Slack in Labor Market and Business Cycles

Takeshi Otsu

Abstract
In this paper, we examine how a focus on an increase in economic slack
contributes to developing clear quantitative guidelines about how to identify arecession. We use monthly labor-market indicators of composite indices of Japan.
Firstly, we find that ‘Index of Non-Scheduled Worked Hours (Industries Covered)’ isa promising variable to identify a recession in Japan. Secondly, the unemployment
rate, found useful to identify the modern U.S. recessions in the literature, does notproduce dates of the turning points consistent with the official reference dates ofJapan.
Key words: economic slack, growth cycles, reference dates
JEL classification: E32

1

Introduction
The study of aggregate fluctuations in economy has been a central subject of

economics since the nineteenth century (see Persons, 1926). Burns and Mitchell(1946) is a compilation of extensive works the NBER researchers undertook at thattime. It has spawned a voluminous literature on the business cycles. One of the key
concepts is the “reference dates,” the dates of peaks and troughs of business cycles.
Burns and Mitchell (1946, pp. 76-77) explained the importance of dating the peaks
and troughs in business-cycle analysis. Romer and Romer (2019) claimed that thedates played an important role in establishing the concept of a recession as arepeated, identifiable phenomenon. The reference dates still remain not only a
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成城・経済研究第 232 号（2021 年 3 月）

starting point of empirical research, but a great concern from the press andpolicymakers. Particularly, significant downturns in economic activity are afundamental motivating concern.
In the business-cycle literature, it is important to distinguish a classical cycle
and a growth one, as pointed out by Pagan (1997). The classical cycle consists ofpeaks and troughs in the levels of aggregate economic activities, often represented
by the gross national product (GDP). On the other hand, the growth cycle exists inthe detrended series, on which the real business-cycle literature focuses. The two
types of cycles show different cyclical timing in general, that is, different dates ofpeaks and troughs. When a series has a cyclical component around a deterministic
upward trend, typical as in economic data, detrending would make its cyclicalpeaks earlier, while delaying its cyclical troughs (see Bry and Boschan, 1971, p. 11).
The reference dates of the business cycle, officially published in the U.S. andJapan, conceptually correspond to the timing of peaks and troughs of the classicalcycle. For example, the NBER focuses on peaks and troughs in the level of
economic activity1). The basic dating procedure, widely used in official agencies
and academic researchers, is developed by the National Bureau of Economic
Research (NBER) in the U.S. It applies the Bry-Boschan procedure (see Bry andBoschan, 1971) to determine turning points of several economic time series selected

as coincident indicators. Dates of these turning points are typically pinned down
by examining a historical diffusion index that shows a share of the number ofseries with a positive change, the so-called expanding series. Pagan (1997, p. 3)
argued against detrending transformation of data, because it is not appropriate to
analyze and interpret the growth cycle, citing the business-cycle characteristics
based on the official reference dates.

1)

The NBER press release on June 8, 2020: http://www2.nber.org/cycles/june2020.pdf

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Slack in Labor Market and Business Cycles

Fabricant (1972) suggested defining a recession as a decline in the proportionof available resources employed in production, or as a widening of the gap
between potential and actual output, rather than as a decline in aggregate economic
activity relative to its trend. Following this line, Romer and Romer (2019) proposedthat the NBER should consider replacing its emphasis on a decline in economic
activity with a focus on a large and rapid rise in economic slack, claiming that itlead to a narrower and more precise definition of a recession that is more firmly
grounded in modern understanding of macroeconomic fluctuations. They showedsome supportive evidence for the United States and Japan in the modern lowgrowth era. They argued that it appeared better suited to identifying episodes ofinterest in settings where trend growth is low as well as more closely
corresponding to how both economists and the public think of a recession.
In this paper, we examine how a focus on an increase in economic slack
contributes to developing clear quantitative guidelines about how to identify arecession. We attempt to nail down what variables are useful for analysis ofrecessions as well as economic slack. We use labor-market data of Japan that hasrecorded a very low rate of economic growth for more than a quarter of century.
We use monthly data of the leading, the coincident, and the lagging indicators,
instead of quarterly data that Romer and Romer (2019) used, to enhancecomparison with the reference dates published in monthly base.
The main findings are as follows. Firstly, ‘Index of Non-Scheduled Worked
Hours (Industries Covered)’ is a promising variable to identify a recession in Japan.
Secondly, the unemployment rate, which Romer and Romer (2019) found useful tofind the modern U.S. recessions, does not produce dates of the turning points
consistent with the official reference dates of Japan.
The rest of the paper is organized as follows. In section 2, we discuss data for
analysis. We use individual indicators of the composite indices, which are related

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成城・経済研究第 232 号（2021 年 3 月）

to the labor market in Japan. Section 3 briefly explains the filtering methods used
to estimate economic slack. In section 4, we compare the estimated dates of peaks
and troughs with the official reference dates to investigate whether we can findeconomic variables useful for recession identification. The final section isallocated to discussion.

