SDS-13
Detecting Information Flows in DominantComponents of Stock Market Returns
Yoko Tanokura, Seisho Satoand Genshiro Kitagawa

October 2019

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

Detecting Information Flows in Dominant Components ofStock Market Returns ∗Yoko Tanokura1,4 , Seisho Sato2 , and Genshiro Kitagawa3
1

Graduate School of Advanced Mathematical Sciences, Meiji University, Tokyo, Japan
2
Graduate School of Economics, The University of Tokyo, Tokyo, Japan
3
Mathematics and Informatics Center, The University of Tokyo, Tokyo, Japan
4
Corresponding author: Yoko Tanokura, e-mail:tanokura@meiji.ac.jp

31 July 2019
Abstract
Political events and economic problems occurred in one country have increasingly inﬂuenced the economies and ﬁnancial markets worldwide since the global economic crisis
driven by the bankruptcy of Lehman Brothers in September 2008. For making both
short-term and long-term investment strategies for investment ﬁrms and pension fundmanagements, and for planning ﬁnancial and economic policies to control the crisis
spillovers, it is indispensable to understand the ﬂuctuation characteristics of ﬁnancial
markets that cause changes of the ﬁnancial and economic environments in short-termand long-term periods. This article aims at detecting information ﬂows for short-termand long-term investments by investigating the ﬂuctuation relationships between dominant components of stock market return, based on long-term daily time series of TOPIX,
S&P500 and DAX Index. We decompose each stock market return into two components,
i.e., the return trend component and the return cyclical component, which are derived
by applying a seasonal adjustment model proposed by Gersch and Kitagawa (1983) andKitagawa and Gersch (1984, 1996) to the stock market index. The dominant component
is found for each of short-term and long-term returns, respectively. Then, we investigate
the relationships of ﬂuctuation between dominant components of stock market returnby conducting generalized power contribution analysis (Tanokura and Kitagawa 2004,
2015). The information ﬂows of the inﬂuential factors are detected. Our ﬁndings indicate the importance of observing mutual relationships of long-term ﬂuctuations, i.e.,
trends, between the three markets and can provide the useful information for building
an investment strategy and making an economic policy.

Keywords: Generalized power contribution, Seasonal adjustment model, Return
decomposition, Time series analysis, Long-term investment∗

This work was supported by MEXT Private University Research Branding Project.

1

1

Introduction

Political events and economic problems occurred in one country have increasingly
inﬂuenced the economies and ﬁnancial markets worldwide since the global economic crisis driven by the bankruptcy of Lehman Brothers in September 2008.
Under these circumstances, in Japan, having a serious problem of aging society
in near future, the awareness of pension has largely increased. For making both
short-term and long-term investment strategies for investment ﬁrms and pension
fund managements, and for planning ﬁnancial and economic policies to control
the above-mentioned spillovers, it is indispensable to understand the ﬂuctuation
characteristics of ﬁnancial markets that cause changes of the ﬁnancial and economic environments in short-term and long-term periods. In particular, since
the prices of ﬁnancial markets ﬂuctuate with both serial correlations and time series correlations, it is signiﬁcant to evaluate the degree and direction of inﬂuence
between markets.
This article aims at detecting information ﬂows for short-term and long-term
investments by investigating the ﬂuctuation relationships between dominant components of stock market return. We focus on the analysis of three stock market
indices such as TOPIX (Japan), DAX Index (Germany) and S&P 500 (US) forthe period from January 4, 1993 to May 31, 2019 (6,890 days).
In ﬁnancial market analysis, technical analysis is practically well known. Murphy deﬁnes technical analysis as the study of market action, primarily through theuse of charts, for the purpose of forecasting future price trends (Murphy 1986).
In a stock market, a trend of the stock price may be regarded as the graduallychanging long-term ﬂuctuations caused by characteristics speciﬁc to the stock.
On the other hand, a cyclical ﬂuctuations around the trend can sensitively be inﬂuenced by those of any other stock prices, and they may behave as only a priceadjustment or may lead to a future change in the trend direction. Therefore,
identifying a trend of price movement is signiﬁcantly important.
We decompose each stock market return into two components, i.e., the return
trend component and the return cyclical component. These components are derived by applying a seasonal adjustment model proposed by Gersch and Kitagawa(1983) and Kitagawa and Gersch (1984, 1996) to the stock market index. The
dominant component is found for each of short-term and long-term returns. Thisimplies the eﬀectiveness of the seasonal adjustment model.
To evaluate the inﬂuence of external information on the stock markets for thelast 26 years, we focus on two inﬂuential indices such as WTI crude oil futures
which is often referred to as a proxy of the oil price, and JPYUSD which isthe foreign exchange rate of the Japanese Yen against the US dollar. Then, weinvestigate the relationships of ﬂuctuation between dominant components of stock2

