Coincident Index and Reference Cycle

Takeshi Otsu

Abstract
This paper seeks an alternative method to make the coincident composite
index that is simpler than the existing method and is consistent with the official
reference cycle as much as possible. We first examine how accurately thecoincident composite index trace the reference cycle, using the business cycle dataof Japan. Then, we investigate aggregation methods to construct the composite
index from individual indicators in terms of normalization of each indicator,smoothing methods, and dating algorithm. The main findings are as follows. First,
use of an outlier trimming procedure and interquartile ranges plays only a minor
role in dating peaks and troughs. Secondly, when filtering methods are used toseasonally adjust series, the Bry-Boschan algorithm can be simplified to a great
extent and does not need the Spencer smoothing, the 12-month moving averages,
and the trimming procedure. Thirdly, the Butterworth filter has the same effects on
dating results as the interquartile ranges. Thus, the smoothing via filtering reduces
variation good enough.
Key words: composite index, coincident indicators; reference dates
JEL classification: E32

1

Introduction
Business cycle has been one of the main macroeconomic subjects in theacademic literature as well as in practice since Burns and Mitchell (1946).
Theoretical developments lead to the real business cycle theory (Kydland and─ 1 ─

成城・経済研究第 224 号（2019 年 3 月）

Prescott, 1982; Prescott, 1986) and the New Keynsian framework (Clarida, Galí, andGartler, 1999, 2000), and spawned various models (see Woodford, 2003; Walsh, 2010).

On the empirical front, a long-lasting concern is to understand the businesscyclical characteristics: identification of the reference dates of turning points,
duration and amplitude of the cycles, and synchronization among regions andindustries. Recent developments and subjects are concisely described in Harding
and Pagan (2016).
There are two types of indices of the business cycles: diffusion and composite
indices. Further, we have two types of diffusion indices in Japan: current andhistorical indices. Each index consists of leading, coincident, and lagging
indicators. The number of indicators to create composite indices are different agent
by agent around the world. For example, Economic and Social Research Institute(ESRI) in Japan currently adopts 9 series to calculate the coincident index, while 4

series are selected at the Conference Board in the United States (the Conference
Board, 2001, p.49).

The diffusion indices are often used to determine turning points of thebusiness cycle, that is, extract the reference cycle (Harding and Pagan, 2016, p.52).
The diffusion indices are typically compiled as follows. First, we determine thestates of expansion and contraction. For example, we might look at the changes ofeach series for the diffusion indices. Then, we count the number of positive (ornegative) changes at each point of time, and divide it by the total number of theseries adopted as an indicator. If more than 50% of the series indicate positive (ornegative) changes, the economic state is considered in an expansion (or contraction,
respectively). Thus, the trough (or peak) can be found at a time, say, t, right before

the share of the positive (negative) changes has just gone beyond 50% at time t+1.
ESRI computes the changes from three months ago for each indicator tocompute the current diffusion indices. It gives 1 for a positive change, 0.5 for no─ 2 ─

Coincident Index and Reference Cycle

change, and 0 for a negative one, sums up these numbers all over the series,
divides them by the number of the series and multiply by 100. Thus, the diffusionindex takes 100 at a point of time when all the series indicate positive changes, 50
in case of no changes at all and 0 in case of negative changes only. On the other
hand, ESRI uses the Bry-Boschan (BB) algorithm to make the historical indices. It
applies the BB algorithm to each series to find turning points. Then, it presupposes
positive changes during the period of a trough to a peak and negative from a peak
to a trough for each series, and obtains the historical index by computing the ratioof the number of the series with positive changes to the total number of the series,
multiplied by 100. Then, an economy is supposed at a peak when a coincident
historical index goes down to less than 50% in the next period, and at a troughwhen it goes up to more than 50%.
While the diffusion indices play an important role in determination of theturning points, the composite indices could be appropriate for analyses of theamplitude of the cycles, rate of change in the cycles, and forecasting. Specifically,
the leading indicators attract a great attention because it is presumed appropriate
for forecasting economic states in the future (Lahiri and Moore, 1991). However, thecoincident and the lagging indices are not much used in academic research, and thecomposite indices derived from them are far less employed. Stock and Watson(1991) estimated the so-called single-index model for the U.S. with four coincident

indicators and compare with the composite coincident index. Harding and Pagan(2006) used them to investigate whether their proposed algorithm could replicate

the NBER (National Bureau of Economic Research) reference cycle.
On the one hand, the lagging composite index rarely draws attention in theliterature, and on the other, the coincident composite index should be useful tounderstand present states of economies. Since the latter is composed of thecoincident indicators, it is also expected to indicate the same peaks and troughs as─ 3 ─

成城・経済研究第 224 号（2019 年 3 月）

the reference cycle. If the coincident composite index accurately follows thereference cycle, we can use it to understand the amplitude of the business cycle
and the magnitude of plunges or booms. Further, it may give a criterion of validityof the business-cycle models and the related econometric models. But, this is not
the case in the literature. For example. Canova (1994, 1998, 1999) investigatedvarious detrending methods with many kinds of quarterly data, adopting thereference cycle as a criterion instead of the composite index. Since the coincident
composite index is complied in a different way from the reference cycle in that theformer is the average of the coincident indicators and the latter is based on the
counts of signs of their changes and other information available, we need toinvestigate how closely the index follows the reference cycle.
This paper attempts to find an alternative method to make the coincident
composite indices that is simpler than the existing method and produces a dating
result consistent with the official reference cycle as much as possible. We startempirical analyses by examining how accurately the coincident composite indicestrace the reference cycle, using the business cycle data of Japan. Then, weinvestigate aggregation methods to compute the composite index from theindividual indicators in terms of normalization of each indicator, smoothing
methods, and dating algorithm.
The main findings are as follows. First, although ESRI uses an outlier
trimming procedure and interquartile ranges for a scaling measure, these devices
play only a minor role in dating peaks and troughs. Secondly, when filtering
methods are used to seasonally adjust series, the Bry-Boschan algorithm can be
simplified to a great extent and does not need the Spencer smoothing, the 12month moving averages, and a trimming procedure.Thirdly, the Butterworth filter
has the same effects on dating results as the interquartile ranges. Therefore, it
reduces variation of the series good enough.

─ 4 ─

Coincident Index and Reference Cycle

The rest of the paper is organized as follows. In section 2, we summarize theaggregation method of the ESRI to make the composite coincident index. Section
3 explains the algorithm of the Bry-Boschan procedure to date peaks and troughs.
It is followed by a brief review of filtering methods used in the paper in section 4.
In section 5, we analyze the coincident indices of Japan. The final section isallocated to discussion.

2

Indicator Aggregation: ESRI Method
In this section, we explain how ESRI (Economics and Social Research Institute)computes the composite indices: the leading, the coincident and the lagging index(Economic and Social Research Institute, 2015). Its method consists of outlier removal

and aggregation, and is common to the three indices. Let x(t), where j indicatesa type of composite indices (j=L: leading, C: coincident, Lag: lagging), i a series
used in each index, and t a point of time. Then, we first calculate a symmetric rateof change as follows:

Step I: Symmetric Rate of Changer(t)=200×x(t)−x(t−1)
x(t)+x(t−1)(1)

Step II: Outlier Removal
Next, outliers are removed. Let Q3 −Q1  an interquartile range of r(t), andoutlier-adjusted rate of change ψ(t)─ 5 ─成城・経済研究

第 224 号（2019 年 3 月）−k(Q3 −Q1 ),ψ(t)= r(t),r(t)<−k
Q3 −Q1 −k≦k(Q3 −Q1 ),k<r(t)≦k
Q3 −Q1 (2)r(t)
Q3 −Q1 

Here, k is a constant threshold value that is set so as to trim 5% at edges ofr (t) ( j=C), where t ranges from January 1980 to the latest December. k takes

a value of 2.02 as of November 2011. This rule is used to remove outliers in thecomputational procedure up until August 2011. A similar rule is used after
September 2011, which is explained later.

Step III: Trend
Two types of averages are used. The first one is called ‘trend of individual series’
by ESRI. It is an avereged outlier-free rate of change of each series in time domain,
computed as follows:μ(t)=

1ψ(τ)
60−s  

(3)

where s is the number of missing values. That is, it is an averaged value over thelast 60 months. The second average is computed across series of the coincident
index:μ (t)=1 × μ (t)
n


(4)

where n is the number of series used to compute the coincident index.

