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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Milda Maria Burzała 
Poznań University of Economics 

The Synchronization of Regional Business Cycles  
with Nationwide Cycles 

A b s t r a c t. This paper attempts to assess the level of synchronization between the business 
cycles of Poland’s regions and those of the country as a whole. The measure of economic activity 
was an index of total industrial output sold, recorded monthly from January 1999 to December 
2008, adjusted for seasonal and random fluctuations. The analysis of dominant business cycles 
was performed using spectral analysis. To assess the synchronization of cycles, characteristics 
of cospectral analysis were used: coefficient of coherence, amplitude intensification and phase 
difference. In the conclusion, an attempt is made to construct a synthetic indicator as a means 
of ranking the regions by degree of business cycle synchronization.  

K e y w o r d s: spectral analysis, cospectral analysis, business cycle, synthetic indicator. 

1. Introduction 
 Comparison of the level of economic activity in Poland’s regions (also 
called provinces or voivodeships) with that of the nation as a whole provides 
a source of information for both the regional authorities and central 
government. From a macroeconomic point of view, the observation of changes 
in a region may be an important factor for evaluating the effectiveness of anti-
cyclic policy and monitoring the channels by which fluctuations spread. 
 The concept of synchronization of fluctuations is understood to relate to 
analysis of particular components of the business cycle in terms of similarity 
of shape, amplitude of fluctuations and relative time lags. To perform such an 
analysis, a measure of economic activity must be adopted. In the empirical 
investigation, the measure used was the monthly recorded indices of total sold 
industrial output from January 1999 to December 2008 (PR_IRt = 
=100 ⋅ PR_IRt / PR_IRt-12), adjusted to eliminate seasonal and random 
fluctuations (Statistica 8 program, Census 2 method). Hence the concept 
of economic activity is associated with changes resulting from the joint effect 



Milda Maria Burzała 

 
62

of growth factors and business cycle fluctuations. Frequency domain analysis 
was carried out, including both spectral and cospectral analysis. 
 Spectral analysis, in relation to the Polish economy, has been primarily used 
in macroeconomic research1, because its use depends on a sufficient length 
of time series2. The fact that the present regional division of Poland has only 
been in existence since 1999 was a significant barrier in applying the spectral 
analysis. 

2. Tools of Spectral and Cospectral Analysis 
 Spectral and cospectral analysis refer to stationary stochastic processes and 
to the frequency domain. Transition from the time domain to the frequency 
domain is accomplished by means of Fourier transformation. In the study, the 
stationarity assumption of the stochastic process was verified using the Dickey–
Fuller (ADF) test. 
 Spectral analysis leads to the determination of a power spectrum, namely 
a spectrum of the considered time series: 

],,[for

)(
2
1cos)(

2
1)(

ππω

τ
π

ωττ
π

ω
τ

ωτ

τ

−∈

== ∑∑
∞

−∞=

−
∞

−∞=

KeKf i

 (1) 

where K(τ) is the autocovariance function, τ = t–s is the distance between the 
two analysed points in time, and ω = 2π/N is the frequency of harmonic 
components. 
 The power spectrum constitutes a distribution of variance of the analysed 
time series and makes it possible to identify the harmonic structure of the series 
and determine the contribution of individual components of the series to the 
variance of the process.   
 Joint spectral analysis enables testing the relations between particular 
frequencies of two time series. The basic magnitude in this case is the cross-
spectrum, which is a distribution of the joint covariance of the two processes 
Kyx(τ): 

].,[for

)(
2
1cos)(

2
1)(

ππω

τ
π

ωττ
π

ω
τ

ωτ

τ

−∈

== ∑∑
∞

−∞=

−
∞

−∞=
yx

i
yxyx KeKf

 (2)

                                                 
1 Cf. for example P. Skrzypczyński (2006, 2008), S. Dudek, D. Pachucki, K. Walczyk (2008); 

include review of research with regard to synchronization of fluctuations, L. Talaga, Z. Zieliński 
(1986). 

2 In the literature it is stated that the minimum length of a time series should be such as to in-
clude approximately 100 observations. 



