SDS-18
Cyclical Components Synthesization
Approach to Construct a Coincident Index
of Business Cycles
Koki Kyo, Hideo Noda
and Genshiro Kitagawa

November 2020

Statistics & Data Science Series back numbers:
http://www.mims.meiji.ac.jp/publications/datascience.html

Cyclical Components Synthesization Approach toConstruct a Coincident Index of Business Cycles
Koki Kyo∗ , Hideo Noda† , and Genshiro Kitagawa‡,§*Faculty of Management and Information Science, Niigata University of Management†Corresponding author, School of Management, Tokyo University of Science,
1-11-2 Fujimi, Chiyoda-ku, Tokyo 102-0071, Japan
E-mail: noda@rs.tus.ac.jp, Phone: +81-3-3288-2570, Fax: +81-3-3288-2498‡Mathematics and Informatics Center, University of Tokyo§Meiji Institute for Advanced Study of Mathematical Sciences (MIMS), Meiji University
November 26, 2020

Abstract
The accurate assessment of business conditions is a long-standing problem
in macroeconomics. To construct a coincident index of growth cycles from agiven set of indicators, we propose a new approach, the cyclical components
synthesization (CCS) approach. We refer to the coincident index of growth
cycles as the index of business cycles (IBC) of coincident economic indications.
The IBC based on the CCS approach has the following properties: (1) its meanis globally stationary; (2) it is constructed as a common factor in the stationary
parts of the selected economic indicators; and (3) its variations are as large as
possible and, therefore, it contains a larger amount of information on businessfluctuations. We examine the performance of the proposed IBC by comparing
it with the composite index and the Nikkei Business Index. The results showthat the CCS approach has a number of advantages over existing methods.
Keywords: Business cycle index; Bayesian modeling; State space model;
Kalman filter; Composite index; Nikkei Business Index

1

1

Introduction

Following the seminal work of Burns and Mitchell (1946), the measurement of business cycles has been recognized as an important issue in macroeconomic studies.
Furthermore, favorable or unfavorable business conditions are of great interest to
many, as the assessment of business conditions has a significant influence on government economic policies. However, conventional indexes may not accurately capturebusiness cycles. Therefore, the question that arises is: what is a better way to
produce useful and reliable indexes? This is a long-standing problem and was oneof the central questions addressed by Stock and Watson (1989). In this paper, we
aim to present an alternative approach to constructing a business cycle index using
coincident economic indicators. Specifically, we attempt to measure growth cycles
and apply the proposed method to Japanese data. As a monthly coincident index ofgrowth cycles in Japan is, to our knowledge, a new development, it may be of broadinterest to macroeconomists.
Conventionally, business conditions are assessed using summary measures for thestate of macroeconomic activity in Japan and the United States (US). The composite
index (CI) and diﬀusion index (DI) are representative summary measures. Althoughboth have the advantage of manageability, they have faced criticism because they
are not based on a strict statistical model. Given this, since the 1980s, there havebeen many studies of statistical methods for business cycle analysis. Pioneering
works in this area are by Stock and Watson (1989, 1991), who developed a statistical method to construct an index of business cycles (IBC) based on a state spacemodel. Stock and Watson (1989, 1991) define the business cycle as a co-movementof macroeconomic variables, and the Stock–Watson index is constructed by extracting the common factor hidden in multiple macroeconomic time series data. Thus,
their proposed model is commonly termed the dynamic factor model, and it was first
applied to analyze the US business cycle. Ohkusa (1992) and Fukuda and Onodera(2001) have applied the Stock–Watson dynamic factor modeling approach to analyze
Japanese business cycles.
The dynamic factor modeling approach has since been extended by Kanoh andSaito (1994), Mariano and Murasawa (2003, 2010), Watanabe (2003), and Urasawa(2014). Kanoh and Saito (1994) extended the dynamic factor model to include
qualitative data from the Short-Term Economic Survey of Enterprises (abbreviated2to Tankan), a statistical survey conducted by the Bank of Japan. Focusing on thecompanies’ judgments about business conditions in Tankan, Kanoh and Saito (1994)
considered that if such judgments reflected an overall assessment of actual businessconditions, then a business index that statistically extracts the actual state of theeconomy from such judgments would be a more appropriate measure of the overall
state of the economy than conventional indexes. They concluded that the peaks andtroughs projected by their proposed index have systematic relationships with thebusiness cycles identified by experts from the Japanese government. Mariano andMurasawa (2003) pointed out that the CI and the Stock–Watson coincident index
have two shortcomings: first, they ignore information contained in quarterly indicators, such as quarterly real gross domestic product (GDP); and second, they lackeconomic interpretation. Therefore, Mariano and Murasawa (2003) extended theStock–Watson coincident index by applying a maximum likelihood factor analysis
to a mixed-frequency series of quarterly real GDP and monthly coincident businesscycle indicators. The resulting index is related to latent monthly real GDP. Furthermore, Mariano and Murasawa (2010) estimated Gaussian vector autoregression(VAR) and factor models for latent monthly real GDP and other coincident indicators using observable mixed-frequency series. For the maximum likelihood estimation of a VAR model, the expectation-maximization algorithm helps to identify agood starting value for a quasi-Newton method. Mariano and Murasawa (2010) concluded that the smoothed estimate of latent monthly real GDP is a natural extensionof the Stock–Watson coincident index. To obtain early estimates of Japan’s quarterly GDP growth in real time, Urasawa (2014) estimated a dynamic factor modelusing mixed-frequency data on GDP, industrial production, employment, private
consumption, and exports. The results of a real-time forecasting exercise suggested
that the model performs well.
Another prominent approach that diﬀers from dynamic factor modeling is theregime-switching modeling approach developed by Hamilton (1989). Whereas dynamic factor modeling is associated with a CI-type index, the regime-switching modeling is associated with a DI-type index. Kim and Nelson (1998) developed a methodthat combined the above two approaches. Watanabe (2003) applied the approach of
Kim and Nelson (1998) to the Japanese economy, and estimated a dynamic Markov
switching factor model using macroeconomic data.
Here, we note a data treatment problem regarding the Stock–Watson dynamic

3

factor modeling approach and its extensions. Specifically, many earlier studies useddiﬀerencing in time series data to obtain stationariness in cases where nonstationary
data (e.g., the mean) was used for convenience. This may result in a loss of significant information. In this paper, we propose an alternative approach to construct a
coincident IBC via the decomposition of time series data into several possible components. In contrast to the Stock–Watson index, we consider an IBC that has thefollowing properties: (1) it is globally stationary in mean; (2) it is constructed using
the common factor of all the relevant coincident indicators; and (3) it has variations
that are as large as possible and, therefore, it contains a larger amount of information on business cycle fluctuations. The IBC based on our proposed approach hasa higher correlation with the cyclical component of GDP than the CI or the Nikkei
Business Index (NBI). That is, the results show that our index performs better thaneither of these indexes.
The remainder of this paper is organized as follows. In Section 2, we present a
brief review of some existing approaches. In Section 3, we explain the frameworkof our new approach for constructing a business cycle index. Then, we describe theparameter estimation procedure in Section 4. In Section 5, we compare the performance of the IBC based on the newly-proposed approach with the CI of coincident
indicators and the NBI using Japanese economic data. In Section 6, we discuss our
new approach. We present our conclusions in Section 7.

