Deviation Cycles in Non-Scheduled Worked Hours

Takeshi Otsu

Abstract
This paper attempts to estimate the probabilities of economic slack in thelabor market of Japan and examines how useful they are to understand the businesscycle. We use Markov switching models to estimate them with a deviation-cycle
data of non-scheduled worked hours. The main findings are as follows. First, asimple Markov switching model generates the smoothed probabilities that areclosely related to the business cycle. Second, inclusion of lagged dependent
variables as explanatory variables does not solve the model misspecification
problem indicated by the conventional specification testing. Third, a Markov
switching model statistically fits the data with frequency components higher thanseasonality better than those without them. The models accepted by specification
testing, however, do not generate the smoothed probabilities consistent with thebusiness cycle.
Key words: economic slack, deviation cycles, Markov switching
JEL classification: C51, E32

1Introduction

The reference dates of business cycle are not only a starting point of empiricalresearch, but a great concern from the press and policymakers. Particularly,
significant downturns in economic activity are a fundamental concern making theheadlines. Recently, Romer and Romer (2019) proposed that the NBER (National
Bureau of Economic Research, U. S. A.) should focus on a large and rapid rise in─ 123 ─

成城・経済研究第 236 号（2022 年 3 月）

economic slack to date peaks and troughs of business cycles, rather than on adecline in economic activity as that the modern definition emphasizes. They
pointed out that the dates played an important role in establishing the concept of arecession as a repeated and identifiable phenomenon.
Furthermore, they claimed that the focus on the economic slack lead to anarrower and more precise definition of a recession that is more firmly grounded inmodern understanding of macroeconomic. They showed some supportive evidence
for the United States and Japan in the modern low-growth era. They argued that it
appeared not only better suited to identifying episodes of interest in settings wheretrend growth is low, but more closely corresponding to how both economists andthe public think of a recession.
Following Romer and Romer (2019), Otsu (2021) attempted to identify
economic indicators of economic slack in Japan, which show peaks and troughs
consistent with the reference dates. It used monthly labor-market indicators ofcomposite indices of Japan. To measure slack in labor market, it used deviation
cycles or growth cycles, that is, the departure from secular trends based on filtering
methods. It used three types of filtering methods: the Christiano-Fitzgerald filter(Christiano and Fitzgerald, 2003), the Hamming-windowed filter (Iacobucci andNoullez, 2005) and the Butterworth filters (e.g. Gomez, 2001; Pollock, 2000). Canova(2007) gives a concise description of these methods.

Otsu (2021) found that ‘Index of Non-Scheduled Worked Hours (Industries
Covered)’ is a promising variable to identify a recession in Japan. It also found thatthe unemployment rate, one of the useful indicators to identify the modern U.S.
recessions in the literature, does not produce dates of the turning points consistent
with the official reference dates of Japan.
To understand how the economic slack deepens, we need to quantify the
degree of slack over time. The economic slack is measured by a deviation cycle as─ 124 ─Deviation Cycles in Non-Scheduled Worked Hours

in Otsu (2021). One way to quantify the slacking degree is to compute probabilitiesof the economy in the economic slack over time, the so-called smoothedprobabilities. Then, this paper uses Markov switching models with a deviationcycle data in the labor market of Japan to compute such probabilities and examines
how useful they are to understand the business cycle.
The main findings are as follows. First, a simple Markov switching model
generates the smoothed probabilities that are closely related to the business cycle.
Second, inclusion of lagged dependent variables as explanatory variables does not
solve the model misspecification problem indicated by the conventional
specification testing. Third, a Markov switching model fits the data with frequency
components higher than seasonality better than those without them in terms of thespecification testing. The models accepted by specification testing, however, do
not generate the smoothed probabilities consistent with the business cycle.
The rest of the paper is organized as follows. In section 2, we briefly review
simple Markov switching models used to be estimated. Section 3 discusses data forour analyses. We use ‘Index of Non-Scheduled Worked Hours’ which is one of theindividual indicators of the composite indices, Japan, and found useful to identify arecession in Japan (see Otsu, 2021). In section 4, we examine the estimation resultsof the Markov switching models. We compute the smoothed probabilities andcompare them with the officially published peaks and troughs of the business cycle
to see whether the estimated probabilities are useful for recession identification.
The final section is allocated to discussion.

2

Markov Switching Model
Following the recommendations in Hamilton (2016), we use the following

simple Markov switching model. Suppose we have M different states. Let y aneconomic time-series variable at time t (t=1, 2, ⋯, T) to be modeled:─ 125 ─

成城・経済研究第 236 号（2022 年 3 月）

ϕ(L)y=μ+ε，i=1, 2, ⋯, M ,(1)ε 〜N (0，σ )，i=1, 2, ⋯, M ,(2)

ϕ(L)=1−∑ϕL ，L y≡y,

(3)



where μ, σ  and ϕ are all unknown constant parameters.
Let the economic state at time t denoted by an unobserved variable, S. The
transition between states is assumed to be governed by a first-order Markov
process that is independent of ε. Then, the transition probability from i to j is:Prob [S= j  S=i]=p，i, j=1, 2, ⋯, M ,

(4)

where


p=1.
∑
(5)

Since the economic state is not observed, we need to infer it from past information.
Let Ω the past information available at time t. Specifically, we set Ω−={y, y, ⋯}. Then, we could calculate the probabilities of being in the state j,
S= j, for t=1, 2, ⋯, T, by iteration as follows:

P [S= j  Ω]=∑ Prob[S= j, S=i  Ω−]

(6)




=∑ Prob[S= j S=i] P [S=iΩ].(7)To find the joint density of y and S, we define the conditional density
function of y for the state i as:f (y  S=i, Ω)=





e
1
exp −  ,
2σ 
 2πσ where

─ 126 ─(8)

Deviation Cycles in Non-Scheduled Worked Hourse=ϕ(L)y−μ，i=1, 2, ⋯, M .