The Synchronization of Regional Business Cycles with Nationwide Cycles 63

Using De Moivre’s lemma, the cross-spectrum can be written in complex form: 

),()(

sin)(
2
1cos)(

2
1)(

ωω

ωττ
π

ωττ
π

ω
τ τ

yxyx

yxyxyx

iqc

KiKf

−=

=−= ∑ ∑
∞

−∞=

∞

−∞=  (3) 

where cyx(ω) is the co-spectrum (the real part of the cross-spectrum); 
           qyx(ω) is the quadrature spectrum (the negative imaginary part of the 
cross-spectrum). 
 In cospectral analysis three characteristics are introduced which form a basis 
for comparison of the course of two processes. In the study, the variable X was 
associated with a series of nationwide output indexes, while the variable Y was 
associated with the corresponding series for a region. 

a) Gain of the variable X with respect to Y is interpreted as the modulus of the 
coefficient β in the regression of variable Y with respect to X for a given 
frequency ω (Gyx(ω) >1 means that the amplitude of process yt for fre-
quency ω is associated with a lower amplitude of process xt): 

].;[

0)(,
)(

])()([
)(

5,022

ππω

ω
ω

ωω
ω

−∈

≥
+

=

for

G
f

qc
G yx

x

yxyx
yx  (4) 

b) The phase difference identifies leads or lags of variable X with respect to Y 
(a positive value represents a lag, a negative value represents a lead) for 
frequency ω: 

].;[
)(
)(

)( ππω
ω
ω

ωφ −∈⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛ −
= for

c
q

arctg
yx

yx
yx  (5) 

c) Coherence is a measure of the fit (R2) in the regression of Y with respect to 
X for frequency ω: 

].;[1)(0,
)()(
)()( 2

22
2 ππωω

ωω
ωω

−∈≤≤
⋅
+

= forK
ff
qc

K yx
yx

yxyx
yx  (6) 

In the empirical analyses we consider only such leads or lags at a given 
frequency which is associated with a high value of the coefficient of coherence 
(for strongly correlated frequency components). 

3. Study Results in Frequency Domain 
 Theoretically, when adjusted for seasonal and random fluctuations, the 
annual indices of industrial output represent changes resulting not only from the 
business cycle, but also the underlying trend. The ADF test applied to the 



Milda Maria Burzała 

 
64

annual indices of output adjusted for seasonal and random fluctuations, in the 
case of the series for each of the analysed regions, indicated a significant 
negativity of the parameter δ in the model (7): 

.___
1

1 ∑
=

−− +Δ+=Δ
k

k
tititt IRPRIRPRIRPR εδδ  (7) 

The ADF test statistics rejects the null hypothesis of unit root at the α = 0.001 
significance level assuming first order autoregression (k=1). The stationarity 
of the time series means that any stochastic trend occurring in the annual indices 
of output can be regarded as a realization of low-frequency fluctuations. 
 For example, figure 1 shows indices of total industrial output sold for 
Poland. Test for stationarity was carried out based on the model: 

._95.0_032.0_ 11
POL
t

POL
t

POL
t IRPRIRPRIRPR −− Δ+−=Δ  

For parameter δ = -0,032 the statistic ADF is equal to -6.128. 
 

Sty-1999 Paź-1999 Lip-2000 Kwi-2001 Sty-2002 Paź-2002 Lip-2003 Kwi-2004 Sty-2005 Paź-2005 Lip-2006 Kwi-2007 Sty-2008 Paź-2008
-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

 
Figure 1. Indices of total industrial output sold for Poland from January 1999 to De-

cember 2008 

 Analysis of the dominant cycles was carried out based on the power spec-
trum. For a clear majority of regions it is possible to indicate one cycle which to 
a significant degree explains the fluctuations in the output indices (shown in 
bold type in Table 1).  
The 40-month cycle turned out to be such a dominant cycle. In the case of the 
nationwide cycle it explains more than 52% of the variability of output indices. 
The effect of the remaining cycles is quite different. For the Wielkopolskie 
region we can notice an equivalent effect from three cycles (120, 60 and 40 
months). Generally it is possible to identify groups of regions where economic 
activity is ruled by long cycles (120, 60 months), medium cycles (40, 30 
months) and short cycles (24, 20 months). The regions with dominating long 
cycles are as follows: Lubuskie (59.8%), Wielkopolskie (59.05%) and 
Mazowieckie (58.27%). Medium cycles dominate in Podkarpackie (70.86%), 
Zachodniopomorskie (70.24%), Świętokrzyskie (63.41%) and Warmińsko-
Mazurskie (62.97). 