2

Brief review of existing approaches and evaluation

2.1

Diﬀusion index, composite index, and the MTV model

There is no uniform set of oﬃcial indicators used to determine business conditions
among developed countries. For example, the CI has a central role in measuringbusiness cycles in the US, the United Kingdom, and Italy, whereas the growth rates
of GDP are emphasized in the measurement of business cycles in Canada.
In Japan, business conditions are typically measured using business cycle indicators such as the CI and DI, which are compiled by the Economic and Social ResearchInstitute (ESRI) of the Japanese Cabinet Oﬃce. Since April 2008, the ESRI has
placed greater emphasis on the CI than on the DI in assessing business conditionsin Japan.
According to the Cabinet Oﬃce, the indicators for indexes of business conditions

4

are re-examined after each business cycle and changed if it is expected that the performance of the indexes will be improved. The DI and CI complied by Japan’s ESRI
consist of three indexes: a leading index (which is constructed based on 11 indicators), a coincident index (based on 10 indicators), and a lagging index (based onnine indicators). The DI represents the ratio of a series that increased in value during the most recent 3-month period. The coincident DI is considered an important
index of peaks and troughs: its 50% line is frequently used to identify the turning
point of a business cycle. However, oﬃcial reference dates are determined by the
ESRI on the basis of much broader information, including other economic indicatorsand professional opinions. The CI uses the same data series as the DI. However,unlike the DI, which only considers the direction of changes in indicators, the CI
takes into account the degree of change. Therefore, the CI is considered useful inmeasuring the speed and magnitude of a business cycle. Originally, the CI and DI
were developed by the National Bureau of Economic Research (NBER) in the US
and the business cycle indicators of the Japanese Cabinet Oﬃce are based on the
NBER calculation method. Although these business cycle indexes are widely used,
it is recognized that they suﬀer from various defects. In particular, they are notderived from any sound statistical methods (Kanoh and Saito, 1994).
Kariya (1988, 1993) proposed a multivariate time series variance component
model known as the MTV model. The basic concept of this model is that it uses
principal component analysis with time series data on macroeconomic variables.
When we apply the MTV model to coincident indicators, a principal component ofthese indicators can be regarded as an IBC. Note that the MTV model may leadto diﬀerent results depending on the component on which it focuses. Therefore,
a definite conclusion cannot be obtained unless there is some objective criterion
justifying a focus on the obtained principal component.

2.2

Approach of Stock and Watson

Stock and Watson (1989, 1991) proposed a more reliable index of business cycles
that measures the state of the economy using a time series model. In this approach,the signal for business fluctuations is generally expressed by a co-movement of selected macroeconomic variables. Thus, Stock and Watson suggested decomposing
macroeconomic variables into common factors and unique (idiosyncratic) factors.
The common factor is considered to be derived by a single unobservable dynamic

5

variable, which is called the state of the economy.
Specifically, a macroeconomic time series xin , for one of m indicators, is expressedas:∆xin = µi + γi ∆cn + uin ,ϕ(L)∆cn = ηn ,di (L)uin = ϵin

(1)
(2)

(i = 1, 2, . . . , m),(3)

where µi is an unknown parameter, γi is the unknown factor loading for the ith
indicator, ∆cn and uin are the common and unique factors of ∆xin , respectively,
with i = 1, 2, . . . , m, and ϕ(L) and di (L) are polynomials with appropriate orders.
Here, L denotes a lag operator, such as Lj ∆cn = ∆cn−j , with ∆ as the diﬀerence
operator defined by ∆ = 1 − L. Moreover, ηn and ϵin are usually assumed to be
Gaussian white noise sequences, which are independent of each other.
Thus, the models in Eqs (2) and (3) are autoregressive (AR) models for ∆cn anduin . To obtain good properties for estimates, it is assumed that ∆cn and uin are
stationary in mean; therefore, ∆xin is also stationary in mean. For most practicalsituations, time series data for the indicators show distinct trends and, therefore, it isnecessary to take the first diﬀerence of the time series xin so that ∆xin is stationary.
Note that the diﬀerence ∆cn for factor cn ensures the correspondence between xin
and cn . When the time series xin is considered stationary, ∆xin in Eq. (1) can be
replaced with xin , and ∆cn in Eqs (1) and (2) can be replaced with cn .
The above dynamic factor models can be expressed by a state space representation. Thus, the Kalman filter algorithm can be applied to estimate the parameters
and the common factor ∆cn and, hence, cn . The estimate for cn is called the Stock–
Watson index (SWI).

2.3

Evaluation

As mentioned in Fukuda (1994) and Komaki (2001), the approaches used to construct business cycle indexes can be classified in terms of whether they consider
deterministic or stochastic trends and the cyclical components of time series data.
Specifically, DI, CI, and MTV modeling approaches are based on a deterministic
perspective. In contrast, dynamic factor modeling is based on a stochastic perspective. An advantage shared by the DI and CI is their simplicity of construction.
However, because they are not based on a clear statistical assumption, there can6be diﬃculties in using them for prediction. The MTV modeling approach is basedon a model involving principal component analysis. However, in general, the data
structure is not clearly defined. Furthermore, principal component analysis often
results in interpretation diﬃculties.
Conversely, the Stock–Watson dynamic factor modeling approach is based on astate space model and the Kalman filter. Therefore, this approach is based on a setof statistical models expressing the structure of the data. However, to satisfy theassumption of stationarity for the data used, many earlier methods take an ad hoc
approach to the treatment of data, such as the use of diﬀerences for the time series.
Another problem with the Stock–Watson approach is the diﬃculty of determining
whether to use diﬀerencing.
Moreover, because of the treatment of the data, it may be diﬃcult to interpret
the SWI results from an economic perspective. Consider here a case, as in Eq. (1),where the diﬀerence in the data is used. Let the polynomial ϕ(L) in Eq. (2) begiven by:ϕ(L) = α0 +

p
∑

αj Lj

j=1

with α0 ̸= 0. Then, from Eq. (2), we can see that the estimated SWI is a result of
the autoregressive integrated moving average model given by:
1
1 ∑αj Lj (cn − cn−1 ) +ηn .
α0 j=1α0pcn = cn−1 −(4)Thus, the behavior of the estimated SWI is very complicated because it is a weighted
random walk process. Therefore, we agree with Mariano and Murasawa (2003) thatthe results of the SWI cannot be interpreted from an economic perspective. Moreover, a further diﬃculty of the Stock–Watson approach is that both the stationary
and nonstationary indicators are included in the coincident indicators.
It is widely accepted that GDP is an inclusive measure of an economy’s performance and, therefore, that it can be used as a real business cycle index. However,using GDP in this way does pose a number of problems, largely due to a lack of statistical suﬃciency and the conflict between the need for immediate reports and the factthat GDP statistics are made quarterly, not monthly, and there is a long lag prior
to their publication. It should be noted that Mariano and Murasawa (2003, 2010)
added GDP together with coincident indicators to construct a business cycle index.
Although this is an excellent concept, the problem inherent in the Stock–Watson

7

approach cannot be completely resolved because it remains based on dynamic factor
modeling.