(9)Then, the joint density of y and S can be written as:f (y, S=i  Ω)= f (y  S=i, Ω)P [S=i  Ω].

(10)

We obtain the marginal density of y by summing over S:

f (y  Ω)=∑ f (y  S= j, Ω)P [S= j  Ω].

(11)



To compute eq.(11), we need stating values for P [S  Ω] in eq.(7). We can
employ the following steady-state probabilities of S. Let the unconditional
probabilities ρ of being in the state i and the transition matrix P*. That is,


ρ

R=

ρ

(12)

⫶

ρ

and

P*=



p

p ⋯ p pp ⋯ p 

⫶

⫶

⋱

⫶

p p ⋯ p





,

(13)



where ∑  ρ=1 and ∑  p=1 for i=1, 2, ⋯, M. Then, a first-orderMarkov-chain property gives:
R=P*R.

(14)

We obtain the steady-state probabilities by solving the eq.(14) under the conditions

of ∑  ρ=1 and R=R. Let the solutions (ρ, ρ, ⋯, ρ ). Then, we set:
─ 127 ─

成城・経済研究第 236 号（2022 年 3 月）

P [S=i  Ω]=ρ，i=1, 2, ⋯, M .

(15)Once we obtain y at the end of time t, we can update the P [S=i  Ω]in the following way:
P [S=i  Ω]==f (y, S=i  Ω)f (y  Ω)f (y  S=i, Ω)P [S=i  Ω]∑  f (y  S= j, Ω)P [S= j  Ω]

(16)

,

(17)

which is known as a filtered probability at time t in the literature. Combing thisupdating formula with eq.(7), we can iteratively compute the marginal density ofy, eq.(11). Then, the log-likelihood function is given byL(λ)=∑ lnf (y  Ω；λ),

(18)



where λ denotes a vector of the unknown constant parameters, (μ，ϕ，p，σ )',i=1, 2, ⋯, M, k=1, 2, ⋯, K, and j=1, 2, ⋯, M −1. The unknown parameters
can be estimated by maximizing the log-likelihood, eq.(18). In the later analysis,
we set M =2. In this case, λ=(μ, μ, ϕ, ϕ, p, p, σ , σ )', k=1, 2, ⋯,K, and the steady-state probabilities in eq.(15) are given as follows:
ρ=

1−p
,
2−p−p

(19)

ρ=

1−p
.
2−p−p

(20)

To draw an inference about the economic states, we might use the filteredprobabilities that are calculated with observed data along with estimates of theconstant unknown parameters. It is possible, however, to obtain a better inference
as more data accumulate. Suppose we have information at time t+h to infer the
economic states at time t. Consider the following joint probabilities:

─ 128 ─Deviation Cycles in Non-Scheduled Worked Hours

Prob[S= j, S=i  Ω]=Prob[S=i  Ω] Prob[S= j  S=i, Ω].

(21)


Let Y={y, y, ⋯, y}. Then, we have:

Prob[S= j  S=i, Ω]
=Prob[S= j  S=i, Y, Ω ]

(22)

=


, S= j  S=i, Ω)f (Yf (Y
 S=i, Ω)

(23)

=


 S=i, S= j, Ω)Prob[S= j  S=i, Ω]f (Y.
f (Y
 S=i, Ω)

Since S is inferred from S with exogenous probabilities in eq.(4), if S affects
only through S, we have:
Y



f (Y
 S=i, Ω)= f (Y S=i, S= j, Ω).

(24)

This gives a substantially simplified expression of eq.(23):Prob[S= j  S=i, Ω]=Prob[S= j  S=i, Ω]. (25)Note that if S affects y∈Y
through its state-dependent lags, that is, ϕ≠0in eq. (3), then, eq. (24) holds only approximately, and eq. (25) involves anapproximation (see Kim, 1994, pp. 9-10, for discussion). If eq.(24) holds either exactly
or approximately good enough, eq.(21) can be written as:
Prob[S= j, S=i  Ω]=Prob[S=i  Ω]Prob[S= j, S=i  Ω]Prob[S=i  Ω]=Prob[S=i  Ω]Prob[S=i  S= j] P [S= j|Ω].
Prob[S=i  Ω]─ 129 ─(26)

成城・経済研究第 236 号（2022 年 3 月）

Then, we can infer S from data observed through t+h:Prob[S= j  Ω]=∑ Prob[S= j, S=i  Ω].

(27)



This is known as an h-time-ahead smoothed inference. The algorithm in eq.(26) isdeveloped by Kim (1994) (also see Ch.4 in Kim and Nelson, 1999). Given Prob[S Ω], the iteration along with eq.(26) and eq.(27) gives rise to the h-time-aheadsmoothed probabilities, Prob[S  Ω], t=t+h−1, t+h−2, ⋯, 1. When
we extend t+h to T, we obtain the full-sample smoother, proposed by Hamilton(1989), which we use in the later analysis. That is,

Prob[S= j  Ω]=∑ Prob[S= j, S=i  Ω].

