Robust Statistics and Business Cycle Dating

Takeshi Otsu

Abstract
In this paper, we investigate the effects of robust statistics introduced byEconomic and Social Research Institute (Japan) on the coincident composite index.
Particularly, we take up interquartile ranges and median-based trimmed mean tosee how these statistics affect business cycle dating results. Further, we examine
what kinds of smoothing methods might make improvement in dating the businesscycle in that they produce closer dates to the official reference dates. The main
findings are as follows. First, the median-based outlier removing procedure andinterquartile ranges play only a minor role in dating peaks and troughs. Secondly,
we find that some filtering methods can simplify the Bry-Boschan algorithm to agreat extent and supersede the Spencer smoothing and the short-term moving
average. Thirdly, the Butterworth filter is most useful in identifying the peaks andthe troughs of business cycles and nullifies necessity of introduction of the robust
statistics and the ad hoc smoothing procedures originally embedded in the BryBoschan procedure.
Key words: composite index, coincident indicators; reference dates
JEL classification: E32

1

Introduction
Diffusion and composite indices are frequently used in empirical analyses ofthe business cycle. Each index consists of leading, coincident, and lagging
indicators. The number of indicators to create composite indices is different agent
─ 93 ─

成城・経済研究第 228 号（2020 年 3 月）

by agent around the world. For example, Economic and Social Research Institute(ESRI) in Japan currently uses 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 used to determine turning points of the businesscycle, that is, extract the reference cycle (Harding and Pagan, 2016, p. 52). The
construction of the indices proceeds as follows. First, we determine the states of
expansion and contraction. For example, we might look at the changes of eachseries for the diffusion indices. Then, we count the number of positive (or negative)changes at each point of time, and divide it by the total number of the series in use.
If more than 50% of the series indicate positive (or negative) changes, the economic
state is considered in an expansion (or contraction, respectively). Thus, the trough (orpeak) 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.
While the diffusion indices play an important role in determination of theturning points, the composite indices would be appropriate for analyses of theamplitude of the cycles, rate of change in the cycles, and forecasting due to theirconstruction. Since April 2008, ESRI in Japan has attached great importance to thelatter, because they could be more informative for economic analyses. It also has
introduced robust statistics into the composite-index compilation to reduce
susceptibility to possible outliers.
Here, a question is whether the composite indices exhibit business cycle dates
reasonably well in the sense that they are consistent with the official reference
dates. As noted in Romer and Romer (2019), the chronology is still important in thebusiness cycle study. Then, for the composite indices to be useful, they are
expected to give the turning points of the business cycle dates well aligned with theofficial reference dates. Specifically, if the coincident composite index accurately

─ 94 ─

Robust Statistics and Business Cycle Dating

follows the reference cycle, we can use it to understand the amplitude of thebusiness cycle and the magnitude of plunges or booms. Further, it may give acriterion of validity of the business cycle models and the related econometricmodels.
This paper investigates the effects of robust statistics introduced by ESRI on
the coincident composite index. Particularly, we take up median-based trimmed
mean and interquartile ranges to see how these statistics affect business cycle
dating results. Further, we examine whether smoothing methods might make
improvement in dating the business cycle in that they produce closer dates to thereference dates. We start empirical analyses by examining how accurately thecoincident composite index traces the reference cycle, using the business cycle
data of Japan. Then, we investigate aggregation methods to compute the composite
index with individual indicators in terms of use of the robust statistics, smoothing
methods, and dating algorithm. Although the coincident indicators are not muchused in academic research, exceptions include Stock and Watson (1991), which
estimated the so-called single-index model for the U. S. with four coincident
indicators and compared the estimates with the composite coincident indicator,and Harding and Pagan (2006) to investigate whether their proposed algorithm
could replicate the NBER (National Bureau of Economic Research) reference cycle.
Since these and other related literature typically study the closeness to the official
dates, we follow suit.
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─ 95 ─

成城・経済研究第 228 号（2020 年 3 月）

is most useful in identifying the peaks and the troughs of business cycles andnullifies necessity of introduction of the robust statistics and the ad hoc smoothing
procedures originally embedded in the Bry-Boschan procedure.
The rest of the paper is organized as follows. In section 2, we summarize theaggregation method of 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 is allocatedto discussion.

