






































DEM_2015_27to48


© 2015 Nicolaus Copernicus University Press. All rights reserved.  
http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2015.002  Vol. 15 (2015) 27−47 

Submitted October 25, 2015  ISSN (online) 2450-7067 
Accepted December 15, 2015 ISSN (print) 1234-3862 

Łukasz Lenart* 

Discrete Spectral Analysis. The Case of Industrial 
Production in Selected European Countries∗∗ 

A b s t r a c t. The aim of this paper is to show the usefulness the discrete spectral analysis in 
identification cyclical fluctuations. The subsampling procedure was applied to construct the 
asymptotically consistent test for Fourier coefficient and frequency significance. The case of 
monthly production in industry in European countries (thirty countries) was considered. Using 
proposed approach the frequencies concerning business fluctuations, seasonal fluctuations and 
trading-day effects fluctuations were recognized in considered data sets. The comparison with 
existing procedures was shown. 

K e y w o r d s: discrete spectral analysis, almost periodic function, frequency identification, 
graphical test.  

J E L Classification: C14, C46, E32. 

Introduction 

 The main part of monthly or quarterly macroeconomic time series con-
cerning industry, trade, service, national accounts, prices, etc. exhibit both: 
seasonal fluctuations and business cycle fluctuations. The problem of analys-
ing these cyclical fluctuations is widely considered in the literature using 
different statistical tools. One popular nonparametric approach is based on 
a spectral analysis.  

                                                 
* Correspondence to Łukasz Lenart, Cracow University of Economics, Department of 

Mathematics, e-mail: lenartl@uek.krakow.pl 
∗∗ This research was supported by Research Grant DEC-2013/09/B/HS4/01945 from the 

National Science Centre 



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

28 

 The spectral analysis of macroeconomic time series is considered mainly 
in continuous counterpart (see Ftiti, 2010; Metz, 2009; Orlov, 2006; Orlov, 
2009; Pakko, 2004; McAdam and Mestre, 2008; Uebele and Ritschl, 2009). 
Under stationarity assumption the continuous function called spectral den-
sity function is defined. Based on spectral density function (and the defini-
tion of harmonizable time series) the popular spectral characteristics are 
defined: modulus of coherency function, dynamic correlation, dynamic cor-
relation on a frequency band, cohesion, cohesion within the frequency band, 
phase shift (see Priestley, 1981; Hamilton, 1994; Croux, 2001 for more de-
tails). These measures are broadly used in analysing the business cycle fluc-
tuations. They are estimated under fundamental assumption that time series 
is zero mean. But this assumption is not supported by any formal statistical 
test in most empirical macroeconomic real data analysis.  
 In this paper the more general assumption is formulated concerning non-
trivial mean function. This general assumption was considered in Lenart, 
2011; Lenart and Pipień, 2013a; Lenart and Pipień, 2013b (in univariate 
case) and recently in Lenart and Pipień, 2015 (in multivariate case) with 
application to macroeconomic time series. In this paper we show that the 
parameters of the discrete spectrum can be identified not only with the sea-
sonal and business cycle fluctuations but additionally with the trading-day 
effect. 
 In the section 2, based on Lenart and Pipień, 2013a and Lenart and 
Pipień, 2015 the model was formulated and illustrative example was pre-
sented. In the next part the empirical analysis was presented. The production 
in industry – monthly data (mining and quarrying; manufacturing; electric-
ity, gas, steam and air conditioning supply) from Feb. 2000 to Dec. 2014 was 
considered. In the first subsection the graphical methods to iden-
tify/recognize the frequency (concerning to business fluctuations, seasonal 
fluctuations and trading-day effect fluctuations) was presented. Such graphi-
cal methods to identify ’periodic phenomena’ in the autocovariance function 
in class of Almost Periodically Correlated time series were presented in 
Hurd and Gerr, 1991 and recently in Lenart, 2011. Finally in the second 
subsection formal statistical test for frequency significance was applied to 
data sets. 