(28)



Finally, we consider the following specification of eq.(4):Prob[S= j  S=i]=Φ(α)，i, j=1, 2, ⋯, M ,(29)where Φ is the standard normal cumulative distribution, and α is a constantparameter. In case of two states, M =2, we only need to estimate p and p in eq.
(4). Thus,
Prob[S=i  S=i]=Φ(α)，3i=1, 2.

(30)

Data
We use the same data as in Otsu (2021): ‘Index of Non-Scheduled Worked

Hours’ (NSWH) that is included in the individual indicators of coincident
composite indices, complied by Economic and Social Research Institute (ESRI)
affiliated with the Cabinet Office, Government of Japan. ESRI routinely examines
and revises the composition of indicators. The latest revision, the 13th revision,
was made in March 2021.
Otsu (2021) used NSWH to estimate economic slack for the period of January─ 130 ─Deviation Cycles in Non-Scheduled Worked Hours

1980 to January 2020. As Romer and Romer (2019, see p. 12) discussed, a recession
can be defined as a sustained 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 business cycles because a recession is interpreted as apart of business cycle. Otsu (2021) used bandpass filters to suppress a secular trend
and noise components and to extract detrended growth cycles, that is, the deviation
cycles. Following Burns and Mitchell (1946), the business cycles are assumed torange from 18 months (1.5 years) to 96 months (8 years), while the secular trend
corresponds to the longer-cycle components and the noise to the shorter ones. The
details are given in Otsu (2021).
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, our data set is susceptible tothis incorrect compilation. 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.
To find how useful the estimated probabilities of economic states are tounderstand the business cycle, we compare the probabilities with the official
reference dates of Japan. Table 1 shows the reference dates of peaks and troughs
identified by ESRI. It also contains periods of expansion, contraction, and durationof 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 for contraction, and 52 for the complete cycle. The reference dates

─ 131 ─

成城・経済研究第 236 号（2022 年 3 月）

are available on the website of ESRI1).
Table 2 gives a summary of descriptive statistics of dependent variables usedin the later analyses. The sample period ranges from February 1980 to January
2020, and the total number of observation is 480. In the second and the thirdcolumn, we compute the statistics with the rate of change of the level data: theoriginal official data and the data with the components higher than the seasonalfrequency being filtered out by a Butterworth filter as in Otsu (2020), respectively.
Both data show the same mean values, but the standard deviation is smaller for thelatter by 43%: the frequencies higher than seasonality give rise to a large variation.
The skewness and the kurtosis indicate that they are away from a normal
distribution. The Jarque-Bera statistic suggests non-normality.
Since the deviation-cycle data could take zero values, we use the firstdifference instead of the rate of change. The fourth column indicates the mean of
the business-cycle components is zero as expected. This is because thecomponents are computed as a deviation from a trend in the filtering process.
Therefore, even if we add the higher frequencies to the business cycle, the mean
would be zero, which is the case as shown in the fifth column. Comparing thestandard deviations in the fourth and the fifth columns, we find that the higher
frequencies contribute to a large variation, as pointed out above. Although theskewness and the kurtosis in the fourth column are smaller in absolute value thanthose in the fifth, they still indicate that the shape of the sample distribution isdifferent from that of a normal curve. In fact, the Jarque-Bera testing rejects thenormality.

1)

Indexes of Business Conditions: https://www.esri.cao.go.jp/en/stat/di/di-e.html, Dec.
2, 2021.

─ 132 ─Deviation Cycles in Non-Scheduled Worked Hours

4

Empirical Analysis

4.1 Preliminary Results
We begin with the estimation results based on the original series of ‘Index of
Non-Scheduled Worked Hours’ for the period from January 1980 to January 2020.
The dependent variable is the rate of change. Table 3 shows Maxmimum
Likelihood estimates of the model when K=0 in section 2. The conventional
wisdom says that estimates of α1 and σ  are statistically significant, while those ofother parameters are either insignificant or marginally significant. Although theestimates of μ and μ are not statistically significant, the positive estimate of μ
suggests the increase (0.12% per month) in the non-scheduled worked hours in thestate one, while the negative one (-1.10% per month) of μ indicates the decrease in
the state two.
Table 3 also shows the specification testing statistics studied by White (1987)
and Hamilton (1996). The testing does not reject the null hypothesis of the firstorder autoregressive (AR(1)) effect in residuals of the state two. The first-order
autoregressive conditional heteroskedasticity (ARCH(1)) in residuals is marginallysignificant in both states. The joint effects of AR(1) and ARCH(1) across states
emerge. These indicate the possibility of misspecification of the econometricmodel.
Table 4 and Table 5 show the estimation results when K=1 and K=2. The
estimates of the constant terms, μ and μ, are either not statistically significant orsignificant only marginally. The estimates of the autoregressive coefficients, ϕj, i,
j=1, 2, are significant in the state two, but marginal in the state one. Since theaverage of the dependent variable is -0.025%, the state one when K=1 indicatesthe increase of 0.151% on average and the state two the decrease of -0.219%.
When K=2, the non-scheduled worked hours increase by 0.160% per month in─ 133 ─