3

Proposed approach

3.1

Model construction

Let yi (n) denote a monthly time series, which is one of m coincident indicators withi = 1, 2, . . . , m. Generally, it is considered that each indicator (or transformed indicator) comprises nonstationary and stationary parts. Furthermore, the nonstationary
part contains a trend component (which expresses the long-term tendency) and aseasonal component (which expresses a patterned variation that appears repeatedly
every year). In contrast, the stationary part is considered to have a cyclical component (which is caused by business cycles) and an irregular component. In Kitagawa(2010) and Kitagawa and Gersh (1984), it is assumed that the cyclical component
can be expressed by a stationary AR model, also called the AR component.
Let g(yi (n)) be a one-to-one transformation of the time series yi (n) for the i-th
indicator. Based on the above consideration, the model for g(yi (n)) can be writtenas:g(yi (n)) = ti (n) + si (n) + ai (n) + εi (n) (i = 1, 2 . . . , m),(5)where ti (n), si (n), and ai (n) denote the trend, seasonal, and cyclical components,
respectively, and εi (n) is the irregular component, also called the observation error.
Note that the function g is determined so that g(yi (n)) can be reasonably expressedin an additive multi-component form, as expressed in Eq. (5). As an example, alogarithmic function is frequently applied to g.
In this paper, we consider a case in which the time series yi (n) has been seasonallyadjusted. Therefore, the seasonal component si (n) does not necessarily need to betaken into account. Thus, the model in Eq. (5) is rewritten as follows:g(yi (n)) = ti (n) + ai (n) + εi (n)(i = 1, 2 . . . , m),(6)

where the observation error εi (n) is a random variable distributed with εi (n) ∼N(0, σi2 ). To obtain meaningful estimates for every component, we employ a Bayesian
approach and treat ti (n) and ai (n) as random variables. Thus, we need to introduce
prior models based on the assumptions about the structure of each component.

8As is typical for economic analysis, we assume that the trend component varies
smoothly over time and that the cyclical component is globally stationary. Therefore, we introduce a second-order stochastic diﬀerence equation as follows:ti (n) = 2ti (n − 1) − ti (n − 2) + ωi (n) (i = 1, 2, . . . , m),(7)where ωi (n) is a random error distributed with ωi (n) ∼ N(0, τi2 ). Furthermore, we
use an AR model for ai (n) as follows:ai (n) =qi∑(i)

βj ai (n − j) + ψi (n) (i = 1, 2, . . . , m),

(8)

j=1
(i)

where qi is the model order, {βj ; j = 1, 2, . . . , qi } are the AR coeﬃcients, and ψi (n)is a Gaussian white noise distributed with ψi (n) ∼ N(0, ηi2 ).
The model in Eq. (6) is an observation model for the time series g(yi (n)), andthe models in Eqs (7) and (8) can be considered as a set of prior models for thetrend ti (n) and cyclical ai (n) components, respectively. Obviously, Eqs (6)–(8) takethe form of a Bayesian linear Gaussian model for the trend and cyclical components.
For the case where the model for each value of i is individually managed, Kitagawa(2010) developed a maximum likelihood method to estimate the parameters σi2 , τi2 ,(i)

ηi2 , and {βj ; j = 1, 2, . . . , qi } based on the Kalman filter algorithm.
As mentioned in Section 1, Stock and Watson (1989, 1991) defined the SWI as
co-movements in a set of macroeconomic variables. Therefore, it is constructed byextracting a common factor that is hidden in multiple macroeconomic time seriesdata.
In contrast to the Stock–Watson approach, we consider that economic cycles canbe organized using the basic classification of short- and long-term cycles. Short-term
cycles generally mean business cycles or business fluctuations. Thus, to simplify the
analysis of business fluctuations, it is necessary to separate business cycles fromother economic cycles in the longer term.
Therefore, we expect that an index for business fluctuations would have thefollowing properties:
1. It is globally stationary in mean.
2. It is based on a common factor from the stationary parts of all of the economic
indicators used.

93. It has variations that are as large as possible and, therefore, it contains a larger
amount of information on business fluctuations.
Thus, to construct an IBC using a common factor extracted from a set of cyclicalcomponents in a time series for coincident indicators, we take the common factorof the cyclical components {ai (n); i = 1, 2 . . . , m}. This can be easily achieved byapplying the method of principal component analysis to the data for {ai (n); i =1, 2 . . . , m}.
To derive an index that is free from the scale of the data, we construct the IBC
based on the results of normalized cyclical components. That is, we normalize thecyclical components by:a∗i (n) =ai (n)
SD{ai (n)}(i = 1, 2, . . . , m).

(9)where SD{ai (n)} is the standard deviation of the time series {ai (n)}. Then, weconstruct the IBC using the first principal component of the normalized cyclicalcomponents {a∗i (n); i = 1, 2, . . . , m} as follows:b(n) =

m∑wi a∗i (n),

(10)

i=1

where {wi ; i = 1, 2, . . . , m} is the principal component loadings for the first principal
component, in which∑m2
i=1 wi = 1.

Hereinafter, the index b(n) defined in Eq. (10) is referred to as the IBC, and themethod to construct the IBC is called the cyclical components synthesization (CCS)
approach. In a procedure to estimate the parameters, the cyclical component canbe estimated as a stationary AR process with mean zero, so that the IBC becomesa stationary process with mean zero. Thus, as with the CI, the IBC can measure
the speed and magnitude of business fluctuations. Moreover, the zero value of the
IBC can be regarded as the midpoint between states of prosperity and depression.

3.2

Modeling and parameter estimation

For the models in Eqs (6)–(8), we make the following assumptions:
1. εi (n), ωi (n), and ψi (n) are independent of one another for all values of i, n.
2. εi (n1 ) and εℓ (n2 ) are independent of each other when i ̸= ℓ or n1 ̸= n2 .
3. ωi (n1 ) and ωℓ (n2 ) are independent of each other when i ̸= ℓ or n1 ̸= n2 .

10

4. ψi (n1 ) and ψℓ (n2 ) are independent when n1 ̸= n2 but may be dependent wheni ̸= ℓ.
(i)

5. The parameters σi2 , τi2 , ηi2 , and {βj ; j = 1, 2, . . . , qi } are constants over time.
Under the above assumptions, we can see that except for ti (n) and ai (n), we
have qi + 3 parameters for each indicator. When the values of these parameters are
given, we can obtain the estimates for ti (n) and ai (n). Hereafter, for each value ofi, based on Bayesian modeling, we refer to ti (n) and ai (n) as stochastic parameters.
(i)

The other parameters, that is, σi2 , τi2 , ηi2 , and {βj ; j = 1, 2, . . . , qi }, are called thehyperparameters.
The fourth assumption is based on the consideration that among the cyclicalcomponents ai (n) with i = 1, 2, . . . , m, there is a common factor to be estimated.
Thus, in principle, ti (n) and ai (n) should be estimated simultaneously for all time
points and all values of i = 1, 2, . . . , m. That is, when the sample size for the timeseries yi (n) is N , it is necessary to estimate {ti (n); n = 1, 2, . . . , N ; i = 1, 2, . . . , m},and {ai (n); n = 1, 2, . . . , N ; i = 1, 2, . . . , m} simultaneously based on the estimates
for the∑miqi + 3m hyperparameters. This is very diﬃcult to do in practice.