Step IV: Standardized Rate of Change
The rate of change is standardized with the interquartile range for each series:─ 6 ─

Coincident Index and Reference Cyclez(t)=

ψ(t)−μ(t)Q3 −Q1 (5)

The average rate of change is computed except missing values of z(t) as follows:Z (t)=

1× z(t)n −s (t) (6)where s (t) is the number of series that have missing values at time t.

Step V: Synthesis
The overall average rate of change is computed as follows:V (t)=μ (t)+Q3−Q1 ×Z (t)

(7)

where


Q3−Q1 =1
 × (Q3  −Q1  )
n


(8)

Step VI: Composite Index
To compute a composite index, the following indexation is used:I (t)=I (t−1)×200+V (t)200−V (t)(9)

where the initial value of I (t) is set to 1. Then, the composite index is obtained asfollows:
CI (t)=I (t)
×100
I

(10)

where I  is an average in the base year.
Since September 2011, a refinement has been made in Step II. The basic idea
is to make outlier adjustments only to series-specific parts of the standardized rateof change, so that a shock common to all the series should be excluded from outlier─ 7 ─

成城・経済研究第 224 号（2019 年 3 月）

removal. First, a time trend for each series is calculated asm(t)=

1r(τ)
60−s  

(11)

where s is the number of series that have missing values at time t. Then, the rate ofchange is standardized with the interquartile range:η(t)=r(t)−m(t)Q3 −Q1 

(12)




Let the median of η(t) across series denoted by η (t). Now, subtracting η (t)from both sides of eq. (12) and rearranging it, we obtain:r(t)=(η(t)−η (t))(Q3 −Q1 )+m(t)+η (t)(Q3 −Q1 ) 
specific partscommon part=r +r (13)
Then, the outlier adjustment is applied to r  as follows:(Q3 −Q1
 ),
−k




r (t)<−k −Q1Q3ψ (t)= r (t),



≦
−k


r (t)

≦k




Q3 −Q1  k (Q3 −Q1 ),


k<

(14)




r (t) Q3
 −Q1  −Q1  an interquartile range of r(t). Further, k is a constant thresholdwhere Q3
value as before that is supposed to trim 5% at edges of r(t) ( j=C), where tranges from January 1985 to the latest December. k takes a value of 2.04 as ofDecember 2015. Then, the outlier-free rate of change, ψ(t) in eq. (2), is obtained

as follows:
+r ψ(t)=ϕ


(15)

─ 8 ─

Coincident Index and Reference Cycle

The rest of the procedure follows as already explained.

3

Bry-Boshcan Procedure
The BB procedure is summarized in Table 1. Watson (1994) found somediscrepancies between the original description by Bry and Boschan (1971) and theFortran program they coded. The description here is modified to be consistent with
the Fortran codes. The procedure presumes to use seasonally adjusted series. In
Step I, outliers, if any, are replaced by the values of the Spencer curve. Here, theoutliers are defined as values whose ratios to (or differences in absolute values from,depending on data) the 15-point Spencer curve are larger than 3. 5 standard

deviations, a threshold value chosen arbitrarily. This Spencer curve is computed as
the 15-month symmetric moving average with particular weights (see Kendall andStuart, 1966, p.458).

Step II starts with the 12-month moving average (MA12, hereafter) of theoutlier-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 a tentative trough. These peaks and troughs arechecked for alternation. For contiguous peaks or troughs, the highest value ischosen for a peak, and the lowest for a trough. If the values are same, we set an
earlier date for a peak, and a later date for a trough, respectively. Note that theMA12 filter is not symmetric: 6 lags and 5 leads. At the ends of the sample, it is an
one-sided filter. Therefore, phase shifts are introduced, which might cause
misinterpretation of timing of economic events.
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 heuristically closer to those of─ 9 ─

成城・経済研究第 224 号（2019 年 3 月）

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 peak, and a later date for a trough.
After alternation check as in Step II, the duration of a peak to peak or a trough totrough (a full cycle) is enforced to be at least 15 months. If the duration is too short,the lower of two peaks or the higher of two troughs are eliminated. If the values are
same, we set an earlier date for a peak, and a later date for a trough, 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 the irregular component. Next, we take theratio 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 thedifference of each component over various time spans. The MCD is the minimum
number of months that gives the ratio less than 1. That is, the MCD is the shortest
months that it takes for the change in the trend-cycle component to dominate thatin the irregular component. The BB procedure confines the MCD between 3 and 6
months. Then, a short-term moving average is computed over the span of MCD,
and used to ensure peaks and troughs within ±6 months as in Step III. Alternationis 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±MCD, whichever is longer (denoted by ‘V. 1’). The second test (‘V. 2’) isalternation check as in Step II. Third (‘V.3’), any turn within less than 6 months
from the ends is removed. In the fourth test (‘V.4’), if the first or the last peak (ortrough) takes a value smaller (or greater) than any value between it and the end of

─ 10 ─

Coincident Index and Reference Cycle

the original series, it is removed. 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 anontrivial 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 15month length, as in Step III. The final test (‘V.6’) is to check whether a phase (peakto trough or trough to peak) duration is at least 5 months. If it is less 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 theprocedure with several steps skipped to see their effects. We also replace the 12month-moving-averaged series with the series smoothed by filtering methods toexamine the importance of smoothness.

4

Seasonal Adjustment via Bandpass filters
To examine effects of seasonal adjustment on determination of the reference

cycle, we use three bandpass filters: Christiano-Fitzgerald filter, Hamming filter,and Butterworth filters. Since the bandpass filters are supposed to extract certain
cycles of a signal, it can also extract cycles longer than seasonal cycles. Here, thedifference between the bandpass filtering and the conventional seasonal
adjustment procedures like X12-ARIMA is whether the cycles shorter than theseasonal cycles are removed or left. The conventional method attempts to remove
the seasonal cycles only, while the bandpass filtering removes all the cyclicalcomponents shorter than and equal to the seasonal cycles. Because our study
concerns business cycles which are supposed to be longer than the seasonal cycles,
we have no good reason to leave the shorter cycles in the series. Further, removalof the shorter cycles could give rise to denoising effects so that arbitrary outlier─ 11 ─

成城・経済研究第 224 号（2019 年 3 月）

removal is less likely to play a great role in dating business cycles.
Before reviewing filtering methods, we first note several criteria to assess
relative performance among those methods in terms of economic analyses. One
criterion is whether a method can extract cyclical components to replicate official
reference dates of the business cycles. Here, the cyclical components obtained byfiltering are considered to be the growth cycle that is supposed to have a close
relation to the business cycle. Canova (1994) examined performance of 11
different detrending methods to replicate NBER dating, assuming that thedetrending removes a secular component. 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. Recently, Otsu (2013) conducted a comparative analysis among band passfilters 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 thatthe Butterworth filters give the business-cycle dates closest to the official reference
dates.
Another criterion is phase shift. That is to say, detrending or transformation
should cause no phase shifts so that it would not change time alignment of events.
In general, use of one-sided filters or statistical models with lagged variables alone
would cause phase shifts, which may lead to misinterpretation of economic events.
Free from phase shifts are two-sided and symmetrical filters such as the BaxterKing (BK) filters (Baxter and King, 1999), the Hamming-windowed filter, and twosided Butterworth filters. Since a large phase shift tends to lead to a large deviationof estimated business-cycle dates from the official ones, this criterion is closely─ 12 ─

