SDS-9
A Robust-filtering Method for Noisy NonStationary Time Series with an Application to
Japanese Macro-consumption
Naoto Kunitomo and Seisho Sato
March 2019

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

A Robust-filtering Method for Noisy
Non-Stationary Time Series with an Application to
Japanese Macro-consumption ∗Naoto Kunitomo †andSeisho Sato ‡March 23, 2019
Abstract
We investigate a new filtering method to estimate the hidden states of randomvariables and to handle multiple non-stationary time series data. It is useful for analyzing small sample non-stationary macro-economic time series inparticular and the method is based on the separating information maximum
likelihood (SIML) developed by Kunitomo, Sato and Kurisu (2018), Kunitomo
and Sato (2017) for estimating the non-stationary errors-in-variables models
and Nishimura, Sato and Takahashi (2019) for a financial application. We generalize the previous results to solve the filtering problem of hidden randomvariables of trend and seasonality, which gives a useful method of handlingmacro-economic time series. We develop the asymptotic theory based on
the frequency domain for non-stationary time series. We apply our filtering
method to analyze quarterly and monthly macro-consumption data in Japan.

Key Words
Noisy Non-stationary time series, Errors-variables models, Measurement Error, Macroeconomic time series, trend, seasonality and noise, Robust-filtering, SIML, FourierInversion, Macro-consumption in Japan∗

2019-5. This is a preliminary memorandum. We thank Akihiko Takahashi for giving a precious
suggestion on the SIML method and the filtering method at the initial stage. The research has
been supported by JSPS-Grant JP17H02513.
†
The corresponding author, School of Political Science and Economics, Meiji University, Kanda-Surugadai 1-1 (Sarugakucho 3rd Building C-106), Chiyoda-ku, Tokyo 101-8301,
naoto.kunitomo@gmail.com‡
Graduate School of Economics, University of Tokyo, Bunkyo-ku,Hongo7-3-1, Tokyo113-0033,
seisho@e.u-tokyo.ac.jp

11. Introduction
There has been a vast amount of published research on the use of statistical timeseries analysis of macro-economic time series. One important feature of macroeconomic time series, which is diﬀerent from the standard time series analysis, is the factthat the observed time series is an apparent mixture of non-stationary components
and stationary components. The second feature is the fact that the measurement
errors in economic time series play important roles because macro-economic data areusually constructed from various sources including sample surveys in major oﬃcialstatistics while the statistical time series analysis often ignored measurement errors.
There is yet third important issue that the sample size of macro-economic data israther small and we have 120, say, time series observations for each series when we
have quarterly data over 30 years. The quarterly GDP series, which is the mostimportant data in Japanese macro-economy for instance, has been published since
1994 by the cabinet oﬃce of Japan. Since the sample size is small, it is important touse an appropriate statistical procedure to extract information on trend and noise(or measurement error) components in a systematic way from data. Since the sample size is small, it is important to use an appropriate statistical procedure to extract
information on trend and noise (or measurement error) components in a systematic
way from data. Some of these aspects have been discussed by Morgenstern (1950),
Granger and Newbold (1977), and Nerlove, Grether and Carvalho (1995) for instance. See https://www.esri.cao.go.jp/index-e.html for the oﬃcial macro-economic(GDP) data published by Cabinet Oﬃce of Japanese Government.
In this study we will investigate a new filtering procedure to estimate the hidden states of random variables, which were non-stationary, and to handle multiple
time series data including small sample economic time series of macro-economic
variables. Kunitomo and Sato (2017), and Kunitomo, Sato and Kurisu (2018) havedeveloped the separating information maximum likelihood (SIML) method for estimating the non-stationary errors-in-variables models. They have discussed theasymptotic properties and finite sample properties of the estimation of unknown
parameters in the statistical models. We utilize their results to solve the filtering
problem of hidden random variables and the resulting method leads to a powerful
new method of handling macro-economic time series.
Earlier and related literature on the non-stationary economic time series analysis
are Engle and Granger (1987) and Johansen (1995), which dealt with multivariate
non-stationary and stationary time series and developed the notion of co-integration
without measurement errors. The problem of our interest is related to their work,
but it has diﬀerent aspects and our focus is on the non-stationarity and measurement
error in the non-stationary errors-in-variable models. Also in econometric literature
the issue of identification of parametric models and the issue of estimation when
the true parameters are around the boundary points have been discussed. There2

are some recent econometric studies on time series in the frequency domain such as
Baxter and King (1999), and Muller and Watson (2018). In this respect, the present
study on the non-stationary errors-in-variables models may give some interpretationon their methods.
In the statistical multivariate analysis, there is also some literature on the errorsin-variables models as Anderson (1984, 2003) and Fuller (1987), but they considered
the multivariate cases of independent observations and the underlying situation is
diﬀerent from ours.
Kitagawa (2010) has discussed the standard statistical filtering methods alreadyknown including the Kalman-filtering and the particle-filtering methods. Since (i)these methods depend on the underlying distributions such as the Gaussian distributions for the Kalman-filtering and (ii) the procedure essentially depends on thedimension of state variables, there may be some diﬃculty to extend to the highdimension cases even when it is fixed, say 10. On the other hand, we expect that
our method has some merits when we need to handle small sample economic times
series with non-stationarity and seasonality with many variables because our method
does not depend on the specific distributions as well as the dimension of the underlying random variables. See Kunitomo, Awaya and Kurisu (2017) for a comparisonof small sample properties of the ML and SIML estimation methods for the errorsin-variables models and Nishimura et al. (2019) for an application of financial datasmoothing. The most important feature of the present procedure is that it may be
applicable to small sample time series data. Also it seems that our new method has
a solid mathematical and statistical foundation based on the spectral decompositionof stochastic processes. As an illustrative application, we use our filtering method
to analyze quarterly and monthly macro-consumption data in Japan.
In Section 2 we give some macro-economic data, which have motivated thepresent study. In Section 3 we define the non-stationary errors-in-variables model.
and the SIML method. Then in Section 4 we introduce the SIML filtering method.
Then in Section 5, we discuss the statistical foundation of the method and in Section
6 we propose a method of choosing the numbers of terms, which is based on prediction, and we give some numerical examples based on a set of simulations. Someempirical examples on macro-consumption data in Japan will be given in Section 7
and concluding remarks are given in Section 8. Some mathematical derivations of
our results will be in Appendix.