market return by conducting generalized power contribution analysis (Tanokura
and Kitagawa 2004, 2015). As a result, the inﬂuential factors are detected. Ourﬁndings indicate the importance of observing mutual relationships of long-term
ﬂuctuations, i.e., trends, between these markets. The information ﬂows of theinﬂuential components of return detected by statistical modeling can provide theuseful information for building an investment strategy and making an economic
policy. As far as we know, there is no researches analyzing long-term ﬁnancial
time series by applying statistical methods.
This article consists of two parts. The following chapter introduces two statistical methods which we utilize in this article, such as a seasonal adjustment
model and a generalized power contribution. Chapter 3 provides the empiricalanalysis for three stock market indices. Each stock market return is decomposed
into the return trend component and the return cyclical component. And then,we detect the dominant component for each of short-term and long-term returns
and investigate the inﬂuential factors in ﬂuctuation of these returns. Finally, we
state our conclusions and future works.

2

Statistical Modeling

2.1

A Seasonal Adjustment Model

This section brieﬂy introduces a seasonal adjustment model proposed by Gersch
and Kitagawa (1983) and Kitagawa and Gersch (1984, 1996).
For an observed time series yn is expressed as
yn = tn + sn + pn + wn .

(1)

A trend component tn is estimated by the following trend component model withthe trend order k∆k tn = vn1 ,vn1 ∼ N (0, τ12 ).

(2)

A seasonal component sn , which forms seasonal ﬂuctuations if it exists, is expressed by the following seasonal component model with a period length p:
p−1
∑

(
)
vn2 ∼ N 0, τ22 .

sn−i = vn2 ,

(3)

i=0

A stationary component pn is estimated by the following stationary AR component model of AR order m:
pn =

m
∑

ai pn−i + vn3 ,vn3 ∼ N (0, τ32 ),

(4)

i=1

which expresses relatively shorter cyclical ﬂuctuations than the gradual long-term
trend component (2). Finally, the distribution of the observation noise wn in (1)3is given bywn ∼ N (0, σ 2 ).
As each component model can be expressed in a state-space model form, theparameter estimations are performed through the composite form of state-spacemodels. The details can be found in Kitagawa (2010).

2.2

Generalized Power Contribution

As a tool to detect ramiﬁcations of price ﬂuctuations, we introduce a generalized
power contribution which extends the concept of the Akaike’s power contribution(Akaike 1968) by decomposing a variance covariance matrix of the noises.
Assume that an l-dimensional stationary time series yn = (yn (1), yn (2), · · · , yn (l))t ,n = 1, · · · , N , is expressed as the following multivariate AR model with order m:
yn =

m
∑

Aj yn−j + vn ,

(5)

j=1

where Aj is an l × l AR coeﬃcient matrix with its (r, s)-component aj (r, s). An
l-dimensional white noise vn satisﬁes the following conditions:E(vn ) = [0, · · · , 0]t , E(vn vnt ) = W, E(vn vht ) = O (n ̸= h), E(vn yht ) = O (n > h).
Here, O is the l × l zero matrix, and W = (σrs ) is a symmetric positive deﬁnite
matrix (i.e., σrs = σsr ) that is referred to as the variance covariance matrix ofthe noises.
The cross spectrum matrix is deﬁned as the l × l matrix P(f ) = (Prs (f )),where the element Prs (f ) is the Fourier transform of the cross-covariance function
Ck (r, s) and is referred to as the cross spectrum. Here, f is a frequency satisfying−1/2 ≤ f ≤ 1/2. The diagonal element Prr (f ) is referred to as the power
spectrum.
It is well known that P(f ) can be obtained asP(f ) = A(f )−1 W(A(f )−1 )∗ ,(6)

where A(f ) is the l × l complex matrix with its (r, s)-component Ars (f ), andA∗ denotes the complex conjugate of a matrix A. Here, Ars (f ) is deﬁned as theFourier transform of the coeﬃcients aj (r, s) in the multivariate AR model (5):Ars (f ) =m∑aj (r, s)e−2πijf ,