Coincident Index and Reference Cycle

related to the first criterion.
The third criterion is stability of the estimated components, so that they would
not change when more observations become available. Then, filtering procedures
had better not be subject to the whole sample. Since most of the procedures
involve estimation of coefficients, time-varying weights, or the Fourier transform,
their resulting components would be susceptible to data updating. Therefore, it is amatter of degree. Otsu (2011b) examined stability of two types of frequencydomain filtering methods, the Hamming-windowed filter and the Butterworth
filters, and one time-varying filtering method in time domain, the CF filtering. It
found that the larger the sample size, the more stable the estimated components
based on the frequency filtering, and that the sample size of 100 for quarterly data
would be good enough to obtain stable estimates in practice. It also showed thatthe Butterworth filters give the most stable estimates among others. Thus, theymight be useful in practice.
The fourth criterion is how much a weight of each cyclical component alters
by detrending or transformation, which is called exacerbation in Baxter and King(1999). When we use finite time-domain filters to approximate the ideal filter,certain components tend to be magnified or reduced as a result of filtering. To
inspect this point, it is useful to look at the frequency response function of thetime-domain filter. Then, it would show oscillations over the frequencies of thepass band and the stop band, indicating magnification and reduction of certain
components. As the filter length gets longer, the oscillations become more rapid
but do not diminish in amplitude. They converge to the band edges or the
discontinuity points of the ideal filter, which is called Gibbs phenomenon. Thisphenomenon is attributed to approximation of infinite sum by truncation. Thisimplies that cutting out a part of the Fourier-transformed series discontinuously as
in Canova (1998, p. 483) would create the same artificial oscillatory behavior in the─ 13 ─

成城・経済研究第 224 号（2019 年 3 月）

estimated components. In light of this criterion, the Butterworth filters and theHamming-windowed filter have a desirable property because they have flat
frequency response functions over the ranges of the pass band and the stop band.
The final criterion is the degree of leakage and compression as discussed in
Baxter and King (1999). That is, detrending or filtering might admit substantial
components from the range of frequencies that are supposed to suppress (leakage),and lose substantial components over the range to be retained (compression). Since
these effects depend on the width of transition bands between the pass and the stopbands, it is better to have narrow transition bands. Otsu (2009, 2010) show that the
Butterworth filters are least afflicted with leakage and compression effects among
others. In the related study, Otsu (2007) examined discrepancies between the ideal
filter and several approximate filters, and found that the Butterworth filters give a
better approximation than other bandpass filters. This also implies that the
Butterworth filters could give rise to the least leakage, compression, andexacerbation effects.
Now we review properties of three methods used later: Christiano-Fitzgerald
filter, Hamming filter, and Butterworth filters. To begin with, we consider thefollowing orthogonal decomposition of the observed series x:x=y+x (16)where y is a signal whose frequencies belong to the interval {[−b, −a]∪
[a, b]}∈[−π, π], while x  has the complementary frequencies. Suppose that
we wish to extract the signal y. The Wiener-Kolmogorov theory of signal
extraction, as expounded by Whittle (1983, Chapter 3 and 6), indicates y can be
written as:y=B(L)x

(17)

─ 14 ─

Coincident Index and Reference Cycle

B(L)=  B L ,

L x≡x

(18)



In polar form, we haveB(e )=

1, for ω∈[−b, −a]∪[a, b](19)0, otherwise

where 0≦a≦b≦π. Theoretically, we need an infinite number of observations,x ’s, to compute y . In practice, the filtering methods approximate y by y , a









filtered series with a finite filter. To estimate y by y , the Christiano-Fitzgerald
filtering is performed in the time domain with truncation at both ends of thesample, while other filtering methods in the frequency under the circularity
assumption. In application to seasonal adjustment, when we set a to zero and b to
the seasonal frequencies concerned, we have power spectra identical to those of
the seasonally adjusted series published officially (see Otsu, 2009, p. 212 and p. 219).
In the later analysis, we set b toπ
. Now, we briefly review three filtering
6

methods mentioned above.

4.1

Christiano-Fitzgerald Filter
Christiano and Fitzgerald (2003) seeked an optimal linear approximation withfinite sample observations. They solved a minimization problem based on themean square error (MSE) criterion in the frequency domain: minimization of aweighted sum of differences between the ideal bandpass-filter’s weights and theirapproximates, using a spectral density of observations as a weight. They derived
optimal filter weights, assuming a difference-stationary process of observed datawith 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.

─ 15 ─

成城・経済研究第 224 号（2019 年 3 月）

They compared variance ratios and correlations between the components extracted
by the CF filters and the theoretical components based on the data generatingprocess of observations. To evaluate the second moments of the theoreticalcomponents, they used the Riemann sum in the frequency domain. They found thatthe 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 the filtered-out
components and the theoretical ones at different leads and lags symmetricallydiminishes as the leads and lags go far away, which might indicate that the degreeof asymmetry was not great. Finally, a CF filter derived under the Random-Walkdata generating process, the so-called Random Walk filter, gives a good
approximation to the optimal filtering that explicitly used the estimatedcoefficients of an optimal moving average process determined empirically.
Therefore, they claimed that we could use the Random Walk filter without
inspecting the data generating process even if the random walk assumption was
false. In the paper, we simply denote it by CF henceforth.
Details of the CF filter are given in Christiano and Fitzgerald (2003) and its
properties are discussed in Iacobucci and Noullez (2005). As argued in Otsu (2015),
the cyclical components extracted by CF might be distorted in magnitude andtiming. The gain function, defined as the modulus of the frequency response
function, shows large ripples over the target ranges, indicating a large distortion inestimating the cyclical components. It also shows leakage effects over higher
frequencies of more than 8 periods per cycle. Further, phase shifts are indicated byvalues of its phase function, defined as arctangent of the ratio of the real-valued
coefficient of the imaginary part of the frequency response function to the real part─ 16 ─

Coincident Index and Reference Cycle

value.
In the paper, we first compute the components of 12 to 2-month cycles, that

2π

is, the frequency range

2π

 12 , 2 , and subtract them from the original series toobtain y .

4.2

Hamming-Windowed Filter
Iacobucci and Noullez (2005) claimed that the Hamming-windowed filter bea good candidate for extracting frequency-defined components. The proposed
filter has a flatter response over the passband than other filters in the literature,such as the HP filter (Hodrick and Prescott, 1997), the BK filter (Baxter and King,
1999), and the CF filter. This means that it has no exacerbation and eliminates

high-frequency components better than the other three filters.
The Hamming-windowed filtering is implemented in the frequency domain.
The procedure is described as follows. First, we subtract, if necessary, the leastsquare regression line to detrend the observation series to make it suitable for the
Fourier transform. Second, we implement the Fourier transform of the detrended
series, Third, we convolve the ideal response with a spectral window to find thewindowed filter response in the frequency domain. The window is the so-called
Tukey-Hamming window (Priestly, 1981, pp.433-442). In the paper, we compute thecomponents with cycles longer than the 13-month cycle, that is, the frequency

 13 , to obtain y , the seasonally-adjusted counterparts.

range 0,

4.3

2π



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

─ 17 ─

成城・経済研究第 224 号（2019 年 3 月）

Butterworth filters are widely used in electrical engineering, and well documented
in standard text books, such as Oppenheim and Schafer (1999) and Proakis andManolakis (2007). The two-sided version guarantees phase neutrality or no phaseshift. It has finite coefficients, and its frequency response is maximally flat overthe pass band: the first (2n−1) derivatives of the frequency response are zero atzero frequency for the nth-order filter. The filter could stationarize an integrated
process of order up to 2n. The order of the filter can be determined so that the edge
frequencies of the pass band and/or the stop band are aligned to some designatedfrequencies. Further, Gomez (2001) pointed out that the two-sided Butterworth
filters could be interpreted as a class of statistical models called UCARIMA (theunobserved components autoregressive-integrated moving average) in Harvey (1989, p.
74). Since the two-sided Butterworth filters are not so often used in the literature,

we present relevant equations to look at them a little bit more closely.
The lowpass filter is expressed as


BFT=



(1+L) (1+L )(1+L) (1+L ) +λ(1−L) (1−L )(20)

where L x=x, and L x=x. Similarly, the highpass filter is expressedas

BFT =λ(1−L) (1−L )(1+L) (1+L ) +λ(1−L) (1−L )(21)

Note BFT+BFT =1, which is the complementary condition discussed by

Pollock (2000, p. 321). Here, λ is the so-called smoothing parameter. We observethat the Butterworth highpass filter in eq. (21) can handle nonstationary
components integrated of order 2n or less. Let ω the cutoff point at which the gainis equal to 0.5. It is shown
λ={tan(ω/2)}



(22)

─ 18 ─

Coincident Index and Reference Cycle

To see this, we replace the L by e  in eq. (20) to obtain the frequency response
function in polar form asψ(e ; λ, n)==1
1+λ(i(1−e )/(1+e ))