2. Two Illustrative Examples
As an illustrative example, we plot the graph of two macro-economic time series inJapan : quarterly (real) consumption and quarterly (real) GDP (1994Q1-2018Q2) as
Figure 2.1. It looks like a simple example of linear regression appeared in the undergraduate textbooks. However, if we draw the time series sequences of these (originaland seasonally unadjusted) macro-data published by the Cabinet oﬃce of Japanese3

Government as Figure 2.1, then we find that the situation is not so simple as it
looks in Figure 2.1. In two time series data, there are clearly trends components,
seasonal components, noise components, and possibly business cycle components.
Although many economists usually use the seasonally adjusted (published) data,
which were constructed by using the X-12-ARIMA program, the eﬀects of filteringof the program are often unknown. The X-12-ARIMA program uses the univariate
ARIMA and reg-ARIMA models and the DECOMP program, which was developedby Kitagawa (2010) and it is a possible choice particularly in Japan, uses the univariate AR model and Kalman filtering. Since each time series are handled by using
diﬀerent filtering procedures (that is, diﬀerent reg-ARIMA models for instance), it
may cause a fundamental problem in their interpretation when the focus of interest
is on the relationships among diﬀerent non-stationary data in particular.
Figure 2.3 gives three diﬀerent macro-consumption data (2002 January - 2016
December), which are observed as monthly time series and widely used by economists
in Japan to judge the current macro-business condition in Japan. The first series
is Kakei-Chosa (the data from monthly consumer-survey), the second one is ShohiDouko (the data from monthly retail), and the third one is DaiSanji-Sangyo (theindex data on commerce). The data construction processes are diﬀerent and quite
complex, and each data reflect diﬀerent aspects of macro-consumption. Since theyshow the similar movements, but there are some diﬀerent aspects of trends andseasonal at the same time. Then we need to unify the monthly consumption series
because we want to judge the business condition by just observing these data forevaluating the state of Japanese macro-economy and making macro-economic policy.
Many economists in both central governments and private sectors usually use theseasonally adjusted data, which were constructed from the quarterly or monthly(original) time series and by using the univariate X-12-ARIMA seasonal adjustment
program. Thus it is important to construct the monthly consumption index, whichis consistent with the published quarterly macro-consumption data.
These two empirical examples and related issues motivated us to develop themultivariate non-stationary errors-in-variables models and the filtering method forthe hidden state variables with measurement errors.

3. The Non-Stationary Errors-in-Variables Model and SIML
Let yji be the i−th observation of the j−th time series at i for i = 1, · · · , n; j =′′1, · · · , p. We set yi = (y1i , · · · , ypi ) be a p × 1 vector and Yn = (yi ) (= (yij )) be an
n×p matrix of observations and denote y0 as the initial p×1 vector. We estimate the′model when the underlying non-stationary trends xi (= (x1i , · · · , xpi )) (i = 1, · · · , n),′
but we have the vector of noise component vi = (v1i , · · · , vpi ), which are independentof xi . We use the non-stationary errors-in-variables representation(3.1)

yi = xi + vi (i = 1, · · · , n),4

11.30
11.25
11.20
11.15

RCONSUMPTION(log)

11.10
11.05

11.60

11.65

11.70

11.75

11.80

RGDP(log)

Figure 2.1 : real-GDP vs. real-Consumption

5125000
110000

RGDP

1995

2000

2005

2010

2015

2010

2015

75000
65000

RCON

Time

1995

2000

2005
Time

Figure 2.2 : real-GDP and real-Consumption

6

1500012000

data$KAKEI

0

50

100

150

100

150

100

150

14000
11000

data$SYOUDOU

Time

0

50

115
90 100

data$SANNJI

Time

0

50
Time

Figure 2.3 : Monthly consumption data7

where xi (i = 1, · · · , n) are a sequence of non-stationary I(1) process which satisfy(x)∆xi = (1 − L)xi = vi

(3.2)

,

with the lag-operator Lxi = xi−1 , ∆ = 1 − L,
(x)
wi =

(3.3)

∞
∑

(x) (x)Cj ei−j ,j=0(x)(x) (x)′(x)

and ei is a a sequence of i.i.d. random vectors with E(ei ) = 0 and E(ei ei ) =
(x)
(x)
Σ(x)
(positive-semi-definite). The coeﬃcients matrices Cj (= ckl (j)) are absoe
∑
(x)
(x)
(x)
lutely summable such that ∞j=0 ∥Cj ∥ < ∞, where ∥Cj ∥ = maxk,l=1,··· ,p |ckl (j)|
(x)

(x)

and Cj = (ckl (j)).
The random vectors vi (i = 1, · · · , n) are a sequence of stationary I(0) process with
(3.4)

vi =

∞
∑

(v) (v)

Cj ei−j ,j=0(v)

where the coeﬃcient matrices Cj∑(v)
are absolutely summable ( ∞j=0 ∥Cj ∥ < ∞,

(v)

(v)

(v)

(v)

(v)

where ∥Cj ∥ = maxk,l=1,··· ,p |ckl (j)| and Cj = (ckl (j))) and ei are a sequence of(v) (v)′(v)
i.i.d. random vectors with E(ei ) = 0, E(ei ei ) = Σ(v)e (non-negative definite).
Let f∆x (µ) and fv (µ) be the spectral density (p × p) matrices of ∆xi and vi(i = 1, · · · , n) as(3.5)

f∆x (µ) = (

∞
∑

∞
∑

(x)
Cj e2πiµj )Σ(x)
e (

j=0

1
1
(x)′
Cj e−2πiµj ), (− ≤ µ ≤ )
2
2
j=0

and
(3.6)

fv (µ) = (

∞
∑

(v)
Cj e2πiµj )Σ(v)
e (

j=0
(x)

∞
∑

1
1
(v)′
Cj e−2πiµj ), (− ≤ µ ≤ )
2
2
j=0

(v)

where we set C0 = C0 = Ip and i2 = −1 (see Chapter 7 of Anderson (1971) for
instance.)
Then the spectral density matrix of the transformed vector process ∆yi (= yi −yi−1 )
is
(3.7)

f∆y (µ) = f∆x (µ) + (1 − e2iµ )fv (µ)(1 − e−2iµ )and we have the key relation at the zero-frequency such that
(3.8)

f∆y (0) = f∆x (0) .
8

We denote the long-run variance-covariance matrices of the trend components andthe stationary components for g, h = 1, · · · , p as
(x)

(3.9)

Ωx = f∆x (0) (= (ωgh )) ,

and
(v)

(3.10)Ωv = fv (0) = (ωgh ) .

When each pair of vectors ∆xi and vi are independently, identically, and normally distributed (i.i.d.) as Np (0, Σx ) and Np (0, Σv ), respectively, and we have′the observations of an n × p matrix Yn = (yi ) and set the np × 1 random vector′′ ′(y1 , · · · , yn ) . Given the initial condition y0 , we have()′′
(3.11)
vec(Yn ) ∼ Nn×p 1n · y0 , In ⊗ Σv + Cn Cn ⊗ Σx ,′where 1n = (1, · · · , 1) and(3.12)



1 0
 1 1

Cn = 
 1 1
 1 ···
1 ···

···
0
1
1
1

0
···
···
1
1


0
0 

0 
.