The Synchronization of Regional Business Cycles with Nationwide Cycles 65

Table 1. Dominant cycles: percentage of explained variability of the time series 

 Length of time 
 Series Regions 

Frequency (duration in months) 
0.008 
(120) 

0.017 
(60) 

0.025 
(40) 

0.033 
(30) 

0.042 
(24) 

0,050 
(20) 

120 POLAND 9.46 6.19 52.10 0.92 15.61 4.42 
120 Dolnośląskie 3.46 8.94 42.43 6.97 5.17 4.51 
120 Kujawsko-Pomorskie 3.99 13.75 40.01 4.70 16.71 8.90 
120 Lubelskie 0.33 6.94 40.48 14.52 11.04 1.35 
120 Lubuskie 15.33 44.46 6.73 16.65 0.44 0.20 
120 Łódzkie 41.14 5.33 20.46 5.79 7.51 1.50 
120 Małopolskie 6.96 3.14 33.43 13.37 3.19 2.82 
118 Mazowieckie* 17.88 40.39 18.58 0.28 7.54 7.69 
60 Opolskie*** xxx 29.66 xxx 37.53 xxx 2.26 
120 Podkarpackie 6.21 9.77 59.26 11.61 3.53 5.87 
120 Podlaskie 13.50 7.94 30.51 8.49 3.35 3.92 
120 Pomorskie 7.11 12.90 10.20 20.38 14.58 25.72 
114 Śląskie** 0.97 12.96 28.62 6.23 16.56 3.04 
120 Świętokrzyskie 5.17 6.90 62.12 1.28 4.67 0.61 
120 Warmińsko-Mazurskie 15.81 2.89 59.74 3.23 14.57 0.11 
120 Wielkopolskie 28.03 31.01 30.41 0.21 3.19 6.73 
120 Zachodniopomorskie 10.02 7.32 67.58 2.66 1.47 1.77 

Note: With a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 
29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; ***60, 30, 20 months. 

Short cycles dominate mainly in Pomorskie (40.30%). In the mentioned above 
regions, the variability of the series of output indices is explained to a much 
higher degree than for Poland as a whole, where long cycles explain hardly 
15.65%, medium cycles 53.02% and short cycles 20.02%. Taking dominant 
cycles as a criterion, the greatest similarity to nationwide activity was displayed 
by those regions where medium-length cycles dominate. Therefore the greatest 
synchronization is observed in the Warmińsko-Mazurskie and Świętokrzyskie 
regions. 
 Cospectral analysis was performed for all cycles lasting from 20 to 120 
months3. The cospectral analysis characteristics are presented in Tables 2, 3 
and 4.  
 The measure of correlation in particular fluctuation bands is the coefficient 
of coherence. The highest values of K2 (which averages 0.933 across the 
regions) is observed for the dominant 40-month cycle and the long-term activity 

                                                 
3 For three regions, some data could not be obtained. In those cases the cycles analysed are 

accordingly shorter. These regions are asterisked in the tables, and the duration of the components 
is given at the foot of the table. The shortest series, for the Opolskie region, allowed only 3 cycles 
to be analysed.  