(23)

1

1+λ{tan(ω/2)}(24)

Here, it is easy to see that eq. (22) holds when ψ(e )=0.5. We also observe ineq. (24) that the first (2n−1) derivatives of ψ(e ) are zero at ω=0; thus, thisfilter is maximally flat. Note that the gain is the modulus of the frequency response
function, and indicates to what degree the filter passes the amplitude of acomponent at each frequency. The Butterworth filters considered here aresymmetric and their frequency response functions are non-negative. Therefore, thegain is equivalent to the frequency response. Then, we can use eq. (24) to specifyω so that the gain at the edge of the pass band is close to one and that of the stopband close to zero. Let the pass band [0, ω ], and the stop band [ω, π], whereω  is smaller than ω. As in Gomez (2001, p. 372), we consider the following
conditions for some small positive values of δ and δ,1−δ<ψ(e ; λ, n)≦1

0≦ψ(eω∈[0, ω ](25)

; λ, n)<δ for ω∈[ω, π]

(26)

for



That is, we can control leakage and compression effects with precision specified
by the values of δ and δ. These conditions can be written as follows:
1+

1+



tan(ω /2)tan(ω/2)tan(ω/2)tan(ω/2) = 1−δ1

(27)





 = δ1(28)

Then, we can solve for the cutoff frequency (ω) and the filter’s order (n), given─ 19 ─

成城・経済研究第 224 号（2019 年 3 月）ω , ω, δ and δ. The closer to zeros both δ and δ, the smaller the leakage andthe compression effects. If n turns out not an integer, the nearest integer isselected.
The Butterworth filters could be based on the sine function. Instead of eq.
(20) and eq. (21), the lowpass and the highpass filters can be written as follows,
respectively.
BFS=

1


1+λ(1−L) (1−L )

BFS =λ(1−L) (1−L )
1+λ(1−L) (1−L )



(29)



(30)

where
λ={2 sin(ω/2)}



(31)

These are the so-called sine-based Butterworth filters. When n is equal to two, eq.
(30) 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 ordertwo (n=2) can be viewed as the HP filter. As in the case of the tangent-based
one, the cutoff point, ω, can be determined with the following conditions:
1+

1+



sin(ω /2)
sin(ω/2)sin(ω/2)
sin(ω/2) = 1−δ1

(32)





 = δ1(33)



We observe that the Butterworth highpass filter in eq. (21) or eq. (30) can handle
nonstationary components integrated of order 2n or less. Thus, the HP filter canstationarize the time series with unit root components up to the fourth order.
Gomez (2001, p. 367) claimed that the BFT would give better approximations toideal low-pass filters than the BFS. A simulation study in Otsu (2007) confirmed it.

─ 20 ─

Coincident Index and Reference Cycle

In the following analysis, we use BFT to extract passband components [0, ω ],

setting ω =

2π
2π
, with the stop band [ω, π] setting ω=. To implement the1312

Butterworth filtering, we need specify two parameter values, n and λ, in eq. (20)or eq. (21). We obtain these values from eqs. (22), (27) and (28) for target
frequency bands, that is, values of ω  and ω with given values of δ and δ. We
set both δ and δ to 0.01. We only use BFS (2nd order) to obtain the HP-filteredpassband components, setting ω=

2π
in eq. (31).
13

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 found that when the cycle period islonger than seven, the matrix inversion is so inaccurate that it is impossible tocontrol leakage and compression effects with a certain precision specified by eq.
(27) and eq. (28), or eq. (32) and eq. (33). Further, the filters at the endpoints ofdata 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
the time-domain filtering.
Alternatively, we can implement the Butterworth filtering in the frequency
domain. In the frequency-domain filtering, cyclical components are computed via
the inverse discrete Fourier transform, using the Fourier-transformed series withthe frequency response function as their weights. In contrast to the time-domain
filtering, the frequency-domain filtering does not introduce any phase shifts, as thetheoretical background of the symmetrical filters dictates. For the frequencydomain procedures to work well, it is required that a linear trend be removed andcircularity be preserved in the time series, which we discuss next.

─ 21 ─

成城・経済研究4.4

第 224 号（2019 年 3 月）

Detrending Method
To obtain better estimates of cyclical components, it is desirable to remove 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), however, this method can produce
spurious periodicity when the true trend is stochastic. Another widely-useddetrending method is the first differencing, which reweighs toward the higher
frequencies and can distort the original periodicity, as pointed out by Baxter andKing (1999), Chan, Hayya, and Ord (1977), and Pedersen (2001).
Otsu (2011a) 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 the linear-regression-based detrending.
Therefore, this detrending method would create less distortion. Let the raw seriesz, t=1, ⋯, T. Then, we compute the drift-adjusted series, x, as follows:x=z−(t+s) μ(34)

where s is any integer and z−z
μ
=
T−1

(35)

Note that the first and the last points are the same values:x=x=

Tz−z+s(z−z)
T−1

(36)

In Christiano and Fitzgerald (2003, p. 439), s is set to −1. Although Otsu (2011a)
suggested some elaboration on the choice of s, it does not affect the 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. (34) would makethe data suitable for filtering in the frequency domain. Since the discrete Fourier─ 22 ─

Coincident Index and Reference Cycle

transform assumes circularity of data, the discrepancy in values at both ends of thetime series could seriously distort the frequency-domain filtering. The eq. (36)implies that this adjustment procedure avoids such a distortionary effect.
A final remark here is that the BB procedure is implemented with trendincluded series. In the business cycle literature, it is important to distinguish aclassical cycle and a growth one, as pointed out by Pagan (1997). The classicalcycle consists of peaks and troughs in the levels of aggregate economic activities,
often represented by the gross national product (GDP). The classical cycle isstudied by Burns and Mitchell (1946), one of the influential seminal works, whichfound that business 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 the cycles show
different dates of the peaks and the troughs. When a series has a cyclicalcomponent around a deterministic upward trend, typical as in economic data,
detrending would make the peaks earlier, while delaying the troughs (see Bry andBoschan, 1971, p. 11). For this reason, the dating based on the growth cycle

generically tends to deviate from that on the classical cycle. Then, Canova (1994,
1999) judged that the estimated dates matched the official dates as long as

deviations were within two or three quarters. The results in Otsu (2013) also showthat the estimated dates of peaks based on the detrended series tend to mark earlier
and those of troughs later than the official dates. Since we only suppress thecyclical components shorter than the seasonal cycle in the paper, we do not havesuch a deviation due to detrending.

4.5

Boundary Treatment
In addition to the detrending method, we make use of another device to─ 23 ─

成城・経済研究第 224 号（2019 年 3 月）

reduce variations of the estimates at ends of the series: extension with a boundary
treatment. As argued by Percival and Walden (2000, p. 140), it might be possible toreduce the estimates’ variations at endpoints if we make use of the so-calledreflection boundary treatment to extend the series to be filtered. We modify thereflection boundary treatment so that the series is extended antisymmetricallyinstead of symmetrically as in the conventional reflecting rule. Let the extended
series f ,
f =



x

if 1≦ j≦T

2x−x if −T+3≦ j≦0(37)

That is, the T−2 values, folded antisymmetrically about the initial data point, areappended to the beginning of the series. We call this extension rule theantisymmetric reflection, distinguished from the conventional reflection.
It is possible to append them to the end of the series. The reason to append theextension at the initial point is that most filters give accurate and stable estimatesover the middle range of the series. When we put the initial point in the middle partof 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 the farthest past in the time series, it does not make sense that theestimate 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 series,f , and extract the last T values to obtain the targeted components, that is,seasonal adjustment factors that are subtracted from the original series to obtainthe seasonally-adjusted series.

─ 24 ─

Coincident Index and Reference Cycle

5

Empirical Analysis

5.1

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

and Social Research Institute (ESRI), affiliated with the Cabinet Office,
Government of Japan. ESRI organizes the Investigation Committee for Business
Cycle Indicators to inspect historical diffusion indexes calculated from selected
series of coincident indexes and other relevant information. To make a historical
diffusion index, the peaks and troughs of each individual time series are dated by
the Bry-Boschan method. Thus, the reference dates correspond to those of peaks
and troughs of the classical cycles, that is, the Burns-and-Mitchell-type cycle
based on the level of aggregate economic activity. Typically, the final
determination of the dates is made about two to three years later.
Table 3 shows the reference dates of peaks and troughs identified by ESRI. It
also contains periods of expansion, contraction, and duration of a complete cycle(trough to trough). There are 15 peak-to-trough phases identified after World War II.