2

Composite Indices: 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 usedin each index (i=1, 2, ⋯, n , where n  is the number of the series used for the jcomposite index), and t a point of time. Then, we first calculate a symmetric rate ofchange as follows:

Step I: Symmetric Rate of Change
To begin with, we compute the following symmetric rate of change:r(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), and─ 96 ─

Robust Statistics and Business Cycle Dating

outlier-adjusted rate of change ψ(t)r(t)<−k
Q3 −Q1 −k(Q3 −Q1 )， ifψ(t)= r(t)，k(Q3 −Q1 )，

if

−k≦

if

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.
Since September 2011, a refinement has been introduced in this step. The
basic idea is to make outlier adjustments only to series-specific parts of thestandardized rate of change, so that a shock common to all the series should be
excluded from outlier removal. First, a time trend for each series is calculated asm(t)=

1
 r(τ)
60−s  

(3)

where s is the number of missing values in the summation. Then, the rate ofchange is standardized with the interquartile range:η(t)=r(t)−m(t)Q3 −Q1 (4)

To compute the common shocks, median values are used. Let the median of η(t)across series denoted by η (t). Now, subtracting η (t) from both sides of eq. (4)and rearranging it, we obtain:r(t)=(η(t)−η (t))(Q3 −Q1 )+m(t)+η (t)(Q3 −Q1 ) 
specific parts=r (t)+r (t)common part(5)
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成城・経済研究第 228 号（2020 年 3 月）

Then, the outlier adjustment is applied to r (t) as follows:



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

(Q3
 −Q1
 ),
k



if

<
k

(6)




r (t) −Q1

Q3




 


where Q3 −Q1  an interquartile range of r (t). Further, k is a constant
threshold value 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:ψ(t)=ψ (t)+r  (t)(7)

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  

(8)

where s is the number of missing values. That is, it is an averaged value over thelast 60 months, and considered as a kind of a trimmed mean due to the trimming
procedure in the previous Step II. The second average is computed across series of
the coincident index:

─ 98 ─

Robust Statistics and Business Cycle Datingμ (t)=1 × μ (t)
n


(9)

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:z(t)=

ψ(t)−μ(t)Q3 −Q1 (10)

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

1× z(t)n −s (t) (11)

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)

(12)

where


Q3−Q1 =1
 × (Q3  −Q1  )
n


(13)

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)where the initial value of I (t) is set to 1. Then, the composite

index is obtained as follows:─ 99 ─

(14)

成城・経済研究CI (t)=

第 228 号（2020 年 3 月）I (t)
×100
I

(15)

where I  is an average in the base year.

3

Dating Algorithm: Bry-Boshcan Procedure
We use the Bry-Boschan (BB) procedure or its modified version to datebusiness cycle. The full BB procedure is summarized in Table 1. Watson (1994)
found some discrepancies between the original description by Bry and Boschan(1971) and the Fortran program they coded. The description here is modified to beconsistent with the Fortran codes. The procedure presumes to use seasonallyadjusted series. In Step I, outliers, if any, are replaced by the values of the Spencer
curve. Here, the outliers are defined as values whose ratios to (or differences inabsolute 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 iscomputed as the 15-month symmetric moving average with particular weights (seeKendall and Stuart, 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, as we discuss later, it introduces phase shifts that might─ 100 ─

Robust Statistics and Business Cycle Dating

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
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’) is─ 101 ─

成城・経済研究第 228 号（2020 年 3 月）

alternation 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

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.