2

Slack in Labor Market: Data
To measure economic slack, we use time series related to labor market,

included in the composite indices of Japan (see Table 2). The reason to use labormarket data is that labor market conditions reflect overall economic slack because
all industries use labor. Although capacity utilization ratio can be used, its
coverage is limited to manufacturing, and removed from the individual indicatorsof the composite indices in October 2011. Therefore, we focus on the labor-related
variables.
Further, Romer and Romer (2019) found that the unemployment rate was
useful to study a recession in terms of economic slack in the U.S. They argued thattrend growth has been relatively steady at a moderately positive level for themodern United States, and that the recessions are all characterized by large andrapid increases in the unemployment rate. Further, they claimed that a focus onsuch a characterization does not alter the chronology of peaks and troughs of the
U. S. business cycles in any important way. It is true, historically, that the
unemployment rate was used extensively to date recessions in the early postwar
period. But, it has played no role in the dating process since it was changed to alagging indicator in 1975.
In Japan, the unemployment rate has been introduced as a lagging indicator in
August 1984. Since the growth rate of Japan is less than 1% on average for the lasttwo decades, a mild shortfall from the growth trend leads to a recession. Therefore,─ 104 ─

Slack in Labor Market and Business Cycles

it is possible for the unemployment rates to characterize recent recessions, and it
would be interesting to reevaluate usefulness of the unemployment in businesscycle analysis.
A caveat is in order. In January 2018, it was revealed that officials at Ministryof Health, Labor and Welfare had incorrectly conducted fundamental statisticalsurvey on labor-related conditions since 2004. Then, in our data set, there are two
indicators that are susceptible to this incorrect compilation. One is ‘Index of NonScheduled Worked Hours’ and the other ‘Index of Regular Workers Employment(Change from Previous Year).’ According to Economic and Social Research Institute(ESRI), affiliated with the Cabinet Office, Government of Japan, it has used thecorrected values published by the Monthly Labor Survey since January 2019 forthe time period from January 2012 onward as a remedy for the faulty data problem.
The earlier part of the series than the correction is connected by a link coefficientmethod. Such a remedy makes these series good enough for our analysis. Thus, we
use them in the later analysis.
In addition, the reason that we choose labor-related variables from theindividual indicators of composite indices is that they are supposed to have strong
relation with business cycles. ESRI routinely examines and revises thecomposition of indicators. The latest revision, the 12th revision, was made in July
2020. All the individual indicators are available since January 1975, amounting tomore than 500 sample points for each series, which would be long enough for ouranalysis.
We attempt to estimate economic slack with series discussed above. As
Romer and Romer (2019, see p. 12) discussed, a recession can be defined as asustained decline in the rate of growth of aggregate economic activity relative to its
long-term trend. Therefore, the ‘slack’ concept is better suited to the growth cycles
than the level cycles. In this interpretation, the growth cycles are deemed detrended

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成城・経済研究第 232 号（2021 年 3 月）

business cycles because a recession is interpreted as a part of business cycle. We
use bandpass filters to suppress a secular trend and noise components and to
extract detrended growth cycles. Following Burns and Mitchell (1946), thebusiness cycles are assumed to range from 18 months (1.5 years) to 96 months (8years), while the secular trend corresponds to the longer-cycle components and thenoise to the shorter ones. The technical details are given in section 3.
We compare dates of turning points in the growth cycles with those of thereference dates to investigate usefulness of economic slack in identification of arecession. The reference dates of business cycles in Japan are determined by ESRI
that organizes the Investigation Committee for Business Cycle Indicators to
inspect historical diffusion indexes calculated from selected series of coincident
indexes and other relevant information. Typically, a final decision on turning
points is made about two to three years later. 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 theclassical cycles, that is, the Burns-and-Mitchell-type cycle based on the level ofaggregate economic activity. As pointed by Canova (1999, 1994), the dates of peaks
and troughs in the growth cycles deviate from those in the level cycles by two or
three quarters, which is confirmed by Otsu (2013). Therefore, it is expected that amonthly comparison would show a deviation by 6 to 9 months.
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.

In these phases, the average period is about 36 months for expansion, 16 forcontraction, and 52 for the complete cycle.
Finally, we also refer to the composite index of consistent indicators forjudgement on usefulness of the growth cycles to find a recession. The composite

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Slack in Labor Market and Business Cycles

coincident index is complied by ESRI, based on individual consistent indicators on
and after 1980, and available from 1985 onward for the 12th-revision data. In our
analysis, we use the indicators in Table 2 from January 1980 to January 2020 toenhance comparison with the composite index. All data are obtained from thewebsite of ESRI2).

3

Departure from Secular Trend: Filtering Methods
In the literature, there are various methods to extract and measure cyclicalinformation. Canova (2007) gives a concise description of methods frequently usedin macroeconomic analyses. We use bandpass filters to compute departure fromsecular trends. They allow us to extract business-cycle components and suppress
secular trends as well as all the cyclical components shorter than and equal to theseasonal cycles. Therefore, it is possible to obtain detrended components less noisy
as much as possible. The cyclical components, if properly extracted, would
incorporate all the turns to be identified as peaks or troughs of businesss cycles.
We use three types of bandpass filters among others: the ChristianoFitzgerald filter (hereafter, CF filter: Christiano and Fitzgerald, 2003), the Hammingwindowed filter (Iacobucci and Noullez, 2005) and the Butterworth filters (e.g. Gomez,
2001; Pollock, 2000). Note that the sine-based Butterworth filter with the second

order is equivalent to the Hodrick-Prescott (hereafter, HP) filter proposed by
Hodrick and Prescott (1997) (see Gomez, 2001, p. 336).
Canova (1994) examined performance of 11 different detrending methods toreplicate NBER dating, assuming that the detrending removes a secularcomponent. Similar analyses are conducted by Canova (1999) with 12 methods
including Hamilton (1989)’s procedure. They found that the HP filter and a2)

Indexes of Business Conditions: https://www.esri.cao.go.jp/en/stat/di/di-e.html, Dec.
23, 2020.