The average period is about 36 months for expansion, 16 for contraction, and 52
for the complete cycle. We compare the reference dates with those of the growth
cycles obtained by filtering methods.
ESRI routinely examines and revises composition of the indicators. Although
the latest revision is made in February 2017, our data are based on the 9th revisionin November 2004, adopted until September 2011, that selected 11 economic
series for the coincident indicators. We use 11 composite coincident indicators ofJapan in monthly basis, retrieved from Nikkei NEEDS CD-ROM (2008). Series
names, as well as mnemonics, are listed in Table 2. The sample period ranges fromJanuary 1980 to January 2008, 337 observations for each series. We choose thisdata set for two reasons. First, it gives a fairly long time series in consistent─ 25 ─

成城・経済研究第 224 号（2019 年 3 月）

composition of the indicators.
Secondly, it is revealed that officials at Ministry of Health, Labor and Welfare
have incorrectly conducted fundamental statistical survey on labor-related conditions since 2004. Then, one of the 11 series, ‘Index of Non-Scheduled Worked
Hours,’ may need correction. This issue is being under investigation as of 27th
January, 2019. Our data may include possibly incorrect data for four years after
2004. It is desirable for the following analyses not only to include business cycles
as many as possible but to avoid contaminated data as much as possible. Thisconsideration leads us to focus on the sample up to January 2008 based on the 9threvision composition.
We note that among the series, ‘Operating Profits’ is available only inquarterly base (end of periods) with seasonal adjustment (X12-ARIMA). We linearly
extrapolate the quarterly data points to make monthly series. All the index-type
data have the base year in 2000.

5.2

Aggregation of Coincident Indicators
To examine to what extent the composite coincident index (CCI) deviatesfrom the reference cycle, we compare the official reference dates with the dates ofpeaks and troughs implied by the CCI. We use the Bry-Boschan algorithm
procedure (see section 3) to identify dates of peaks and troughs of the CCI, because
ESRI uses it to calculate the diffusion index that gives fundamental information todetermine the reference dates. Here, we use the coincident indicators seasonallyadjusted by the official agents, so that we can exclude influence of seasonal
adjustment on dating results.
In the first (‘Official Ref. Dates’) and the second (‘Official CCI’) columns ofTable 4, we find that dates of peaks differ between the official reference cycle andthe official CCI, except May 1997. The official CCI identifies November 1981 as a

─ 26 ─

Coincident Index and Reference Cycle

peak, while the official peak date indicates February 1980. Since the compositionof coincident indicators is routinely revised, the set of indicators used in 1980 isdifferent from that of the paper. This would be one reason for the discrepancy. Yet,
there are other reasons as well.
As already mentioned, ESRI uses the historical diffusion indices (coincident
indicators) to determine the reference cycle. However, only publicly available are

the materials used at the committee after 2002 onward. Thus, we alternatively use
the current diffusion index (Nikkei NEEDS CD-ROM, 2008) to examine thedeviation between the reference cycle and the composite indices. We find that theindex took 92 on average during February 1979 to February 1981. This might giverise to the official peak date, February 1980. During May 1980 to May 1981, thecurrent diffusion index took less than 50 points. It reached 54.5 in June 1981,
marked 100 in August, then down to 54. 5 in December and to less than 50
afterwards. The official CCI seems pick up these small bumps. There would be
two reason for this discrepancy. First, the Bry-Boshcan algorithm uses the moving
averages and the Spencer smoothing: the former become asymmetric at endpoints
and the latter uses averages of the initial or the last four points as observations at
endpoints. Thus, it may introduce distortion in dating computation. Secondly, iteliminates peaks within 6 months at endpoints in Step IV (see Table 1). Therefore, it
never identifies February 1980 as a peak since our data set begins from January
1980. Then, we do not pay much attention to the deviation from the reference dates
in early 1980s in the following analysis.
To check our aggregation program, we attempt to replicate the official CCI by
aggregating the coincident indicators (seasonally adjusted series) published by theofficial agents, according to the procedure described in section 2. Note that the firstquartile in eq. (5), Q1 , is set to the 84th value of 336 rates of changes, r(t) ineq. (1), in ascending order, and the third quartile, Q3 , to the 253d value. The
─ 27 ─

成城・経済研究第 224 号（2019 年 3 月）

results are shown in Table 4 and Table 5. The dates in the third column (‘Aggr.
Indicators’) in Table 4 match well with the dates given by the official CCI. Only

difference is observed in 1981. In the fourth column (‘Aggr. Ind. (No Trim)’), we do
not implement the outlier removal (the threshold value: 2.02) to see its effect. Then,the peak in 1997 becomes two months earlier, March instead of May. Although wefind similar effects later in the paper, it is fair to say that the outlier removal has
only a limited role in dating the reference cycle.
As for the troughs, the results are shown in Table 5. the official CCI deviatesfrom the reference dates in 1993 by two months and in 1999 by one month. The
aggregation of the coincident indicators gives dating results similar to the official
CCI with or without outlier removal. The deviation from the reference dates is 6
months for the peaks from 1985 to 2000 and 3 months for the troughs from 1983 to
2002. A large deviation in the troughs is observed for the aggregated index, but
this is mainly due to the deviation in the early 1980s: February 1983 versus
October 1982. If we exclude it, the deviation reduces to 3 months. Therefore, it canbe said that the computed index yields the dating results equivalent to those thatthe official CCI does.
We now examine whether the dates of the turning points depend on thelocation and the scaling measures in aggregation of coincident indicators. ESRI
uses the 5-year averages defined in eq. (3) and the interquartile ranges tostandardize the rate of change of each indicators in eq. (5). These quantities seem
preferred because they are supposed to be insensitive to outliers. The thirdcolumns in Table 4 and in Table 5 give the results of the corresponding case. When
we use the sample mean instead of the 5-year trend, we have the results in thesecond (‘Interquartile’ of ‘Sample Mean’) and the third (‘Standard Deviation’ of ‘Sample
Mean’) columns in Table 6 and Table 7. These results are very similar to those theofficial CCI gives in Table 4 and Table 5. It is interesting to see that use of the─ 28 ─

Coincident Index and Reference Cycle

sample mean makes the dating results closer to those of the official CCI, and thatwhether we use the interquartile ranges or the standard deviation does not matter.
In addition, the fourth column (‘Standard Deviation’ of ‘CM: 5-year Trend’) in Table 7
indicates that use of the 5-year trend introduces the deviation from the date of
February 1983. Here, we have no evidence to encourage the use of the interquartile
ranges and the 5-year trend.

5.3

Smoothing Methods and Bry-Boshcan Procedure
ESRI uses seasonally adjusted series. The conventional seasonal adjustment

attempts to remove seasonal frequencies only, leaving all the higher frequencies inthe series. In terms of economic analysis, there is no sound reason that economic
data should include the frequencies higher than seasonal ones. If an economic
theory neither presumes seasonality or difference of seasonality among economic
variables nor designates specific forms of econometric models with seasonal
effects, it is most likely not to intend to explain the fluctuation shorter thanseasonality. Further, if the main concern is about the business cycle, we may
remove such a shorter cyclical movement in data.
In this section, we use filtering methods discussed in section 4, instead of the
X12-ARIMA method, to remove all the frequencies higher than seasonality. Eachof the 11 coincident indicators is filtered and aggregated to make a composite
coincident index. Otsu (2011b) examined the performance of the filtering methods
used in the paper to extract the seasonal components, and found that they are very
useful and the corresponding ‘seasonally-adjusted’ series are smoother than theseries with the X12-ARIMA seasonal adjustment.
In Table 8 and Table 9, we use the tangent-based Butterworth filter. The
results of ‘Case 1’ are obtained with the Bry-Boschan (BB) procedure, skipping
the steps of I, III and IV in Table 1. We also note that these results are exactly same─ 29 ─