To mitigate the diﬃculties in estimating hyperparameters, we develop a new estimation technique combining the maximum likelihood method and principal component analysis. A key quantity is the ratio of variances for error in the cyclicalcomponent to that in the trend component, that is, for i = 1, 2, . . . , m, the proportion:
λi =

τi2
ηi2

(11)

is a key parameter for the present problem. Introducing this parameter, we cancontrol the relative variations in the trend and cyclical components and, therefore,
maintain the balance between the variances of errors in the models for the twocomponents.
As mentioned in the preceding subsection, the first objective of this study is toobtain the IBC in Eq. (10) with a larger variance, and the second is to ensure thegoodness of fit of the models to the data. Thus, we have two criteria for the modelingand parameter estimation: the first criterion is the variance of the constructed IBC
and the second is the likelihood of the parameters. In the present paper, we firstestimate the ratio λi by maximizing the contribution rate (CR) of the IBC, which is11

constructed as a principal component. Then, we estimate the other hyperparameters
by maximizing the likelihood based on the constructed model.
Moreover, as will be shown in the following section, a set of initial estimates
for the cyclical components ai (n) is necessary in an algorithm for parameter estimation, even though these initial estimates are tentative. As the initial estimates
for ai (n)(i = 1, 2, . . . , m), we will use the diﬀerences between yi (n) and its moving
average with term 2L + 1:(0)bi (n)aL∑
1= yi (n) −yi (n + ℓ)
2L + 1 ℓ=−L(12)bi (n)(0) = 0.
for n = L + 1, L + 2, . . . , N − L; otherwise, we will use a

3.3Standard score of a business cycle

The scale of the IBC is not essential for its application because it may change according to the data used. Thus, it is necessary to determine a standard to compare
diﬀerent IBC scales. To ensure easy understanding and application, we utilize theconcept of a standard score (e.g., as used to evaluate students) and define the standard score of a business cycle (SSBC) as:
SSBC(n) = 50 +10b(n)SD{b(n)}with SD{b(n)} being the standard deviation of b(n). Because the mean of b(n) iszero, we can see that the mean and the standard deviation of SSBC(n) are 50 and
10, respectively. That is, the level of the SSBC is 50, representing the midpoint
between the states of prosperity and depression.
To avoid conceptual confusion, we mainly focus on the IBC below.

4

Method for parameter estimation

4.1

Estimating the trend and cyclical components

To simplify the estimation problem, we tentatively assume that the parameters canbe estimated individually for the model of each indicator with i = 1, 2, . . . , m. To
express the model in a state space representation, we define the state vector basedon related quantities in the trend and cyclical components as:Tx(i)n = (ti (n), ti (n − 1), ai (n), ai (n − 1), · · · , ai (n − qi + 1)) ,12

and we use the following matrices:

2 −1 0
···
 1 0
0
···


(i)
(i)
 0 0 β
β2
1

 .
..
 .
.
1
0
F (i) =  .
 .
.
 .
..
0
1
 .
 .
..
..
..
 .
.
 .
.
.
0 0
0
···





··· ··· 0··· ··· 0 (i) · · · · · · βqi ··· ···
..
.
.. ..
.
.
0
1

1
 0

 0




0
0 , G = 

.


.. 
 ..
. 

 ..
.. 

 .
. 
0
0





T

0
1
 0 
0 



 1 
1 






 0 
0 
, H =  .
..  .. 
 . 
. 



 .. 
.. 


.
. 
0
0

Moreover, to reduce the number of parameters, which will be estimated using numeric methods, we follow Kitagawa (2010) and combine:εei (n) =εi (n),σie i (n) =ωωi (n)
,
σi

τi2
,
σi2

ηei2 =

with:τei2 =

ψei (n) =ψi (n)
,
σi

ηi2
.
σi2

(13)

Then, the models in Eqs (6)–(8) can be expressed by the following state space
representation:
(i)

x(i)
= F (i) xn−1 + Gv (i)
n
n ,

(14)

yi (n) = Hx(i)n + εi (n),(15)

e i (n), ψei (n))T . In the state space model comprising Eqs (14) and (15),where v n = (ω(i)

the time-varying stochastic parameters ti (n) and ai (n) are included in the state
(i)

(i)

vector xn . Therefore, their estimates can be obtained from the estimate of xn ,under the assumption that the AR model order qi and the hyperparameters, τei2 , ηei2 ,
(i)

(i)

(i)

β (i) = (β1 , β2 , and . . . , βqi )T , are given.
(i)

(i)

Let x0 denote the state at the initial time point and let Yk= {yi (1), yi (2), . . . , yi (k)}(i)

denote a set of observations for yi (n) up to the time point k. Assume that x0 ∼
(i)

(i)

(i)

(i)

N(x0|0 , C 0|0 ). It is well known that the density function f (xn |Yk ) for the state
(i)

(i)

xn , conditional on Yk , is Gaussian and, therefore, it is only necessary to obtain
(i)

(i)

(i)

(i)

(i)

the mean xn|k and the covariance matrix C n|k of xn with respect to f (xn |Yk ).
(i)

(i)

When we are given the values of qi , τei2 , ηei2 , and β (i) , the distribution N(x0|0 , C 0|0 )(i)for x0 , and a set of observations for yi (n) up to the time point N , then the estimates(i)for the state xn can be obtained using the well-known Kalman filter (for n =1, 2, . . . , N ) and fixed-interval smoothing (for n = N − 1, N − 2, . . . , 1) recursively
as follows (see Kitagawa, 2010):

13

[Kalman filter]
(i)

(i)

(i)

(i)

xn|n−1 = F (i) xn−1|n−1 ,C n|n−1 = F (i) C n−1|n−1 (F (i) )T + GQi GT .
L(i)= C n|n−1 H T (HC n|n−1 H T + Ri )−1 ,
n
(i)

(i)

(i)

(i)

(i)

xn|n = xn|n−1 + L(i)n (yn − Hxn|n−1 ),
(i)

(i)

C n|n = (I − L(i)n H)C n|n−1 .
[Fixed-interval smoothing]P (i)= C n|n (F (i) )T (C n+1|n )−1 ,
n
(i)

(i)

(i)

(i)

(i)

(i)

(i)

= xn|n + P (i)n (xn+1|N − xn+1|n ),

(i)

(i) T
= C n|n + P (i)n (C n+1|N − C n+1|n )(P n ) .

xn|N
C n|N

(i)

(i)

Here, Ri = 1, Qi = diag(τei2 , ηei2 ), and I denotes an identity matrix.
(i)

(i)

(i)

Then, the posterior distribution of xn can be defined using xn|N and C n|N .
Subsequently, the estimates for the time-varying stochastic parameters ti (n) andai (n) can be obtained because the state space model described by Eqs (14) and (15)(i)

incorporates ti (n) and ai (n) in the state vector xn . Hereinafter, the estimates ofbi (n), respectively.
ti (n) and ai (n) are denoted by tbi (n) and a

4.2

Estimating the hyperparameters

For i = 1, 2, . . . , m, Eqs (11) and (13) lead to:
λi =

τei2
.
ηei2

Thus, we have:τei2 (λi ) = λi ηei2 ,that is, τei2 corresponds with λi one-to-one for the given value of ηei2 . Therefore,
we regard λi as a hyperparameter instead of τei2 . When the values of qi , σi2 , and(i)λi , together with the time series data YN = {yi (1), yi (2), . . . , yi (N )}, are given, alikelihood function for the hyperparameters ηei2 and β (i) , which is derived from thealgorithm of the Kalman filter, is as follows:(i)f (i) (YN |σi2 , τei2 (λi ); ηei2 , β (i) ) =

N
∏

(i)

fn(i) (yi (n)|Yn−1 , σi2 , τei2 (λi ); ηei2 , β (i) ),

n=1

14

(i)

(i)

where fn (yi (n)|Yn−1 , σi2 , τei2 (λi ); ηei2 , β (i) ) is the conditional density function of yi (n)
(i)

(i)

given the past history Yn−1 = {yi (n − 1), yi (n − 2), . . .} and λi . Assume that Y0={yi (0), yi (−1), . . .} is an empty set. Then, we have:
(i)

(i)

(i)

f1 (yi (1)|Y0 , σi2 , τei2 (λi ); ηei2 , β (i) ) = f1 (yi (1)|σi2 , τei2 (λi ); ηei2 , β (i) ).
(i)

By taking the logarithm of f (i) (YN |σi2 , τei2 (λi ); ηei2 , β (i) ), the log-likelihood is obtainedas:(i)

LLi (ηei2 , β (i) |σi2 , λi ) = log f (i) (YN |σi2 , τei2 (λi ); ηei2 , β (i) )
=

N
∑

(i)

log fn(i) (yi (n)|Yn−1 , σi2 , τei2 (λi ); ηei2 , β (i) ).