0
1 n×n

We return to consider the general case of (2.1)-(2.4) and use the Kn −transformation′that from Yn to Zn (= (zk )) by()(3.13)
Zn = Kn Yn − Ȳ0 , Kn = Pn C−1
n ,
where

(3.14)


10 ··· 00
 −1 10 ··· 0 



−1 1
0 ··· C−1,n =  0
 0
0 −1 10 
000 −1 1n×n

and(3.15)√(n)(n)
Pn = (pjk ) , pjk =

]
[
1
1
2
2π
(k − )(j − ) .
cos
2n + 1
2
2
n + 12′′−1= Pn Dn Pn and Dn is a diagonal
By using the spectral decomposition C−1
n Cn
2k−1
matrix with the k-th element dk = 2[1 − cos(π( 2n+1 ))] (k = 1, · · · , n) and we write[ ()]
2k − 1∗2 π(3.16)
akn (= dk ) = 4 sin(k = 1, · · · , n) .
2 2n + 1

9Then the separating information maximum likelihood (SIML) estimator of Σ̂x when(x)
wi are i.i.d. vectors can be defined byn
1 ∑′Gm = Σ̂x,SIM L =zk zk ,
mn k=1m(3.17)

where we set m = mn = [nα ] (0 < α < 1).
The estimation of the variance-covariance matrix Σv when vi are i.i.d. vectors havebeen discussed by Kunitomo and Sato (2017). Some consistent estimators of Σv
have been developed.

4. The SIML Filtering Method
4.1

The Basic Filtering

We consider the general filtering procedure based on the Kn −transformation (3.13).
Because the elements of the resulting n×p random matrix Zn by this transformation
take real values in the frequency domain, it is easy to interpret their roles. We
consider the inversion of a transformation of orthogonal frequency processes. Let ann × p matrix′X̂n = Cn Pn Qn Pn C−1
n (Yn − Ȳ0 )

(4.1)
and

Zn = Pn C−1
n (Yn − Ȳ0 ) .

(4.2)

The stochastic process Zn are the orthogonal decomposition of the original timeseries Yn and Qn is an n × n filtering matrix. We give explicit forms of usefulexamples including the trend filtering procedure and the seasonal filtering procedure
for macro-time series. Although we can interpret the existing filtering procedures
within our general framework, first it is useful to give linear filtering procedures.
(i) General Linear Filtering : Let an n × n diagonal matrix
(4.3)

Qn =

n
∑

′

wi,n ei ei

i=1′

and ei = (0, · · · , 1, · · · ) (i = 1, · · · , n) are the unit vectors and wi,n (i = 1, · · · , n)are some non-negative constants. The trivial case is when we take wi,n = 1 (i =1, · · · ), we have the identity matrix. There are several useful linear filtering cases
and we will give two cases as the trend filtering and the seasonal filtering by choosing
10

wi,n = 1 or 0 for some i′ s.
(ii) Trend Filtering : Let an m × n choice matrix Jm = (Im , O), and let also n × p
matrix′′X̂n = Cn Pn Jm Jm Pn C−1
n (Yn − Ȳ0 )(4.4)

and and an n × n matrix
′

(4.5)

Q n = P n Jm Jm P n .

We will construct an estimator of n×p hidden state matrix Xn in the lower frequency
parts by using the inverse transformation of Zn by deleting the estimated noise parts.
(See Nishimura, Sato and Takahashi (2019) as an example.) For this purpose, let
the [m + (n − m)] × [m + (n − m)] partitioned matrix()
P11 P12Pn =P21 P22
and
(4.6)

′

Pn Jm Jm Pn =

(

′

P11
′
P12

)

(
(P11 , P12 ) = In −

′

P21
′
P22

)
(P21 , P22 ) .

′′After some calculations (see the Appendix), the (j, j )-th element of Qn = Pn Jm Jm Pn (=(qj,j ′ )) is given by[]2mπ
sin 2n+1(2j − 1)12m(4.7) qj,j =+,π
2n + 1 2n + 1 sin 2n+1
(2j − 1)
[
]
′
′
2mπ
2mπ
sin 2n+1(j + j − 1) sin 2n+1
(j − j )1′
qi,j ′ =+(j ̸= J ) .
′′ππ2n + 1 sin 2n+1 (j + j − 1) sin 2n+1 (j − j )We evaluate MSE of X̂n (n × p) and then in the simple case
(4.8)

′

E[tr(X̂n − Xn ) (X̂n − Xn )]
′
′ −1
′
′
′
′
= tr(Σv )tr[Jm Pn Cn Cn Pn Jm × Jm Pn C−1
n C n P n Jm ]
′
′
′
′ −1
′
∗′
+tr(Σx )tr[J∗m Pn Cn Cn Pn J∗m × J∗m Pn C−1
n C n P n Jm ] ,where Jm = (Im , O) (m × n matrix) and J∗m = (O, In−m ) ((n − m) × n matrix).
(iii) Seasonality Filtering : We consider the filtering based on the Kn −transformationin (3.13). We consider the inversion of some frequency parts of the random matrix
Zn . The leading example is the seasonal frequency in the discrete time series and11we take s (> 1) being a positive integer. Let an m2 × [m1 + m2 + (n − m1 − m2 )]choice matrix Jm1 ,m2 ,n = (O, Im2 , O), and let also n × p matrix′′X̂n = Cn Pn Jm1 ,m2 ,n Jm1 ,m2 ,n Pn C−1
n (Yn − Ȳ0 )(4.9)

and an n × n matrix′(4.10)

Qn = Pn Jm1 ,m2 ,n Jm1 ,m2 ,n Pn .

As an example, when we have the seasonal frequency s (> 1), we can take m1 =[2n/s] − [m/2] and m2 = m. For instance, we take s = 4 for quarterly data and′s = 12 for monthly data. After some calculations, the (j, j )-th elemnt of Qn =′Pn Jm1 ,m2 ,n Jm1 ,m2 ,n Pn (= (qj,j ′ )) is given by(4.11) qj,jqi,j ′[]
1 +m2 )π1 )πsin 2(m2n+1(2j − 1) − sin 2(m
(2j
−
1)
2m2
1
2n+1
=
+
,
π
2n + 1 2n + 1
sin 2n+1(2j − 1)[′′
1 1+m2 )π1 )πsin 2(m2n+1(j + j − 1) − sin 2(m(j + j − 1)
1
2n+1
=
π
2n + 1
sin 2n+1(j + j ′ − 1)]′′
1 +m2 )π1 )πsin 2(m2n+1(j − j ) − sin 2(m
(j
−
j
)
′
2n+1
+
(j ̸= j ) .
′π
sin 2n+1 (j − j )We note that when m1 = 0 and m2 = m, the resulting formula become to those in
the basic case.
We evaluate MSE of Ŝn , which is X̂ (n × p) and then in the simple case′E[tr(Ŝn − Sn ) (Ŝn − Sn )]
′
′
′
′ −1
′
′
= tr(Σv )tr[Jm1 ,m2 ,n Pn Cn Cn Pn Jm1 ,m2 ,n × Jm1 ,m2 ,n Pn C−1
n Cn Pn Jm1 ,m2 ,n ]
′