成城・経済研究第 236 号（2022 年 3 月）

the state one and decreases by -0.096% in the state two. When K=0 and K=1,
the worked hours change more rapidly when decrease than when increase. On thecontrary, the average decreasing rate in the state two is, in absolute value, smaller
than the average increasing rate in the state one.
In Table 4, the specification testing shows no AR(1) effects in residuals, but
marginal statistical significance in ARCH(1) and the joint effects. When K=2 in
Table 5, it indicates no AR(1) in the state two and no ARCH(1) in the state one, but
marginal significance of AR(1) effect in the state one and of ARCH(1) in the state
two. The joint testing indicates the presence of AR(1) and ARCH(1) effects. The
statistical testing gives mixed results on the misspecification. It seems, however,that the model with K=1 is comparatively better specified among the three cases.
To examine effects of high-frequency components on estimates, we separate
the series into components with lower than 13 months per cycle and those withhigher than 12 months per cycle, that is, seasonality. We use a bandpass filter, thetangent-based Butterworth filter, to extract a designated frequency band. In Table
6, we suppress the components that have frequencies higher than seasonality.
Since the NSWH, officially published, is a seasonally adjusted series, thesuppressed components are supposed to consist of all frequencies less than one
year. Then, the specification testing tells that the model is misspecified. In Table 7,
we only use the higher frequency components to estimate the model. Then, AR(1)effect is found in the state one, and the joint testing of AR(1) and ARCH(1)indicates marginal statistical significance. But, other statistics show no
misspecification. Therefore, the higher-frequency components make thespecification testing less likely to reject the null hypothesis of no misspecification.
Figure 1 shows the smoothed probabilities of the increasing NSWH index,
based on estimates in Table 3 through Table 6. The shaded areas in the figure
indicate the periods from peaks to troughs of the economy. Thus, we expect to─ 134 ─Deviation Cycles in Non-Scheduled Worked Hours

observe that the probability of the increasing index is high at the peak, that is, thestart of an area, and getting lower and lower for some period toward the trough,may it not be monotonically. The first panel shows the smoothed probabilities
when K=0. They well correspond to the movement of the index. In addition, it
indicates that the economy is likely to be in the state of decreasing NSWH, during
the recession periods of 1991 to 1993 and 2008 to 2009. Note that the smoothedprobabilities of eq. (25) in section 2 do not involve any computational
approximation in this case. In the second panel of K=1, the smoothedprobabilities give a similar result, but a better description for the periods of 1997 to1999 and 2000 to 2002 because they clearly indicate the economy is in the state
two, the decreasing NSWH.
In the third panel, the smoothed probabilities fluctuate a lot due to smallertransition probabilities: p=0.8404 and p=0.8303. It is hard to judge whether

they correspond to the business cycle. In the fourth panel, we suppress thefrequency components higher than seasonality. In the state one, the NSHW index
changes on average by -0.031 points when the mean of the dependent variables is-0.020, while in the state two by 0.003. Thus, the state two would indicate theincreasing state, and its smoothed probabilities appear in the fourth panel. They
move quite differently from those in other panels. It is hard to understand theirrelation with the business cycle or the index because the economy is likely to be in
the state two (increasing worked hours) during the recession after 2000.

4.2 Deviation Cycles
In Table 8, we apply the Markov switching model to the business-cycle
components, which are extracted with a Butterworth filter as in Otsu (2021), theparameter estimates are statistically significant at 5% level. The specification
testing indicates possibility of model misspecification. Table 9 shows the─ 135 ─

成城・経済研究第 236 号（2022 年 3 月）

estimation result when the model have a lagged dependent variable as an
explanatory variable. The estimates are statistically significant, except those of theconstant terms. But, the specification testing rejects the null hypotheses of noAR(1) and no ARCH(1) effects: there is no improvement in model specification.
When the higher frequency components are included, as in the previous
section 4.1, the model specification somewhat improves. The specification testingin Table 10 accepts no AR(1) effect in state one, and no ARCH(1) effect in state
two. But, it still indicates misspecification possibilities. In Table 11, even when alagged dependent variable is added to the model, the misspecification remains.
Further, many of the parameter estimates are statistically not significant.
Since the estimate on ϕ is not significant, we estimate the model with aconstraint of ϕ=0 in Table 12. Then, the specification testing only indicates

ARCH(1) effect in state one, and marginally rejects the joint hypotheses of noAR(1) and no ARCH(1) effects. In experiments, we used more lagged dependent
variables as explanatory variables, but we only obtained unreliable estimates andeven no normal convergence.
Figure 2 shows the smoothed probabilities computed with estimates in Tables
8 - 10 and Table 12. In the first panel, the smoothed probabilities are very high,almost 1, at the peaks. Although they are also high at the troughs, they sharply godown once and then go up toward the troughs. Therefore, it is possible to interpretthat the smoothed probabilities predict the end of recessions. The graph in the firstpanel is comparable with that of the second panel in Figure 1, but gives even better
description of the 1980s. The second panel shows that the smoothed probabilities
move almost oppositely to those of the first panel. Thus, the probabilities seem lag
behind the business cycle except the period of the 1990s. Note that the graphs in
the first and the second panels are drawn based on the estimates of the models of
which the conventional specification testing indicates misspecification.
─ 136 ─Deviation Cycles in Non-Scheduled Worked Hours

The smoothed probabilities in the third and the fourth panels only capture therecession from February 2008 to March 2009. In other periods, the smoothedprobabilities indicate that the economy is in the state of rising non-scheduled
worked hours. That is, they have no relation with the business cycle. Although thespecification testing tells that the models are less likely to be misspecified,particularly in Table 12, the generated smoothed probabilities seem not useful forthe business-cycle analyses.
In sum, when we estimate a simple Markov switching model estimated withthe filtered data that have only the business-cycle components, the specification
testing rejects the null hypotheses of no misspecification. But the estimates
generates the smoothed probabilities which are closely related to the businesscycle. Further, inclusion of higher frequency components make the specification
testing less likely to indicate misspecification of the econometric models that
produce the smoothed probabilities not easily understood in terms of the businesscycle.

