3. Algorithm for the quantification of business cycles, their synchronization and harmonization, and its application

Measuring business cycles is necessary and usually the starting point of their research. Measurement in this context means quantifying the following characteristics of a business cycle (2002):


Figure 2. Stylized recession phase (Harding and Pagan [16]).

<sup>1</sup> The idea of the "common cycles" is presented by A. S. Blinder and S. Fischer [2].

Harding and Pagan [16] graphically showed and explained the ratio of the peak (A) and the trough (C) as an example of a stylized recession (Figure 2). The height of the triangle in this graph is the amplitude, and the base is the duration. In this article, the authors also rightly notice that the knowledge of these two elements for any cycle makes it possible to calculate the area of the triangle and thereby estimate (say) the total loss in the output from the peak to the trough.

For the purposes of quantification of the synchronization and harmonization of business cycles on the macroeconomic level and further cross-country comparison, we suggest the following algorithm, presented in Figure 3.

At the first stage of our algorithm, we extract trend, cyclical, and irregular components of the initial seasonal and calendar adjusted time series of quarterly indicators of gross value added by the types of activity (sectors) by country. Examples of implementation of the proposed algorithm of band-pass filtering and the following analysis are based on the seasonal and calendar adjusted time series of quarterly indicators of gross value added by the types of activity (sector) represented by Eurostat<sup>2</sup> and Rosstat<sup>3</sup> . The indicators used are presented in fixed prices and converted to the natural logarithm, so they reflect the relative growth of the value added, forming the output of gross domestic product (GDP).

According to the approach of Baxter and King [1], the ideal band-pass filter should satisfy the six requirements:


Сonsequently, at the macroeconomic level, the economic cycle is an integral result of business cycles of different economic activities that are at different or equal (to a greater or lesser extent)

According to this idea, we included following two points in the practical part of this article:

1. Explanation of the applied algorithm for measuring the business cycle in relation to different types of economic activities by countries and conducting cross-country analysis.

2. Interpretation of synchronization and harmonization of business cycles as economic defi-

Measuring business cycles is necessary and usually the starting point of their research. Measurement in this context means quantifying the following characteristics of a business cycle (2002):

nitions and their evaluation at the macroeconomic level.

The idea of the "common cycles" is presented by A. S. Blinder and S. Fischer [2].

Figure 2. Stylized recession phase (Harding and Pagan [16]).

• The duration of the cycle and its phases • The amplitude of the cycle and its phases • Any asymmetric behavior of the phases • Cumulative movements within phases

1

3. Algorithm for the quantification of business cycles, their synchronization and harmonization, and its application

phases of the "common"<sup>1</sup> cycle.

32 Statistics - Growing Data Sets and Growing Demand for Statistics

<sup>2</sup> Eurostat: http://ec.europa.eu/eurostat

<sup>3</sup> Russian Federal State Statistics Service (Rosstat): http://www.gks.ru

Figure 3. An algorithm for quantifying business cycles and their synchronization and harmonization at the macroeconomic level.

We applied two filters for calculations, which are widely represented in the economic literature: the Hodrick-Prescott (HP) filter and the Baxter-King (BK) filter. These filters, calculated in the "mFilter" package of R-CRAN [26], largely meet the above requirements.

Figure 4. Comparison of business cycles extracted by filters HP (gray line) and BK (black line): Germany, France, Grease,

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and Russia.

Figure 4 shows that HP and BK filters on quarterly data give very close results.

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization… http://dx.doi.org/10.5772/intechopen.75580 35

Figure 4. Comparison of business cycles extracted by filters HP (gray line) and BK (black line): Germany, France, Grease, and Russia.

We applied two filters for calculations, which are widely represented in the economic literature: the Hodrick-Prescott (HP) filter and the Baxter-King (BK) filter. These filters, calculated in

Figure 3. An algorithm for quantifying business cycles and their synchronization and harmonization at the macroeco-

the "mFilter" package of R-CRAN [26], largely meet the above requirements. Figure 4 shows that HP and BK filters on quarterly data give very close results.

34 Statistics - Growing Data Sets and Growing Demand for Statistics

nomic level.

The difference of the application of HP and BK filters is opened by Baxter and King [1]. As they note, the BK filter is easier in design and gives more exact results for data sampled at other-thanquarterly frequencies. However, the HP filter provides a longer cycle curve, because it is not based on moving average calculation (we can see it on Figure 3: the black line is shorter than the gray line).