′

′

′

′

′

−1
∗
+tr(Σx )tr[J∗m1 ,m2 ,n Pn Cn Cn Pn J∗m1 ,m2 ,n × J∗m1 ,m2 ,n Pn C−1
n Cn Pn Jm1 ,m2 ,n ] ,where Jm1 ,m2 ,n = (O, Im2 , O) (m2 × n matrix) and an (n − m2 ) × n matrix
[
]
Im1 O
O
∗
Jm1 ,m2 ,n =.
O O In−m1 −m2(iv) Non-linear Filtering : The method of filtering procedure based on (4.1) and(4.2) is simple, but very general. There can be several extensions and a combinationof trend filtering and seasonal filtering is such an example, which can be written as(4.3) for instance. There can be possible extensions of the linear filtering procedures
to non-linear filtering procedures based on (4.1) and (4.2) and other filtering methods
in the existing literature can be interpreted as special cases of (4.3).
12

4.2

An Extended Errors-in-Variables Model

Now we will consider the meaning of the SIML filtering. For this purpose we extend
the basic model of (3.1) by adding seasonal components and investigate the additive
decomposition modelyi = xi + si + vi (i = 1, · · · , n),(4.12)

where we take positive integers s (s > 1), N , and n = sN for the resulting simplicity
and arguments and si (i = 1, · · · , n) are a sequence of non-stationary process which
satisfy(s)∆si = (1 − L)si = vi ,(4.13)
where

(s)
vi =

(4.14)∞∑(s) (s)Csj ei−sj ,

j=0
(s)

(s)

(s) (s)′and ei is a a sequence of i.i.d. random vectors with E(ei ) = 0 and E(ei ei ) =Σ(s)e (non-negative definite).
Let f∆x (µ), f∆s (µ), and fv (µ) be the spectral density (p × p) matrices of ∆xi , ∆si
and vi (i = 1, · · · , n) as(4.15)f∆x (µ) = (

∞
∑

(x)
Cj e2πiµj )Σ(x)
e (

∞
∑

1
1
(x)′
Cj e−2πiµj ) (− ≤ µ ≤ ) ,
2
2
j=0

j=0

(4.16)f∆s (µ) = (

∞
∑

(s)
Csj e2πiµsj )Σ(s)
e (

j=0

∞
∑

1
1
(s)′
Csj e−2πiµsj ) (− ≤ µ ≤ ) ,
2
2
j=0

and
(4.17)

fv (µ) = (

∞
∑

∞
∑

(v)
Cj e2πiµj )Σ(v)
e (

j=0
(x)

(s)

1
1
(v)′
Cj e−2πiµj ) (− ≤ µ ≤ )
2
2
j=0

(v)

where we set C0 = C0 = C0 = Ip as normalizations and i2 = −1 (see Chapter
7 of Anderson (1971) for instance.)
Then the p × p spectral density matrix of the transformed vector process, which areobservable, the spectral density of the diﬀerence series ∆yi (= yi − yi−1 ) can be
represented as(4.18)f∆y (µ) = f∆x (µ) + f∆s (µ) + (1 − e2πiµ )fv (µ)(1 − e−2πiµ ) .
13

We denote the long-run variance-covariance matrices of the trend components andthe stationary components for g, h = 1, · · · , p as
(4.19)

1(x)(s)Ωx = f∆x (0) (= (ωgh )) , Ωs = f∆s ( ) (= (ωgh )) ,
s

and
(v)

(4.20)Ωv = fv (0) = (ωgh ) .

5. A Statistical Foundation
At the first glance, the SIML filtering procedure might be seen as an ad-hoc statisticalprocedure without any mathematical foundation. However, on the contrary, there
is a rather solid statistical foundation.
2π
Let θjk = 2n+1
(j − 12 )(k − 12 ) ,1(n)
pjk = √
(eiθjk + e−iθjk )2n + 1and we write
(5.1)

(n)

∆λ z

(n)
(λk ) =n∑(n) (n)(n)
pjk rj , rj = yj − yj−1 ,j=1(n)which is actually (the real-valued) Fourier-transformation. Then ∆λ z(n) (λk ) (k =(n)
1, · · · , n) are the (real-valued) Fourier-transformation of data at the frequency λk (=(k − 1/2)/(2n + 1)), which is a (real-part of) estimate of the orthogonal incremental
process z(λ).
For the development of statistical inferences, we have the next result by using
the CLT (central limit theorem) for discrete and (ergodic) stationary time series.
See the Appendix for the derivations.
Theorem 5.1 : Let rj (j = 1, · · · , n) be an ergodic stationary stochastic process′with Γ(h) = E(rj rj−h ) and
(5.2)

∞
∑

∥Γ(h)∥ < ∞ .

h=0

∑
(n)
(n) (n)(n)(i) Let ∆λ z(n) (λk ) = nj=1 pjk rj and rj be an ergodic stationary sequence withE[rj ] = 0 and the (symmetrized real) spectral density matrix(5.3)

fSR (λ) = Γ(0) +∞∑cos(2πhλ)[Γ(h) + Γ(−h)] ,

h=1

14

is the positive definite and bounded (real-valued and symmetrized) spectral matrix.
(n)(n)
Also assume that λk → s, λk′ → t and 0 < s < t < 21 .
Then as n −→ ∞
]
[
[
]
(n)
∆λ z(n) (λk )fSR (s)0w(5.4)−→ N2p 0, [
] .
(n)
0
fSR (t)∆λ z(n) (λk′ )

(5.5)

∑n

∑[tn]

(ii) Let Zn (t) − Zn (s) = √1nn −→ ∞

k=[sn](n) (n)j=1 pjk rj

for 0 < s < t < 1. Then asw

Zn (t) − Zn (s) −→ Np [0, FSR (t) − FSR (s)] ,where FSR (t) is the p × p (symetrized real) spectral distribution matrix
∫ t
(5.6)
FSR (t) =fSR (λ)dλ .
0This theorem covers the basic model with (3.1)-(3.4) and the extended modelwith (4.12)-(4.14) with the moment conditions because ∆yi are stationary in thesecases. The spectral density in the basic model is given by (3.7) while the spectral
density in the extended model is given by (4.18). We immediately find that the longrun variance-covariance matrix of hidden states in the basic model can be estimatedby (3.17).
Since the asymptotic variance-covariance matrix of the orthogonal random vec(n)tors ∆λ z(n) (λk ) is the (symmetrized real) spectral density matrix, it should be
estimated consistently. Although there can be many ways to solve the problem, it
may be natural to use the estimation method as (3.17) at the zero frequency.
The SIML estimator of fSR (t) (0 < 12 ) can be defined by(5.7)

1
Gm (t) =mn[2nt]+[ m2n ]

∑

(n)

(n)

(n)

(n)

′

(∆t zk (λk ))(∆t zk (λk )) .

k=[2nt]−[ m2n ]+1

Then we have the next result.
Theorem 5.2 : Let rj (j = 1, · · · , n) be an ergodic stationary stochastic process′with Γ(h) = E(rj rj−h ) with (5.2) and (5.3). Assume the fourth order moment
conditions supj≥1 [∥rj ∥4 ] < +∞. Assume (3.1)-(3.4) and in (5.7) we set mn =[nα ] (0 < α < 1). Then for any t ∈ (0, 21 ), as n −→ ∞
(5.8)

p

Gm (t) −→ fSR (t) .
15

Furthermore, it is possible to show the asymptotic normality of the SIML estimator
under the condition that 0 < α < 0.8. (Kunitomo and Sato (2017) have discussed
the case of t = 0.) Furthermore, it may be possible to develop the consistentestimation of spectral components of non-stationary time series in the form of (4.7).