1. Model specification 

    Let tY  be macroeconomic time series (index, gross data) with possible: 
seasonal fluctuations with period T , business fluctuations and trading-day 



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

29

fluctuations. Let us denote the natural logarithm: )(ln=
~

tt YY . Based on Le-
nart and Pipień, 2013a we assume that  

),(),(=)
~

( ttfYE t µβ +  (1) 

 where )(tµ  is almost periodic function (ap in short) of the form 
tiemt ψ

ψ
ψµ )(=)( ∑

Ψ∈

, where  

.)(
1

lim=)(
1=

ti
n

tn
et

n
m ψµψ −

→∞
∑  

We assume that the set of frequencies 0}=|)(:|)[0,2{= /∈Ψ ψπψ m  is finite 
and unknown. This set Ψ  can be decompose in natural way via: 

321= Ψ∪Ψ∪ΨΨ , where 1Ψ  corresponds to business fluctuations, 

1}0,1,=,/{22 −⊂Ψ TkTkπ  corresponds to seasonal fluctuations and 3Ψ  is 
a set of remaining frequencies (corresponding to interaction between sea-
sonal and business fluctuations and frequencies corresponding to trading-day 
effects). Equivalently, the model (1) can be written via:  

),()(),(=)
~

( 21 tttfYE t µµβ ++  (2) 

 where )(1 tµ  is a periodic function with period T  which represents the sea-

sonal fluctuations and tiemt ψ

ψ
ψµ )(=)(

2\
2 ∑

ΨΨ∈

. The function )(1 tµ  can be 

equivalently represented by the vector (sequence of seasonal values) 
'

121 ][= −Tµµµ Kµ , and )(= 121 −+++− TT µµµµ K .  

The sequence 1
~~

= −− ttt YYX  represents the monthly log growth rate. If 
),( βtf  is a polynomial of order one then  

),(~)(~=)( 21 ttXE t µµ +  (3) 

 where 1)()(=)(~
111 −− ttt µµµ  is periodic function that corresponds to sea-

sonal pattern and tiemttt ψ

ψ
ψµµµ )(~=1)()(=)(~

2\
222 ∑

ΨΨ∈

−− . The sequence 

1
~~

=' −− ttt YYX  represents the annual log growth rate. If ),(βtf  is a polyno-
mial of order one then  

),('~=)'( 2 tXE t µ  (3) 



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

30 

 where tiemttt ψ

ψ
ψµµµ )('~=12)()(=)('~

2\
222 ∑

ΨΨ∈

−− . In future work we as-

sume that the autocovariance function of the time series }:{ Z∈tXt  (or 

}:'{ Z∈tX t ) is a periodic function with the same periodT . This is a natural 
generalization of the assumption concerning second order stationarity. This 
assumption follows from the natural hypothesis that for monthly data at 
some months the variability can be higher than at another month. With no 
loss of generality we assume that exists natural m  such that msn = . Then 
the time series }:{ Z∈tXt  can be represented as a second order stationary T
-valued time series with almost periodic mean function. More precisely the 

time series T
tsststt XXXY ][= 1)(21)(1 K−+−+  is T  values second order station-

ary time series with almost periodic mean function. This mean function can 
be decomposed (in natural way) to two main parts: periodic function (that 
corresponds to seasonal frequencies 2Ψ ) and almost periodic function (that 

corresponds to frequencies 31 Ψ∪Ψ ) 

 The natural estimator of kµ  (k=1,2,...,T–1) based on sample 

},,,{ 21 nXXX K  where mTn =  has the following form  

.
1

=ˆ 1)(
1=

, Tjk

m

j
nk X

m −+∑µ  (4) 

The estimator '
1,2,1, ]ˆˆˆ[=ˆ nTnnn −µµµ Kµ  is asymptotically normally distrib-

uted with known variance covariance matrix.  
Theorem 2.1 Assume that there exist constants 0>δ , ∞∆ <  and ∞<K  
such that   

a) ,<||sup 2 ∆+∈ δ||Z tt X   

b) Kk
k

≤+
∞

∑ δ
δ

α 2

0=

)( .  

Then  

)(0,)ˆ( 1 Σ→− −T

d

nm Nµµ  (5) 

   
Proof. The proof can be found in the Appendix.  