5

Discussion
This paper attempts to estimate the smoothed probabilities of economic slackin the labor market of Japan and examines how useful they are to understand thebusiness cycle. We use Markov switching models to estimate them with adeviation-cycle data. The main findings are as follows. First, a simple Markov
switching model generates the smoothed probabilities that are closely related to thebusiness cycle. Second, inclusion of lagged dependent variables as explanatory
variables does not solve the model misspecification problem indicated by theconventional specification testing. Third, a Markov switching model fits the datawith frequency components higher than seasonality better than those without themin terms of the specification testing. The models accepted by specification testing,─ 137 ─

成城・経済研究第 236 号（2022 年 3 月）

however, do not generate the smoothed probabilities consistent with the businesscycle.
Two final remarks are in order. First, our results suggest that statistical andeconomic criteria might contradict each other. In another word, a good model in astatistical sense does not necessarily generate meaningful implication in terms of
economics.
As the American Statistical Association (ASA) noted in the statement on
statistical significance and p-values (Wasserstein and Lazar, 2016), we should not
drive final conclusions solely from results of the null-hypothesis significancetesting (NHST). That is, p-values above or below 0.05 neither prove nor disproveof the reality of anything. Therefore, we need further investigations, without any
preconceived belief, on the usefulness of the estimated smoothed probabilities interms of economics: the most pernicious effect of NHST is the delusive belief thatstatistical significance constitutes proof, as pointed out by Matthews (2021).
Secondly, it is known that statistical conditions, which the standard
asymptotic distributional theory assumes, are not satisfied in statistical testing ofMarkov switching models, particularly on numbers of states: the models areplagued with unidentified nuisance parameters and identically zero scores under
the null hypothesis. Thus, nonstandard tests are developed in the literature that
includes Hansen (1992), Garcia (1998), Cho and White (2007) and Qu and Zhuo
(2021). Then, it is interesting to see whether such a nonstandard testing leads tofinding statistical models that produce the smoothed probabilities allowing
sensible economic interpretations.
Finally, although we find a simple Markov switching model producessmoothed probabilities interpretable in terms of the business cycle, we shouldmake sure that such a simple econometric model works well for other businesscycle indices or indicators. These are left for the ongoing research.

─ 138 ─Deviation Cycles in Non-Scheduled Worked Hours

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」.

References
Burns, A. F. and W. C. Mitchell, 1946, Measuring Business Cycles, NBER Studies in
Business Cycles No. 2, NBER, New York.
Canova, F., 2007, Methods for Applied Macroeconomic Research, Princeton University
Press, New Jersey.
Cho, J. S. and White, H., 2007, “Testing for regime switching,” Econometrica, 75, 16711720.
Christiano, L. J. and T. J. Fitzgerald, 2003, “The band pass filter,” International Economic
Review 44(2), 435-465.
Garcia, R., 1998, “Asymptotic null distribution of the likelihood ratio test in Markov
switching models,” International Economic Review 39(3), 763-788.
Gomez, V., 2001, “The use of Butterworth filters for trend and cycle estimation ineconomic time series,” Journal of Business and Economic Statistics 19(3), 365-373.
Hamilton, J. D., 1989, “A new approach to the economic analysis of nonstationary timeseries and the business cycle,” Econometrica 57, 357-384.
Hamilton, J. D., 1996, “Specification testing in Markov-switching time-series models,”
Journal of Econometrics 70, 127-157.
Hamilton, J. D., 2016. “Macroeconomic Regimes and Regime Shifts,” Ch.3, in Handbookof Macroeconomics, Volume 2A, edited by John B. Taylor and Harald Uhlig,
Amsterdam: Elsevier, 163-201.
Hansen, B. E., 1992, “The likelihood ratio test under nonstandard conditions: Testing theMarkov switching model of GNP,” Journal of Applied Econometrics 7, S61-S82.
Erratum, 1996, 11, 195-198.
Iacobucci, A. and A. Noullez, 2005, “A frequency selective filter for short-length timeseries,” Computational Economics 25, 75-102.
Jarque, C. M. and A. K. Bera, 1987, “A test for normality of observations and regression
residuals,” International Statistical Review 55(2), 163-172.
Kim, C. J., 1994, “Dynamic linear models with Markov-switching,” Journal of
Econometrics 60, 1-22.
Kim, C. J., and C. R. Nelson, 1999, State-Space Models with Regime Switching, MIT
Press, Cambridge, M. A.
Matthews, R., 2021, “The p-value statement, five years on,” Sig- nificance, 18(2), 16-19.