(7)

j=0

where a0 (r, r) = −1, and a0 (r, s) = 0 for r ̸= s (Whittle 1963; Akaike andNakagawa 1988). For simplicity, denoting A(f )−1 as B(f ) = (brs (f )), (6) isgiven byP(f ) = B(f ) W B(f )∗ .
4(8)

As the original deﬁnition in Akaike (1968) assumes that the components of
the noise vn are mutually uncorrelated: σrs = 0, r ̸= s, W is diagonal. Therefore,
from (8) the power spectrum of the r-th component yn (r) of yn at a frequency f
can be simply expressed as
Prr (f ) =l∑brs (f )σss brs (f )∗ ≡

s=1

l
∑

|brs (f )|2 σss .

(9)

s=1

That is, the power spectrum Prr (f ) of yn (r) is composed of l noise inﬂuences, andthe degree of inﬂuence from the s-th noise component vn (s) on the ﬂuctuationof yn (r) is evaluated by |brs (f )|2 σss for s = 1, . . . , l. Therefore, Akaike’s power
contribution is deﬁned asrrs (f ) =|brs (f )|2 σss,
Prr (f )(10)

which expresses the proportion of the ﬂuctuation of yn (r) caused by the s-th noise
component vn (s) at a frequency f .
For a general case, we consider decomposing W into a sum of matrices withrank one. We assume that the common inﬂuence of all the variables is derived
from the smallest correlation coeﬃcient. Then, W is expressed as a sum of at
most l(l + 1)/2 terms:
W=

l−2 ∑
k+1
∑

ql−(k+1)+j,j IHj (k) ItHj (k) +k=0 j=1l∑

qjj IHj (l−1) ItHj (l−1) ,

(11)

j=1

where IHj (k) = [ijk (1), . . . , ijk (l)] is an l-dimensional vector, of which k components are 0 and (l − k) components are either 1 or −1, depending on the signsof correlations for k = 0, . . . , l − 1; j = 1, . . . , k + 1. Here, Hj (k), the suﬃx ofIHj (k) , is a subset Hj (k) = {hj,1 , . . . , hj,k } of H = {1, . . . , l} and indicates thecomponents of 0 of IHj (k) . Note that the last term of (11) can be expressed asdiag{q11 , . . . , qll }.
Then, by (8),P(f ) =

l−2 ∑
k+1
∑

ql−(k+1)+j,j B(f ) IHj (k) ItHj (k) B(f )∗k=0 j=1+B(f ) diag{q11 , . . . , qll } B(f )∗ .

(12)

Therefore, the power spectrum of its r-th component is expressed as
Prr (f ) =

l−2 ∑
k+1
∑

ql−(k+1)+j,j

k=0 j=1

+

l
∑

l
∑

l
∑

crjk (h)crjk (n)∗h=1,h̸=r n=1,n̸=r

qjj |brj (f )|2 ,

(13)

j=1

where crjk (h) = ijk (h) brh (f ).
5(13) implies that the power spectrum Prr (f ) can generally be decomposed into
two terms. The ﬁrst term expresses the l(l − 1)/2 common inﬂuences resulting
from correlations between l noise components, and the second one does the linﬂuences resulting from the diagonal matrix of the noises. We refer to the ﬁrst
term as correlated noise and the second term as independent noise. Note that(13) becomes (9) when qrs = 0 for r ̸= s.
Finally, the generalized power contribution is deﬁned as

l
l
∑
∑



q
crjk (h) crjk (n)∗ l−(k+1)+j,j
h=1,h̸=r n=1,n̸=rPrr (f )r̃rjk (f ) =(k= 0, . . . , l − 2; j = 1, . . . , k + 1)


2


q
|b
(f
)|

l−(k+1)+j,j rjPrr (f )(k = l − 1; j = 1, . . . , l).

(14)

As the generalized power contribution simultaneously measures the degree ofinﬂuence between various combinations of the noises, multi-directional causations
between variables can be evaluated. The details can be found in Tanokura andKitagawa (2015).

3

Empirical Analysis

3.1

Data

We use the time series data of three stock market indices such as TOPIX (Japan),
DAX Index (Germany) and S&P 500 (US) for the period from January 4, 1993
to May 31, 2019 (6,890 days).
To evaluate the inﬂuence of external information on the stock markets for thelast 26 years, we focus on two inﬂuential indices such as WTI crude oil futures
which is often referred to as a proxy of the oil price, and JPYUSD which is theforeign exchange rate of the Japanese Yen against the US dollar. Data source is
Bloomberg LP. Hereafter, we refer to TOPIX, DAX Index, S&P 500, WTI crude
oil futures and JPYUSD as TPX, DAX, SP, WTI and JPY, respectively.
The missing price on a market holiday is taken the price on the previous
trading day.