In the traditional statistical time series analysis for a stationary discrete (vector) process r∗k with the spectral distribution F, it has been known that there
exists a right-continuous orthogonal increment (vector, complex-valued) processz∗ (λ) (−1/2 ≤ λ ≤ 1/2) such that
∫
∗
(5.9)
rk =
ei2πkν dz∗ (ν) (k = 1, · · · , n).
(−1/2,1/2](The topic here goes back to Doob (1953), but see Hannan (1971) or Brockwell andDavis (1990).) The trend component and seasonal component of (real-valued) timeseries in our setting can be defined by∫(u)cos(2πkν)w(u, ν)z(ν) (k = 1, · · · , n)(5.10)
rk =(0,1/2]for u = x or u = s, where w(u, ν) is the indicator function of some frequencies
around zero (for trend) and seasonal frequency (for seasonality), respectively, andz(ν) 0 < ν ≤ 1/2 is the right-continuous orthogonal increment (real-valued) process,which is the limiting continuous process of discrete time series.
(u)

Since z(ν) is not observed with finite data, the (real-valued) estimate of rk (i.e.
the hidden components of rk ) from data can be represented as
∫
k−1
k
(u,n)
(5.11) rk=cos(2πkν)wn (u, ν)dz(n) (ν) (<ν≤, k = 1, · · · , n),2n2n(0,1/2]where wn (u, ν) is a measurable function and z(n) (ν) is the estimated orthogonal
process at the frequency ν ∈ [0, 1/2] from data where we have abused some notations
such as z(n) (0) = 0. Hence the ln -th component of (4.1) with (4.2) and (4.3) could
be approximated and written as]
l n [∫∑(u,n)(n)(5.12)
x̂ln =
cos(2πkν)wn (u, ν) dz (ν) ,k=1(0,1/2]

when we take ln = [t n] (0 < t < 1) and choose the kernel function w( u, ν) appropriately in the frequency domain. Hence these representations could be interpreted as
16the (real) Fourier inverse of Kn − transformation of finite observations of time seriesin (5.1).
From this interpretation, we find that there may be an interesting representation
problem of (discrete time and continuous time) stationary processes and orthogonal
incremental stochastic processes, which is closely related to the method of data
analysis we are investigating.

6

Prediction and Model Selection

6.1

Prediction

The prediction problem can be solved as follows. For the simplicity, we consider theh−period ahead prediction of yj (j = n + h, h ≥ 1) given the information available(n)
at j = n. It is natural to use the incremental vectors rn+s (s ≥ 1) available at
j = n. Then by using the orthogonal processes and ignoring the sampling errors of
estimating them (that is n should be large enough), we have an expression of theprediction error as(6.13)

e(h) =

h
∑

∫ 1/2
rj =

j=1

−1/2

[e−i(n+1)ν + · · · + e−i(n+h)ν ]w(u, ν)z̃(dν) .

Hence the prediction MSE is given as(6.14)′∫ 1/2 [

E[e(h)e (h)] =−1/2

sin hν
ssin ν2]2w(u, ν)f (ν)dν ,where f (ν) is the p × p spectral density matrix for rj . When h = 1 and w(u, ν) = 1
for ν ∈ (−1/2, 1/2), we have the familiar expression of one-period prediction MSE
in time series analysis as
∫ 1/2
′
′
(6.15)
E[e(1)e (1)] =f (ν)dν = E[rn+1 rn+1 ] ,−1/2

which corresponds the variance-covariance matrix of innovation vector at j = n.
When p = 1, we find that
[∫
]
∫
1/2

(6.16)−1/2

1/2

f (ν)dν ≥ exp

log[f (ν)]dν

,

−1/2

which is the lower bound of the one-step ahead prediction. (See Chapter 3 of Hannan(1070).) Therefore the prediction MSE of our predictor is slightly greater than thelower bond of prediction MSE. When the spectral density is flat, two prediction
MSE are the same.
17

6.2

Model Selection

When we have estimates of the state variables xi (i = 1, · · · , n), the estimates ofnoise components are v̂i = yi − x̂i (i = 1, . . . , n). Then an estimated MSE of the
one-step ahead prediction errors based on the SIML-smoothing or filtering is givenby(6.17)′

PMSEn (h) = E[(yn+h − yn )(yn+h − yn ) |Fn ] ,where Fn is the σ−field (information) available at n.
Then one may try to minimize the estimated h-step prediction MSE by choosing anappropriate m. It may be reasonable to choose h = 2, 3 for the estimation of trend
while h = 4, 8 for the estimation of seasonal from our limited experiments.

6.3

Numerical Experiments

We have done several Monte Carlo experiments on the problem of choosing anappropriate m. For this purpose, we first use the simple model of trend plus noisemodel, xi = xi−1 + ut and yi = xi + vi (i = 1, · · · , n). The criterion function is theprediction MSE given by(6.18)

1
PMSE∗n (h) =n∑(yi − x̂i )2 .
h i=n−h+1

We give some results on the trend filtering as Tables 6.1-6.3 by taking h = 2, · · · , 8
and n = 80, 120, 200, 300. In our simulations, as n increases, we have larger optimal
choice of m. Also as h increases, the optimal choice m decreases. When we havelong-horizon with h, it may be natural to use small number of lower frequencies. σxand σv , the optimal choice of m could be stable.

18

Table 6.1 : Optimal Choice of m
(σx = 0.3 , σv = 0.05)n80
h=2 12
h=3 8
h=4 6
h=5 5
h=6 4
h=7 4
h=8 3

120
19
12
9
7
6
5
4

200
32
21
15
13
10
9
8

400
65
42
32
26
21
18
16

Table 6.2 : Optimal Choice of m
(σx = 0.3 , σv = 0.4)n80
h=2 13
h=3 8
h=4 6
h=5 5
h=6 4
h=7 4
h=8 3

120
19
13
9
8
6
5
5

200
33
21
16
13
11
9
8

400
66
43
32
26
21
18
16

Table 6.3 : Optimal Choice of m
(σx = 0.3 , σv = 1.0)n80h=2 13
h=3 9
h=4 6
h=5 5
h=6 4
h=7 4
h=8 4

120
20
13
10
8
7
6
5

200
33
22
16
13
11
9
8

400
67
44
33
26
22
19
16

As the second example, we use the simple model of seasonal-plus-noise model. xi =xi−1 + ut , si = xi × SAi , SAi = SAi−4 , SAj = U (−0.75, 0.75) (j = 1, 2, 3), SA4 =−(SA1 + SA2 + SA3 ) and yi = si + vi (i = 1, · · · , n). We have used the filters on[n/2 − m∗1 , n/2 + m∗1 ] and [n − m∗2 , n]. (The notations are slightly diﬀerent from theones in Section 5.) The criterion function is the prediction MSE given by(6.19)