─ 139 ─

成城・経済研究第 236 号（2022 年 3 月）

Otsu, T., 2020, “Robust statistics and business cycle dating,” Seijo University Economic
Papers 228, 93-136, The Economic Institute of Seijo University (Japan).
Otsu, T., 2021, “Slack in labor market and business cycles,” Seijo University Economic
Papers 232, 101-140, The Economic Institute of Seijo University (Japan).
Pollock, D. S. G., 2000, “Trend estimation and de-trending via rational square-wave
filters,” Journal of Econometrics 99, 317-334.
Qu, Z. and F. Zhuo, 2021, “Likelihood ratio-based tests for Markov regime switching,”
The Review of Economic Studies 88(2), 937-968.
Romer, C. D., and D. H. Romer, 2019, “NBER business cycle dating: history andprospect,” prepared for the session NBER and the Evolution of Economic Research,
1920-2020 at the ASSA Annual Meeting, San Diego, California, January 2020.
Wasserstein and Lazar, N. A., 2016, “The ASA’s statement on p-values: context, process,and purpose,” American Statistician, 70(2), 129-133.
White, H., 1987, “Specification testing in dynamic models,” in: T. F. Bewley, eds.,
Advances in Econometrics, Fifith World Congress, Vol. I, Chpt. 1, Cambridge
University Press, Cambridge, 1-58.

─ 140 ─Deviation Cycles in Non-Scheduled Worked Hours

Table 1 Reference Dates of Business Cycles: Japan
Date (month, year)
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*
May,
2020*
Peak

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, November 30, 2021.

Table 2 Descriptive Statistics of Dependent Variables
Rate of Change (%)
Statistic

Official Data
(S.A.)*

mean-0.0202(standard error)(0.0556)
standard deviation
1.2185(standard error)(0.1593)skewness-0.7454(standard error)(0.1111)
kurtosis6.5430(standard error)(0.2201)
maximum value
3.2508minimum value-7.1763
Jarque-Bera Stat.**295.5017(p-value)(0.0000)
Sample Period: Feb. 1980 to Jan. 2020

Passed Band[0, 2π/13]-0.0202(0.0318)
0.6962(0.0627)-1.3371(0.1111)9.0725(0.2201)
2.3093-3.7469
880.5278(0.0000)

First Difference
Passed Band
[2π/96, 2π/18]

0.00000.0000(0.0440)(0.0525)0.96501.1495(0.0728)(0.1235)0.0017-0.4994(0.1111)(0.1111)
3.94475.2054(0.2201)(0.2201)3.42093.4941-3.3436-5.7733
17.8505
117.2285(0.0001)(0.0000)# of obs.: 480

Note: * Seasonally Adjusted series. ** See Jarque and Bera (1987).

─ 141 ─Passed Band
[2π/96, 2π/2]成城・経済研究第 236 号（2022 年 3 月）

Table 3 Estimation of Markov Switching Model: lag K=0
Parameter
Estimate (1)μ
0.118955μ-1.096310
α2.214343α1.307382σ
1.009709σ1.936229
Log-Likelihood value: -739.8542
Implied transition probability of staying

Standard Error (2)*0.175755
1.267497
0.425981
0.822758
0.137952
1.071101in state one: 0.9866;

Specification Testing**
Null Hypothesisχ  stat.
No AR(1) in residuals
in state one
0.7468in state two
13.9268
No ARCH(1) in residuals
in state one
5.1498in state two
6.4100No AR(1) and ARCH(1) in both states
20.2367(1)/(2)0.677-0.865
5.198
1.589
7.319
1.808

p-value
0.49850.38710.00000.11210.00000.0707# of obs.: 480
in state two: 0.9045

d.f.
1
1
1
1
4

p-value
0.38750.00020.02320.01130.0004

Note: The dependent variable is the rate of change.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

Table 4 Estimation of Markov Switching Model: lag K=1
Parameter
Estimate (1)μ
0.145106ϕ-0.235354
μ-0.206045ϕ
0.533912α
1.914620α1.397630σ0.938806σ1.431189
Log-Likelihood value: -726.5472
Implied transition probability of staying

Standard Error (2)*0.073330
0.123192
0.207374
0.186637
0.323686
0.298379
0.062638
0.203039
in state one: 0.9722;

Specification Testing**
Null Hypothesisχ  stat.
No AR(1) in residuals
in state one
1.2557in state two
1.2306
No ARCH(1) in residuals
in state one
4.7044in state two
6.5763
No AR(1) and ARCH(1) in both states
11.8272(1)/(2)
1.979-1.910-0.994
2.861
5.915
4.684
14.9887.049

p-value
0.04780.05610.32040.00420.00000.00000.00000.0000# of obs.: 479
in state two: 0.9189

d.f.
1
1
1
1
4

p-value
0.26250.26730.03010.01030.0187

Note: The dependent variable is the rate of change.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

─ 142 ─Deviation Cycles in Non-Scheduled Worked Hours
Table 5 Estimation of Markov Switching Model: lag K=2
Parameter
Estimate (1)μ
0.143496ϕ-0.445443
ϕ-0.228365μ-0.076363
ϕ
0.460431
ϕ
0.326100α0.996271
α0.955174σ
0.963335σ0.994711
Log-Likelihood value: -720.9461
Implied transition probability of staying