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成城・経済研究第 232 号（2021 年 3 月）

frequency domain filter as an approximation to the Butterworth filter (see Canova,
1998, p. 483) would be the most reliable tools to reproduce the NBER dates. Otsu
(2013) conducted a comparative analysis among bandpass filters such as theChristiano-Fitzgerald filter (Christiano and Fitzgerald, 2003), the Hammingwindowed 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.
Now we review properties of the three filters in turn: Christiano-Fitzgerald
filter, Hamming-windowed filter, and Butterworth filters. To begin with, we
consider the following orthogonal decomposition of the observed series x:〜x=y+x (1)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 ofsignal extraction, as expounded by Whittle (1983, Chapter 3 and 6), indicates y canbe written as:y=B(L)x(2)

B(L)=  B L ，L x≡x

(3)



In polar form, we haveB(e )=

1, for ω ∈[−b, −a]∪[a, b](4)

0, otherwise

where 0≦a≦b≦π. In the business-cycle literature, the values of a and b areoften set to the frequencies that correspond to 8 and 1.5 years, respectively. In case2π

of monthly data we use later, the frequency range is set to─ 108 ─2π

 96 , 18 .

Slack in Labor Market and Business Cyclesy

Theoretically, we need an infinite number of observations, x’s, to compute. 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 the sample, while other filtering
methods in the frequency domain are implemented under the circularity
assumption.
Since details of the CF filter and the Hamming-windowed filter are given in
Christiano and Fitzgerald (2003) and in Iacobucci and Noullez (2005), we only
briefly review them. As for Butterworth filters, we describe them in a little detail.
Then, we discuss detrending and boundary treatment.

3.1

Christiano-Fitzgerald Filter
Christiano and Fitzgerald (2003) sought 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.
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 is─ 109 ─

成城・経済研究第 232 号（2021 年 3 月）

relatively 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.
As argued in Otsu (2015), the cyclical components extracted by CF might bedistorted in magnitude and timing. Its gain function, defined as the modulus of thefrequency response function, shows large ripples over the target ranges, indicatinga large distortion in estimating the cyclical components. The CF filter also shows
leakage effects (see Baxter and King, 1999, p. 580) over higher frequencies of morethan 8 periods per cycle. Further, phase shifts are indicated by values of its phasefunction, defined as arctangent of the ratio of the real-valued coefficient of theimaginary part of the frequency response function to the real part value.

3.2

Hamming-Windowed Filter
Iacobucci and Noullez (2005) claimed that the Hamming-windowed filter be agood 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), andthe CF filter. This means that it has no exacerbation (see Baxter and King, 1999, p.

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Slack in Labor Market and Business Cycles

580) 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).

3.3

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
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

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成城・経済研究第 232 号（2021 年 3 月）


BFT=



(1+L) (1+L )(1+L) (1+L ) +λ(1−L) (1−L )(5)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 )(6)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.(6) can handle nonstationary components
integrated of order 2n or less. Let ω the cutoff point at which the gain is equal to0.5. It is shown
λ={tan(ω/2)}



(7)

To see this, we replace the L by e  in eq.(5) to obtain the frequency response
function in polar form asϕ(e ; λ, n)==1
1+λ(i(1−e )/(1+e ))

(8)

1

1+λ{tan(ω/2)}(9)Here, it is easy to see that eq.(7) holds when ϕ(e )=0.5. We also observe ineq.(9) that the first (2n−1) derivatives of ϕ(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.(9) to specify ωso that the gain at the edge of the pass band is close to one and that of the stop band

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Slack in Labor Market and Business Cycles

close to zero. Let the pass band [0, ω ], and the stop band [ω, π], where ω  issmaller than ω. As in Gomez (2001, p. 372), we consider the following conditions
for some small positive values of δ and δ,1−δ<ϕ(e , λ, n)≦1

forω∈[0, ω ]

(10)0≦ϕ(e , λ, n)<δ for ω∈[ω, π](11)That is, we can control leakage and compression (see Baxter and King, 1999, p. 580)
effects with precision specified by the values of δ and δ. These conditions can be
written as follows:tan(ω /2) 1=tan(ω/2)
1−δ

1+





1+

 tan(ω /2)  = δ1

(12)tan(ω/2) 

(13)





Then, we can solve for the cutoff frequency (ω) and the filter’s order (n), givenω , 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.(5)and eq. (6), 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 )



(14)



(15)

where
λ={2 sin(ω/2)}



(16)

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成城・経済研究第 232 号（2021 年 3 月）

These are the so-called sine-based Butterworth filters. When n is equal to two, eq.
(15) is the HP cyclical filter, derived in King and Rebelo (1993, p. 224). Thus, as
pointed out by Gomez (2001, p. 336), the sine-based Butterworth filter with 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) 1=sin(ω/2)
1−δ

(17)



sin(ω/2)  1=sin(ω/2)
δ

(18)





We observe that the Butterworth highpass filter in eq. (6) or eq. (15) 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 better approximations
to ideal low-pass filters than the BFS. A simulation study in Otsu (2007) confirmedit.
In the paper, we apply the Butterworth filters to extraction of components
over a certain band [ω, ω], where ω is smaller than ω. The bandpass filter isobtained as the difference between two highpass filters in eq.(6), or two lowpass
filters in eq.(5) with different values of λ, as in Baxter and King (1999, p. 578).
Suppose a lowpass filter has the pass band [0, ω ] and the stop band [ω, π].