MSEn =n∑(yi − ŝi )2 .

i=n+1−h19

Table 6.4 : Optimal Choice of m∗1 and m∗2(σx = 0.001 , σv = 0.01)n80(m∗1 , m∗2 ) 120(m∗1 , m∗2 ) 240(m∗1 , m∗2 )
h=4
14,17
22,20
34,22
11,19
17,20
27,21
h=8
h=12
16,18
20,21
24,21
Table 6.5 : Optimal Choice of m∗1 and m∗2(σx = 0.001 , σv = 0.5)n80(m∗1 , m∗2 ) 120(m∗1 , m∗2 ) 240(m∗1 , m∗2 )
h=4
6,6
10,9
20,17
h=8
4,3
5,5
10,10
4,3
4,3
7,7
h=12
Table 6.6 : Optimal Choice of m∗1 and m∗2(σx = 0.001 , σv = 1.2)n80(m∗1 , m∗2 ) 120(m∗1 , m∗2 ) 240(m∗1 , m∗2 )
h=4
6,5
10,9
20,17
h=8
4,3
5,4
10,4
h=12
4,2
4,3
6,6
We give some results on the seasonal filtering as Tables 6.4-6.6 by taking h = 4, 8, 12
and n = 80, 120, 240 because we investigate the non-stationary seasonal components.
In our simulations, as n increases, we have larger optimal choice of m1 and m2 , butthey are not large in Tables 6.5 and 6.6. As h increases, the optimal choice m may
be gradually decreasing.

7. An Application to Japanese Macro-consumption
We have applied our filtering method to the analysis of Japanese quarterly (real)consumption-GDP data as the first example and three monthly consumption dataas the second example, which have been discussed in Section 2. All figures in thissection are gathered in the Appendix B.
First, we calculate the transformation of the original quarterly consumption data,
which show the non-stationarity and it may be a typical macro-economic variable.
Then we calculate the realized Zn as Figure 7.1 from the diﬀerenced consumption
data. In this case the Zn series gives a strange form mainly because the nonstationarity and seasonality. Since there are clear seasonal components in the original series, we calculated the realized Zn (Figure 7.2) and the empirical cumulative20

distribution of Zn2 (Figure 7.3), which correspond to the normalized sample spectral
distribution. Because we have quarterly macro-data, we have a large up and downaround 50, which corresponds to the seasonal frequency at s = 4. The empirical
spectral density has a abrupt change at this frequency. Since it has been a practicein time series data analysis to use seasonal diﬀerencing in the Box-Jenkins method,
we calculated the realized Zn (Figure 7.4) after seasonal diﬀerencing. Although thespectral contribution around the seasonal frequency, there are some rather wild fluctuations at many other frequencies. Because we have some diﬃculty to interpretthe resulting time series, it may not be possible to justify the seasonal diﬀerencing
procedure. In our analysis we simply use the diﬀerencing and then use the frequency
domain analysis.
In Figure 7.5, we have investigated the analysis of real GDP. We have chosenm = [n.99 ] and delete the seasonal frequency around 48-52 and some high-frequency
part. It means that we delete 5 data around the seasonal frequency and several highfrequencies, which correspond to the aliasing eﬀects frequency and this procedure
was necessary to obtain stable empirical results. Then we compared the filtered timeseries by our method and the oﬃcial (published) seasonally adjusted time series. We
have found that the diﬀerences of these two time series are rather small and they
are often of negligible magnitude. Although our filtering procedure is quite simple,
this empirical example shows the usefulness of our method developed in the present
study.
As the second example, we have analyzed three consumption (monthly) timeseries and the quarterly consumption time series. As we have seen in Section 2, threemacro-consumption series have similarities and some diﬀerences. In our example, ourgoal is to construct the monthly consumption index, which is close to the observed
quarterly consumption time series. Because there are non-stationary trend, seasonaland measurement errors, it may not be obvious to construct such consumption index
by the existing statistical tool.
Let Yi (i = 1, · · · , n) be the target (quarterly) time series and Zkt (k = 1, 2, 3; t =3(i − 1) + j, j = 0, 1, 2) be the k-th monthly time series. (t = 0 is the initial period.)
Then the criterion function is
]2
[
n
3
∑
∑
(T )
(T )
(7.1)
MSE(m, m1 , m2 , m3 , w1 , w2 , w3 ) =∆Ŷi −
wj ∆Zji
,
i=1
(T )

(T )

(T )

j=1

where ∆Ŷi
= Ŷi − Ŷi−1 , (the trend part of the estimated ∆Yi because we
(T )
(T )
(T )
observe the quarterly data on Yi ) ∆Zji = Zji − Zj,i−1 (the trend parts of ∆Zji ),and wj (j = 1, 2, 3) are (unknown) weight coeﬃcients and m, mj (j = 1, 2, 3) are
the numbers of trend filtering. In the above formulation we need to measure theprediction errors based on diﬀerenced data because we have non-stationary trends
and seasonal.
21

By using the least squares method, we minimized the MSE criterion with respect
to the underlying parameters. The estimated wj (j = 1, 2, 3) are 3.69, 5.19 and1.64 (while the measurement units are diﬀerent), but their magnitude are aboutcomparable to the published quarterly consumption level at 2002Q1), which arestatistically significant with 1%. The optimal choice of m = 29 while m1 = 36, m2 =23 and m3 = 33. In our limited experiments, we have found some improvements ofprediction errors by choosing diﬀerent m and mj (j = 1, 2, 3).
The black curves are the original series and the red curves are estimated trend
curves in Figures 7.6, 7.7 and 7.8. By taking relatively large mj (j = 1, 2, 3) we canrecover the cycle components of each series, which are the key role to be used as
the indicators of macro-business condition. In Figure 7.9, the green curve shows thepredicted value calculated from the latest observed (quarterly) data plus the predicted monthly part based on the estimated parameters. Since there is no monthly
observation of quarterly published consumption, we draw their latest (quarterly)level by the black curve and the estimated SIML (filtered) values by the red curve.
One notable problem may be the introduction of consumption tax in 2014 April andthere was a sharp deviation of trend mainly because the black curve is the quarterly
observed macro-consumption. In the present study we did not have a focus on thisevent, but it could be handled by some additional complication such as using adummy variable. Overall, we have found that while our procedure is relatively simple in comparison to the X-12-ARIMA seasonal adjustment with reg-ARIMA model,
the predictive results are satisfactory. In Figure 7.10, we have drawn the prediction
errors in terms of the diﬀerenced value Yi (i = 1, · · · , n) based on our procedure.
This figure illustrates the usefulness of the procedure because the macro-economic
time series are non-stationary with measurement errors.