3.2

Return Decomposition

We apply the seasonal adjustment model proposed by Kitagawa and Gersch toeach log-transformed variable. Here, we suppose no seasonality for each index.
Therefore, the following decomposition is performed for each yn :log yn = tn + pn + wn ,6

wn ∼ N (0, σ 2 ).

(15)

We refer to the trend component tn as KG-trend. Taking 1-day diﬀerence of (15),log yn − log yn−1 = tn − tn−1 + pn − pn−1 + wn − wn−1
is obtained. Therefore, the daily return rn is decomposed as
rn = rtn + rcn ,(16)

where rtn = tn − tn−1 and rcn = (pn − pn−1 ) + (wn − wn−1 ) is referred to as thereturn trend component and the return cyclical component, respectively.
For a long-term return, we deﬁne the k-day return (k > 1) as
rn (k) = log yn − log yn−k .

(17)For example, we take k = 260 as 260-day return, i.e., 1-year return, k = 780 as
3-year return and k = 1, 300 as 5-year return. Then, as rn (k) = rn + rn−1 + · · · +rn−k+1 , it can be decomposed asrn (k) =k−1∑(rtn−i + rcn−i ) = rtn (k) + ctn (k).

(18)

i=1

Here, we refer to rtn (k) and rcn (k) as k-day return trend component and k-day
return cyclical component, respectively. Note that rtn (k) is the k-day diﬀerenceof the KG-trend.

3.3

Characteristics of KG-trend

Figure 1 shows the decomposition of the log-transformed TPX. By minimizing
AIC (Akaike 1998), the trend order two and the AR order two are selected.
The top graph shows the log-transformed TPX and its KG-trend, the middle
one shows the stationary AR component, and the bottom one does the noise.
Although the KG-trend does not look like a smooth trend curve due to the data
length of 26 years long, the smooth KG-trend can be found in the short-term
period, as shown in Figure 2 for the detailed period of June 1, 2018 to December
24, 2018.
Similarly, the other four variables are decomposed respectively, as shown in
Figure 3. The trend order two and the AR order two are also selected for all
indices by AIC.

7

Figure 1: The decomposition of the log-transformed TOPIX (TPX): The index
and the KG-trend (top), the stationary AR component (middle) and the noise(bottom).

Figure 2: The log-transformed TOPIX(TPX) and the KG-trend for the detailed
period of June 1, 2018 to December 24, 2018.

8

Figure 3: The decomposition of the log-transformed S & P 500(SP), DAX Index(DAX), WTI crude oil futures (WTI) and JPYUSD (JPY) from top to bottom.

9Generally, stock prices always show zigzag movements even though they are in
an up-trend or a down-trend. In order to indicate a relatively long-term trend of
price movements, a simple moving average for the period of k-day is practicallyoften used. However, as it is based on past prices, the longer k is for taking
averages, the later a peak or a bottom of moving averages appears, which canbe seen in the 20-day moving average (blue line) and the longer 60-day moving
average (green line) for TPX in Figure 4. On the other hand, we found that the
KG-trend (orange line) which is exponential-transformed corresponding to theoriginal TPX, appropriately captures a peak or a bottom of the price movements.
This is one of the merits of KG-trend.

Figure 4: Comparison of a KG-trend with moving averages: the 20-day moving
average (blue line), the 60-day moving average (green line) and the KG-trend(orange line) of TPX (black line).