(16)

n=1

As given by Kitagawa (2010), when using the Kalman filter, the conditional
(i)

(i)

density fn (yi (n)|Yn−1 , σi2 , τei2 (λi ); ηei2 , β (i) ) has a normal density given by:(i)
fn(i) (yi (n)|Yn−1 , σi2 , τei2 (λi ); ηei2 , β (i) ) = √

{

1

exp
(i)

(i)

−

(yi (n) − ybn|n−1 )22πσi2 wn|n−1(i)2σi2 wn|n−1

}

,
(17)

(i)
(i)
where ybn|n−1 is the one-step ahead prediction for yn and wn|n−1 is the variance ofthe predictive error. They are given, respectively, by:
(i)

(i)

ybn|n−1 = Hxn|n−1 ,

(i)

(i)

wn|n−1 = HC n|n−1 H T + σi2 .

Thus, for a given value of λi , the estimates of the hyperparameters ηei2 and β (i)can be obtained numerically using the maximum likelihood method. That is, thehyperparameters can be estimated numerically by maximizing LLi (ηei2 , β (i) |σi2 , λi )in Eq. (16) together with Eq. (17). Furthermore, based on Kitagawa (2010), thebi2 for σi2 is obtained analytically by:
estimate σ
(i)

bi2 =
σ

N (y (n) − ybn|n−1 )
i
1 ∑
.
(i)
N n=1
w
2

(18)

n|n−1

It is notable that we can increase the eﬃciency of estimating β (i) when we apply thefast recursive estimation method proposed by Kyo and Noda (2011). Additionally,2b i denote the estimates of τe2 and λi , respectively. Then, from the abovelet τbei and λisettings, the estimates for ηi2 and τi2 are given, respectively, by:
2

be ,
bi2 η
ηbi2 = σib i ηb2 .
τbi2 = λ
i

15

(19)

It is obvious that the estimates for the hyperparameters ηei2 and β (i) depend on
the value of λi . Furthermore, from the algorithm of the Kalman filter, we can seethat the estimates for the elements of the state vector are a set of functions of λi .
That is, under a given value of λi , we can estimate the hyperparameters ηei2 andβ (i) . Thus, as a function of λi , the estimate for the cyclical component ai (n) canbe obtained using the Kalman filter and fixed-interval smoothing. Assume thatthe estimates of the cyclical components for the other indicators (except that of
the i-th) are given in advance. Then, we apply the principal component method fornormalized estimates of the cyclical components and we can obtain the first principal
component as the IBC b(n), as defined in Eq. (10).
As mentioned in subsection 3.1, as a signal of business fluctuations, we expectthat our IBC will have larger variations. Therefore, we estimate the hyperparameterλi by numerically maximizing the variance Var{b(n)} of b(n). In the same way, thevalue of the AR model order qi is determined by the maximization of Var{b(n)}.

4.3

The algorithm

To summarize the scheme for parameter estimation, the algorithm for the CCS
approach is obtained as follows:
1. Assume s = 0, set an appropriate value for L, and then give the initial estimates:bi (n)(0) ; n = 1, 2, . . . , N ; i = 1, 2, . . . , m}{afor the cyclical components using Eq. (12), with N being the length of thetime series data.
2. For i = 1, 2, . . . , m, perform the following steps.
3. Replace the value of s with s + 1.
(a) Estimate the hyperparameters ηei2 and β (i) numerically by maximizing the
log-likelihood in Eq. (16) together with Eq. (17) under the given valueof λi .
(b) Compute the estimate of σi2 using Eq. (18).
bi (n)(s) ; n = 1, 2, . . . , N } for the cyclical compo(c) Obtain the estimates {anent ai (n) by the Kalman filter and a fixed-interval smoothing algorithm.

16

b∗i (n)(s) ; n = 1, 2, . . . , N } for {abi (n)(s) ;(d) Compute the normalized values {an = 1, 2, . . . , N } using Eq. (9).
(e) Perform the principal component analysis for:b∗1 (n)(s) , ab∗2 (n)(s−1) , . . . ab∗m (n)(s−1) )T
(a



∗
(s)
∗
(s)
∗

b1 (n) , ab2 (n) , ab3 (n)(s−1) , . . . ab∗m (n)(s−1) )T
(a



 (a
b∗1 (n)(s) , . . . , ab∗i (n)(s) ,a∗ (n)(i,s) =∗
bi+1 (n)(s−1) , . . . ab∗m (n)(s−1) )T
a




b∗1 (n)(s) , . . . , ab∗m−1 (n)(s) , ab∗m (n)(s−1) )T

 (a

 b∗
(s)
∗
(s)
∗
b2 (n) , . . . , abm (n)(s) )T(a1 (n) , a

if i = 1,
if i = 2,
if 2 < i < m − 1,
if i = m − 1,
if i = m,Treat {a∗ (n)(i,s) ; n = 1, 2, . . . , N } as a set of m-variate time series data,
and then obtain:b(n)(i,s) = wT a∗ (n)(i,s)
as the first principal component of a∗ (n)(i,s) with w = (w1 , w2 , . . . , wm )T
being the vector of the principal component loadings, and Var{b(n)(i,s) }being its variance.
(f) Estimate λi and qi by numerically maximizing Var{b(n)(i,s) }.
4. When the value of Var{b(n)(m,s) } almost converges to a fixed level, proceed to
the next step; otherwise, return to the beginning of step 2.
5. End the calculation and take the estimates corresponding to the values of λiand qi as the final results.
6. Calculate the estimates for ηi2 and τi2 using Eq. (19).

5

Evaluation of the IBC for the analysis of Japanesebusiness cycles

5.1

Comparing the performance with the coincident CI

The main aim of the coincident CI is to measure the speed and magnitude of economic fluctuations in a particular period. It is designed to be a useful tool to analyze
current business conditions. Thus, it is important that we compare the performanceof the IBC with that of the coincident CI.
Figure 1 shows the coincident CI in Japan from January 1985 to December 2015.
We can observe that the CI varies around a horizontal level of 100; that is, the CI
is almost stationary in mean for the observed period.