Milda Maria Burzała 

 
66

represented by the 120-month cycle (average 0.830 across the regions). The 
most weakly correlated are the 30-month cycles (average K2 across the regions: 
0.451). The results obtained confirm certain intuitive suppositions that the 
differentiation in economic activity may refer to shorter cycles. In the long term 
a higher coherence can be expected between regional and nationwide economic 
activity. Analysing the various fluctuation bands, certain groups of regions can 
be identified where K2 ≥ 0.95 (shown in bold type in Table 2). For short,  
24- and 20-month cycles there are far fewer such regions than for long cycles 
(120 and 60 months), which confirms the intuitive assumptions referred to 
above. Taking as a criterion the average value of K2 for all frequencies, it is 
possible to identify the regions with high average coefficients of coherence 
(Mazowieckie and Wielkopolskie, with averages of 0.913 and 0.894 
respectively) and with relatively low average coefficients of coherence 
(Lubuskie and Lubelskie, with averages of 0.448 and 0.491 respectively). The 
averages given are a measure of the mean correlation between the cycles 
distinguished for a given region and the nationwide cycles.  

Table 2. Values of coefficient of coherence 

 Length of time 
Series Regions     

Frequency (duration in months) 
0.008 
(120) 

0.017 
(60) 

0.025 
(40) 

0.033 
(30) 

0.042 
(24) 

0,050 
(20) 

120 Dolnośląskie 0.817 0.809 0.970 0.071 0.862 0.891 
120 Kujawsko-Pomorskie 0.579 0.511 0.964 0.639 0.966 0.921 
120 Lubelskie 0.219 0.691 0.948 0.038 0.822 0.227 
120 Lubuskie 0.733 0.616 0.671 0.253 0.267 0.145 
120 Łódzkie 0.982 0.361 0.940 0.461 0.938 0.667 
120 Małopolskie 0.975 0.938 0.963 0.203 0.749 0.930 
118 Mazowieckie* 0.960 0.737 0.897 0.972 0.977 0.934 
60 Opolskie*** Xxx 0.967 Xxx 0.912 xxx 0.662 
120 Podkarpackie 0.980 0.970 0.974 0.027 0.660 0.577 
120 Podlaskie 0.955 0.856 0.984 0.553 0.827 0.692 
120 Pomorskie 0.827 0.718 0.851 0.191 0.665 0.583 
114 Śląskie** 0.644 0.870 0.964 0.360 0.917 0.887 
120 Świętokrzyskie 0.923 0.909 0.993 0.526 0.912 0.100 
120 Warmińsko-Mazurskie 0.964 0.060 0.975 0.746 0.967 0.437 
120 Wielkopolskie 0.991 0.827 0.963 0.740 0.930 0.912 
120 Zachodniopomorskie 0.993 0.986 0.995 0.531 0.707 0.573 

Note: With a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 
29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; ***60, 30, 20 months. 

 Gain coefficients make it possible to compare the amplitudes of cycles ob-
served in a region with the amplitude of nationwide cycles within particular 
bands of fluctuations. Analysing the values given in Table 3, it is noticed that 



The Synchronization of Regional Business Cycles with Nationwide Cycles 67

the average coefficients of intensification for 120- and 60-month cycles (1.141 
and 1.296 respectively) indicate that on average, the amplitude of long-term 
fluctuations within regions is higher than the amplitude of nationwide fluctua-
tions.  

Table 3. Gain coefficients 

 Length of time  
Series Regions  

Frequency (duration in months) 
0.008 
(120) 

0.017 
(60) 

0.025 
(40) 

0.033 
(30) 

0.042 
(24) 

0.050 
(20) 

120 Dolnośląskie 0.859 1.408 1.318 0.509 0.841 1.301 
120 Kujawsko-Pomorskie 0.711 1.165 1.139 1.262 1.354 1.739 
120 Lubelskie 0.180 1.029 1.126 0.432 1.026 0.387 
120 Lubuskie 1.558 2.115 0.509 1.095 0.228 0.226 
120 Łódzkie 1.822 0.519 0.555 0.687 0.619 0.463 
120 Małopolskie 0.951 0.830 0.902 0.818 0.518 0.816 
118 Mazowieckie* 1.273 1.646 0.602 0.565 0.674 1.070 
60 Opolskie*** xxx 1.486 xxx 1.733 xxx 0.496 
120 Podkarpackie 0.974 1.344 1.246 0.314 0.529 0.925 
120 Podlaskie 0.792 0.639 0.524 0.672 0.322 0.524 
120 Pomorskie 1.147 1.339 0.635 1.144 1.213 2.304 
114 Śląskie** 0.473 1.609 1.312 1.073 1.701 1.282 
120 Świętokrzyskie 1.163 1.600 1.737 1.184 0.836 0.240 
120 Warmińsko-Mazurskie 1.457 0.256 1.234 1.215 1.106 0.399 
120 Wielkopolskie 1.985 1.856 0.883 0.662 0.531 1.202 
120 Zachodniopomorskie 1.773 1.888 1.959 1.426 0.489 0.771 

Note: With a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 
29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; 60, 30, 20 months.  