成城・経済研究第 224 号（2019 年 3 月）

as those of the full BB procedure. This implies that the outlier removal and the
Spencer smoothing have no effect on the results. The columns of ‘Case 2’ show
the results when we further remove Step V.1-V.2 in the BB procedure. Thus, thedata processing in the BB procedure is limited to the asymmetric 12-monthmoving average. As for the peaks, the dates fit better with the official reference
dates, whether we use the sample mean or the 5-year trend, while we observe notmuch improvement for dating the troughs. The deviation is 3 months for the peaks,
and 7 months for the troughs. This contrasts with the results of the official
seasonally-adjusted data in Table 4 and Table 5, in which the dates of the troughsshow smaller deviations than those of the peaks.
In experiments, we confirm that the choice of the scaling measures, theinterquartile ranges or the standard deviation, has no influence on the results.
Again, neither the central measures nor the scaling measures does not play animportant role. In sum, the simple mean and the standard deviation is goodenough, and the 12-month moving average with the simplified BB procedure gives
dating results that are close enough to the official reference dates, when we use the
Butterworth filter for the seasonal adjustment.
The Hamming-windowed filter produces similar results as the Butterworth
filter. The second column of ‘Case 1’ in Table 10 only shows two-month deviationin 1991. Interestingly, when we use the interquartile range as a scaling measure,
the result gets closer to those of the second column in Table 8. This implies that the
Butterworth filter works as the interquartile range does. Although the equivalent
result of ‘Case 1’ is also obtained with the full BB procedure, skipping Step V.1
and V.2 in Table 1 gives a different result to the third column of ‘Case 2’, but it isquite similar to the result in the corresponding column in Table 8. Only difference
between the third columns in both tables is in that choice of the scaling measures
has some effects on the dating result in case of the Hamming-windowed filter. In─ 30 ─

Coincident Index and Reference Cycle

the fourth and the fifth column, we find that use of the 5-year trend does not showmuch improvement in dating. The results of the troughs in Table 11 are also
equivalent to those in Table 9, specifically in the third column, the ‘Case 2’ of thesample mean. Again, we cannot find clear evidence for the 5-year trend, theinterquartile ranges, and the Spencer smoothing.
Turning to the Christiano-Fitzgerald filter, we find in the columns of ‘Case 1’
in Table 12 and Table 13 that the dating results depend on the choice of the scalingmeasures. Using the interquartile ranges makes the dating results closer to those of
the Butterworth filter in Table 8 and Table 9. In contrast, the results of ‘Case 2’ areless likely to be susceptible to the scaling choice, and comparable to those of the
Butterworth filter. The deviation tends to be slightly larger, but it is fair to say that
it is same in magnitude. Finally, the Hodrick-Prescott filter gives rise to the resultsthat depend on the choice of the scaling measures and that get closer to those of the
Butterworth when the series are standardized with the interquartile ranges. The
deviation tends to be larger compared with the results of other filtering methods.
To investigate the effects of the outlier removal with a threshold value of 2.
02, we have repeated the same calculation of Table 8 through Table 15 with thecomposite indices computed with the outlier removing process. The maindifference is found in the latter half of 1990s. When we use the sample mean andthe standard deviation as the location and the scaling measures, respectively, we
obtain a peak in May 1997 with all the filtering methods but the HP filter, and atrough in February 1999 with all the filters. The former is exactly same as that of
the reference cycle, and the latter is different from the reference date by only one
month. Thus, the outlier removal plays a certain role in the late 1990s.
In our final experiment, we use the composite coincident indices compliedwith the filtered indicators, instead of the 12-month moving average (MA12), in
Step II of Table 1, so that the BB procedure becomes free from phase shifting.

─ 31 ─

成城・経済研究第 224 号（2019 年 3 月）

Table 16 and Table 17 show the results. First, we find new dates of peaks andtroughs in 1995. The peaks are dated in January with the Butterworth, theHamming-windowed, and the CF filters, and in March with the HP filter. Since we
had the Great Hanshin-Awaji Earthquake in 17th Janurary, 1995, it is reasonable toidentify January 1995 as a peak. The troughs are identified in July with the
Butterworth and the Hamming-windowed filters, June with the CF filter, andAugust with the HP filter. It is certainly arguable that these dates should be
suppressed. Secondly, we find that these results are identical to those of ‘Case 1’ inTable 8 through Table 15, except 1995. Thus, it can be interpreted that phase-shifteffects by the MA12 are offset by the Step V. 1 and V. 2 in Table 1. Third,
implementing the outlier removal with the threshold value of 2.02 makes a slightchange in the results in 1986 and 1999. It is observed that the dates in these years
get closer to the reference dates.
In short, when we use either the Butterworth or the Hamming-windowed
filters, the dating results do not depend on the choice of the central and the scalingmeasures. On the other hand, the Christiano-Fitzgerald filter and the HodrickPrescott filter produce the dating results that are susceptible to the choice, inparticular, of the scaling measures. Use of the interquartile ranges makes the dating
results closer to those of the Butterworth filter. In addition, the Spencer smoothing
and the 12-month moving average in the BB procedure does not play much role
when the filtering methods are used. Further, it does no harm to skip the final steps
in the BB procedure to ensure peaks and troughs within the short-term periods(Step V.1 and V.2 in Table 1), specifically with the Butterworth filter. Finally, we

note that we have equivalent results over the different filtering methods when we
use the sample mean as a central measure and skip Step I, III through V.2.

─ 32 ─

Coincident Index and Reference Cycle

6

Discussion
This paper attempts to find an alternative method to make the coincident

composite index that is simpler than the existing method and is consistent with theofficial reference cycle as much as possible. We examine how accurately thecoincident composite index trace the reference cycle, using the business cycle dataof Japan. Further, we investigate aggregation methods to construct the composite
index from the individual indicators in terms of normalization of each indicator,smoothing methods, and dating algorithm.
The main findings are as follows. First, although ESRI uses an outlier
trimming procedure and interquartile ranges for a scaling measure, these play only
a minor role in dating peaks and troughs, possibly, in the late 1990s. Secondly,
when filtering methods is used to seasonally adjust series, the Bry-Boschanalgorithm can be simplified to a great extent and does not need the Spencer
smoothing, the 12-month moving averages, and a trimming procedure. Thirdly, theButterworth filter has the same effects on dating results as the interquartile ranges.
Therefore, it reduces variation of the series good enough.
These findings suggest that it is possible to simplify the compilation processof the composite indices. Since the composite indices await various kinds of
economic analyses for vairous purpose, simplicity and clarity in the compilation isdesirable. Although ESRI uses X12-ARIMA for seasonal adjustment, it involves
arbitrariness in selecting ARIMA models, setting parameters, judgements on
statistical significance of estimates. Moreover, different economic variables
require different X12-ARIMA models. Then, X12-ARIMA could distort therelation among economic variables, as pointed out by Sims (1974) and Wallis(1974). In contrast, the filtering methods only require frequencies or periods per

cycle to be preserved or suppressed, which might be given by economic analyses,

─ 33 ─

成城・経済研究第 224 号（2019 年 3 月）

and make it possible to filter any variable with exactly same parameter values.
When we use the Butterworth or the Hamming-windowed filter to seasonallyadjust series, we do not have to use the 5-year trend as a location measure to
standardizing the economic indicators. It is not clear why we should use the 5-year
trend. The judgement is rather subjective. Further, the filtering also reduces thenecessity of the interquartile ranges as a scaling measure, indicating that thestandard deviation works well. The threshold value is set as arbitrarily as theconventional significance levels in statistics, such as 10%, 5%, and 1%. We alsofind that the outlier removal seems not play an important role. There is no firm
grounds that we should remove 5% at edges of the standardized series. and that In
many cases, the so-called outliers in economic variables give some clues tounderstand important economic phenomena or effects of exogenous variables.
Then, careless outlier removal would be harmful. These findings indicate that we
could use a simple frequency-domain filtering and a conventional normalization to
construct a composite coincident index.
Concerning the Bry-Boschan algorithm, we can get rid of the Spencer and themoving-average smoothing, so that we may avoid arbitrariness accompanied by
the former in determination of polynomial orders and phases shifts introduced by
the latter. All what we need is to determine the minimum duration of phases andcycles and the enforcement rules of alternation of peaks and troughs. We do notneed such repeated processes as in the original BB procedure to determine thedates of peaks and troughs.
In the paper, we have found and discussed a possibility of simplifying theaggregation and the dating algorithm to identify peaks and troughs of the businesscycle. However, we need to investigate samples in other periods and othercountries to come to a final conclusion. Further, we may need to investigate otherdating rules suggested in the literature (see Webb, 1991; Harding and Pagan, 2016).