 



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

31

 The problem Ψ∈ψ  is equivalent to 0=|)(~| /ψm  (or 0=|)('~| /ψm ). The 

natural estimator of Fourier coefficient based on sample nXXX ,,, 21 K  has 
the following form:  

.
1

=)(ˆ
1=

ti
t

n

t
n eX

n
m ψψ −∑  (6) 

This estimator is asymptotically normally distributed (see Lenart and Pipień, 
2013a). The frequencies from the set 2\ ΨΨ  are estimated in nonparametric 
way using procedure introduced in Lenart and Pipień, 2013a. This procedure 
is based on subsampling approach applied for (nonstationary) almost peri-
odically correlated time series (for more details see to Lenart, 2013; Lenart 
and Pipień, 2013a). Note that using similar technique the problem of testing 
seasonality effect (in nonparametric framework) in macroeconomic time 
series was considered in Lenart and Pipień, 2013b. The multivariate case of 
the model (1) was considered in Lenart and Pipień, 2015, where the testing 
procedure for common length of the business cycle was considered in se-
lected European countries.  
     To show the impact of frequencies from the sets: 1Ψ , 2Ψ , 3Ψ  to the 
dynamic of time series we consider the following illustrative example. 
Example 2.1 Let us consider time series }:{ Z∈tXt  of the form:  

,)(=

)(

t

t

ti
t emX ηψ

µ

ψ

ψ
+∑

Ψ∈
444 3444 21

 (7) 

where }:{ Z∈ttη  is zero mean time series with finite variance on integer 
line. It is assumed (in this example) that the considered time series is not 
necessary second order stationary. To explain the impact of the frequencies 
from the sets: 1Ψ , 2Ψ , 3Ψ  (under decomposition 321= Ψ+Ψ+ΨΨ ) we 
consider four cases (T1–T4): 
T1  ∅Ψ =1 , ∅Ψ =3 , ,5}1,2,=:/12{2=2 KkkπΨ , with  ttt εηη +−10.95=

, where }:{ Z∈ttε  is gaussian white noise with zero mean and variance 

equal one. In this case only seasonal pattern (with period 12=T ) is 
present  in mean function. 

T2 {0.15}=1Ψ , ∅Ψ =3 , ,5}1,2,=:/12{2=2 KkkπΨ , with  

ttt
t εηπη +− −1|))

6
(sin|0.5(0.95= , where }:{ Z∈ttε  is gaussian white 

noise with zero mean and variance equal one. In this case considered 



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

32 

time series has periodic autocovariance function with period equal to 
12 and seasonal pattern (in mean function) with length of the season 
equal 12. In the mean function we put additional frequency (equal to 
0.15) that corresponds to cyclical fluctuations with period approx. to 
3.5 years.  

T3 ∅Ψ =1 , 0.1}/3{2=3 +Ψ π , ,5}1,2,=:/12{2=2 KkkπΨ , with 

ttt εηη +− −10.6= , where }:{ Z∈ttε  is gaussian white noise with zero 
mean and variance equal one. In this case the additional frequency 

0.1/32 +π  was put in mean function.  
T4   {0.15}=1Ψ , 0.1}/3{2=3 +Ψ π , ,5}1,2,=:/12{2=2 KkkπΨ , with  

ttt εηη +−10.5= , where }:{ Z∈ttε  is gaussian white noise with zero 
mean and variance equal one. In this case we consider the case where 
both set: 1Ψ  and 3Ψ  are not empty.   

 
Figure 1. The realizations of }:{ Z∈ttη  for considered cases, 180=n  

-3

-2

-1

1

2

3

T3

-2

-1

1

2

3

T4

-8

-6

-4

-2

2

4

T1

-2

2

4

T2



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

33

 
Figure 2. The realizations of }:{ Z∈tXt  for considered cases, 180=n  

 
Figure 3. The estimated magnitude of Fourier transforms |)(ˆ| ψnm  for considered 

cases, ][0,πψ ∈     

-5

5

10

15

T3

-10

-5

5

10

T4

-15

-10

-5

5

10

15

T1

-10

-5

5

10

15

T2

0.5 1 1.5 2 2.5 3

0.25

0.5

0.75

1

1.25

1.5

1.75

T3

0.5 1 1.5 2 2.5 3

0.5

1

1.5

T4

0.5 1 1.5 2 2.5 3

0.5

1

1.5

T1

0.5 1 1.5 2 2.5 3

0.5

1

1.5

T2



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

34 

 In all cases we clearly observe peaks at seasonal frequencies 
,5}1,2,=:/12{2 Kkkπ  (see Figure 3). Note that the Fourier coefficient at 

frequency π  equals zero. In the case T1 considered time series is second 
order stationary and the peak close to zero corresponds to second order 
properties of considered time series. In the case T2 and T4 the peak at fre-
quency 0.15 is observed while  the frequency from the set 3Ψ ( 0.1/32 +π ) 

can be recognized easily only in the case T4.  