Here, ω  indicates a frequency at which the cycle is longer by some periods thanat ω and corresponds to ω  in eq.(12), while ω corresponds to ω in (13). This
lowpass filter has the cutoff frequency of ω and the order of n determined in eq.
(12) and (13). Let λ the corresponding value of λ. Similarly, another lowpass filter
has the pass band [0, ω] and the stop band [ω , π]. Here, ω  indicates afrequency at which the cycle is shorter by some periods than at ω. In short, we
assume that ω <ω<ω<ω . ω corresponds to ω  in (12), and ω 

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Slack in Labor Market and Business Cycles

corresponds to ω in (13). The filter has the cutoff frequency of ω and the orderof n. Then, the value of λ is λ. The bandpass filter, BFT (λ, n, λ, n), canbe obtained as
BFT (λ, n, λ, n)=BFT(λ, n)−BFT(λ, n)(19)The corresponding frequency response is expressed ash(ω; λ, n, λ, n)=ϕ(e ; λ, n)−ϕ(e ; λ, n) (20)
We can obtain the bandpass filter for the sine-type, BFS (λ, n, λ, n), and its
frequency response in a similar manner.
Alternatively, we sequentially apply the highpass filter with a lower cutoff
frequency to a series, and then further apply the lowpass filter with a higher cutoff
frequency to the filtered series. Although Pedersen (2001, p. 1096) reported that thesequential filtering has less distorting effects than use of the linear combination of
the filters, the empirical results in the following sections do not change whether we
use the difference method (the linear combination) or the sequential method. Yetanother method is to convert the lowpass filter to the bandpass filter by thefrequency transformation, described in a standard textbook (e. g. Proakis andManolakis, 2007, p. 733), and explicitly obtain the bandpass filter (see Gomez, 2001, p.
371). This filter, however, has only one order parameter, implicitly assuming n isequal to n. But, the values of n and n are very different in fact (see Otsu, 2015).
Therefore, we would not use the transformation method later in the paper. Here,
we use the difference method, because it is easy to control leakage andcompression effects at a specific frequency.
We need specify two parameter values, n and λ, in eq. (5) or eq. (6) toimplement the Butterworth filtering. We obtain these values from eqs.(7), (12) and(13) for target frequency bands, that is, values of ω  and ω with given values of δ─ 115 ─

成城・経済研究第 232 号（2021 年 3 月）

and δ. We set both δ and δ to 0.01.
In the paper, we attempt to extract cyclical components with periods per cycleof 1.5 years to 8 years. In terms of a period per cycle (p), a frequency (ω) isexpressed as

2π. Therefore, using the notation in the previous section, the targetpband, [ω, ω], is

2π

2π 96 , 18  in months. Following Otsu

(2015), we set ω  to2π
2π
and ω  to. In this case, the transition bands are
132
12
2π

2π

2π

 132 , 96  and2π

 18 , 12 , respectively. Setting ω to ω in eq.(12) and ω to ω in eq.(13), we








find n and ω. λ is obtained from eq.(7). Similarly, we find n and λ by settingω to ω  in eq.(12) and ω  to ω in eq.(13), together with eq.(7). In a similar way,
we compute the parameter values of the sine-based Butterworth filter from eq.(16),
eq.(17) and eq.(18).
Two remarks are in order. As is always the case, the sine-based filter
commands a higher order than the tangent-based on under the same precision
values of δ and δ. In addition, as already mentioned, the well-known HP filter isviewed as the sine-based Butterworth filter with an order of two. This implies thatthe HP filter either does not preserve the precision or requires a very wide
transition band. In the literature, it is pointed out that it might mislead researchers
to false empirical results (Harvey and Jaeger, 1993), or it could generate spurious
business-cycle dynamics (Cogley and Nason, 1995). In the paper, we use the HP filter
for completeness.
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

─ 116 ─

Slack in Labor Market and Business Cycles

period is longer than seven, the matrix inversion is so inaccurate that it is
impossible to control leakage and compression effects with a certain precision
specified by eq.(12) and eq.(13), or eq.(17) and eq.(18). Further, the filters at theendpoints 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 trans-form, 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.

3.4

Detending 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 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 (2011) found that the drift-adjusting method employed by Christiano andFitzgerald (2003, p. 439) could preserve the shapes of autocorrelation functions andspectra of the original data better than the linear-regression-based detrending.

─ 117 ─

成城・経済研究第 232 号（2021 年 3 月）

There-fore, 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(21)

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

(22)

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

Tz−z+s(z−z)
T−1

(23)

In Christiano and Fitzgerald (2003, p. 439), s is set to −1. Although Otsu (2011)
suggested some elaboration on the choice of s, it does not 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.(21) would makethe data suitable for filtering in the frequency domain. Since the discrete Fouriertransform assumes circularity of data, the discrepancy in values at both ends of thetime series could seriously distort the frequency-domain filtering. The eq. (23)implies that this adjustment procedure avoids such a distortionary effect.