─ 34 ─

Coincident Index and Reference Cycle

These are left for the future research.

Acknowledgements

A part of this work is financially supported by Tokubetsu Kenkyu

Josei (Seijo University) under the title of「景気動向指数の作成法と頑健手法に関する研究」.

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─ 38 ─

Coincident Index and Reference Cycle

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-month leads 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 troughwith 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 as in Step II within 6 months, using XOSP. Modify if necessary.
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 a peak and a later one for atrough.

IV

Dating with short-term moving average:
1. Spencer filtering:
Use the Spencer curve of the original series (X) as the trend-cycle component, and compute theirregular component by the difference between X and the Spencer curve.
Find the minimum number of months (MCD, Months of Cyclical Dominance) over which theaverage rate of change in the trend-cycle component exceeds the average change in the irregularcomponent. 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 the span 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 ifnecessary.
4. Enforcement of alternation: Check alternation as in Step II.

V

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 theoriginal 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) than values closer toend.
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 5 months.

─ 39 ─

成城・経済研究第 224 号（2019 年 3 月）

Table 2 Coincident Indicators: Japan (9th Revision: Nov. 2004 - Sept. 2011)
Series Name
1.
2.
3.
4.
5.
6.

Mnemonic (NEEDS)*

Index of Industrial Production (Mining and Manufacturing)Index of Producer’s Shipments(Producer Goods for Mining and Manufacturing)
Large Industrial Power Consumption, mil. kwh.
Index of Capacity Utilization Ratio (Manufacturing)Index of Non-Scheduled Worked Hours (Manufacturing)

7.

Index of Producer’s Shipment(Investment Goods Excluding Transport Equipment)
Retail Sales Value (Change From Previous Year, %)8.

Wholesale Sales Value (Change From Previous Year, %)9. Operating Profits, thou. mil. yen (All Industries)
10. Index of Sales in Small and Medium Sized Enterprises(Manufacturing)
11. Effective Job Offer Rate (Excluding New School Graduates)

IIP00P001(@)
IIP00S255(@)
CELL9(@)
IIP00O01(@)
HWINMF00(HWINMF05@)
IIP00S204(IIP00SINV@)ZCSHVB20(ZCSHVB20V)
ZCSHVB00(ZCSHVB00V)ZBOAS@**SMSALE@ESRAO(@)* “@” indicates seasonally-adjusted series.
** Only quarterly series are available. A linear-interpolation is used to obtainmonthly series.

Table 3 Reference Dates of Business Cycles in Japan
Dates (month, year)
Peak
TroughJune,
1951
October,
1951
January,
1954
November, 1954
June,
1957
June,
1958
December, 1961
October,
1962
October,
1964
October,
1965
July,
1970
December, 1971
November, 1973
March,
1975
January,
1977
October,
1977
February, 1980
February, 1983
June,
1985
November, 1986
February, 1991
October,
1993
May,
1997
January,
1999
November, 2000
January,
2002
February, 2008
March,
2009
March,
2012
November, 2012

Number of Periods (in months)
Expansion
Contraction
Duration
─
4
─
27
10
37
31
12
43
42
10
52
24
12
36
57
17
74
23
16
39
22
9
31
28
36
64
28
17
45
51
32
83
43
20
63
22
14
36
73
13
86
36
8
44

Source: Indexes of Business Conditions, Economic and Social Research Institute,Cabinet Office, Government of Japan, July 24, 2015.

─ 40 ─

Coincident Index and Reference Cycle

Table 4 Comparison with Reference Dates: Peaks, Official S.A. Series
Official Ref. Dates
Year
Month
1980
2
1985

6

1991
2
1997
5
2000
11
2008
2
Deviation***

Official CCI
Year
Month

Aggr. Indicators*
Year
Month

1981
1985
1990

11
5
10

1981
1985
1990

12
5
10

1981
1985
1990

12
5
10

1997
2000

5
12

1997
2000

5
12

1997
2000

3
12

6 months

Aggr. Ind. (No Trim)**
Year
Month

6 months

8 months

Note: * Compiled from coincident indicators (S.A. series).
** Outlier removal (threshold value: 2.02) is not implemented.
*** Deviation from the reference dates, sum of absolute values from 1985 to 2000.

Table 5 Comparison with Reference Dates: Troughs, Official S.A. Series
Official Ref. Dates
Year
Month
1977
10

1983
1986
1993

2
11
10

1999
1
2002
1
2009
3
Deviation***

Official CCI
Year
Month

Aggr. Indicators*
Year
Month

1981

5

1981
1982

5
10

1981
1982

5
10

1983
1986
1993
1998

2
11
12
12

1986
1993
1998

11
12
12

1986
1993
1998

11
12
12

2002

1

2002

1

2002

1

3 months

7 months

Aggr. Ind. (No Trim)**
Year
Month

7 months

Note: * Compiled from coincident indicators (S.A. series).
** Outlier removal (threshold value: 2.02) is not implemented.
*** Deviation from the reference dates, sum of absolute values from 1983 to 2002.

─ 41 ─

成城・経済研究第 224 号（2019 年 3 月）

Table 6 Comparison with Reference Dates: Peaks, Alternative Measures in eq.(5)Official Ref. Dates
Year
1980

Month
2

1985

6

1991
2
1997
5
2000
11
2008
2
Deviation*

Central Measure(CM): Sample Mean
Standard Deviation
Interquartile
Year
Month
Year
Month

CM : 5-year Trend, eq.(3)Standard Deviation
Year
Month

1981
1985
1990

11
5
10

1981
1985
1990

11
5
10

1981
1985
1990

11
5
10

1997
2000

5
12

1997
2000

5
12

1997
2000

5
12

6 months

6 months

6 months

Note: 1. May (1997) is altered to March with no outlier removal (threshold value: 2.02).
2. Same results without Step I, III & IV and Spencer smoothing in BB proc. (Table 1)* Deviation from the reference dates, sum of absolute values from 1985 to 2000.

Table 7 Comparison with Reference Dates: Troughs, Alternative Measures in eq.(5)Official Ref. Dates
Year
1977

1983
1986
1993

Month
10

2
11
10

1999
1
2002
1
2009
3
Deviation*

Central Measure(CM): Sample MeanInterquartile
Standard Deviation
Year
Month
Year
Month
1981

5

1981

5

1983
1986
1993
1998

2
11
12
12

1983
1986
1993
1998

2002

1

2002

3 months

CM: 5-year Trend, eq.(3)Standard Deviation
Month
Year
1981
1982

5
10

2
11
12
12

1986
1993
1998

11
12
12

1

2002

1

3 months

7 months

Note: 1. Same results are obtained with or without outlier removal (threshold value: 2.02).
2. Same results without Step I, III & IV and Spencer smoothing in BB proc.(Table 1)* Deviation from the reference dates, sum of absolute values from 1983 to 2002.

─ 42 ─

Coincident Index and Reference Cycle

Table 8 Variants of BB Procedure: Peaks, Butterworth (tangent based) Filter
Official Ref. Dates
Year
1980

Mon.
2

1985

6

1991
2
1997
5
2000
11
2008
2
Deviation*

Central Meas.: Sample Mean
Case 1Case 2
Year
Mon.
Year
Mon.

CM: 5-year Trend, eq.(3)Case 1Case 2
Mon.
Year
Mon.
Year

1981
1985
1990

9
7
11

1981
1985
1990

9
7
11

1981
1985
1990

12
6
12

1997
2000

4
11

1997
2000

4
11

1997
2000

5
10

5 months

1981
1985

11
6

1991
1997
2000

1
4
10

3 months

5 months

3 months

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* Deviation from the reference dates, sum of absolute values from 1985 to 2000.

Table 9 Variants of BB Procedure: Troughs, Butterworth (tangent based) Filter
Official Ref. Dates
Year
1977

Mon.
10

Central Meas.: Sample Mean
Case 2
Case 1
Year
Mon.
Year
Mon.