4

Removing Seasonality: Filtering Methods
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 methods attempt 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, removal

─ 102 ─

Robust Statistics and Business Cycle Datingof the shorter cycles could give rise to denoising effects so that arbitrary outlier
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 bandpass
filters such as the Christiano-Fitzgerald filter (Christiano and Fitzgerald, 2003), theHamming-windowed filter (Iacobucci and Noullez, 2005) and the Butterworth filters(e.g. Gomez, 2001; Pollock, 2000), using Japanese real GDP data. It showed that the

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 deviation

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成城・経済研究第 228 号（2020 年 3 月）

of estimated business cycle dates from the official ones, this criterion is closely
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 ChristianoFitzgerald filtering. It found that the larger the sample size, the more stable theestimated 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 that the Butterworth filters give the most stable estimates among
others. Thus, they might be useful in practice.
The fourth criterion is how a weight of each cyclical component alters bydetrending 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─ 104 ─

Robust Statistics and Business Cycle Datingin Canova (1998, p. 483) would create the same artificial oscillatory behavior in theestimated 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) showed thatthe 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)

─ 105 ─

成城・経済研究第 228 号（2020 年 3 月）



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 domain are implemented
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 following analyses, we set b toπ
. Now, we briefly
6

review three filtering 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.

─ 106 ─

Robust Statistics and Business Cycle Dating

They compared variance ratios and correlations between the components extracted
by the Christiano-Fitzgerald filters and the theoretical components based on thedata generating process of observations. To evaluate the second moments of thetheoretical 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 isrelatively more important. Further, they claimed that the time-varying weights
should not introduce severe nonstationarity in the filter approximation because thevariance ratios 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 indicatethat the degree of asymmetry was not great. Finally, one of the ChristianoFitzgerald filters derived under the Random-Walk data generating process, the socalled Random Walk filter, gives a good approximation to the optimal filtering that
explicitly used 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 generating process even if the random walkassumption 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. Its 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. The CF filter also shows leakage effects overhigher frequencies of more than 8 periods per cycle. Further, phase shifts areindicated by values of its phase function, defined as arctangent of the ratio of thereal-valued coefficient of the imaginary part of the frequency response function to─ 107 ─

成城・経済研究第 228 号（2020 年 3 月）

the real part value.
In the paper, we first compute the cyclical components between 12-month2π

and 2-month cycles, that is, the frequency range

2π

 12 , 2 , and subtract themfrom the original series to obtain 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 responseover the passband than other filters in the literature, suchas the HP filter (Hodrick and Prescott, 1997), the BK filter (Baxter and King, 1999), andthe CF filter. This means that it has no exacerbation and eliminates highfrequency 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 de-trended
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) has proposed the tangent-based Butterworth filters in the two-sided expression, which are called rational square-wave filters. The one-sided

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Robust Statistics and Business Cycle Dating

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 Butter-worth
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 ) +(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)

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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(efor ω∈[0, ω ](25)

; λ, n)<δ for ω∈[ω, π]

(26)



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

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Robust Statistics and Business Cycle Datingω , 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) canhandle nonstationary components integrated of order 2n or less. Thus, the HP
filter can stationarize the time series with unit root components up to the fourthorder.

Gomez (2001, p. 367) claimed

that

the

BFT

would

give

betterapproximations to ideal low-pass filters than the BFS. A simulation study in Otsu

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(2007) confirmed it. In the following analysis, we use BFT to extract passband

components [0, ω ], setting ω =ω=2π
, with the stop band [ω, π] setting
13

2π. To implement the Butterworth filtering, we need specify two parameter
12

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) toobtain the HP-filtered passband components, setting ω=

2π
in eq. (31).
13

Turning to implementation, we can implement the Butter-worth 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 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. (27) and eq. (28), or eq. (32) and eq. (33). Further, the filters at
the endpoints of data have no symmetry due to the finite truncation of filters. Thisimplies 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.

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4.4

Extracting Seasonal Components
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 difference, 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 do-main. Since the discrete Fourier─ 113 ─

成城・経済研究第 228 号（2020 年 3 月）

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. It shows different business cycle dates
from those of the classical cycles. When a series has a cyclical component arounda determinis-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.
In addition to the detrending method, we make use of another device toreduce 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-called─ 114 ─

Robust Statistics and Business Cycle Datingreflection 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

2x−x if

1≦ j≦T−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.