17

120
110
100
90
80
1985

1990

1995

2000

2005

2010

2015

Figure 1: Time series data for the CI in Japan (January 1985 to December 2015);
Source: ESRI, Cabinet Oﬃce, Government of JapanThe coincident CI comprises the quantitative changes in indicators that are sensitive to business cycle movements. The following 10 indicators were used to construct
the CI:
C1: Index of industrial production (mining and manufacturing)C2: Index of producers’ shipments (producer goods for mining and manufacturing)C3: Index of producers’ shipments of durable consumer goods
C4: Index of nonscheduled working hours (industries covered)
C5: Index of producers’ shipments (investment goods excluding transport equipment)
C6: Retail sales value (change from previous year)
C7: Wholesale sales value (change from previous year)
C8: Operating profits (all industries)
C9: Index of shipments in small and medium-sized enterprises
C10: Eﬀective job oﬀer rate (excluding new graduates)
The functions used to transform the indicators in the model in Eq. (6) are
given in Table 1. To express each function g(·) using the additive multi-components
form in Eq. (6) and to equalize the error variance for each component over time, a18

Table 1: Functions for transforming the indicators
Name of indicator
Indicator
Function g(·)
Name of indicator
Indicator
Function g(·)

C1
y1
log(y1 )
C6
y6
y6

C3
y3
log(y3 )C8y8log(y8 )

1990

2000

2010

C3: log(y3)
4.6
4.0

1980

1990

2000

2010

2010

1990

2000

2010

2010

−10
1980

1990

2000

2010

1980

C8: log(y8)

1990

2000

2010

C9: log(y9)

4.3

4.6

4.9

10.0 11.0 12.0

0 20
−30

1980

2000

10

5.0
4.2
2000

C7: y71990

C6: y6

4.6

4.4 4.6 4.8

1990

1980

C5: log(y5)30

C4: log(y4)

1980

C5
y5
log(y5 )C10
y10
log(y10 )

3.4

4.4
4.0

1980

C4
y4
log(y4 )C9y9log(y9 )C2: log(y2)3.8 4.2 4.6

C1: log(y1)C2y2
log(y2 )
C7
y7
y7

1980

1990

2000

2010

1980

1990

2000

2010

−0.8 −0.2

0.4

C10: log(y10)

1980

1990

2000

2010

Figure 2: Time series data for g(yi (n)) (January 1975 to June 2016); Source foryi (n): ESRI, Japanese Government’s Cabinet Oﬃce

logarithmic transformation is employed for most indicators, excluding C6 and C7.
(Note that for C6 and C7, the identical transformation is used because there aresome negative values in the data sets, which means that a logarithmic transformation
cannot be applied.)
Figure 2 shows the transformed time series data for each indicator for the period
January 1975 to June 2016. Note that to correspond with the end time point of the
CI series, in the following analysis we only used part of the data set, namely thedata for the period January 1975 to December 2015.

19

As mentioned in the algorithm in subsection 4.3, the computation for parameter
estimation is based on a set of initial estimates of the cyclical components. The
value of L used in Eq. (12) to obtain the initial estimates of the cyclical components
is determined as L = 12.
The computation for parameter estimation is achieved by repetition. Figure 3
shows the variation of the CR for the first principal component with respect to thenumber of repetitions. It can be seen that the CR converges to a limited larger value

0.68 0.69 0.70 0.71 0.72

CR

(approximately 0.725) and it steadies from the sixth repetition.

2

4

6

8

10

Number of repetition

Figure 3: The contribution rate for each iteration

For reference purposes, the estimated hyperparameters at each iteration are listed
in Table 2.
Table 2: Main results for hyperparameter estimation
i
qi
bi2
σ
τbi2
ηbi2
i
qi
bi2
σ
τbi2
ηbi2

1
10
3.75 × 10−8
5.23 × 10−8
3.03 × 10−4
62
5.36 × 10−2
5.58 × 10−5
5.48

2
10
8.74 × 10−8
4.24 × 10−8
3.12 × 10−4712
7.19 × 10−4
1.58 × 10−4
7.19

3
10
1.04 × 10−4
6.42 × 10−8
1.30 × 10−387
1.69 × 10−5
2.35 × 10−7
1.07 × 10−3204
1
4.80 × 10−6
1.90 × 10−7
1.39 × 10−4
91
6.11 × 10−8
1.69 × 10−7
2.53 × 10−452
1.03 × 10−7
3.79 × 10−7
4.81 × 10−41014.90 × 10−5
1.02 × 10−6
4.63 × 10−4

C2

C3

3.8

3.6

4.0

4.2

4.4

4.0 4.2 4.4 4.6

4.6

C1

1980

1990

2000

2010

1980

1990

2000

2010

1990

2000

2010

10 15
2000

2010

1980

2010

1990

2000

2010

4.6

C9

4.2

4.4

10.5 11.0 11.5
2000

2010

5
1990

C8

−2 0 2 4 6

1990

2000

0
1980

C7

1980

1990

C6

4.2 4.4 4.6 4.8

4.65
4.50

1980

1980

C5

4.80

C4

1980

1990

2000

2010

1980

1990

2000

2010

−0.6

−0.2

0.2

C10

1980

1990

2000

2010

Figure 4: Estimated trend components (January 1975 to December 2015)

Figure 4 shows the estimated trend components for each transformed indicator.
We can see that the behavior of each trend is distinct and, therefore, it may be
diﬃcult to extract a common factor from the estimated trend components.
Figure 5 shows estimated cyclical components for each transformed indicator.
The correlation matrix for the estimated cyclical components is given in Table 3.
(Please note that because the matrix is symmetric, only the elements below thediagonal are shown.) This table illustrates that there is a very high positive correlation between pairs among the estimated cyclical components. Thus, it is possible
to extract a common factor using the first principal component composed of theestimated cyclical components with a very high CR.
Thus, to construct the IBC, we use principal component analysis based on thecorrelation matrix shown in Table 3. The largest eigenvalue for the correlation
matrix is 7.247, and the corresponding eigenvector is:
w = (0.363, 0.341, 0.296, 0.335, 0.336, 0.164, 0.294, 0.338, 0.342, 0.309)T .
From these results, we can see that the CR for the first principal component is 72.47%

21

C2

C3

1980

1990

2000

2010

−0.2
−0.5

−0.3

−0.1

−0.3 −0.1

0.1

0.1

0.1

C1

1980

1990

2000

2010

2000

2010

2000

2010

2010

1990

2000

2010

C9

0.00

−1.5
2000

1980

−0.15

−0.5

10
−10

1990

2010

0 5
1990

C8

−30

1980

2000

−10
1980

C7

0.10

1990

1990

C6

−0.2

−0.05
−0.20

1980

1980

C5

0.0

0.10

C4

1980

1990

2000

2010

1980

1990

2000

2010

−0.4

−0.1 0.1

C10

1980

1990

2000

2010

Figure 5: Estimated cyclical components (January 1975 to December 2015)

Table 3: Correlation matrix for the estimated AR components

C1
C2
C3
C4
C5
C6
C7
C8
C9
C10

C1
1
0.942
0.827
0.865
0.880
0.358
0.707
0.876
0.919
0.769

C2

C3

C4

C5

C6

C7

C8

C9

C10

1
0.746
0.794
0.750
0.324
0.709
0.852
0.900
0.622

1
0.667
0.653
0.289
0.523
0.702
0.688
0.587

1
0.813
0.375
0.646
0.798
0.788
0.800

1
0.344
0.657
0.767
0.851
0.851

1
0.429
0.376
0.297
0.331

1
0.732
0.735
0.572

1
0.774
0.737

1
0.694

1

because the sum of the variances for each of the normalized cyclical components is
10. Moreover, w is equivalent to the vector of the principal component loadings for
the IBC. An element in the vector of the principal component loadings expresses
how the IBC depends on the corresponding indicator. For example, w1 = 0.363 is22the maximum element and w9 = 0.342 is the second maximum element in the vector
of principal component loadings. Hence, we can say that the index of industrial
production (C1) has the largest eﬀect on the IBC, the index of shipments in small
and medium-sized enterprises (C9) has the second largest eﬀect, and so on.
Based on the normalized estimates for the cyclical components and the vector ofprincipal component loadings w, the estimated IBC, say b(n), is computed using Eq.
(12). Figure 6 shows the time series of the estimated IBC for the period January1975 to December 2015. Here, the estimated IBC is temporarily called the IBC-10
because it is obtained based on the 10 indicators. It can be observed that the IBC10 varies around zero, and then drops suddenly in the first half of 2009 after the

−10

−5

0

5

collapse of the Lehman Brothers.