As a KG-trend is estimated as a smoother which is fully utilized the priceinformation during the whole period, it updates its own past values when a new
information is added. Let us examine how a KG-trend changes. We compare the
KG-trend for the whole period of January 4, 1993 to May 31, 2019 (6,890 days)
with the one for Period C of January 4, 1993 to June 30, 2008 (4,041 days). In
fact, the KG-trend for Period C was estimated based on the price informationfrom January 4, 1993 to June 30, 2008.
The top graph in Figure 5 shows the KG-trend for the whole period (orangeline) and the one for Period C (blue line) for TPX. As the diﬀerences between
them in Period C are too small to distinguish the orange line from the blue one,the diﬀerence: the KG-trend for the whole period minus the KG-trend for Period
C is shown in the bottom graph. The diﬀerence was in a signiﬁcantly small rangeof (−0.0005, 0.0005) until the end of May 1, 2008. As shown in the KG-trend
for the whole period (orange line) of the top graph, the bankruptcy of Lehman
Brothers occurred after Period C. As the KG-trend for Period C does not reﬂect
this bankruptcy, since then, the diﬀerence has widen downward to −0.00559 at
the end date of Period C.
Similarly, in Figure 6, the diﬀerence: the KG-trend for the whole period
10minus the KG-trend for Period C for SP, DAX, WTI and JPY is shown, respectively. It is intriguing that the diﬀerence for JPY has fallen in the small rangeof (−0.003, 0.003) through Period C. It is because the KG-trend for the whole
period exhibited the similar transitions with the KG-trend for Period C for awhile after Period C. In fact, the sharp downtrend of JPY started at the middle
August and lasted until the end of January, 2009.

Figure 5: KG-trend for the whole period (orange line) and the one for Period
C (January 4, 1993 to June 30, 2008; blue line) (top), and the diﬀerence: theKG-trend for the whole period minus the KG-trend for Period C (bottom) for
TPX.

Figure 6: Diﬀerence: the KG-trend for the whole period minus the KG-trend for
Period C for SP, DAX, WTI and JPY (from top to bottom).

From the point of view of long-term investment, the diﬀerence of KG-trends
at the end date of the period is small enough to ignore. Therefore, in this sense,
the stability of KG-trend is presumed.
11

3.4

Properties of Return Decomposition

Now we observe some outstanding properties on components of return as statedin (16) and (18). Figure 7 shows the return trend component (red line) andthe return cyclical component (blue line) of the daily return for TPX, SP, DAX(left), WTI and JPY (right). It is found that the ﬂuctuation range for return
trend component is almost one tenth of that for return cyclical component for
all indices. This implies the ﬂuctuation of return cyclical component dominatesthat of daily return. In fact, the correlation coeﬃcient between return cyclicalcomponent and daily return for TPX, SP, DAX, WTI and JPY is signiﬁcantly
high values of 0.985, 0.989, 0.985, 0.986 and 0.983, respectively.

Figure 7: Return trend component (red line) and return cyclical component (blueline) of daily return for TPX, SP, DAX (left), WTI and JPY (right).

Let us consider the case of long-term returns. For example, Figure 8 shows
the return trend component (red line) and the return cyclical component (blueline) of the 1-year (260-day) return for TPX, SP, DAX, WTI and JPY. Unlike
the above-mentioned case of daily returns, the return trend component largely
swings and the ﬂuctuation range is larger than that of return cyclical component
for all indices. In fact, the correlation coeﬃcient between return trend component and 1-year return for TPX, SP, DAX, WTI and JPY signiﬁcantly increases
to 0.993, 0.992, 0.993, 0.991 and 0.992, respectively. Also, those between return
cyclical component and 1-year return for TPX, SP, DAX, WTI and JPY largely
12decreases to 0.152, 0.166, 0.154, 0.176 and 0.166, respectively.

Figure 8: Return trend component (red line) and return cyclical component (blueline) of 1-year return for TPX, SP, DAX (left), WTI and JPY (right).

The next Table 1 compares the transition of correlation coeﬃcients between
return trend component and the return of 5-day, 20-day, 60-day, 3-year (780day) and 5-year (1, 300-day) in the upper row, with that between return cyclicalcomponent and the each return for TPX, SP, DAX, WTI and JPY in the lower
row. It is noteworthy that the longer period of return is, the larger the correlationcoeﬃcient of return trend component becomes. That is, long-term returns areclosely related to the behavior of return trend component. Instead, the smaller
the correlation coeﬃcient of return cyclical component becomes. It is clariﬁedthat the return cyclical component plays an important role in ﬂuctuations ofdaily return, while the trend component becomes more signiﬁcant in ﬂuctuationsof longer term return.
Note that we do not consider the time diﬀerence between stock markets in thisarticle. In practice, the New York market is open after the Tokyo stock market
was closed on the same day. Table 2 compares the correlation coeﬃcients ofreturn trend component of 1-year return between markets. We divide the whole
period of analysis into the four sub-periods as follows. Period C which stated
before is divided into Period G: January 4, 1993 to October 17, 2000 and Period
H: October 18, 2000 to June 30, 2008. The rest of the whole period is divided
13