8. Concluding Remarks
When the observed non-stationary time series contain noises, it may be diﬃcult
to disentangle the eﬀects of trends and the noises. For instance, in many macro-times
series we observe non-stationary trend, non-stationary seasonality and stationary
cycles with measurement errors. In this paper we investigate a new procedure to decompose of time series into non-stationary trend components, seasonal components
and stationary noise (or measurement errors) components. One important conclusion is that it is useful to transform the observed time series by Kn −transformationand investigate the transformed Zn series, which is based on Kunitomo and Sato(2017), Kunitomo et al. (2018) and Nishimura et al. (2019). We can investigate
the information of noisy-time series such as macro-economic variable by looking attheir frequency and re-cover the traditional spectral distribution by the squaredZn −variables.
As an illustrative empirical example, we have used our filtering method proposedin this paper to analyze quarterly and monthly macro-consumption data in Japan.
22

We have applied our method to construct the monthly consumption index, whichis consistent with the published or oﬃcial (GDP-)consumption quarterly data. Although the problem is practically complicated, we have shown that our method gives
a useful result for practical purposes.
There can be several interesting problems developed by our approach in this paper. Since it is easy to handle the Kn −transformation and the transformed Zn −data,
it may be straight-forward to develop a new way to determine the number of trend
factors and seasonal factors. Since there are many important empirical applications
such as the examples mentioned by Baxter and King (1999), and Müller and Watson(2018), we have currently investigating several empirical examples with trends andseasonality.

References[1] Brockwell, P. and R. Davis (1990), Time Series : Theory and Methods, 2nd
Edition, Wiley.
[2] Baxter, H. and R. King, ”Measuring Business Cycles : Approximate BandPass Filters for Economic Time Series,” it Review of Economics and Statistics,
Vol.81-4, 575-593.
[3] Doob, J.L (1953), Stochastic Processes, Wiley.
[4] Durrett, R. (1991), Probability : Theory and Examples, Duxbury Press.
[5] Engle, R. and C.W.J. Granger (1987), “Co-integration and Error Correction,”
Econometrica, Vol.55, 251-276.
[6] Fuller, W. (1987), Measurement Error Models, John-Wiley.
[7] Granger, C. W. J. and Newbold, P. (1977), Forecasting Economic Time Series,
Academic Press.
[8] Johansen, S. (1995), Likelihood Based Inference in Cointegrated Vector Autoregressive Models, Oxford UP.
[9] Hannan, E.J. (1970), Multiple Time Series, Wiley.
[10] Hayashi, F. (2000), Econometrics, Princeton University Press.
[11] Kitagawa, G. (2010), Introduction to Time Series Analysis, CRC Press.
[12] Kunitomo, N. , S. Sato and D. Kurisu (2018), Separating Information Maximum
Likelihood Estimation for High Frequency Financial Data, Springer.

23[13] Kunitomo, N. and S. Sato (2017), ”Trend, Seasonality and Economic Time
Series : the Non-stationary Errors-in-variables Models,” SDS-4, MIMS, Meiji
University, http://www.mims.meiji.ac.jp/publications/2017-ds.
[14] Kunitomo, N., N. Awaya and D. Kurisu (2017), ”Some Properties of Estimation Methods for Structural Relationships in Non-stationary Errors-in-Variables
Models,” SDS-3, MIMS, Meiji University.
[15] Morgenstern, Oskar (1950). On the accuracy of economic observations, (1966,
revised edition), Princeton University Press.
[16] Nerlove, M., D.M. Grether and J.L. Carvalho (1995), Analysis of Economic
Time Series : A Synthesis, Revised Edition, Academic Press.
[17] Müller, U. and M. Watson (2018), ”Long-run Covariability,” Econometrica,
86-3, 775-804.
[18] Nishimura, G. K., Seisho Sato, Akihiko Takahashi (2019), ”Term Structure
Models During the Global Financial Crisis: A Parsimonious Text Mining Approach,” Asic-Pacific Financial Markets.

24

APPENDIX A : Mathematical Derivations
In this Appendix, we give some details of derivations which we have omitted in theprevious sections.
(i) On (4.7) and (4.11) :2π
Let θjk = 2n+1
(j − 12 )(k − 12 ) (j, k = 1, · · · , n) . We use the relation that
θjk + θj ′ ,k =

2π
1
2π
1
′
′
(j + j − 1)(k − ) , θjk − θj ′ ,k =(j − j )(k − ) .
2n + 12
2n + 1
2

Then we have
(A.1)

4

∑

[cos θjk cos θj ′ ,k ]

k∈In

∑

=

[e

i(θjk +θ ′ )j ,k−i(θjk +θ ′ )

+e

j ,k

]+

k∈In

∑

i(θjk −θ ′ )

[e

j ,k

+e

−i(θjk −θ ′ )
j ,k

],

k∈In

where In = [1, · · · , m] (or In = [m1 + 1, · · · , m1 + m2 ]) is the index set for j and k.
For In = [m1 + 1, · · · , m1 + m2 ], by re-writing
2π
1 2π
′
′
(j + j − 1) +(j + j − 1)(k − m1 ) ,θjk + θj ′ ,k = (m1 − )2 2n + 1
2n + 1
and

1 2π
2π
′
′
θjk − θj ′ ,k = (m1 − )(j − j ) +(j − j )(k − m1 ) ,2 2n + 1
2n + 1the summation of the first two terms becomes′

1 − ei 2n+1 (j+j −1)m2
2π

′

2π
i(m1 + 21 ) 2n+1
(j+j −1)e

×′

1 − ei 2n+1 (j+j −1)
2π

′

+e

′

1 − e−i 2n+1 (j+j −1)m22π2π−i(m1 + 12 ) 2n+1
(j+j −1)×′

1 − e−i 2n+1 (j+j −1)
2π′

For the last two terms, we need to evaluate each terms when (i) j = j and (ii)′′j ̸= j , separately. By using the similar calculations, when j ̸= j the summation oflast two terms becomes′

2πi(m1 + 21 ) 2n+1
(j−j )

e

2π

×

′

1 − ei 2n+1 (j−j )m22π′

1 − ei 2n+1 (j−j )′2π−i(m1 + 12 ) 2n+1
(j−j )

+e

′

2π

×

′

1 − e−i 2n+1 (j−j )m2′

1 − e−i 2n+1 (j−j )2π.

