV

Skip BB.III-V.2

Year

Month

Year

Month

Year

Month

Year

Month

1957
1961

6
12

1957

5

1957

5

1957

7

1964
1970
1973

10
7
11

1962
1964

3
12

1962
1964

3
12

1962
1964

3
12

1973

12

1974

1

1974

1

1980
1981
1985
1991
1995
1997
2000

2
10
5
5
4
5
12

1980
1981
1985
1991
1995
1997
2000

2
10
5
5
4
5
12

1980
1981
1985
1991

4
11
6
2

1997
2000
2002

6
10
11

2003

1

2003

1

1977
1980

1
2

1985
1991

6
2

1997
2000

5
11

2008

2

Table 6 Comparison with Reference Dates: Troughs, IIP (official S.A.) w/o MA12Official Ref. Dates

Skip BB.III

Skip BB.III-IV

Skip BB.III-V.2

Year

Month

Year

Month

Year

Month

Year

Month

1958
1962
1965
1971
1975
1977

6
10
10
12
3
10

1958
1962
1965

6
12
5

1958
1962
1965

6
12
5

1958
1962
1965

5
10
5

1975

3

1975

3

1975

3

1980
1982

8
10

1980
1982

8
10

1980
1982

10
11

1986

8

1986

8

1986

9

1994
1995
1998

1
9
8

1994
1995
1998

1
9
8

1994

1

1998

10

2001

11

2001

11

2001

12

2003

8

2003

5

1983
1986
1993

2
11
10

1999

1

2002

1

2009

3

―１１８―

Abridged Bry-Boschan Procedure

Table 7 Comparison with Reference Dates: Peaks, IIP (S.A. by Butterworth filters)
Official Ref. Dates

BB proc.

Skip BB.III (-IV)

Skip BB.III-V.2

No X12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

Year

Month

1957
1961

6
12

1957

8

1957

8

1957

8

1957

8

1962

4

1964

10
1965

1

1970
1973

7
11

1977
1980

1
2

1985
1991
1997
2000

6
2
5
11

2008

2

1973

11

1973

11

1973

11

1973
1976

11
12

1980
1981

4
11

1980
1981

4
11

1980

5

1980
1981

4
11

1985
1991
1997
2000

7
1
5
9

1985
1991
1997
2000

7
1
5
9

1982
1985
1991
1997
2000

1
9
3
5
9

1985
1991
1997
2000
2002
2004
2006

7
1
5
9
9
6
8

2007

6

* The III-V.2 steps in BB procedure are skipped with no outlier adjustment.

Table 8 Comparison with Reference Dates: Troughs, IIP(S.A. by Butterworth filters)
Official Ref. Dates

BB proc.

Skip BB.III (-IV)

Skip BB.III-V.2

No X12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

Year

Month

1958
1962
1965
1971
1975
1977

6
10
10
12
3
10

1958

5

1958

5

1958

5

1958
1962
1965

5
10
6

1975

4

1975

4

1975

6

1980
1982

12
11

1980
1982

12
11

1980
1982

12
10

1975
1977
1980
1982

4
5
12
11

1986

9

1987
1993
1998

1
12
11

1987
1993
1998

1
12
11

1993
1998

12
12

1987
1993
1998

1
12
11

2001

12

2001

12

2001

12

2002

1
2003
2004

3
12

1983
1986

2
11

1993

10

1999

1

2002

1

2009

3

* The III-V.2 steps in BB procedure are skipped with no outlier adjustment.

―１１９―

成城・経済研究

第２
１
５号 （２
０
１
７年１月）

Table 9 Comparison with Reference Dates: Peaks, IIP (S.A. by Hamming filter)
Official Ref. Dates

BB proc.

Skip BB.III (-IV)

Skip BB.III-V.2

No X12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

Year

Month

1957
1961

6
12

1957

7

1957

7

1957

8

1957

7

1962

4

1964

10
1965

1

1970
1973

7
11

1977
1980

1
2

1985
1991
1997
2000

6
2
5
11

2008

2

1973

11

1973

11

1973

11

1973
1976

11
12

1980
1981

4
11

1980
1981

4
11

1980

5

1980
1981

4
11

1985
1991
1997
2000

7
3
6
10

1985
1991
1997
2000

7
3
6
10

1982
1985
1991
1997
2000

1
9
3
5
9

2007

7

2007

7

2007

6

1985
1991
1997
2000
2002
2004
2007

7
3
6
10
10
5
7

* The III-V.2 steps in BB procedure are skipped with no outlier adjustment.