Standard Error (2)*0.121809
0.166390
0.122723
0.082629
0.131414
0.084314
0.863142
0.322274
0.071872
0.159658
in state one: 0.8404 ;Specification Testing**
Null Hypothesisχ  stat.
No AR(1) in residuals
in state one
4.0309in state two
0.4196
No ARCH(1) in residuals
in state one
2.2403in state two
6.5791No AR(1) and ARCH(1) in both states
16.5823(1)/(2)
1.178-2.677-1.861-0.924
3.504
3.868
1.154
2.964
13.4046.230

p-value
0.23880.00740.06280.35540.00050.00010.24840.00300.00000.0000# of obs.: 478
in state two: 0.8303

d.f.
1
1
1
1
4

p-value
0.04470.51710.13450.01030.0023

Note: The dependent variable is the rate of change.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

 13 

Table 6 Estimation of Markov Switching Model: lag K=1, passband = 0,
Parameter
Estimate (1)μ-0.011628ϕ
0.941869μ
0.022683
ϕ
0.989060α
1.828202α1.418384σ
0.066488σ0.283458
Log-Likelihood value : 361.8797
Implied transition probability of staying

Standard Error (2)*0.005579
0.012344
0.025617
0.034801
0.108519
0.185042
0.004929
0.036483
in state one : 0.9662 ;Specification Testing**
Null Hypothesisχ  stat.
No AR(1) in residuals
in state one
288.86232
in state two
177.31453
No ARCH(1) in residuals
in state one
130.00234
in state two
39.72080
No AR(1) and ARCH(1) in both states
427.82944

2π(1)/(2)-2.08476.3020.885
28.42016.8477.665
13.4907.770

p-value
0.03710.00000.37590.00000.00000.00000.00000.0000# of obs.: 479
in state two: 0.9220

d.f.
1
1
1
1
4

p-value
0.00000.00000.00000.00000.0000Note: The dependent variable is the rate of change.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

─ 143 ─

成城・経済研究第 236 号（2022 年 3 月）

Table 7 Estimation of Markov Switching Model: lag K=1, passband =Parameter
Estimate (1)μ-0.005808
ϕ-0.348756μ-0.006848
ϕ-0.096412
α1.614109α0.730825σ0.720943σ1.532536
Log-Likelihood value: -621.9115
Implied transition probability of staying

Standard Error (2)*0.036867
0.0551340.150284
0.169189
0.423039
0.256415
0.071872
0.193351
in state one : 0.9467;

Specification Testing**χ  stat.
in state one
9.9882in state two
0.0092
No ARCH(1) in residuals
in state one
2.1377in state two
1.4615
No AR(1) and ARCH(1) in both states
12.6773

Null Hypothesis
No AR(1) in residuals

 12 , 2 2π2π(1)/(2)-0.158-6.326-0.046-0.5703.816
2.850
10.0317.926

p-value
0.87480.00000.96370.56880.00010.00440.00000.0000# of obs.: 479
in state two: 0.7676

d.f.
1
1
1
1
4

p-value0.0016
0.92340.14370.22670.0130Note: The dependent variable is the first difference.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

Table 8 Estimation of Markov Switching Model: lag K=0, passband =Parameter
Estimate (1)μ
0.093496μ-0.289540
α
2.031778α1.487239σ0.284500σ1.035998
Log-Likelihood value: -266.6184
Implied transition probability of staying

Standard Error (2)*0.021771
0.117931
0.105717
0.108531
0.011534
0.073706
in state one: 0.9789;

Specification Testing**χ  stat.
in state one
329.2961
in state two
140.6161
No ARCH(1) in residuals
in state one
135.8444
in state two
60.3336
No AR(1) and ARCH(1) in both states
446.1956

Null Hypothesis
No AR(1) in residuals

 96 , 18 2π2π(1)/(2)4.295-2.45519.21913.70324.66614.056p-value
0.00000.01410.00000.00000.00000.0000# of obs.: 480
in state two: 0.9315

d.f.
1
1
1
1
4

p-value
0.00000.00000.00000.00000.0000Note: The dependent variable is the first difference.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

─ 144 ─Deviation Cycles in Non-Scheduled Worked Hours

Table 9 Estimation of Markov Switching Model: lag K=1, passband =Parameter
Estimate (1)μ-0.009204
ϕ
0.947211
μ
0.028452ϕ
0.997802α
1.884400α1.272363σ
0.061765σ0.238799
Log-Likelihood value: 461.3215
Implied transition probability of staying

Standard Error (2)*0.006683
0.038573
0.037744
0.034484
0.318420
0.1271200.012973
0.058994
in state one: 0.9702;

Specification Testing**χ  stat.
in state one
326.0702
in state two
166.5462
No ARCH(1) in residuals
in state one
161.5971
in state two
61.3015
No AR(1) and ARCH(1) in both states
440.6322
Null Hypothesis
No AR(1) in residuals

 96 , 18 2π2π(1)/(2)-1.37724.5570.754
28.9355.918
10.0094.761
4.048

p-value
0.16840.00000.45100.00000.00000.00000.00000.0001# of obs.: 479
in state two: 0.8984

d.f.
1
1
1
1
4

p-value
0.00000.00000.00000.00000.0000Note: The dependent variable is the first difference.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

Table 10 Estimation of Markov Switching Model: lag K=0, passband =Parameter
Estimate (1)μ
0.039724μ-4.638722
α2.843678α
0.632870σ
1.066225σ
0.912002
Log-Likelihood value: -720.6579
Implied transition probability of staying