In turn, the short cycles (24 and 20 months, with average values 0.799 and 
0.884 respectively) are characterized by a higher amplitude of nationwide 
fluctuations. Closest to unity are the average gain coefficients for medium-term 
fluctuations (40 and 30 months, average values 1.045 and 0.924 respectively), 
which indicate the greatest degree of synchronization with nationwide 
fluctuations. 
For each band of fluctuations it is also possible to indicate the region displaying 
the greatest synchronization with the nationwide cycle: 

− Podkarpackie and Małopolskie for the 120-month cycle; 
− Lubelskie for the 60-month cycle; 
− Małopolskie for the 40-month cycle; 
− Śląskie for the 30-month cycle; 
− Lubelskie for the 24-month cycle; 
− Mazowieckie for the 20-month cycle. 



Milda Maria Burzała 

 
68

Table 4. Phase difference in radians (in months) 

 Length 
of time 
series 

Regions     
Frequency (duration in months) 

0.008 
(120) 

0.017 
(60) 

0.025 
(40) 

0.033 
(30) 

0.042 
(24) 

0,050 
(20) 

120 Dolnośląskie -0.563  
(-10.8) 

0.485 
(4.6) 

0.017 
(0.1) 

0.527 
(2.5) 

0.652 
(2.5) 

1.058 
(3.4) 

120 Kujawsko-Pomorskie -1.490 
(-28.5) 

1.099 
(10.5) 

0.048 
(0.3) 

0.045 
(0.2) 

1.043 
(4.0) 

0.654 
(2.1) 

120 Lubelskie -0.760 
(-14.5) 

1.099 
(10.5) 

0.421 
(2.7) 

-0.502 
(-2.4) 

0.123 
(0.5) 

2.339 
(7.4) 

120 Lubuskie -1.215 
(-23.2) 

0.497 
(4.7) 

0.722 
(4.6) 

0.278 
(1.3) 

1.299 
(5.0) 

1.762 
(5.6) 

120 Łódzkie -1.766 
(-33.7) 

-1.147 
(-11.0) 

0.556 
(3.5) 

1.194 
(5.7) 

1.428 
(5.5) 

0.323 
(1.0) 

120 Małopolskie -0.212 
(-4.0) 

0.338 
(3.2) 

0.170 
(1.1) 

-0.706 
(-3.4) 

-0.132 
(-0.5) 

-0.026 
(-0.1) 

118 Mazowieckie* -0.635 
(-11.9) 

-0.682 
(-6.4) 

-0.081 
(-0.5) 

-0.097 
(-0.5) 

-0.087 
(-0.3) 

-0.424 
(-1.3) 

60 Opolskie*** xxx 
 

0.145 
(1.4) 

xxx 
 

-0.420 
(-2.0) 

xxx 
 

-0.221 
(-0.7) 

120 Podkarpackie -0.444 
(-8.5) 

-0.087 
(-0.8) 

0.072 
(0.5) 

1.038 
(5.0) 

2.261 
(8.6) 

-0.972 
(-3.1) 

120 Podlaskie -0.624 
(-11.9) 

0.419 
(4.0) 

0.743 
(4.7) 

0.677 
(3.2) 

1.563 
(6.0) 

2.140 
(6.8) 

120 Pomorskie -0.615 
(-11.7) 

0.797 
(7.6) 

0.531 
(3.4) 

0.740 
(3.5) 

-1.431 
(-5.5) 

1.626 
(5.2) 

114 Śląskie** -0.002 
(-0.04) 

0.449 
(4.1) 

0.018 
(0.1) 

0.881 
(4.0) 

-0.404 
(-1.5) 