5

Empirical Analysis

5.1

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

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成城・経済研究第 228 号（2020 年 3 月）

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 consistentcomposition of the indicators.
Secondly, it is revealed that officials at Ministry of Health, Labor and Welfare
had incorrectly conducted fundamental statistical survey on labor-related
conditions since 2004. Then, one of the 11 series, ‘Index of Non-Scheduled─ 116 ─

Robust Statistics and Business Cycle Dating

Worked Hours,’ may need correction. Our data may include possibly incorrect
data for four years after 2004. It is desirable for the following analyses not only toinclude business cycles as many as possible but to avoid contaminated data asmuch as possible. This consideration leads us to focus on the sample up to January
2008 based on the 9th-revision 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

Composite Coincident Index and Replication
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. To begin with, we use the coincident indicators
seasonally adjusted by the official agents, so that we can exclude influence ofdifferent 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 apeak, 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

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成城・経済研究第 228 号（2020 年 3 月）

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 1980, 90. 9 inFebruary, down to 81.8 in March and 86.4 in April. This might give rise to theofficial peak date of February 1980. During May 1980 to May 1981, the current
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 sharply down to 18.2 in January
1981, less than 50 afterwards up until July 1982. The official CCI seems pick up
these small bumps.
There would be two reasons 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 thelast four points as observations at endpoints. Thus, it may introduce distortion in
dating computation. Secondly, it eliminates peaks within 6 months at endpoints in
Step IV (see Table 1). Therefore, it never identifies February 1980 as a peak since
our data start in January 1980. Then, we do not pay much attention to the deviationfrom 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. (10), 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
results are shown in Table 4 and Table 5. 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.

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Robust Statistics and Business Cycle Dating

The dates in the third column (‘Aggr. Indicators’) in Table 4 match well withthe dates given by the official CCI. Only difference is observed in 1981. In the
fourth column (‘Aggr. Ind. (Median)’), we use eq. (3) - (7) instead of eq. (2) to see
the effect of the median-based outlier removal. Then, the peak in 1997 becomestwo months earlier, March instead of May. Although we find similar effects laterin the paper, it is fair to say that the median-based outlier removal has only alimited 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 in-dicators gives dating results same as the official
CCI with or without the median-based outlier removal. A large deviation in thetroughs is observed for the aggregated index, but this is mainly due to the deviationin the early 1980s: February 1983 versus October 1982. If we exclude it, thedeviation reduces to 3 months. Therefore, it can be said that the computed index
yields the dating results equivalent to those that the 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. (8) and the interquartile ranges tostandardize the rate of change of each indicators in eq. (10). These quantities seem
preferred because they are supposed to be insensitive to outliers. When we use thesample mean instead of the 5-year trend, we have the results in the second(‘Interquartile’ of ‘Sample Mean’) and the third (‘Standard Deviation’ of ‘Sample Mean’)columns in Table 6 and Table 7. These results are same as those the official CCI
gives in Table 4 and Table 5. It is interesting to see that use of the sample mean
with the standard deviation produces such identical results. We can say that theinterquartile ranges do not give better results than the standard deviation. In
addition, the fourth column (‘Standard Deviation’ of ‘CM: 5-year Average’) in Table 7

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成城・経済研究第 228 号（2020 年 3 月）

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

Effects of Smoothing 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 analyses, there is no sound reason that economic
data should include the frequencies higher than seasonal ones. If an economic
theory neither presumes effects 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 than seasonality. Further, if
the main analytical concern is about the business cycle, we may remove all thecyclical components shorter than seasonality beforehand.
In this section, we use filtering methods discussed in section 4, instead of the
X12-ARIMA method used by the official agents, to remove all the frequencies
higher than seasonality. The 11 coincident indicators are separately filtered andaggregated to make a composite coincident index. Otsu (2011b) examined theperformance of these filtering methods and found that they are very useful inextracting the seasonal components and that the corresponding ‘seasonallyadjusted’ series are smoother than the seasonally-adjusted series with the X12
ARIMA.
In the following analyses, we use the 5-year average in eq. (8) as a central
measure. Similar results are obtained when we alternatively use a sample mean. 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 ofI, III and IV in Table 1. We also note that these results are exactly same as those of