1980

1990

2000

2010

Figure 6: Time series for IBC-10 (January 1975 to December 2015)

To illustrate how the IBC-10 correlates with the coincident CI, we create a scatter
diagram (see Figure 7) for the period January 1985 to December 2015. Note thatthe correlation coeﬃcient between these two indexes is 0.8966. This implies that
there is a very high positive correlation between the IBC-10 and the CI.
An important issue is: how well does the IBC-10 express the state of businessfluctuations? It is diﬃcult to examine the performance of a business index because
we have no objective standard that expresses the level of business fluctuations. However, as mentioned in subsection 2.3, it is widely accepted that GDP is an inclusive
measure of economic performance and Mariano and Murasawa (2010) used real GDP
to measure business cycles. Thus, we use real GDP as an expediential reference forbusiness cycles.

23

120
110
100
80

90

CI

−10

−5

0

5

IBC−10Figure 7: Scatter diagram of IBC-10 and CI (January 1985 to December 2015)

Similar to the model in Eq. (1), we employ the following model:log(yn ) = tn + sn + an + εn ,εn ∼ N(0, σ 2 ),where yn denotes the quarterly time series data on real GDP in Japan, and tn , sn , andan express the trend, seasonal, and cyclical components, respectively. Furthermore,
εn is the irregular component. We use the following trend component, seasonal
component, and AR models, respectively:
tn = 2tn−1 − tn−2 + vn1 ,vn1 ∼ N(0, τ12 ),sn = −(sn−1 + sn−2 + sn−3 ) + vn2 ,

an =

q
∑

βj an−j + vn3 ,vn2 ∼ N(0, τ22 ),vn3 ∼ N(0, τ32 ),j=1

to express the priors for each component. Then, the logarithm of real GDP (logGDP) can be decomposed into the trend, seasonal, and cyclical components together
with the irregular component. A detailed explanation of the modeling and parameter
estimation can be found in Kitagawa (2010).

24

11.6511.80

data for log−GDP

1995

2000

2005

2010

2015

2010

2015

2010

2015

2010

2015

11.65

11.75

trend component

1995

2000

2005

−0.02

0.04

seasonal component

1995

2000

2005

−0.06

0.00

Cyclical component

1995

2000

2005

Figure 8: Results for decomposition of log-GDP in Japan (Q1, 1994 to Q1, 2015)

Figure 8 shows the results for the decomposition of log-GDP in Japan for the
first quarter (Q1) of 1994 to Q1 of 2015.
Here, we take the cyclical component of log-GDP as a measure of a business cycle,
abbreviated to cyclical log-GDP. Thus, the performance of an IBC can be measured
using its correlation with the cyclical log-GDP. We compare the performance of the
IBC-10 and CI by comparing their correlation coeﬃcients with the cyclical log-GDP
from Q1 1994 to Q1 2015. Note that for the correspondence to the quarterly GDP
data, we take the average for the 3 months as the value for each quarter for the
IBC-10 and CI. The line graphs for the time series data on the cyclical log-GDP,25

the IBC-10, and the CI are shown in Figure 9. The graphs show that the behaviors
for these three time series have a high level of similarity.

−0.06

0.00

Cyclical log−GDP

1995

2000

2005

2010

2015

2010

2015

2010

2015

−5 −3 −1

1

IBC−10

1995

2000

2005

80

100

CI

1995

2000

2005

Figure 9: Cyclical log-GDP, IBC-10, and CI (Q1, 1994 to Q1, 2015

Figure 10 shows scatter diagrams for the cyclical log-GDP with the CI (leftdiagram) and the IBC-10 (right diagram). The correlation coeﬃcients of the cyclical
log-GDP with the CI and IBC-10 are 0.7181 and 0.8524, respectively. Thus, the IBC10 has a higher correlation with the cyclical log-GDP than the CI, which implies
that the IBC-10 performs better than the coincident CI.

5.2

Comparing the performance with the NBI

Now, we compare the performance of the proposed IBC, based on the CCS approach,with a Stock–Watson index. The NBI is constructed based on the Stock–Watson
dynamic factor model by the Nihon Keizai Shimbun Inc. in Japan. Thus, we cancompare the performance of our CCS approach with the Stock–Watson approach,using the dynamic factor modeling method, by comparing the performance of the
IBC with that of the NBI.

26

2
−3

−1

IBC−10

0

1

110
100
80

−5

90

CI

−0.06

−0.02

0.02

−0.06

Cyclical log−GDP

−0.02

0.02

Cyclical log−GDP

Figure 10: Scatter diagrams of the cyclical log-GDP with CI and IBC-10 (Q1, 1994
to Q1, 2015)

Figure 11 shows a set of monthly time series data for the NBI for January 1973
to June 2016. This data set was obtained from the Nihon Keizai Shimbun, Inc.
website (http://www.nikkei.com/biz/report/nkidx/). From the figure, we cansee that the NBI time series is not globally stationary in mean; that is, a trend
component exists in the NBI series. Thus, compared with the IBC, the time series

70 80 90110

for NBI is diﬃcult to explain because its behavior is somewhat complicated.

1980

1990

2000

2010

Figure 11: Time series data for NBI (January 1973 to June 2016); Source: Nihon
Keizai Shimbun, Inc.

The NBI is constructed using time series data for the following four indicators:

27

NBI-y1: Index of industrial production (mining and manufacturing), C1
NBI-y2: Index of nonscheduled working hours (industries covered), C4
NBI-y3: Eﬀective job oﬀer rate (excluding new graduates), C10
NBI-y4: Total retail and wholesale sales value (ratio to the same month of theprevious year, in %)Note that the first three indicators, NBI-y1, NBI-y2, and NBI-y3, are the same as
C1, C4, and C10, respectively. Because these indicators were used to construct thecurrent CI and IBC-10, which we have illustrated in Figure 2, we need only show thedata for NBI-y4. The monthly time series data for NBI-y4 from January 1980 to June
2016 are shown in Figure 12. Note that this data set is obtained from the website of
the Ministry of Economy, Trade, and Industry (http://www.meti.go.jp/statistics/index.html).

4.4 4.5 4.6 4.7 4.8

The logarithmic transformation is applied to all four indicators.

1980

1990

2000

2010

Figure 12: Time series data for NBI-y4 (January 1980 to June 2016); Source: Ministry of Economy, Trade and Industry

To compare the performance of the IBC with the NBI under the same conditions,
we construct the IBC following our CCS approach. The IBC is based on the data
for the above four indicators for January 1980 to June 2016, corresponding withthe period of the NBI-y4 data. The initial settings are very similar to those forconstructing the IBC-10. The main estimates for the hyperparameters are given in
Table 4 for the models of g(yi (n))(i = 1, 2, 3, 4).
Figures 13 and 14 show the estimated trend and cyclical components for all
indicators.