0.143 
(0.4) 

120 Świętokrzyskie -0.268 
(-5.1) 

0.520 
(5.0) 

0.280 
(1.8) 

0.526 
(2.5) 

0.693 
(2.6) 

-1.892 
(-6.0) 

120 Warmińsko-Mazurskie -2.058 
(-39.3) 

-1.711 
(-16.3) 

-0.266 
(-1.7) 

-0.068 
(-0.3) 

1.070 
(4.1) 

1.186 
(3.8) 

120 Wielkopolskie 0.438 
(8.4) 

0.139 
(1.3) 

-0.163 (-
1.0) 

-0.129 (-
0,6) 

-0.676 (-
2.6) 

-0.594 (-
1.9) 

120 Zachodniopomorskie 0.302 
(5.8) 

-0.070 
(-0.7) 

-0.170 
(-1.1) 

-0.429 
(-2.0) 

0.379 
(1.4) 

-2.003 
(-6.4) 

Note: With a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 
29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; 60, 30, 20 months. 

The phase differences presented in Table 4 enable analysis of the leads and lags 
of nationwide components of the business cycle relative to the corresponding 
components for a region. It should be noted that for most regions (apart from 
Wielkopolskie and Zachodniopomorskie), the 120-month cycle, connected with 
long-term activity, displays a lag with respect to nationwide activity. The 
longest lags are found in this fluctuation band (more than three years for the 
Warmińsko-Mazurskie region). For each component of the cycle it is possible 
to indicate the regions where economic activity runs in parallel with nationwide 
activity (without leading or lagging). However these results are different from 
those obtained using amplitude of fluctuations as a criterion. The greatest 



The Synchronization of Regional Business Cycles with Nationwide Cycles 69

number of regions, for which the phase difference in radians is smaller in 
absolute value than 0.1, can be indicated as having a dominant 40-month cycle. 
Such a phase difference is associated with lags (leads) in economic activity by 
up to one month. These are the regions Dolnośląskie, Kujawsko-Pomorskie, 
Mazowieckie, Podkarpackie and Śląskie. 

4. Indicator of Synchronization of Fluctuations  
 To summarise the study, an attempt was made to build a synthetic indicator 
as a basis for assigning ranks to each region depending on the degree 
of synchronization of fluctuations. For this purpose three characterictics 
of cospectral analysis were used, together with a cycle pattern perfectly 
synchronized with the national cycle. Such a cycle would be a cycle observed in 
a region having K2 = 1, G(ω) = 1 and φ(ω) = 0 (maximum correlation, no 
intensification of fluctuations and no phase difference). 

Table 5. Synthetic indicators and ranks for individual regions 

Regions 

Partial synthetic indicators 
Final syn-

thetic indica-
tor 

Ranks Frequency (duration in months) 
0.008 
(120) 

0.017 
(60) 

0.025 
(40) 

0.033 
(30) 

0.042 
(24) 

0,050 
(20) 

Dolnośląskie 0.179 0.239 0.094 0.519 0.181 0.265 0.175 2 
Kujawsko-Pomorskie 0.431 0.367 0.055 0.197 0.251 0.309 0.191 4 

Lubelskie 0.586 0.270 0.110 0.546 0.085 0.755 0.220 7 
Lubuskie 0.407 0.487 0.341 0.320 0.633 0.742 0.429 15 
Łódzkie 0.468 0.505 0.213 0.435 0.322 0.298 0.388 14 

Małopolskie 0.051 0.113 0.062 0.420 0.228 0.075 0.148 1 
Mazowieckie* 0.174 0.353 0.149 0.135 0.103 0.101 0.235 11 
Opolskie*** xxx 0.156 xxx 0.277 Xxx 0.276 0.226 9 

Podkarpackie 0.077 0.111 0.082 0.657 0.559 0.303 0.185 3 
Podlaskie 0.158 0.201 0.233 0.334 0.455 0.532 0.255 12 
Pomorskie 0.184 0.297 0.220 0.420 0.373 0.708 0.436 16 
Śląskie** 0.257 0.264 0.095 0.364 0.265 0.131 0.196 6 