2. Empirical analysis 

 We consider production in industry – monthly data (mining and quarry-
ing; manufacturing; electricity, gas, steam and air conditioning supply) from 
Feb 2000 to Dec 2014. We consider monthly and annual rate of change 
(gross data and data adjusted by working days). The production in industry 
for thirty countries were considered (with ordinal numbers): Belgium (1); 
Bulgaria (2); Czech Republic (3); Denmark (4); Germany (5); Estonia (6); 
Greece (7); Spain (8); France (9); Croatia(10); Italy (11); Cyprus (12); Lat-
via (13); Lithuania (14); Luxembourg (15); Hungary (16); Malta (17); Neth-
erlands (18); Austria (19); Poland (20); Portugal (21); Romania (22); Slove-
nia (23); Slovakia (24); Finland (25); Sweden (26); United Kingdom (27); 
Norway (28); Former Yugoslav Republic of Macedonia, (29); Serbia (30);  

2.1. Graphical methods for frequency identification 

 In this section we introduce two graphical methods (visual methods) for 
recognise cyclical fluctuations. The first method is based directly on observ-
ing the estimates of magnitude of Fourier coefficient |)(| ψm  (as a function 
of frequency). In this graphical method we try to recognise maximum local 
values (peaks). The second method is based on Theorem 2.1. In this method 
we consider only the m-o-m change cantered at each month by appropriate 
estimates nk,µ̂  of seasonal coefficient kµ . The aim of this method is to rec-

ognize any cyclical fluctuations around estimates nk,µ̂ .  

 In the first step we consider monthly rate of change (gross data – data 
not adjusted by working days). For each considered economies the test pre-
sented in Lenart and Pipień, 2013b reject the null hypothesis concerning 
empty set of seasonal frequencies 2Ψ . In this case the estimated magnitude 
of Fourier coefficients (for ][0,πψ ∈ ) clearly support the existence of the 
peaks at seasonal frequencies (see Figure 8). The peaks at frequencies that 
correspond to business cycle fluctuations are not observed jointly  for  consi- 



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

35

 
 
Figure 4. Cantered seasonal values at their estimates (January–June). Data not ad-

justed by working days 

January

February

March

April

May

June

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,4

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

0,3

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

0,3

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

0,3

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

36 

 
Figure 5. Cantered seasonal values at their estimates (July–December). Data not 

adjusted by working days 

 

July

August

September

October

November

December

-0,15

-0,1

-0,05

0

0,05

0,1

0,15

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,4

-0,2

0

0,2

0,4

0,6

0,8

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,4

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

37

 
Figure 6. Cantered seasonal values at their estimates (January–June). Data adjusted 

by working days 

January

February

March

April

May

June

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

0,3

0,4

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

0,3

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,3

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

38 

 
Figure 7. Cantered seasonal values at their estimates (July–December). Data ad-

justed by working days 

July

August

September

October

November

December

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,4

-0,2

0

0,2

0,4

0,6

0,8

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,3

-0,2

-0,1

0

0,1

0,2

0,3

0,4

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

-0,2

-0,1

0

0,1

0,2

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

39

dered countries. But the additional frequency that belongs to the set 3Ψ  is 

clearly observed. This frequency is approx. to 2.19 and is connected with 
trading-day effect in considered data. This frequency is a main theoretical 
frequency associated with a monthly trading-day effect (see Ladiray, 2012, 
page 262). The peak at this frequency is observed for almost all considered 
data sets also in the case of annual rate of change (see Figure 10). When the 
estimates of Fourier coefficient |)(ˆ| ψnm  are considered for data adjusted by 
working days (monthly and annual rate of change) the peaks at frequency 
approx. 2.19 are not observed. 
 