Correlation coeﬃcientswith:
TPX return trend component
SP return trend component
DAX return trend component
WTI return trend component
JPY return trend component
TPX return cyclical component
SP return cyclical component
DAX return cyclical component
WTI return cyclical component
JPY return cyclical component

5-day
return
0.515
0.486
0.528
0.515
0.560
0.923
0.937
0.920
0.924
0.911

20-dayreturn
0.874
0.849
0.880
0.873
0.890
0.675
0.725
0.674
0.677
0.663

60-dayreturn
0.966
0.954
0.965
0.966
0.970
0.338
0.393
0.347
0.339
0.320

3-yearreturn
0.998
0.998
0.998
0.995
0.998
0.089
0.092
0.091
0.128
0.092

5-yearreturn
0.998
0.998
0.998
0.997
0.998
0.080
0.077
0.082
0.104
0.086

Table 1: Correlation coeﬃcient of return trend component with the return of 5day, 20-day, 60-day, 3-year (780-day) and 5-year (1, 300-day) for TPX, SP, DAX,WTI and JPY in the upper row, and that of return cyclical component with thereturn of 5-day, 20-day, 60-day, 3-year (780-day) and 5-year (1, 300-day) for TPX,
SP, DAX, WTI and JPY in the lower row.

into Period I: July 1, 2008 to February 15, 2013 and Period J: February 16, 2013
to May 31, 2019.
For reference, the lowest row shows the correlation coeﬃcient between TPX
and SP on the same day. Compared with the correlation coeﬃcient between TPX
and SP (1-day before), it is interesting that there were no signiﬁcant diﬀerences
between them. As a whole, the correlation coeﬃcients reached the highest in
Period I reﬂecting the spillover of crises. It is notable that the correlation coeﬃcients of TPX with SP and DAX, were not so high in Period G, that is, the1990’s.
Correlation coeﬃcients
TPX and SP (1-day before)
DAX and TPX
DAX and SP
TPX and SP

Period G
0.299
0.281
0.543
0.295

Period H
0.535
0.563
0.874
0.532

Period I
0.813
0.757
0.890
0.815

Period J
0.712
0.701
0.713
0.701

Whole period
0.590
0.558
0.782
0.587

Table 2: Correlation coeﬃcient of return trend component of 1-year return between stock market indices: TPX and SP (1-day before), DAX and TPX, DAX
and SP, and TPX and SP for Period G: January 4, 1993 to October 17, 2000,
Period H: October 18, 2000 to June 30, 2008, Period I: July 1, 2008 to February
15, 2013 and Period J: February 16, 2013 to May 31, 2019.

On the other hand, the correlation coeﬃcient of return cyclical component ofdaily return between stock market indices appears diﬀerently as shown in Table 3.
The correlation coeﬃcients between DAX and TPX were always the lowest level
through all sub-periods. Unlike in the case of return trend component, compared
the correlation coeﬃcient between TPX and SP (1-day before) with that between
TPX and SP (the lowest row), it is found that the time diﬀerence makes sense eventhough the correlation is relatively not so high. Similar with the case of return
14trend component in Table 2, the correlation coeﬃcients reached the highest in
Period I reﬂecting the spillover of crises.
Correlation coeﬃcients
TPX and SP (1-day before)
DAX and TPX
DAX and SP
TPX and SP

Period G
0.294
0.224
0.290
0.045

Period H
0.378
0.175
0.572
0.102

Period I
0.548
0.281
0.672
0.096

Period J
0.458
0.229
0.504
0.065

Whole period
0.422
0.223
0.526
0.092

Table 3: Correlation coeﬃcient of return cyclical component of daily return between stock market indices: TPX and SP (1-day before), DAX and TPX, DAX
and SP, and TPX and SP for Period G: January 4, 1993 to October 17, 2000,
Period H: October 18, 2000 to June 30, 2008, Period I: July 1, 2008 to February
15, 2013 and Period J: February 16, 2013 to May 31, 2019.

For daily investment, the time diﬀerence should be considered, however, weleave the further investigation for a future work.