Standard Error (2)*0.049567
0.429257
0.300776
0.674055
0.039122
0.192502
in state one: 0.9978;

Specification Testing**χ  stat.
in state one
0.0761in state two
12.7162
No ARCH(1) in residuals
in state one
7.2408in state two
0.3163
No AR(1) and ARCH(1) in both states
283.6656

Null Hypothesis
No AR(1) in residuals

 96 , 2 2π2π(1)/(2)0.050-10.806
9.454
0.939
27.2544.738

p-value
0.42290.00000.00000.34780.00000.0000# of obs.: 480
in state two: 0.7366

d.f.
1
1
1
1
4

p-value
0.78270.00040.00710.57380.0000Note: The dependent variable is the first difference.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

─ 145 ─

成城・経済研究第 236 号（2022 年 3 月）

Table 11 Estimation of Markov Switching Model: lag K=1, passband =Parameter
Estimate (1)μ
0.037448ϕ-0.104763
μ-0.123096ϕ
0.552941α2.164467α1.204721
σ0.984642σ1.614181
Log-Likelihood value: -714.0702
Implied transition probability of staying

Standard Error (2)*0.100012
0.182284
0.249174
0.220038
1.037087
0.446357
0.107822
0.631742
in state one: 0.9848;

Specification Testing**χ  stat.
in state one
6.6071in state two
3.1374
No ARCH(1) in residuals
in state one
7.0750in state two
0.2480No AR(1) and ARCH(1) in both states
13.6935

Null Hypothesis
No AR(1) in residuals

 96 , 2 2π2π(1)/(2)0.374-0.575-0.494
2.513
2.087
2.699
9.132
2.555

p-value
0.70810.56550.62130.01200.03690.00700.00000.0106# of obs.: 479
in state two: 0.8858

d.f.
1
1
1
1
4

p-value
0.01020.07650.00780.61850.0083

Note: The dependent variable is the first difference.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

Table 12 Estimation of Markov Switching Model: K=1, ϕ=0, passband =Parameter
Estimate (1)μ
0.015548μ-0.087890
ϕ
0.598596α2.605718α1.549390σ
1.021331σ1.750022
Log-Likelihood value: -714.0702
Implied transition probability of staying

Standard Error (2)*0.056555
0.328173
0.248715
0.557654
0.851682
0.053686
0.202081
in state one: 0.9954;

Specification Testing**χ  stat.
in state one
1.7539in state two
1.9011No ARCH(1) in residuals
in state one
9.0109in state two
0.7520No AR(1) and ARCH(1) in both states
11.1123

Null Hypothesis
No AR(1) in residuals

 96 , 2 2π2π(1)/(2)0.275-0.268
2.407
4.673
1.819
19.0248.660

p-value
0.78340.78880.01610.00000.06890.00000.0000# of obs.: 479
in state two: 0.9394

d.f.
1
1
1
1
4

p-value
0.18540.16800.00270.38580.0253

Note: The dependent variable is the first difference.
* Heteroskedastic consistent estimates. ** See White (1987) and Hamilton (1996).

─ 146 ─Deviation Cycles in Non-Scheduled Worked Hours

Figure 1 Smoothed Probabilities: Non-Scheduled Worked Hours (Seasonally Adjusted)
150

Explanatory variables : no lagged depedent variable.
1

0.6

100

0.4
Official Data
Smoothed(K=0)
Peak to TroughProbability

Index

0.8

0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)
150

Explanatory variables : a lagged depedent variable K = 1.
1

0.6

100

0.4
Official Data
Smoothed(K=1)
Peak to TroughProbability

Index

0.8

0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)

150

Explanatory variables : lagged depedent variables K = 2
1

0.6

100

0.4
Official Data
Smoothed(K=2)
Peak to TroughProbability

Index

0.8

0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)
150

Explanatory variables : a lagged dependent var. K = 1, passband [0, 2 /13].
1

0.6

100
Official Data
Smoothed(K=1)
Peak to Trough

0.4
0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)─ 147 ─Probability

Index

0.8

成城・経済研究第 236 号（2022 年 3 月）

Figure 2 Smoothed Probabilities: Non-Scheduled Worked Hours (Business Cycle)
150

Explanatory variables : no lagged depedent variable, passband [2 /96, 2 /18].
1

0.6

100
Official Data
Smoothed(K=0)
Peak to Trough

0.4

Probability

Index

0.8

0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)
150

Explanatory variables : a lagged depedent variable K = 1, passband [2 /96, 2 /18].
1

0.6

100
Official Data
Smoothed(K=1)
Peak to Trough

0.4

Probability

Index

0.8

0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)

150

Explanatory variables : no lagged depedent variable, passband [2 /96, 2 /2].
1

0.6

100
Official Data
Smoothed(K=0)
Peak to Trough

0.4

Probability

Index

0.8

0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)
150

Explanatory variables : a lagged dependent var. K = 1, 11 = 0, passband [2 /96, 2 /2].
1

0.6

100
Official Data
Smoothed(K=1)
Peak to Trough

0.4
0.2

50
0
81-01 83-01 85-01 87-01 89-01 91-01 93-01 95-01 97-01 99-01 01-01 03-01 05-01 07-01 09-01 11-01 13-01 15-01 17-01 19-01

Date (year-month)─ 148 ─Probability

Index

0.8