     

     

     

     

     

     
Figure 8. The estimated magnitude of Fourier transforms |)(ˆ| ψnm  for m-o-m data 

– not adjusted by working days, ][0,πψ ∈   

 Using second graphical method, the trading-day effect is clearly ob-
served. The Figure 4 and 5 presents the results of mean subtraction sepa-
rately at each different month. Obtained results clearly show the common 
fluctuations (around their estimates) for almost each countries. Note that the 
phase of this fluctuations (above or below mean) differs in different consecu-
tive month at the same year (compare for example April with May in the 
same years). Therefore, such fluctuation seems not to be the effected by 

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Belgium

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Bulgaria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Czech Republic

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Denmark

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Germany

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Estonia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Greece

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Spain

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

France

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Croatia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Italy

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Cyprus

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Latvia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Lithuania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Luxembourg

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Hungary

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Malta

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Netherlands

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Austria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Poland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Portugal

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Romania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovenia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovakia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Finland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Sweden

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

United Kingdom

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Norway

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Former Yugoslav Republic of 

Macedonia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Serbia



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

40 

business cycles. Finally, these fluctuations are not observed for the data after 
working day adjustment (see Figure 6 and 7).  
 

     

     

     

     

     

     
Figure 9. The estimated magnitude of Fourier transforms |)(ˆ| ψnm  for m-o-m data 

– adjusted by working days, ][0,πψ ∈    

 

2.2. Testing for frequency significance 

 In this part we consider formal statistical tools to detect significant fre-
quencies in mean function for considered data sets. We use nonparametric 
test presented in Lenart, 2013; Lenart and Pipień, 2013a and we compare our 
results with parametric estimates obtained by so-called contraction method 
(CM in short) introduced by Li and Song, 2002. Based on Lenart, 2013; 
Lenart and Pipień, 2013a we consider the following testing problem:  

Ψ∈
Ψ∈/

ψ
ψ

:

:

1

0

H

H
 (8) 

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Belgium

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Bulgaria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Czech Republic

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Denmark

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Germany (until 1990 former 

territory of the FRG)

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Estonia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Ireland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Greece

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Spain

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

France

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Croatia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Italy

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Cyprus

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Latvia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Lithuania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Luxembourg

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Hungary

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Malta

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Netherlands

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Austria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Poland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Portugal

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Romania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovenia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovakia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Finland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Sweden

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

United Kingdom

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Norway

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Former Yugoslav Republic of 

Macedonia, the



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

41

with the test statistics |)(ˆ| ψnmn  and critical values of the test calculated 

by subsampling approach (we take nb 2.5= ). Based on the test (8) and 
estimation procedure proposed in Lenart, 2013 the set of frequency was es-
timated. In our analysis we consider both: gross data and data adjusted by 
working days for both m-o-m and y-o-y case.  
 To compare our results we use parametric CM. In this case the number 
of iterations is 100 on the interval ),0(π  (with the resolution equal to 300). 

The bandwidth parameters are:  96.0=1η  for 20<m , 98.0=1η  for 40<m ,

99.0=1η  for 60<m , 995.0=1η  for 80<m . 
 

     

     

     

     

     

     
Figure 10. The estimated magnitude of Fourier transforms |)(ˆ| ψnm  for y-o-y data – 

not adjusted by working days, ][0,πψ ∈   

 In the case of m-o-m data sets method based on subsampling procedure 
clearly confirm the existence of seasonal frequencies for data not adjusted 
and adjusted by working days (see Figure 12 (a)–(b)). In the case of data not 
adjusted by working days the significant frequency approx. 2.19 is observed 
for most considered countries, while in the case of data adjusted by working 
days this frequency is clearly not significant on considered significance 
level. In the case of y-o-y data sets not adjusted by working days the method 

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Belgium

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Bulgaria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Czech Republic

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Denmark

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Germany

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Estonia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Greece

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Spain

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

France

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Croatia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Italy

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Cyprus

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Latvia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Lithuania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Luxembourg

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Hungary

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Malta

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Netherlands

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Austria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Poland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Portugal

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Romania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovenia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovakia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Finland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Sweden

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

United Kingdom

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Norway

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Former Yugoslav Republic of 

Macedonia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Serbia



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

42 

based on subsampling procedure also confirm the significance of frequency 
approx. 2.19 for most considered countries (see Figure 12 (c)–(d)). In the 
case of data adjusted by working days the frequency approx. 2.19 is clearly 
not significant (under considered significance level).  
 