3.5

Boundary Treatment
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-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

─ 118 ─

Slack in Labor Market and Business Cycles

series f  ,
f =



x

if 1≦ j≦T

2x−x if −T+3≦ j≦0(24)

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.

4

Empirical Analysis

4.1

Dating Algorithm
To identify dates of peaks and troughs, we use a modified version of the Bry-

Boschan (BB) method developed by Bry and Boschan (1971). Otsu (2017)
investigated whether the bandpass filtering could simplify the BB algorithm andfound that it would be a good substitute for smoothing procedures involved in the
BB procedure, such as the 12-month moving average, Spencer filtering, and theshort-term moving average with a span of months defined by Months of Cyclical─ 119 ─

成城・経済研究第 232 号（2021 年 3 月）

Dominace (see Bry and Boschan, 1971, p. 25). Therefore, it is possible to substantiallyshorten the procedure. Table 1 shows the modified procedure. In the following
analysis, we do not put back the deterministic linear trend in Step I, because our
purpose is not to examine changes in the level of economic activity, but those ineconomic slack. Furthermore, we use a couple of filtering methods other than thetangent-based Butterworth filter for comparison.

4.2

Comparison with Reference Dates
Figure 1 through Figure 5 show the business-cycle components of the fiveseries. On the whole, they move differently from the official coincident composite
index (CCI) after the last official trough of November 2012. In addition, Figure 1 of
‘New Job Offers (Excluding New School Graduates)’ shows dates of a peak and atrough in the latter half of 2000s that are quite different from those of the official
CCI. Further, the cyclical component of ‘Index of Regular Workers Employment’
tends to show more turning points than those of other series, as shown in Figure 4.
In Figure 5 of ‘Unemployment Rate,’ the troughs tend to lag behind those of theofficial reference cycle. It may not be surprising that the two coincident indicators,‘Index of Non-Scheduled Worked Hours’ and ‘Index of Effective Job Offer Rate,’
produce the cyclical components that are better aligned with the official CCI, as
shown in Figure 2 and Figure 3.
Table 4 and 5 show the estimated dates of peaks and troughs with ‘New Job
Offers (Excluding New School Graduates).’ In the first (‘Official Ref.’) and the second(‘Official CCI’) columns of Table 4, we find that dates of peaks differ between theofficial reference cycle and the official CCI, except March 2012. When we use theoriginal BB procedure instead of the abridged procedure, we have a similar result
based on the 9th-revision data as shown in Otsu (2019), except that it identified
May’s in 1985 and in 1997 as peaks instead of July in 1985 and March in 1997.

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Slack in Labor Market and Business Cycles

On the other hand, the official dates of troughs are better identified in Table 5.
Although no trough is pinned down around February 1983 that is identified in Otsu
(2019), it would be fair to say that the different dating procedures produce a

qualitatively similar result as a whole. Therefore, the main difference between theofficial reference dates and the turning points of the official CCI in Table 4 and 5comes from the fact that the dating committee uses more information than the CCI.
Turning to the peak dates of the business-cycle components, we find moredates identified than the official reference dates. They do not identify the official
peaks in 1980, 1985, 2008, and 2018. Since the sample period starts from 1980, it
is not surprising that the first peak is not identified due to insufficient data. Theirdeviations from the other peaks of the reference cycle are one or two months at
most.
The last row shows the average deviation in months. For each filtering
method, it is computed by averaging the number of months that is the sum ofabsolute differences between the dates of the reference-cycle peaks and the nearest
dates of the peaks in the estimated cycle. It ranges from around 5 to 6 months.
Table 5 shows the dates of troughs. In contrast with the case of peaks, we findthat dates of troughs in the official reference cycle and the official CCI match well,
except in February 1983 and January 1999. The estimated business-cycle
components identify the troughs in 1986, 1993, and 2002, but fail to identify otherofficial troughs. The average deviations, similarly computed as in the case ofpeaks, are different across the filtering methods, ranging from 2.7 to 7.3 months.
Table 6 and 7 show the results for ‘Index of Non-Scheduled Worked Hours.’
The cyclical components well identify the official dates of peaks after 1997, while
better identifying those of troughs. The average deviations in the peak dates arearound 4 months for the cases of the Butterworth, the Hamming-windowed, andthe CF filters, and less than 3 months for the HP filter. They mark smaller─ 121 ─