CM: 5-year Trend, eq.(3)Case 1Case 2
Year
Year
Mon.
Mon.

1981

2

1981

1

1983
1986
1993

1
11
12

1999
2001

4
12

1983
1986
1993

2
11
10

1983
1986
1993

1
11
12

1999

1

1999
2001

4
12

2002
1
2009
3
Deviation*

7 months

1981
1982

4
12

1986
1993

10
12

1999

3

2002

1

7 months

7 months

1982

12

1986
1993

10
12

1999

2

2002

1

6 months

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* Deviation from the reference dates, sum of absolute values from 1983 to 2002.

─ 43 ─

成城・経済研究第 224 号（2019 年 3 月）

Table 10 Variants of BB Procedure: Peaks, Hamming-Windowed Filter
Official Ref. Dates
Year
1980

Mon.
2

1985

6

1991
2
1997
5
2000
11
2008
2
Deviation**Central Meas.: Sample Mean
Case 2
Case 1
Year
Mon.
Year
Mon.

CM: 5-year Trend, eq.(3)Case 1Case 2
Year
Year
Mon.
Mon.

1981
1985
1990

1981
1985
1990

10
7
11

1981
1985
1990

1997
2000

5
11

1997
5
2000
10
(2007
6)*
3 months

10
1981
11
6(7)* 1985
6
12(11)*
1991
1
1997
5
1997
4
2000
11
2000
10
(2007
6)*
2(4) months
3 months

4 months

12612

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when scaling measure is the interquartile range.
** Deviation from the reference dates, sum of absolute values from 1985 to 2000.

Table 11 Variants of BB Procedure: Troughs, Hamming-Windowed Filter
Official Ref. Dates
Year
1977

Mon.
10

Central Meas.: Sample Mean
Case 1Case 2
Year
Mon.
Year
Mon.

CM: 5-year Trend, eq.(3)Case 2
Case 1
Year
Mon.
Year
Mon.

1981

2

1981

2

1983
1986
1993

2
10
12

1999
2001

4
12

1983
1986
1993

2
11
10

1983
1986
1993

2
10
12

1999

1

1999
2001

4
12

2002
1
2009
3
Deviation**

7 months

1981
1982

4
12

1986
1993

10
12

1999

3

2002

1

7 months

7 months

(1981
1982

3)*
12

1986
1993

10
12

1999

2

2002

1

6 months

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when scaling measure is the standard deviation.
** Deviation from the reference dates, sum of absolute values from 1983 to 2002.

─ 44 ─

Coincident Index and Reference Cycle

Table 12 Variants of BB Procedure: Peaks, Christiano-Fitzgerald Filter
Official Ref. Dates
Year
1980

Mon.
2

1985

6

1991
2
1997
5
2000
11
2008
2
Deviation**Central Meas.: Sample Mean
Case 1Case 2
Year
Mon.
Year
Mon.

CM: 5-year Trend, eq.(3)Case 1Case 2
Mon.
Year
Mon.
Year

1981
1985
1990

8(9)*
6(7)*
10

1981
1985
1990

11
6
12

1981
1985
1990

9
6(7)*
10

1981
1985
1990

12
6
12

1997
2000

5(4)* 1997
10(11)* 2000

4
10

1997
2000

5
1997
10(11)* 2000

5
10

5(6) months

4 months

5(5) months

3 months

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when scaling measure is the interquartile range.
** Deviation from the reference dates, sum of absolute values from 1985 to 2000.

Table 13 Variants of BB Procedure: Troughs, Christiano-Fitzgerald Filter
Official Ref. Dates
Year
1977

Mon.
10

1983
1986
1993

2
11
10

1999

1

2002
1
2009
3
Deviation**Central Meas.: Sample Mean
Case 1Case 2
Year
Mon.
Year
Mon.

CM: 5-year Trend, eq.(3)Case 2
Case 1
Year
Mon.
Year
Mon.

1981
1982

1
12

1981
1982

4
12

1981
1982

1
12

1986
1993

11
12

1986
1993

10
12

1986

11

1999
2001

3(4)*
12

1999

3

1994
1999
2001

1
3(4)*
12

2002

1

7(8) months

7 months

8(9) months

1982

12

1986
1993

10
12

1999

2(3)*

2002

1

6(7) months

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when scaling measure is the interquartile range.
** Deviation from the reference dates, sum of absolute values from 1983 to 2002.

─ 45 ─

成城・経済研究第 224 号（2019 年 3 月）

Table 14 Variants of BB Procedure: Peaks, Hodrick-Prescott Filter
Official Ref. Dates
Year
1980

1985

Mon.
2

6

1991
2
1997
5
2000
11
2008
2
Deviation***Central Meas.: Sample Mean
Case 1Case 2
Year
Mon.
Year
Mon.

CM: 5-year Trend, eq.(3)Case 2
Case 1
Mon.
Year
Mon.
Year

1981

10

1981

12

1981

11

1985
1990

6(7)*
10

1985

6

1985
1990

7
10

1997
2000

4(3)*
10

1997
2000

4(3)*
10

6(8) months

1991
1**
1997
4
2000
9(10)*
2007
6
4(4) months

7(8) months

1982
1985
1990

1
6
12

5
1997
2000
10
6
2007
3 months

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when scaling measure is the interquartile range.
** The date of December 1990 is obtained instead of January 1991, when scalingmeasure is the interquartile range.
*** Deviation from the reference dates, sum of absolute values from 1985 to 2000.

Table 15 Variants of BB Procedure: Troughs, Hodrick-Prescott Filter
Official Ref. Dates
Year
1977

1983
1986
1993
1999

Mon.
10

2
11
10
1

2002
1
2009
3
Deviation**Central Meas.: Sample Mean
Case 2
Case 1
Year
Mon.
Year
Mon.
1981

2

1983
1986
1993
1999
2001

1
11
12
5
12

8 months

1981
1982

CM: 5-year Trend, eq.(3)Case 1Case 2
Year
Year
Mon.
Mon.

4
12

1986
1993
1999

10
12
2(3)*

2002

1

6(7) months

1983
1986
1993
1999
2001

1
11
12
1(5)*
12

4(8) months

1982

11(12)*

1986
1993
1999

10
12
2(3)*

2002

1

7(9) months

Note: Case 1 Skip I, III and IV in BB procedure (Table 1).
Case 2 Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when scaling measure is the interquartile range.
** Deviation from the reference dates, sum of absolute values from 1983 to 2002.

─ 46 ─

Coincident Index and Reference Cycle

Table 16 Bry-Boschan Procedure without Internal Smoothing: Peaks
Official Ref. Dates
Year
Mon.
1980
2
1985

6

1991

2

1997
2000

5
11

Butterworth
Year
Mon.

HammingYear
Mon.

CF Filter
Year
Mon.

HP Filter
Mon.
Year

1981
1985
1990

9
7
11

1981
1985
1990

10
6
12

1981
1985
1990

8
6
10

1981
1985
1990

10
7(6)*
10

1995
1997
2000

1
4
11

1995
1997
2000

1
5
11

1995
1997
2000

1
5
10

1995
1997
2000

3
4
10

Note: 1. Sample mean for central measure, standard deviation for scaling measure.
2. Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when outliers are removed (threshold of 2.02).

Table 17 Bry-Boschan Procedure without Internal Smoothing: Troughs
Official Ref. Dates
Mon.
Year
1977
10

1983
1986
1993

2
11
10

1999

1

2002

1

Butterworth
Year
Mon.

HammingYear
Mon.

CF Filter
Year
Mon.

HP Filter
Mon.
Year

1981

2

1981

2

1981
1982

1
12

1981

2

1983
1986
1993
1995
1999
2001

1
11
12
7
4(3)*
12

1983
1986
1993
1995
1999
2001

2
10
12
7
4(3)*
12

1986
1993
1995
1999
2001

11
12
6
3
12

1983
1986
1993
1995
1999
2001

1
11
12
8
5(1)*
12

Note: 1. Sample mean for central measure, standard deviation for scaling measure.
2. Skip I and III through V.2 in BB procedure (Table 1).
* The number in ( ) shows a month when outliers are removed (threshold of 2.02).

─ 47 ─