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Robust Statistics and Business Cycle Dating

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, the dataprocessing in the BB procedure is limited to the asymmetric 12-month moving
average in this case.
In comparison of Table 8 with Table 4, we find that the Butterworth-basedsmoothing gives rise to business cycle dates closer to the official reference dates
than the official CCI data, whether we use the interquartile ranges or the standard
deviation as a scaling measure. Interestingly, ‘Case 2’ shows a better result,
implying that the internal smoothing procedures other than 12-month moving
average do more harm than good. In contrast, we see a slightly worse result in
dating troughs, comparing Table 9 with Table 5. However, this might be due toaggregation of individual indicators that could be different from those used by
ESRI. If we compare the results in Table 9 with the ‘Aggr. indicators’ in Table 5,
the Butterworth smoothing still shows improvement by 2 months.
The Hamming-windowed filter produces similar results as the Butterworth
filter. The second and the fourth column labelled as ‘Case 1’ in Table 10 showeven closer dates, because the peak in 1997 is identified exactly same as in theofficial reference date. The results of the troughs in Table 11 are also equivalent tothose in Table 9, but marginally worse due to the discrepancy either in 1986 or
1999. In addition, we find in both of Table 10 and Table 11 that the interquartileand the standard deviation do not make difference in dating. Finally, comparing
the ‘Case 1’ with the ‘Case 2’ in Table 10, the’Case 2’ gives the dates closer to theofficial reference dates. Therefore, we find no important roles of the Spencer
filtering and the short-term moving average (MCD) embedded in the original BB
procedure. Again, we cannot find clear evidence that the outlier removals and theinterquartile ranges play an important role in dating.

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Turning to the results of the Christiano-Fitzgerald filter in Table 12, we findsimilar results for the peak dating as in Table 8. The choice between theinterquartile and the standard deviation does not make difference, while the
Spencer and the MCD smoothing do more harm than good: they make the dates
further deviate from the official ones. These findings are strengthened for thetroughs, as shown in Table 13. We also find that the trough dating with theChristiano-Fitzgerald filter is more dependent on the choice of the scaling
statistics, and the smoothing procedures, comparing with that of the Butterworth
filter in Table 9.
Finally, the Hodrick-Prescott filter gives rise to the results that depend onwhether we use the Spencer and the MCD smoothing or not, but not on the choice
between the interquartile ranges and the standard deviation. As for the peaks, thesmoothing procedures embedded in the original BB procedure make a large
deviation from the reference dates. In contrast, these procedures give marginallybetter results for the troughs. These findings imply that the Hodrick-Prescott filter
produces noisier series than other filters.
So far, we find that the interquartile ranges, the Spencer smothing, and theshort-term moving average (MCD) do not play much role in dating the businesscycles when we apply the smoothing methods in section 4 to the coincident
composite indices. Now, we investigate the role of the outlier removal (a thresholdvalue of 2.02). We skip Step II in section 2 and use r (t) in eq. (1) for ψ (τ) in eq.

(8). That is to say, we abandon the trimmed mean procedure. In addition, we use
the standard deviations instead of the interquartile ranges in Step IV and V.
The results for the peaks are shown in Table 16. We find that the filtering
methods give peak dates closer to the official ones than the official CCI in Table 4:
the discrepancy is only 3 months. As for the troughs, the discrepancy is 6 months
in Table 17 for all the filtering methods, which is larger than that of the official
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Robust Statistics and Business Cycle Dating

CCI in Table 5. But, the third column (‘Aggr. Indicators’) of Table 5 suggests that
this is mainly because that our data set is different from those used by ESRI. Thatis, the filtering methods at least produce comparable results without the robust
statistics and the outlier removals.
In summary, when we use the filtering methods to make seasonal adjustment,
it does not matter whether we use the interquartile ranges or the standard
deviations. Secondly, the internal smoothing in the original BB procedure, such as
the Spencer and the MCD, does more harm than good. Thirdly, we find that thebusiness cycle dates get closer to the official reference dates without the medianbased outlier removal. Finally, the Butterworth filter, among others, gives better
results in the sense that it replicates the official reference dates more closely.