成城・経済研究第 232 号（2021 年 3 月）

deviations less than and equal to 2.5 months after 1997 onward. As for the troughs,
the deviations are less than 2 months, slightly smaller for the cases of theHamming-windowed and the HP filters.
In Table 8, the peak in 2008 is not identified by the Butterworth and the HP
filters with ‘Index of Effective Job Offer Rate.’ However, in general, the peaks
after 1991 are well identified. The average deviations are slightly larger than thosein Table 6. As for the troughs in Table 9, the reference dates are not well identified,
and the average deviations range from 3.6 to 8.0 months, much larger than those in
Table 7.
The ‘Index of Regular Workers Employment’ seems not useful to identify theturning points of the official reference cycle. As shown in Table 10 and 11, theaverage deviations are much larger than those of the other indicators. Particularly,
they are substantially large in case of the troughs: more than 10 months. The‘Unemployment Rate’ does not identify the official reference dates, either. In
Table 12 and 13, the peaks in 2000, 2008, and 2012 are well identified with thecyclical components obtained by the Hamming-windowed and the CF filters.
However, the average deviations range from 3.0 to 4.9 months for the peaks andfrom 5.1 to 7.4 months for the troughs. That is, the unemployment rate produces achronology of the turning points that is rather different from those of the official
reference cycle. In contrast to the finding of Romer and Romer (2019), it implies
whether a focus on economic slack or on the level of economic activity gives rise
to a different result.
In sum, the variable, ‘Index of Non-Scheduled Worked Hours,’ can be used
to identify the turning points of the business cycles based on economic slack.
Other variables may not be useful to identify a recession on the supposition that theofficial reference dates are true.
Finally, we examine how useful the magnitude of deviation from secular─ 122 ─

Slack in Labor Market and Business Cycles

trends is for recession identification. As discussed above, ‘Index of NonScheduled Worked Hours’ gives rise to the dates of turning points closer to theofficial reference dates compared to the other series. Then, we show deviation of
the business-cycle components from the secular trends for this series in Table 14
and 15.
In Table 14, the CF and the HP filters indicate negative values, -0.04 and-0.09, respectively. This implies that the economy reaches the peaks even when the
non-scheduled worked hours are less than a trend, that is, a ‘normal’ level. In manycases, however, the peaks come up when they go above the normal level.
Particularly, they are well beyond the normal level on the dates close to the official
reference dates. A similar finding comes from Table 15: the economy reaches thetroughs when the index is well below the normal level. In general, a large deviation
is marked around the official reference dates. Therefore, the index can be used as
an indicator of economic slack to locate recessions. Note that the magnitude of
deviation varies across the different filters. Thus, a further investigation is required
to set a specific value of criterion to determine peaks and troughs.

5

Discussion
This paper has examined how a focus on an increase in economic slack

contributes to developing clear quantitative guidelines about how to identify arecession. We use labor-market data of Japan that has recorded a very low rate of
economic growth for a long time. We use monthly data of the leading, thecoincident, and the lagging indicators, instead of quarterly data used by Romer andRomer (2019), to enhance comparison with the reference dates published inmonthly base.
The main findings are as follows. Firstly, ‘Index of Non-Scheduled Worked
Hours (Industries Covered)’ is a promising variable to identify a recession in Japan.

─ 123 ─

成城・経済研究第 232 号（2021 年 3 月）

Secondly, the unemployment rate, which Romer and Romer (2019) found useful tofind the modern U.S. recessions, does not produce dates of the turning points
consistent with the official reference dates of Japan.
Note that, in compilation of the coincident composite index, the Investigation
Committee for Business Cycle Indicators, ESRI, decided to replace ‘Index of NonScheduled Worked Hours (Industries Covered)’ with ‘Total Hours Worked (Industries
Covered)’ times ‘Employee [by number of persons engaged in non-agriculturalindustries]’ provided by Ministry of Health, Labour and Welfare, Japan, at themeeting held on July 30, 2020. The new series is to be supplied after January 2021
onward. The committee has argued that the former series shows a downward trend
due to social policies introduced to improve the work-life balance in Japan, andthat it should be replaced by some other variables that would better indicate as awhole how firms adjust employment and hours worked. Thus, one of theremaining problems is how such a replacement contributes to identifying
recessions in terms of economic slack.
Another problem is to find a quantitative criterion, that is, the magnitude of
deviation from the normal level, to pin down peaks or troughs. It crucially depends
on how to set the normal level, and might give a very different picture of thebusiness cycles, compared to one drawn by the official reference cycle. These
problems are left for future research.

Acknowledgements

A part of this work is financially supported by Tokubetsu Kenkyu

Josei (Seijo University) under the title of「景気後退の定量的同定に関する研究」.

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Slack in Labor Market and Business Cycles

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Chan, H. C., J. C. Hayya, and J. K. Ord, 1977, “A note on trend removal methods: Thecase of polynomial regression versus variate differencing,” Econometrica 45, 737744.
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Hodrick, R. J. and E.C. Prescott, 1997, “Post-war U.S. business cycles: An empiricalinvestigation,” Journal of Money, Credit, and Banking 29, 1-16. Also in Discussion
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Economics 41, 94-128.
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成城・経済研究第 232 号（2021 年 3 月）

Table 1 Abridged Bry-Boschan Procedure
Step

Procedure

I

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

II

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

─ 128 ─

Slack in Labor Market and Business Cycles

Table 2 Labor Market Indicators of Japan
Series Name
1.
2.
3.
4.
5.

Mnemonic (ESRI)

New Job Offers (Excluding New School Graduates, persons)Index of Non-Scheduled Worked Hours (Industries Covered, 2015=100)
Effective Job Offer Rate (Excluding New School Graduates, times)
Index of Regular Workers Employment (Change from Previous Year, %)Unemployment Rate (Inverted Scale, %)L3 (leading)
C4 (coincident)
C9 (coincident)
Lg2 (lagging)Lg6 (lagging)

Note: Seasonally-adjusted series of the 12th-revision composite indices, July 30, 2020.