Table 10 Comparison with Reference Dates: Troughs, IIP (S.A. by Hamming filter)
Official Ref. Dates

BB proc.

Skip BB.III (-IV)

Skip BB.III-V.2

No X12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

Year

Month

1958
1962
1965
1971
1975
1977

6
10
10
12
3
10

1958

4

1958

4

1958

5

1958
1962
1965

4
10
6

1975

4

1975

4

1975

6

1980
1982

12
11

1980
1982

12
11

1980
1982

12
10

1975
1977
1980
1982

4
5
12
11

1986

9

1987
1993
1998

1
12
12

1987
1993
1998

1
12
12

1993
1998

12
12

1987
1993
1998

1
12
12

2001

12

2001

12

2001

12

2002

1
2003
2004

4
10

1983
1986

2
11

1993

10

1999

1

2002

1

2009

3

* The III-V.2 steps in BB procedure are skipped with no outlier adjustment.

―１２０―Abridged Bry-Boschan Procedure

Table 11 Comparison with Reference Dates: Peaks, IIP (S.A. by CF filter)
Official Ref. Dates

BB proc.

Skip BB.III (-IV)

Skip BB.III-V.2

No X12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

Year

Month

1957
1961

6
12

1957

5

1957

5

1957

8

1957

5

1962

4

1964

10
1965

4

1970
1973
1977
1980

7
11
1
2

1985
1991
1997
2000

6
2
5
11

2008

2

1973

9

1973

9

1973

11

1973

9

1980
1981

5
9

1980
1981

5
9

1980

5

1980
1981

5
9

1985
1991
1997
2000

7
6
6
9

1985
1991
1997
2000

7
6
6
9

1982
1985
1991
1997
2000

1
9
3
5
9

1985
1991
1997
2000
2002
2004

7
6
6
9
9
8

* The III-V.2 steps in BB procedure are skipped with no outlier adjustment.

Table 12 Comparison with Reference Dates: Troughs, IIP (S.A. by CF filter)
Official Ref. Dates

BB proc.

Skip BB.III (-IV)

Skip BB.III-V.2

No X12, No Spencer*

Year

Month

Year

Month

Year

Month

Year

Month

Year

Month

1958
1962
1965
1971
1975
1977

6
10
10
12
3
10

1958

8

1958

8

1958

5

1958
1962
1965

8
10
9

1975

3

1975

3

1975

6

1975

3

1980
1982

12
12

1980
1982

12
12

1980
1982

12
10

1980
1982

12
12

1986

9

1987

1

1987

1

1987

1

1993

12

1998

12

1994
1998

1
12

2001

12

2002

1
2003
2005

3
2

1983
1986
1993

2
11
10

1999

1

2002

1

2009

1994
1998

1
12

1994
1998

1
12

2001

12

2001

12

3

* The III-V.2 steps in BB procedure are skipped with no outlier adjustment.

―１２１―

成城・経済研究

第２
１
５号 （２
０
１
７年１月）

Table 13 Comparison with Reference Dates: Peaks, IIP (Bandpassed by Butterworth)Sine-based filter
Official Ref. Dates

BB proc.

Tangent-based filter

No MA12, No Spencer*

Any combination**

Year

Month

Year

Month

Year

Month

Year

Month

1957
1961

6
12

1957

8

1957

8

1957

8

1964
1970
1973
1977
1980

10
7
11
1
2

1964
1970
1973

11
8
11

1962
1964
1970
1973

4
11
8
11

1962
1964
1970
1973

4
11
8
11

1985
1991
1997
2000

6
2
5
11

1980
1981
1985
1991
1997
2000
2007

3
11
3
3
6
10
7

1980
1981
1985
1991
1997
2000
2007

3
11
3
3
6
10
7

1980
1981
1985
1991
1997
2000
2007

3
11
3
3
6
10
7

2008

2

* Same results without III-V.2 steps in BB procedure.
** Same results with or without filtering, or skipping III-V.2 steps. No outliers detected.

Table 14 Comparison with Reference Dates: Troughs, IIP (Bandpassed by Butterworth)Sine-based filter
Official Ref. Dates

BB proc.