6

Discussion
In this paper, we investigate the effects of robust statistics introduced by

ESRI (Japan) on the coincident composite index. Particularly, we take upinterquartile ranges and median-based trimmed mean to see how these statistics
affect business cycle dating results. Further, we examine what kinds of smoothing
methods might make improvement in dating the business cycle in that they
produce closer dates to the official reference dates.
The main findings are as follows. First, the median-based outlier removing
procedure (trimmed mean) and interquartile ranges play only a minor role in datingpeaks and troughs. Secondly, we find that some filtering methods can simplify the
Bry-Boschan (BB) algorithm to a great extent and supersede the Spencer
smoothing and the short-term moving average (MCD). Thirdly, the Butterworth
filter is most useful in identifying the peaks and the troughs of business cycles andnullifies necessity of introduction of the robust statistics and the ad hoc smoothing
procedures originally embedded in the BB procedure.

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These findings suggest that it is possible to simplify and clarify thecompilation process of composite indices and indicate that we could use a simple
frequency-domain filtering and a conventional normalization to construct acomposite index. Since composite indices await various kinds of economic
analyses for various purpose, simplicity and clarity in the compilation aredesirable. In many cases, the so-called outliers in economic variables give someclues to understand important economic phenomena or effects of exogenous
variables. Then, careless outlier removal would be harmful.
Concerning the Bry-Boschan (BB) algorithm, we can get rid of the Spencer
and the short-term moving average smoothing, so that we may avoid arbitrarinessaccompanied 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 and cycles and the enforcement rules of alternation of peaks andtroughs. We do not need such repeated processes as in the original BB procedure
to determine the dates of peaks and troughs.
Furthermore, 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,
and make it possible to filter any variable with exactly same parameter values.
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. Similar findings appear in Otsu (2019). However, there are more to be done
to reach final conclusion. Future research includes investigation of data in other─ 124 ─

Robust Statistics and Business Cycle Dating

periods and other countries, and other dating rules suggested in the literature (seeWebb, 1991; Harding and Pagan, 2016).

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」.

References
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132, 59-79.
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Robust Statistics and Business Cycle Dating

Otsu, T., 2019, “Coincident index and reference cycle,” Seijo University Economic
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成城・経済研究第 228 号（2020 年 3 月）

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 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) 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.

─ 128 ─

Robust Statistics and Business Cycle Dating

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.

─ 129 ─

成城・経済研究第 228 号（2020 年 3 月）

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

Aggr. Ind. (Median)**
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

6 months

8 months

Note: * Aggregating 11 coincident indicators, removing outliers(threshold value: 2.02).
** Outlier removal (threshold value: 2.02) based on median values, eq. (6).
*** 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

Aggr. Ind. (Median)**
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

7 months

Note: * Aggregating 11 coincident indicators, removing outliers (threshold value: 2.02).
** Outlier removal (threshold value: 2.02) based on median values, eq.(6).
*** Deviation from the reference dates, sum of absolute values from 1983 to 2002.

─ 130 ─

Robust Statistics and Business Cycle Dating

Table 6 Comparison with Reference Dates: Peaks, Alternative Measures in eq. (10)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 Average, eq. (8)Standard Deviation
Year
Month

1981
1985
1990

12
5
10

1981
1985
1990

11
5
10

1981
1985
1990

11
5
10

1997
2000

3
12

1997
2000

3
12

1997
2000

3
12

8 months

8 months

8 months

Note: 1. Outlier removal (threshold value: 2.02) based on median values, eq. (6).
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. (10)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
1982

5
10

1981

5

1986
1993
1998

11
12
12

1983
1986
1993
1998

2002

1

2002

7 months

CM: 5-year Trend, eq. (8)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. Outlier removal (threshold value: 2.02) based on median values, eq. (6).
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.

─ 131 ─

成城・経済研究第 228 号（2020 年 3 月）

Table 8 Seasonal Adjustment by Butterworth (tangent based) Filter: Peaks
Official Ref. Dates
Year
1980

Mon.
2

1985

6

1991
1997
2000

2
5
11

2008
2
Deviation*

Scaling Meas.: Interquartile
Case 2
Case 1
Year
Mon.
Year
Mon.