Table 3 Reference Dates of Business Cycles: 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
October,2018*

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
71
─
─

* Provisional date, as of July 30, 2020.
Source: Indexes of Business Conditions, Economic and Social Research Institute,Cabinet Office, Government of Japan, July 30, 2020.

─ 129 ─

成城・経済研究第 232 号（2021 年 3 月）

Table 4 Month at Peaks: New Job Offers
Year

Official
Ref.
2

Official
CCI*

1980
1981
1984
1985
6
7
1989
1990
10
1991
2
1997
5
3
2000
11
12
2006
2007
8
2008
2
2012
3
3
2013
2014
3
2017
12
2018
10
2019
Deviation (avg. months)**

Business Cycle Components obtained by:
Butterworth
Hamming
CF
HP
10
3

10
4

9
7

8
8

3

1

2
3
11
12

2
3
11
12

1
3
11
12

4
8

3

2

3
10

6
6.0

5
5.3

4
4.9

6
5.0

12
4
10
12

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1985.

Table 5 Month at Troughs: New Job Offers
Year

Official
Ref.

Official
CCI*

1980
1981
1982
1983
2
1986
11
11
1989
1993
10
12
1999
1
2001
12
2002
1
2003
2009
3
3
2012
11
11
2013
2015
2016
5
Deviation (avg. months)**

Business Cycle Components obtained by:
Butterworth
Hamming
CF
HP
8
11
1
9
9
9
9
12

12

10
8

1
7

10
7

12
11
10
6

12
12
10
5

3

3

4

7

7

7

8

6

1
4

7.3

6.9

2.7

2
4
4.6

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1983.

─ 130 ─

Slack in Labor Market and Business Cycles

Table 6 Month at Peaks: Index of Non-Scheduled Worked Hours
Year

Official
Ref.
2

Official
CCI*

1980
1981
1984
1985
6
7
1988
1989
1990
10
1991
2
1997
5
3
2000
11
12
2004
2007
8
2008
2
2010
2012
3
3
2013
2014
3
2017
12
2018
10
2019
Deviation (avg. months)**

Butterworth
10
3

Business Cycle Components obtained by:
Hamming
CF
7
12
12
4
5

HP
9
3

9
2
11

9
5
10
3

1
3
10
2

2
3
10
1

3
6
3

2
7
1

3
7
1
12

1

1

7
4.4

7
4.1

4
10
2
2
8
1
3

4
3.6

5
2.4

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1985.

Table 7 Month at Troughs: Index of Non-Scheduled Worked Hours
Year

Official
Ref.

Official
CCI*

1981
1982
1983
2
1986
11
11
1987
1989
1993
10
12
1998
12
1999
1
2001
12
2002
1
2005
2009
3
3
2011
2012
11
11
2016
5
2017
Deviation (avg. months)**

Butterworth

1

Business Cycle Components obtained by:
Hamming
CF
3
5
12
1

HP
3
1

1

1

12

11
12

1
3
10
12

12

11

1
12

6
5
3
11
11

9
5
3
12
6

6
5
5
12
6

1.1

1.4

1.1

3
12
4
5
3
12
9
1.6

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1983.

─ 131 ─1
8
11

成城・経済研究第 232 号（2021 年 3 月）

Table 8 Month at Peaks: Index of Effective Job Offer Rate
Year

Official
Ref.
2

Official
CCI*

1980
1981
1982
1984
1985
6
7
1990
10
1991
2
1997
5
3
2000
11
12
2001
2004
2006
2007
8
2008
2
2012
3
3
2013
2014
3
2017
12
2018
10
2019
Deviation (avg. months)**

Butterworth
7
11

Business Cycle Components obtained by:
Hamming
CF

HP

12

9

1

3

2

4
4
12

4
3

4
4
12

1
3

3
7
11

12

1
11
11

5

7
1
5
11

6
12

2
4

5
2

4
5.1

4
4.4

2
4.1

1
2.9

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1985.

Table 9 Month at Troughs: Index of Effective Job Offer Rate
Year

Official
Ref.

Official
CCI*

1980
1981
1982
1983
2
1986
11
11
1987
1993
10
12
1999
1
2001
12
2002
1
2003
2005
2009
3
3
2012
11
11
2013
2015
2016
5
Deviation (avg. months)**

Butterworth
12

Business Cycle Components obtained by:
Hamming
CF

HP

9

3
10

5
9

2
11

4
12
7

4
11
2

5
11
1

4
11
2

6

5

8

8

8

4
8

8

3
11

3
10

2
7

8.0

3.6

3

9
3.6

5.9

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1983.

─ 132 ─

Slack in Labor Market and Business Cycles

Table 10 Month at Peaks: Index of Regular Workers Employment
Year

Official
Ref.
2

Official
CCI*

1980
1981
1985
6
7
1989
1990
10
1991
2
1993
1997
5
3
2000
11
12
2001
2004
2007
8
2008
2
2011
2012
3
3
2014
3
2015
2017
12
2018
10
Deviation (avg. months)**

ButterworthBusiness Cycle Components obtained by:
Hamming
CF

HP

7
1

5
2

4
8

4
9

12
8

1

5
3
1
4
10
5
12

7
2

9

1
8

4
6

8

1

2

3
8

2
6

4
2
9
8

4
3
9
7

2
9
7

3
10
7

5.3

5.3

6.6

6.0

3
8
12
12

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1985.