     

     

     

     

     

     
Figure 11. The estimated magnitude of Fourier transforms |)(ˆ| ψnm  for y-o-y data – 

adjusted by working days, ][0,πψ ∈     

 The procedure proposed by Li and Song, 2002 gives similar results for 
frequency approx. 2.19. But for the frequency that corresponds to business 
cycle fluctuations the CM gives different results (in comparison with method 
based on subsampling). In the case of CM for almost all countries the fre-
quencies that correspond to business cycle fluctuations are included in esti-
mated set. It should be emphasized that in the case of subsampling the non-
parametric test is applied – based on resampling procedure, while in the case 
of CM the only estimation procedure (based on parametric model) is used 
without any test procedure (see estimation procedure in Li and Song, 2002). 
Therefore Figure 12 a)–d) presents results of testing procedure while e)–f) 
only results of the estimation procedure.  
 
 

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Belgium

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Bulgaria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Czech Republic

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Denmark

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Germany

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Estonia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Greece

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Spain

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

France

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Croatia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Italy

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Cyprus

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Latvia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Lithuania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Luxembourg

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Hungary

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Malta

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Netherlands

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Austria

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Poland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Portugal

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Romania

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovenia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Slovakia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Finland

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Sweden

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

United Kingdom

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Norway

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Former Yugoslav Republic of 

Macedonia

0,0 0,3 0,6 0,9 1,3 1,6 1,9 2,2 2,5 2,8 3,1

Serbia



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

43

 
(a) subsampling for m-o-m data, not 

adjusted by working days 

 
(b) subsampling for m-o-m data;  ad-

justed by working days 

 
(c) subsampling for y-o-y data, not ad-

justed by working days 

 
(d) subsampling for y-o-y data,  adjusted 

by working days 

 
(e) CM for y-o-y, data not adjusted by 

working days 

 
(f) CM for y-o-y, data adjusted by work-

ing days 
 
Figure 12. (a)–(d) Estimated frequency sets (dots). Horizontal axis – estimated fre-

quency; Vertical axis – the ordinal number of considered country. (a)–(d) 
– nonparametric approach based on subsampling methodology introduced 
in Lenart, 2013; (e)–(f) CM introduced by Li and Song, 2002  

0

5

10

15

20

25

30

0,00 0,52 1,05 1,57 2,10 2,62 3,14

0

5

10

15

20

25

30

0,00 0,52 1,05 1,57 2,10 2,62 3,14

0

5

10

15

20

25

30

0,00 0,52 1,05 1,57 2,10 2,62 3,14

0

5

10

15

20

25

30

0,00 0,52 1,05 1,57 2,10 2,62 3,14

0

5

10

15

20

25

30

0,00 0,52 1,05 1,57 2,10 2,62 3,14

0

5

10

15

20

25

30

0,00 0,52 1,05 1,57 2,10 2,62 3,14



Łukasz Lenart 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

44 

Conclusions 

 The discrete spectral analysis can be useful supplementary analysis to 
usual spectral analysis based on continuous spectrum. Presented methods 
(graphical and formal statistical test) are useful in recognizing frequencies 
that corresponds to business fluctuations, seasonal fluctuations and trading-
day effects. One of the main open problems is a formal statistical test for 
trading-day effect. Corresponding test for business fluctuations was intro-
duced in Lenart and Pipień, 2013a and in Lenart and Pipień, 2013b for sea-
sonal pattern.  Note that in the literature the popular so-called visual test (see 
Ladiary, 2012) for trading-day frequencies plays a central role.  

Appendix 

Proof of the Theorem 2.1. We start from auxiliary Lemma.  
Lemma A.1 If assumptions of the Theorem 2.1 hold then for any 

1},{1,2,, 21 −∈ Tkk K  the limit  














−− −+−+

∞→
∑∑ ))((

1
lim

21)2(211)1(1
1=21=1

kTjkkTjk

m

j

m

jm
XX

m
E µµ           (9) 

 exists and it’s finite.   
  