Scaling Meas.: Std. Dev.
Case 1Case 2
Year
Year
Mon.
Mon.

1981
1985
1990

9
7
11

1981
1985
1990

12
6
12

1981
1985
1990

9
7
11

1981
1985
1990

12
6
12

1997
2000

4
11

1997
2000
2007

5
10
6

1997
2000

4
11

1997
2000
2007

5
10
6

5 months

3 months

5 months

3 months

Note: Central measure is the past 5-year average in eq. (8).
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 Seasonal Adjustment by Butterworth (tangent based) Filter: Troughs
Official Ref. Dates
Year
1977

Mon.
10

Scaling Meas.: Interquartile
Case 2
Case 1
Year
Mon.
Year
Mon.

Scaling Meas.: Std. Dev.
Case 1Case 2
Year
Year
Mon.
Mon.

1981

1

1981

1

1983
1986
1993

1
11
12

1999
2001

2
12

1983
1986
1993

2
11
10

1983
1986
1993

1
11
12

1999

1

1999
2001

2
12

2002
1
2009
3
Deviation*

5 months

1981
1982

4
12

1986
1993

10
12

1999

1

2002

1

5 months

5 months

1981
1982

3
12

1986
1993

10
12

1999

1

2002

1

5 months

Note: Central measure is the past 5-year average in eq. (8).
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.

─ 132 ─

Robust Statistics and Business Cycle Dating

Table 10 Seasonal Adjustment by Hamming-Windowed Filter: Peaks
Official Ref. Dates
Year
1980

Mon.
2

1985

6

1991
1997
2000

2
5
11

2008
2
Deviation*

Scaling Meas.: Interquartile
Case 1Case 2
Year
Mon.
Year
Mon.

Scaling Meas.: Std. Dev.
Case 1Case 2
Mon.
Year
Year
Mon.

1981
1985
1990

10
7
11

1981
1985
1990

12
6
12

1981
1985
1990

10
7
11

1981
1985
1990

12
6
12

1997
2000

5
11

1997
2000
2007

5
10
6

1997
2000

5
11

1997
2000
2007

5
10
6

4 months

3 months

4 months

3 months

Note: Central measure is the past 5-year average in eq. (8).
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 11 Seasonal Adjustment by Hamming-Windowed Filter: Troughs
Official Ref. Dates
Year
1977

Mon.
10

Scaling Meas.: Interquartile
Case 1Case 2
Year
Mon.
Year
Mon.

Scaling Meas.: Std. Dev.
Case 2
Case 1
Mon.
Year
Mon.
Year

1981

2

1981

2

1983
1986
1993

2
10
12

1999
2001

3
12

1983
1986
1993

2
11
10

1983
1986
1993

2
10
12

1999

1

1999
2001

3
12

2002
1
2009
3
Deviation*

6 months

1981
1982

4
12

1986
1993

10
12

1999

2

2002

1

6 months

6 months

1981
1982

4
12

1986
1993

10
12

1999

2

2002

1

6 months

Note: Central measure is the past 5-year average in eq. (8).
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.

─ 133 ─

成城・経済研究第 228 号（2020 年 3 月）

Table 12 Seasonal Adjustment by Christiano-Fitzgerald Filter: Peaks
Official Ref. Dates
Year
1980

Mon.
2

1985

6

1991
2
1997
5
2000
11
2008
2
Deviation*

Scaling Meas.: Interquartile
Case 1Case 2
Year
Mon.
Year
Mon.

Scaling Meas.: Std. Dev.
Case 1Case 2
Mon.
Year
Mon.
Year

1981
1985
1990

9
7
10

1981
1985
1990

12
6
12

1981
1985
1990

9
6
10

1981
1985
1990

12
6
12

1997
2000

5
11

1997
2000

5
10

1997
2000

5
10

1997
2000

5
10

5 months

3 months

5 months

3 months

Note: Central measure is the past 5-year average in eq. (8).
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 13 Seasonal Adjustment by Christiano-Fitzgerald Filter: Troughs
Official Ref. Dates
Year
1977

Mon.
10

Scaling Meas.: Interquartile
Case 1Case 2
Year
Mon.
Year
Mon.