28

Table 4: Main results for hyperparameter estimation
i
qi
bi2
σ
τbi2
ηbi2

1
10
3.31 × 10−8
4.40 × 10−7
3.31 × 10−42
102.74 × 10−5
2.42 × 10−7
7.03 × 10−53
125.41 × 10−8
2.68 × 10−7
5.41 × 10−4

NBI−y2

4.55

4.3

4.4

4.65

4.5

4.75

4.6

4.85

NBI−y14
18.35 × 10−5
2.83 × 10−6
2.53 × 10−4

1980

1990

2000

2010

1980

1990

2000

2010

NBI−y4−0.6 −0.4 −0.2 0.0 0.2

4.60 4.65 4.70 4.75 4.80

NBI−y3

1980

1990

2000

2010

1980

1990

2000

2010

Figure 13: Estimated trend components (January 1980 to June 2016)

NBI−y2−0.15−0.2 −0.1

−0.05

0.0

0.05

0.1

NBI−y1

1980

1990

2000

2010

1980

1990

2000

2010

NBI−y4−0.2 −0.1−0.25 −0.15 −0.05

0.0

0.05

0.1

NBI−y3

1980

1990

2000

2010

1980

1990

2000

2010

Figure 14: Estimated cyclical components (January 1980 to June 2016)

Figure 15 shows the estimated time series of the IBC, termed the IBC-4.
Figure 16 shows a scatter diagram for the IBC-4 and NBI from January 1980 to

29

4
2
0
−2
−4
−6
−8
1980

1990

2000

2010

Figure 15: Time series for IBC-4 (January 1980 to June 2016)

June 2016. Note that the correlation coeﬃcient between these two indexes is 0.6509.
These results indicate that the correlation coeﬃcient is not very high, which may be

−4
−8

NBI

0

2

4

a result of the trend component of the NBI.

90

100

110

120

IBC−4

Figure 16: Scatter diagram for IBC-4 and NBI (January 1980 to June 2016)

We compare the performance of the IBC-4 and NBI by using the cyclical logGDP as a reference. However, to obtain the stationary part of the NBI and, thus, toensure that the NBI has a higher correlation with the cyclical log-GDP, we remove
the trend component from the NBI series.

30

The top graph in Figure 17 shows the quarterly time series data for the NBI,
which are obtained as the averages over the corresponding 3 months from Q1 1994
to Q1 2015. From this graph, we can see a declining trend in the NBI time series.

90

100

110

Original NBI

1995

2000

2005

2010

2015

2010

2015

2010

2015

103.0

104.0

105.0

Trend of NBI

1995

2000

2005

−15

−5

5

Adjusted NBI

1995

2000

2005

Figure 17: Result for decomposing the NBI (Q1, 1994 to Q1, 2015)

Let yn and xn be the values of the NBI and the trend at time point n (quarter),respectively. Then, we can compute the adjusted NBI by:z n = yn − x n ,(n = 1, 2, . . . , N1 )with N1 = 85 being the length of the time series data from Q1 1994 to Q1 2015.
Thus, to ensure that the short-term variations in the adjusted NBI remain as powerful as possible, we express the trend of the NBI as a straight line:
xn = a + bn,(n = 1, 2, . . . , N1 ).

Then, we estimate the parameters a and b by maximizing the correlation coeﬃcient
between the adjusted NBI and the cyclical log-GDP under the condition that:N1∑

zn = 0.

n=131

b = 105.31 and bb = −0.0295.
Note that the estimates for the parameters are a

The graphs for the estimates of the trend in the series of the NBI and the adjusted
NBI are plotted in Figure 17.
Thus, we can employ correlation analysis for the cyclical log-GDP with the NBI,
adjusted NBI, and IBC-4. Figure 18 shows the scatter diagrams of the cyclical logGDP with the NBI (left diagram), adjusted NBI (center), and IBC-4 (right). Notethat the correlation coeﬃcients of the cyclical log-GDP with these three indexes are
0.8277, 0.8355, and 0.8466, respectively. That is, the correlation coeﬃcient of thecyclical log-GDP with the adjusted NBI is higher than that with the NBI, and thecorrelation coeﬃcient with the IBC-4 takes the highest value. These results reveal

0
−4
−6

−10

−0.02

0.02 0.04

Cyclical log−GDP

−8

−15

90

−0.06

−2

IBC−4

0
−5

100
95

NBI

105

Adjusted NBI

5

2

110

10

4

115

that our CCS approach performs well.

−0.06

−0.02

0.02 0.04

Cyclical log−GDP

−0.06

−0.02

0.02 0.04

Cyclical log−GDP

Figure 18: Scatter diagrams of the cyclical log-GDP with all indexes (Q1, 1994 to
Q1, 2015)

6

Discussions

To summarize Sections 4 and 5, our results show that our CCS approach has many
advantages when compared with existing approaches. These are discussed below.
1. Our approach is systematic. As previously mentioned, when using the Stock–
Watson dynamic factor modeling approach, it is necessary to transform thedata to be stationary by diﬀerencing the time series data. However, it is often
diﬃcult to determine whether it is appropriate to do this and, even whendiﬀerencing is necessary, there is no clear criterion to determine the orderof diﬀerence. Thus, the Stock–Watson dynamic factor modeling approach is32somewhat ad hoc regarding the use of data. Moreover, the MTV modelingmethod is ad hoc in the interpretation of results. As shown in Figure 4, under
the CCS approach, the model in Eq. (2) can generally be applied for indicatorswith various types of trends, so long as these are smooth. Even where the timeseries data are stationary in mean, it can be ensured that the estimates for thetrend component are obtained as a constant.
2. The results of our approach are clear and easy to apply. As shown in Eq. (4),in most cases following the Stock–Watson approach, the behavior of an estimated IBC is complicated because it comprises stationary and nonstationary
parts. In contrast, the structure of the IBC is very simple, ensuring greater
understanding because of global stationarity.
3. Our approach has a strong performance in analyzing and predicting businessconditions. The results shown in the preceding section imply that the IBC
can better explain the variation in GDP than the CI and NBI. Moreover, themodels using the CI approach do not have clear structures, making it diﬃcult
to predict future business conditions. In contrast, each cyclical component of
our approach is estimated using the Kalman filter, which is well known for its
good predictive abilities.
In addition, if we use the model in Eq. (5) instead of that in Eq. (6) and add
a prior model for the seasonal component, we can process the time series data inadvance without the need for a seasonal adjustment. Thus, our CCS approach can
be widely applied as a more general method.

7

Conclusions

In this paper, to construct a coincident index of growth cycles from a given setof indicators, we proposed an alternative approach, which we refer to as the CCS
approach, to develop an IBC of coincident economic indications.
The framework of the CCS approach can be summarized as follows: (1) We used
the same time series data as the CI and DI compiled by the ESRI; (2) Seasonallyadjusted data were decomposed into a trend, cyclical, and irregular components; and(3) We constructed the IBC based on the first principal component of the normalized
estimates for the cyclical components.

33

We examined whether the constructed coincident IBC performs better than existing indexes. The correlation coeﬃcients of the cyclical component of real GDP
were 0.7181 and 0.8524 for the CI and IBC, respectively, which indicates that the
IBC performed better than the CI. Focusing on the adjusted NBI and IBC, thecorrelation coeﬃcients of the cyclical component of GDP were 0.8355 and 0.8466,
respectively, indicating that IBC performed better than the NBI. Therefore, theresults show that the CCS approach has a number of advantages over existing methods.
Acknowledgments
This work was supported in part by a Grant-in-Aid for Scientific Research (B)(18H03210) and a Grant-in-Aid for Scientific Research (C) (19K01583) from theJapan Society for the Promotion of Science.

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