3.5

Fluctuation Characteristics of Each Component

In the previous section, we clarify that the dominant component for daily returns is the return cyclical component and that for 1-year returns is the return
trend component. In order to detect information ﬂows of the above-mentioned
dominant components, we investigate the ﬂuctuation relationships of component
between three stock market indices.
Now we conduct the generalized power contribution analysis to each component of daily return and 1-year (260-day) return, respectively. A ﬁve-variate ARmodel is ﬁtted to each component of TPX, SP, DAX, WTI and JPY for each
sub-periods of G: January 4, 1993 to October 17, 2000, H: October 18, 2000 toJune 30, 2008, I: July 1, 2008 to February 15, 2013 and J: February 16, 2013 to
May 31, 2019, and generalized power contributions are calculated.
Figure 9 shows the graph matrix of the generalized power contributions % of
return trend component of 1-year return for three stock markets with power spectra. From top row to bottom row, the four sub-periods are shown, and from left
column to right column, the generalized power contributions % of return trend
component of 1-year return for TPX, SP and DAX. Each graph shows each proportion of contributor within the power spectrum (white line) at each frequency.
For example, in Period G (the 1990s), the contribution from the index itself occupied 60 − 80 % for TPX, while that for SP and DAX was only 20 %, respectively.
Interestingly, this is the lowest contribution from the index itself among four subperiods for SP and DAX. This implies the global diversiﬁcation already started
in the US and Germany. Furthermore, from the middle graph in Period G, it
is very impressive that TPX was involved in all correlated noise contributing to
SP, and all the contributions are mostly constant at any frequency. On the other15

hand, from the top left graph, it is noteworthy that TPX was independent or
self-contained in terms of ﬂuctuations of 1-year return. The number of concerned
indices of correlated noise such as TPX+DAX is mostly two for Period G, whilemore concerned indices of correlated noise contributed largely since then (i.e.,the 2000s). There can be found some common noise contributions among countries and among sub-periods. As a whole, after the occurrence of the bankruptcy
of Lehman Brothers in Period I, the contribution from various correlated noises
increased.

Figure 9: Generalized power contribution % and log-transformed power spectrum(rhs; white line) of return trend component of 1-year return for TPX, SP and
DAX for the sub-periods of G, H, I and J (from top to bottom).

16

Next, consider the case of daily return. In Figure 10, the graph matrix of thegeneralized power contributions % of return cyclical component of daily returnfor three indices with power spectra are shown. From top row to bottom row, thefour sub-periods are shown, and from left column to right column, the generalized
power contributions % of return cyclical component of daily return for TPX, SP
and DAX. Each graph shows each proportion of contributor within the power
spectrum (white line) at each frequency. Compared with the previous case of
1-year return trend component, the magnitude of power spectrum signiﬁcantly
decreased for all indices according to the right hand scale. The contribution fromindex itself keeps consistently at least 20 % for all indices. In particular, in thecurrent Period J, SP received the constantly 30 % self-contribution. This can bereﬂected by the recent bullish US stock market which has updated its highest
record.
As a summary of the generalized power contribution analysis, Figure 11 shows
the transitions of top 3 contributors with power contribution % of 1-year return
trend component of TPX, SP and DAX (left) and those of daily return cyclicalcomponent of TPX, SP and DAX (right) for the sub-periods of G, H, I and J (fromtop to bottom). Each power contribution % is calculated for all frequency domain.
For daily return cyclical component (right), the top 1 contributor was the index
itself for all sub-periods. On the other hand, for 1-year return trend component(left), the top 1 were various kinds of noises. In Period G (the 1990s), for 1year return trend component, it is astonishing that TPX was concerned with all
three indices as not only the independent noise but also correlated noises. Then,in Period H (the 2000s), TPX lost its own contribution. This implies that thedependence of TPX on external information strengthened in ﬂuctuations of 1-year
return. Despite that, it is interesting that the independent noise of TPX largely
contributed to other two indices. On the other hand, SP strongly contributed to
all three indices, especially, DAX. As for daily return cyclical component (right),
ALL-TPX largely contributed to all three indices in Period H. This implies that
TPX was isolated in ﬂuctuations of daily return. In Period I which was after
the bankruptcy of Lehman Brothers, ALL-WTI largely contributed to TPX andDAX for 1-year return trend component (left), and to all three markets for return
cyclical component (right). That is, WTI was isolated in ﬂuctuations of bothdaily and 1-year returns. The contribution strength of SP remained in Period I.
However, in the current Period J which was after the European debt crisis, the top
3 contributors were concentrated on DAX, TPX+DAX+JPY and TPX+DAX for
all three markets for 1-year return trend component (left).

17Figure 10: Generalized power contribution % and log-transformed power spectrum (rhs; white line) of return cyclical component of daily return for TPX, SP
and DAX for the sub-periods of G, H, I and J (from top to bottom).