Świętokrzyskie 0.106 0.259 0.231 0.284 0.171 0.772 0.227 10 
Warmińsko-Mazurskie 0.422 0.756 0.106 0.152 0.191 0.515 0.192 5 

Wielkopolskie 0.317 0.298 0.66 0.194 0.240 0.166 0.222 8 
Zachodniopomorskie 0.243 0.242 0.271 0.331 0.285 0.490 0.272 13 

Note: With a number of observations less than 120, the length of the analysed cycles changes: * 118, 59, 39.3, 
29.5, 23.6, 19.7; ** 114, 57, 38, 28.5, 22.8, 19; 60, 30, 20 months. 

For each region in each band of fluctuations, the modulus of the distance from 
the pattern was analysed. These distances were normalized according to the 
simple formula: modulus of observed distance divided by maximum distance. 
This approach meant that the range of variability could be normalized and the 



Milda Maria Burzała 

 
70

distances made independent of the units used. The partial synthetic indicator in 
a given band is the arithmetic mean of the normalized distances. The final 
synthetic indicator for a given region, encapsulating information from all bands, 
was based on a weighted average, where the weights reflected the degree 
of variability explained by each component of the cycle. The values of the 
synthetic indicators and the ranks assigned to each region are presented in Table 
5. The smaller the value of the synthetic indicator, the smaller the distance from 
the model. Hence a rank of 1 denotes the highest degree of synchronization. 

5. Summary 
 Summing up the results of the spectral and cospectral analysis, it is possible 
to identify one cycle which is dominant both for nationwide activity and for the 
majority of the regions. This is the 40-month cycle, which also displays the 
greatest synchronization when nationwide changes are compared with the 
changes observed in the regions. However, the classification of the regions 
varies depending on the criterion adopted for comparison. 
The indicator constructed in subsection 4 can take values from 0 to 1. The final 
values of the indicator appearing in the table lie within the interval  
[0.148, 0.436]. The relative narrowness of this interval and the values of the 
synthetic indicator (< 0.5) indicate a high level of synchronization between the 
business cycles recorded in the regions and those of the country as a whole. 

References 
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gospodarki z krajami strefy euro w kontekście struktury tych gospodarek, 
http://www.nbpnews.pl/r/nbpnews/Pliki_PDF/NBP/Publikacje/analityczne/irg_sghP.pdf 

Skrzypczyński, P. (2006), Analiza synchronizacji cykli koniunkturalnych w strefie euro, 
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Skrzypczyński, P. (2008), Wahania aktywności gospodarczej w Polsce i strefie euro, Materiały 
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Talaga, L., Zieliński, Z. (1986), Analiza spektralna w modelowaniu ekonometrycznym, PWN, 
Warsaw. 

Zeliaś, A. (1988), Metody statystyki międzynarodowej, PWE, Warsaw. 

Synchronizacja cykli koniunkturalnych województw  
z cyklami ogólnokrajowymi 

Z a r y s  t r e ś c i. W artykule podjęto próbę oceny stopnia synchronizacji cykli koniunkturalnych 
województw Polski z cyklami ogólnokrajowymi. Miernikiem aktywności gospodarczej były 
rejestrowane miesięcznie od stycznia 1999 do grudnia 2008 indeksy produkcji sprzedanej prze-
mysłu ogółem oczyszczone z wahań sezonowych i przypadkowych. Analizę dominujących cykli 
koniunkturalnych przeprowadzono z wykorzystaniem analizy spektralnej. Do oceny synchroniza-
cji cykli wykorzystano charakterystyki analizy kospektralnej: współczynnik koherencji, wzmoc-



The Synchronization of Regional Business Cycles with Nationwide Cycles 71

nienie amplitudy oraz przesunięcie fazowe. W podsumowaniu artykułu podjęto próbę budowy 
miernika syntetycznego, który był podstawą przypisania rang poszczególnym województwom ze 
względu na stopień synchronizacji cyklu koniunkturalnego.  

S ł o w a  k l u c z o w e: analiza spektralna, analiza kospektralna, cykl koniunkturalny, miernik 
syntetyczny. 



 