Table 11 Month at Troughs: Index of Regular Workers Employment
Year

Official
Ref.
2
11

Official
CCI*

1983
1986
11
1987
1989
1990
1992
1993
10
12
1995
1998
12
1999
1
2001
12
2002
1
2003
2006
2009
3
3
2010
2012
11
11
2013
2014
2015
2016
5
2018
Deviation (avg. months)**

Butterworth
10

Business Cycle Components obtained by:
Hamming
CF
10
9

10

10

10

9

7
10

HP
9

11
6
9
11

10

7

10

8

11

10

10

1
6

2
6

5

1
12

1

1

2

1
12

10

12

12

4
1
1
12
10
8
10.3

2
9
8
10.4

2
8
8
11.1

8
9
10.3

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1983.

─ 133 ─

成城・経済研究第 232 号（2021 年 3 月）

Table 12 Month at Peaks: Unemployment Rate (Inverted Scale)Year

Official
Ref.
2

Official
CCI*

1980
1982
1985
6
7
1988
1990
10
1991
2
1992
1995
1997
5
3
2000
11
12
2004
2005
2007
8
2008
2
2011
2012
3
3
2013
2014
3
2017
12
2018
10
2019
Deviation (avg. months)**

ButterworthBusiness Cycle Components obtained by:
Hamming
CF

1
4

2
5

7

4

2
2
9
12

1
1
9
11

HP

2
6
8
3

1
5

9
1
8
12
8

5
2
8
11

4

4

4
10

2
9

2
8

8

4

1

7

11
2

3

5
7
3.9

4.9

7

5

3.0

3.7

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1985.
Table 13 Month at Troughs: Unemployment Rate (Inverted Scale)
Official
Official
Year
Ref.
CCI*
1981
1983
2
1984
1986
11
11
1987
1989
1990
1991
1993
10
12
1994
1995
1998
12
1999
1
2001
12
2002
1
2003
2005
2006
2009
3
3
2012
11
11
2013
2016
5
2017
2018
2019
Deviation (avg. months)**

Butterworth
1

Business Cycle Components obtained by:
Hamming
CF
2
3
3
2

HP
2
5

5
4

6

7
3
12

4

7

1

4
10

4
11

3
11

5
3
12

2

2

2

3

3

3

3

2
11

8

1
10
10

2
10
12

5

3

2
12

1
7.4

5.1

5.1

5

9
3
4

Note: * Composite Coincident Indicator. Base year = 2015.
** Between the official dates and the nearest estimates on and after 1983.

─ 134 ─5.4

Slack in Labor Market and Business Cycles

Table 14 Values at Peaks: Index of Non-Scheduled Worked Hours
Official Ref.
Year
Month
1980
2
1981
1984
1985
6
1988
1989
1990
1991
2
1997
5
2000
11
2004
2007
2008
2
2010
2012
3
2013
2014
2017
2018
10
2019

Business Cycle Components obtained by:
Butterworth
Hamming
CF
HP
1.19
2.62
1.49
-0.04
-0.09
4.40
4.72
4.96
2.94
1.35
3.04
9.23
6.53
8.15
6.86
10.31
9.17
8.52
7.22
2.58
3.15
3.24
3.38
1.56
2.01
2.88
1.47
7.80
0.31
1.78

7.36
2.13
2.78

4.63

4.13

0.53

0.66

6.37
3.78
4.04
2.79

6.80
1.18
0.62
2.77

2.44

1.25

Note: Values of deviation from secular trends. Base year = 2015.

Table 15 Values at Troughs: Index of Non-Scheduled Worked Hours
Official Ref.
Year
Month
1981
1982
1983
2
1986
11
1987
1989
1993
10
1998
1999
1
2001
2002
1
2005
2009
3
2011
2012
11
2016
2017

Business Cycle Components obtained by:
Butterworth
Hamming
CF
HP
0.51
-0.58
-0.49
-3.15
-1.32
-2.05
-3.71
-11.25-10.05-10.63

-9.17
-4.42

-8.76
1.02
-8.00
-5.85

-5.42

-4.14

-3.92
-4.61

-0.65
-13.77
-1.26
-0.67
-2.35

-1.55
-13.21
0.53
-0.54
-3.09

-0.96
-11.28
-0.64
-1.82
-0.75

-3.12
-6.99
-0.28
-14.94
-1.28
-0.88
-2.17

Note: Values of deviation from secular trends. Base year = 2015.

─ 135 ─

-7.05
2.81
-7.63

成城・経済研究第 232 号（2021 年 3 月）

Figure 1 Slackness: New Job Offers

─ 136 ─

Slack in Labor Market and Business Cycles

Figure 2 Slackness: Index of Non-Scheduled Worked Hours (Industries Covered)─ 137 ─

成城・経済研究第 232 号（2021 年 3 月）

Figure 3 Slackness: Index of Effective Job Offer Rate

─ 138 ─

Slack in Labor Market and Business Cycles

Figure 4 Slackness: Index of Regular Workers Employment

─ 139 ─

成城・経済研究第 232 号（2021 年 3 月）

Figure 5 Slackness: Unemployment Rate

─ 140 ─