Tangent-based filter

No MA12, No Spencer*

Any combination**

Year

Month

Year

Month

Year

Month

Year

Month

１958
1962
1965
1971
1975
1977

6
10
10
12
3
10

1958

4

1965
1971
1975

9
7
5

1958
1962
1965
1971
1975

4
9
9
7
5

1958
1962
1965
1971
1975

4
9
9
6
5

1981
1982

1
11

1981
1982

1
11

1981
1982

1
11

1987
1993
1998

2
11
10

1987
1993
1998

2
11
10

1987
1993
1998

2
11
10

2001

11

2001

11

2001

12

1983
1986

2
11

1993

10

1999

1

2002
2009

1
3

* Same results without III-V.2 steps in BB procedure.
** Same results with or without filtering, or skipping III-V.2 steps. No outliers detected.

―１２２―Abridged Bry-Boschan Procedure

Table 15 Comparison with Reference Dates: Peaks, IIP (Bandpassed by filters)
Official Ref. Dates

Hamming filter

CF filter

HP filter

Year

Month

Year

Month

Year

Month

1957
1961

6
12

1957

9

1957

9

3
12

3

10

1962
1964

1962

1964

1965
1970
1973
1977
1980

1
6
12
1
2

1982
1985
1991
1997
2000

1
3
5
6
10

2004
2007

6
7

1970
1973
1977
1980

7
11
1
2

1985
1991
1997
2000

6
2
5
11

2008

1970
1973
1977
1980

7
12
1
2

1982
1985
1991
1997
2000
2002
2004
2007

1
3
5
6
10
12
6
6

Year

Month

1964

12

1970
1973

8
11

1980
1981

4
11

1985
1991
1997
2000

5
6
6
10

2004

7

2

Note: same results with or without use of MA 12, Spencer curve, and III-V.2 steps.

Table 16 Comparison with Reference Dates: Troughs, IIP (Bandpassed by filters)
Official Ref. Dates

Hamming filter

CF filter

HP filter

Year

Month

Year

Month

Year

Month

1958
1962
1965
1971
1975
1977

6
10
10
12
3
10

1958
1962
1965
1971
1975
1977

4
8
8
9
4
6

1958
1962
1965
1971
1975
1977

3
8
7
9
4
6

1981
1982

1
10

1981
1982

1
10

1987
1993
1998

2
10
9

1987
1993
1998

2001

12

2003
2005

5
3

1983
1986

2
11

1993

10

1999

1

2002

1

2009

Year

Month

1965
1971
1975

8
8
4

1980

12

1982

12

2
9
9

1987
1993
1998

1
10
9

2001

11

2001

12

2005

4

2005

1

3

Note: same results with or without use of MA 12, Spencer curve, and III-V.2 steps.

―１２３―

成城・経済研究
Table 17

第２
１
５号 （２
０
１
７年１月）

Abridged Bry-Boschan Procedure

Step

Procedure

I

Business cycle extraction :Extracting business-cycle components by a two-sided tangent-based Butterworth filter.
The deterministic linear trend is put back as described in Section 3.4.

II

Dating with the business-cycle components:
1. Identification of peaks and troughs:
Find the maximum (peak) or the minimum (trough) of the extracted components
within ±6 months (leads and lags).
2. Enforcement of alternation:
Ensure the peaks and the troughs are alternate. If not, choose a peak with a greater
value and a trough with a smaller value. If the values are same, choose an earlier
peak and a later trough.
3. Elimination of turns within 6 months at endpoints:
Eliminate peaks and troughs within 6 months of beginning and end of series.
4. Enforcement of the first and last peak (or trough) to be extrema:
Eliminate peaks (or troughs) at both ends of series which are lower (or higher) thanvalues closer to end.
5. Enforcement of the minimum cycle duration:
Check if the peak-to-peak and the trough-to-trough cycles are less than 15 months. If
not, eliminate lower peaks (or higher troughs), or if equal, a later peak and an earlier
trough.
6. Enforcement of the minimum phase duration:
Eliminate phases (peak to trough or trough to peak) whose duration is less than 5months.

―１２４―Abridged Bry-Boschan Procedure

Figure 1 Seasonally Adjusted Data: Index of Industrial Production of Japan

―１２５―

成城・経済研究

第２
１
５号 （２
０
１
７年１月）

Figure 2 Bandpassed Data: Index of Industrial Production of Japan―１２６―