Proof of the Lemma. Notice that  

),(cov
||

1=

),(cov
1

=

))((
1

21

1

1=

1)2(21)1(1
1=21=1

21)2(211)1(1
1=21=1

Tkk

m

m

TjkTjk

m

j

m

j

kTjkkTjk

m

j

m

j

XX
m

XX
m

XX
m

E

τ
τ

τ

µµ

+

−

+−

−+−+

−+−+








 −














−−

∑

∑∑

∑∑

             (10) 

Using now Theorem 3, page 9 form Doukhan, 1994 we get inequality

|)(|8|),(cov| 12
22

21
TkkXX Tkk τα δ

δ

τ +−∆≤ +
+ . Since (10) is a Feyèr’s sum at a 

point 0=0x  the limit for ∞→m  exists. This finish the proof of the 

Lemma.  



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

45

  We use the same technique as presented in Lenart, 2013 in proof of 
Theorem 2.1. Let us consider the decomposition:  

.))ˆ(())ˆ(ˆ(=)ˆ(
)(2)(1

4444 34444 214444 34444 21

nr

n

nr

nnn EmEmm µµµµµµ −+−−  

 Using the same arguments it is sufficient to show the convergence:  

)(0,)( 11 Σ→ −T

d

nr N  (11) 

 and  

0.)(2 →nr  (12) 

To show (11) it is enough to use the Cramér-Wold device, Theorem 3.3.1 
from Guyon, 1995 and Lemma 5.1. To prove (12) notice that for any 

1,1,2,= −Tk K  we have  

0.
1

=
1

2
)(~1

)(~1
=

)1)((~1
=

)(
1

=|)ˆ(|

2\

)1)((

2\1=

2
1=

1)(
1=

,

→









−
≤

−+

−−

∑

∑∑

∑

∑

ΨΨ∈

−+

ΨΨ∈

−+

m
O

e
m

m

em
m

Tjk
m

XE
m

mEm

Ti

Tjki
m

j

m

j

kTjk

m

j
knk

ψ
ψ

ψ

ψ

ψ

ψ

µ

µµµ

 

This completes the proof.  

References  
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ables: theory and empirics, The Review of Ecomonics and Statistics, 83(2), 232–241, 
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Analiza spektrum dyskretnego na przykładzie produkcji  
przemysłowej w wybranych krajach europejskich 

Z a r y s  t r e ś c i. Artykuł przedstawia analizę spectrum dyskretnego dla produkcji przemy-
słowej wybranych gospodarek europejskich. Analizie poddano dane o częstotliwości mie-
sięcznej. Przyjęto założenie o prawie okresowej postaci funkcji wartości oczekiwanej wokół 



Discrete Spectral Analysis. The Case of Industrial Production… 

DYNAMIC ECONOMETRIC MODELS 15 (2015) 27–47 

47

długookresowej tendencji rozwojowej. W artykule przedstawiono dwie metody graficzne 
identyfikacji wahań cyklicznych (koniunkturalnych, sezonowych oraz wynikających z różnej 
liczby dni roboczych w miesiącu). Pierwsza z nich bazuje na identyfikacji najwyższych war-
tości estymatora współczynnika Fouriera, jako funkcji częstotliwości. Metoda ta może być 
stosowana dla miesięcznej oraz rocznej stopy wzrostu. Druga metoda graficzna może być 
zastosowana jedynie dla miesięcznej stopy wzrostu.  Bazuje ona na graficznej identyfikacji 
wahań cyklicznych w danych po odjęciu od nich wartości estymatora współczynników sezo-
nowych. W kolejnej części artykułu przedstawiono wyniki formalnego testu istotności często-
tliwości w oparciu o prace: Lenart, 2013 oraz Lenart oraz Pipień, 2013a. Warto zaznaczyć, iż 
test ten pozwolił na formalne zidentyfikowanie częstotliwości korespondującej do wahań 
związanych z elektem różnej liczby dni roboczych w kolejnych miesiącach roku.  

S ł o w a  k l u c z o w e: analiza spectrum dyskretnego, funkcja prawie okresowa, identyfika-
cja częstotliwości, test graficzny.  




	Pusta strona