18

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Figure 11: Top 3 contributors with power contribution % of return trend component of 1-year return of TPX, SP and DAX (left) and those of return cyclicalcomponent of daily return of TPX, SP and DAX (right) for the sub-periods of
G, H, I and J (from top to bottom).

In Period J, the outstanding contribution of TPX for 1-year return trend
component may be recovered as the contribution of SP to DAX disappeared.
Moreover, the correlated noise of TPX+DAX+JPY commonly contributed toboth 1-year return trend component and daily return cyclical component.
Our ﬁndings indicate the importance of observing the mutual relationshipsof long-term ﬂuctuations, i.e., trends, between these markets. The information
ﬂows of inﬂuential component to return detected by statistical modeling can be
useful in building an investment strategy and making an economic policy.

4

Conclusions

Aiming at detecting information ﬂows for short-term and long-term investments
by investigating the ﬂuctuation relationships between dominant components ofstock market return, we conducted the analysis on the three stock market indicessuch as TOPIX (Japan), DAX Index (Germany) and S&P 500 (US) for the period
from January 4, 1993 to May 31, 2019 (6,890 days).
Compared with moving averages in practical technical analysis, we showed
one of the merits of the KG-trend which is extracted from a stock market index
by a seasonal adjustment model proposed by Gersch and Kitagawa (1983) andKitagawa and Gersch (1984, 1996). In fact, we decomposed each stock market
return into the return trend component and the return cyclical component whichwere derived by applying the seasonal adjustment model to the stock market
index. It was found that the return cyclical component was dominant in daily
19return and so was the return trend component in 1-year return. This implies theeﬀectiveness of applying the seasonal adjustment model. Then, to evaluate theinﬂuence of external information on the stock markets for the last 26 years, we
focused on two inﬂuential indices such as WTI crude oil futures which is often
referred to as a proxy of the oil price, and JPYUSD which is the foreign exchange
rate of the Japanese Yen against the US dollar. We investigated the relationshipsof ﬂuctuation between dominant components by conducting generalized power
contribution analysis (Tanokura and Kitagawa 2004, 2015).
In the 1990s, for return trend component of 1-year return, TPX was concerned with all three market indices not only as the independent noise but also
as correlated noises. Then, in the 2000s, TPX lost its own contribution, however, the independent noise of TPX largely contributed to other two markets.
This implies that TPX strengthened the dependency on external information but
remained isolated in terms of 1-year return ﬂuctuation. On the other hand, inthe 2000s, SP strongly contributed to all three indices in ﬂuctuations of 1-year
return.
As for return cyclical component of daily return, ALL-TPX largely contributed to all three indices in the period of early 2000s before the Lehman shock.
This implies that TPX was isolated even in ﬂuctuations of daily return. In the
period before the bankruptcy of Lehman Brothers, ALL-WTI largely contributed
to TPX and DAX for return trend component of 1-year return, and to all three
indices for return cyclical component of daily return. This reﬂects the current oil
market situation. Because WTI were isolated in ﬂuctuations of both daily and 1year returns. The contribution strength of SP remained after the Lehman shock.
However, in the current period after the European debt crisis, the top 3 contributors were concentrated on DAX, TPX+DAX+JPY and TPX+DAX for all threemarkets for return trend component of 1-year return. As the contribution of SP
to DAX disappeared and the correlated noise of TPX+DAX+JPY commonly
contributed to both return trend component and return cyclical component, thestrength of SP contribution decreased. The outstanding contribution of TPX for
return trend component of 1-year return in the 1990s may be recovered.
Our ﬁndings in this article indicate the importance of observing the mutual relationships of long-term ﬂuctuations which is trends, between these stock
markets. The information ﬂows of inﬂuential component to return ﬂuctuations
detected by statistical modeling can provide the useful information for building
an investment strategy and making an economic policy.
Finally, following future works can be considered. As a KG-trend is estimatedas a smoother which is fully utilized the price information during the whole period,
it updates its own past values when a new information is added. Therefore, theexistence of the diﬀerence between the past KG-trend and the updated KG-trend20

close to the end of the past period will be investigated. In addition, the inﬂuenceof the time diﬀerence between the New York market and the Tokyo stock market
on the same day should be investigated. Utilizing the information ﬂows, the asset
allocation between three markets can be further analyzed. This can be practicallyuseful.

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