When j = j , θjk − θj ′ ,k = 0 and the summation of last two terms become m2 . Hence25.

by using the relation′

1 − ei 2n+1 (j+j −1)m2
2π

′

2π
i(m1 + 12 ) 2n+1
(j+j −1)

×

e

2π

+e
′

′

1 − e−i 2n+1 (j+j −1)m2
2π

′

2π
(j+j −1)−i(m1 + 12 ) 2n+1×′

1 − e−i 2n+1 (j+j −1)
2π

′

.

ei 2n+1 2 (j+j −1)(m1 ) − ei 2n+1 2 (j+j −1)(m1 +m2 )
2π

=

′

1 − ei 2n+1 (j+j −1)

1

2π

′

1

′

ei 2n+1 (− 2 )(j+j −1) − ei 2n+1 ( 2 )(j+j −1)
1

2π

2π

′

′

e−i 2n+1 2 (j+j −1)(m1 ) − e−i 2n+1 2 (j+j −1)(m1 +m2 )
2π

+

1

1

2π

′

1

′

e−i 2n+1 (− 2 )(j+j −1) − e−i 2n+1 ( 2 )(j+j −1)
2π

1

2π

1

′

′

and the corresponding results for j − j (there are two cases when (a)j = j and′(b)j ̸= j ), we have the result.
(ii) On (4.8) : When Yn − Ȳ0 = Xn + Vn , we re-write
′

′

′

−1
(A.2) X̂n − Xn = Cn Pn Jm Jm Pn C−1
n (Xn + Vn ) − Cn Pn Pn Cn Xn
′
′
′
′
−1
= Cn Pn Jm Jm Pn C−1
n Vn + Cn Pn [In − Jm Jm ]Pn Cn Xn .

In the simple case, we have
(A.3)

′

′

E[tr(X̂n − Xn ) H H(X̂n − Xn )]′′′′−1′ ′ ′
= trHCn Pn Jm Jm Pn C−1
n [tr(Σv )In ]Cn Pn Jm Jm Pn Cn H′′′′
−1′ ′ ∗′ ∗+trHCn Pn J∗m J∗m Pn C−1
n [tr(Σx )In ]Cn Pn Jm Jm Pn Cn H′′′′
−1′ ′ ′= tr(Σv )trHCn Pn Jm Jm Pn C−1
n Cn Pn Jm Jm Pn Cn H′′′′
−1′ ′ ∗′ ∗+tr(Σx )trHCn Pn J∗m J∗m Pn C−1
n Cn Pn Jm Jm Pn Cn H ,′′where J∗m = (Om , In−m ) and In = Jm Jm + J∗m J∗m .

(iii) On Theorems 5.1 and 5.2 : Basically, we apply CLT (Theorem 7.6 ofDurrett (1991) for instance) on the sequence of ergodic stationary (discrete) timeseries ∆yi and we give an outline of our derivations. Hence we need to confirm thatthe resulting variance-covariance terms correspond to those of the limiting Gaussian
random variables.
For this purpose, we evaluate
(A.4)
n
[
] [ 1 ] ∑′−iθiθ(n) (n)(n) (n)E ∆λ z (λk )∆λ z (λk ) =(eiθjk +e−iθjk )(e j′ k′ +e j′ k′ )E[rj rj ′ ] .
2n + 1′j,j =126

′When k ̸= k , it can be shown that the right-hand side terms are bounded by using′
the straight-forward and tedious calculations. When k = k , the right-hand side
consists of four terms which are associated with−iθ ′(eiθjk + e−iθjk )(e j′ k + e
iθ

j

i(θ

+θ

−i(θ

+θ

)

i(θ

) = e jk j′ k + e jk j′ k + e jk
= (1) + (2) + (3) + (4) (, say) .

−θ ′

j k

−i(θjk −θ ′ )

+e

j k

The sums of each terms of (1) and (2) are bounded. Under the assumption of
stationarity of rj , The dominant terms of (A.2) are (3) and (4), and then they areapproximately to
(A.5)
k
k
nΓ(0) + n cos 2π()[Γ(1) + Γ(−1)] + n cos 2π()[Γ(2) + Γ(−2)] + · · · .
2n + 12n + 1
We take k/(2n + 1) −→ t (0 ≤ t < 1/2). Then we have (5.3) and (5.4).
Next, we consider
(A.6)
n
∑
1 ∑
1 ∑
1
′
iθ
−iθ
(n) (n)(n) (n)∆λ z (λk )∆λ z (λk ) =(eiθjk +e−iθjk )(e j′ k′ +e j′ k′ )rj rj ′ .
m k∈Im k∈I 2n + 1 ′mmj,j =1

There are four terms, but the first two terms are stochastically bounded. We find
that
(A.7)
] ∑
n
n [
] ′
∑[
∑
k
−
1/2
′
i(θjk −θ ′ )−i(θjk −θ ′ )j k + ej k
2 cos 2πs(
rj rj ′ =rs+j ′ rj ′ ] .
e) [2n + 1′′sj =1j,j =1

Under the assumptions we have made, we find that for any s
1∑′p
rs+j ′ rj ′ −→ Γ(s) .
n ′
n

(A.8)

j =1

(See Chapter 8 of Anderson (1971) and Brockwell and Davis (1990) for instance.)
Therefore, we can show that (A.4) devided by n is approximately as(A.9) Γ(0) + cos 2π(kk)[Γ(1) + Γ(−1)] + cos 2π()[Γ(2) + Γ(−2)] + · · · .
2n + 12n + 1

By taking k/(2n + 1) −→ t (0 ≤ t < 1/2) as n −→ ∞, we have (5.8). The rigorousarguments of derivations could be tedious, but they are straight-forward and wehave omitted them.

27

APPENDIX B : Some Figures
In this Appendix B, we give figures used in Section 7. As we have explained in
Section 2, all data are oﬃcial data published by ESRI (Economic and Social ResearchInstitute), Cabinet Oﬃce of Japan and Statistics Bureau, Ministry of Internal Aﬀairs
and Communications. They are available from the government oﬃcial Web-cite :e.stat.

28

6e+05
4e+05
RCON

2e+05
0e+00
−2e+05
0

20

40

60

80

100

freq

Figure 7.1 : Real-Consumption (original series)

29

0
RCON

−5000
−10000
0

20

40

60

80

100

freq

Figure 7.2 : Real-Consumption (diﬀerencing series)

30

1.0
0.8
0.6
0.0

0.2

0.4

Fn

0

20

40

60

80

100

freq

Figure 7.3 : Real-Consumption (empirical spectral distribution)

31

3000
2000
1000
RCON

0
−1000
−3000
0

20

40

60

80

freq

Figure 7.4 : Real-Consumption (seasonal diﬀerencing)32real GDP (unadjusted, 1994.Q1−2018.Q2)

n=98

11.70
11.65
11.60
11.55

log(gdp)

11.75

11.80

original series
official S.A.seriessimlfilter(alpha=0.99,z[48:52]<− 0)

0

20

40

60
Time

Figure 7.5 : Quarterly real-GDP

33

80

100

16000
15000
14000
13000
12000
0

50

100
Time
KAKEI

Figure 7.6 : Kakei-Chosa Series

34

150

14000
13000
12000
11000
0

50

100
Ti
me
SYOUDOU

Figure 7.7 : Shoudou Series

35

150

115
110
105
100
95
90
0

50

100
Ti
me
SANNJI

Figure 7.8 : Sanji Series

36

150

300000
SA, monthly

290000280000270000
0

50

100
Time
SHOUHI

Figure 7.9 : Consumption Series

37

150

1500
1000
500
0
−1500

−1000

−500

difference

0

50

100

150

Time
Diff(SHOUHI trend)

Figure 7.10 : Predicted and realized Consumption Series (in diﬀerencing)38