Scaling Meas.: Std. Dev.
Case 2
Case 1
Mon.
Year
Mon.
Year

1981
1982

1
12

1981
1982

1
12

1986

11

1994
1999
2001

1
2
12

1983
1986
1993

2
11
10

1986

11

1999

1

1994
1999
2001

1
3
12

2002
1
2009
3
Deviation*

8 months

1982

12

1986
1993

10
12

1999

2

2002

1

6 months

7 months

1982

12

1986
1993

10
12

1999

1

2002

1

5 months

Note: Central measure is the past 5-year average in eq. (8).
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.

─ 134 ─

Robust Statistics and Business Cycle Dating

Table 14 Seasonal Adjustment by Hodrick-Prescott Filter: Peaks
Official Ref. Dates
Year
1980

Mon.
2

1985

6

1991
1997
2000

2
5
11

2008
2
Deviation*

Scaling Meas.: Interquartile
Case 1Case 2
Year
Mon.
Year
Mon.

Scaling Meas.: Std. Dev.
Case 1Case 2
Mon.
Year
Mon.
Year

1981
1985
1990

10
7
10

1981
1985
1990

12
6
12

1981
1985
1990

10
7
10

1981
1985
1990

12
6
12

1997
2000

4
10

1997
2000
2007

5
10
6

1997
2000

4
10

1997
2000
2007

5
10
6

7 months

3 months

7 months

3 months

Note: Central measure is the past 5-year average in eq. (8).
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 15 Seasonal Adjustment by Hodrick-Prescott Filter: Troughs
Official Ref. Dates
Year
1977

Mon.
10

Scaling Meas.: Interquartile
Case 2
Case 1
Year
Mon.
Year
Mon.

1982
1983
1986
1993
1999

2
11
10
1

2002
1
2009
3
Deviation*

1983
1986
1993
1999
2001

1
11
12
1
12

4 months

Scaling Meas.: Std. Dev.
Case 1Case 2
Year
Year
Mon.
Mon.
1981

2

1983
1986
1993
1999
2001

1
11
12
1
12

12

1986
1993
1999

10
12
1

2002

1

5 months

4 months

1982

12

1986
1993
1999

10
12
1

2002

1

5 months

Note: Central measure is the past 5-year average in eq. (8).
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.

─ 135 ─

成城・経済研究第 228 号（2020 年 3 月）

Table 16 Dating Peaks: No Outlier Removal and No Robust Statistics
Official Ref. Dates
Year
Mon.
1980
2

1985

6

1991
2
1997
5
2000
11
Deviation*

Butterworth
Year
Mon.

HammingYear
Mon.

CF Filter
Year
Mon.

1981

12

1981

12

1981

12

1985
1990

6
12

1985
1990

6
12

1985
1990

6
12

1997
5
2000
10
3 months

1997
5
2000
10
3 months

1997
5
2000
10
3 months

HP Filter
Year
Mon.

1982
1985
1990

1
6
12

1997
5
2000
10
3 months

Note: 1. The past 5-year average for central measure, standard deviation for scalingmeasure.
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 17 Dating Troughs: No Outlier Removal and No Robust Statistics
Official Ref. Dates
Mon.
Year
1977
10

1983
1986
1993

2
11
10

1999
1
2002
1
Deviation*

Butterworth
Year
Mon.

HammingYear
Mon.

CF Filter
Year
Mon.

HP Filter
Mon.
Year

1982

12

1981
1982

3
12

1982

12

1982

11

1986
1993

10
12

1986
1993

10
12

1986
1993

10
12

1986
1993

10
12

1999
2
2002
1
6 months

1999
2
2002
1
6 months

1999
2
2002
1
6 months

1999
2
1
2002
6 months

Note: 1. The past 5-year average for central measure, standard deviation for scalingmeasure.
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.

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