Because we use the quarterly data by country, the HP filter is more suitable for business cycle extraction as a basis of further analysis.

At stage 2 of our algorithm, we will evaluate the delays and correlations between the "total" business cycle reflected by the GDP curve and the business cycles of various economic activities (sectors) presented by added value curves, and on this basis, we will estimate the "common" cycle as the result of aggregating specific cycles by sectors. This approach corresponds to Mitchell's definition of business cycle presented above in this paper.

According to our approach, the difference between "total" and "common" cycle consists in the following:

"Total" cycle is a cyclical curve of any macroeconomic statistical indicator chosen for the purposes of measurement and analysis of business cycle (usually GDP or in our case the gross value added as a sum of added value by sectors (analogue of GDP). "Common" business cycle is a result of aggregation of different phases of sector's business cycles. The modern scientific problem is measurement of leading or lagging sector's business cycles relatively "common" cycle.

Coincidence of "total" and "common" cycles confirms the correctness of choosing the statistical indicator for measurement of business cycle on the macroeconomic level.

actions as "mining" and "public administration and defense and social security," demonstrate

Table 1. Spearmen correlation coefficients of "total" business cycle estimated by gross value added and business cycles

on different types of activity (sectors) with different lags of influence (in quarters of year) (Germany).

Types of activity (sectors) according to NACE2 Code Lags of correlation with the "total" cycle in quarters

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization…

Information and communication J 0.568 0.573 – – 0.281 Financial and insurance activities K 0.545 0.386 0.231 – 0.293 Real estate activities L – – 0.266 0.316 0.307 Professional, scientific, and technical activities M–N 0.680 0.680 – – 0.418

Agriculture, forestry, and fishing A – – 0.343 0.221 – Industry (except construction) B–E 0.840 0.855 0.320 – 0.310 Manufacturing C 0.857 0.863 0.311 – 0.338 Construction F 0.492 0.487 –––

Lag = 0 Lag = 2 Lag = 4 Lag = 6 Lag = 8

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37

G–I 0.803 0.802 – 0.265 0.419

O–Q – – 0.321 0.307 –

R–U 0.479 0.471 –––

On the basis of the results of the previous stage of analysis, we can create a "common" cycle model as an aggregate of business cycles of various types of activity, taking into account their correlation and lagging effect interconnection with "total" cycle, which in this case plays the role of "connection bridge" between the business cycles of different types of activity and

This conceptual approach determines the possibility of solving the problem of "Common"

The factor loadings to "common" cycle are presented in Table 3. This and the following tables are the copies of the reports on the performed calculations in the software package of statistical analysis "STATISTICA," which gives an opportunity to imagine the practical side of the

The general cycle corresponds to the first principle component, since it is characterized by significant statistical relationships<sup>4</sup> with most types of activity, i.e., manifests its effect of the aggregated cycle by Mitchell's definition of business cycle. In both cases the first principle

cycle estimation with the methodology of principle component analysis.

component will explain about 40% of the variance of the initial feature space.

a statistically significant lagging "countercyclical" effect.

"common" cycle as a latent variable.

The statistically significant correlations are marked in red.

Wholesale and retail, transport, accommodation, and

Public administration, defense, education, human

Arts, entertainment, and recreation: other service

Note: Dash indicates a statistically insignificant correlation

food service activities

health, and social work

activities

research (Tables 3–5).

4

The results are presented by two countries (Germany and Russia) because of the limitation of an article volume.

The results presented in Table 1 make it possible to conclude that in Germany the sectors enter the general cycle not simultaneously. Analyzed types of activities can be divided into three groups:


Similar analysis for Russia (Table 2) led to the conclusion that the majority of sectors (D, F, G, H, I, J, K, N, O) show simultaneous and "pro-cyclical" oscillations with the "total" cycle. Such


Table 1. Spearmen correlation coefficients of "total" business cycle estimated by gross value added and business cycles on different types of activity (sectors) with different lags of influence (in quarters of year) (Germany).

actions as "mining" and "public administration and defense and social security," demonstrate a statistically significant lagging "countercyclical" effect.

On the basis of the results of the previous stage of analysis, we can create a "common" cycle model as an aggregate of business cycles of various types of activity, taking into account their correlation and lagging effect interconnection with "total" cycle, which in this case plays the role of "connection bridge" between the business cycles of different types of activity and "common" cycle as a latent variable.

This conceptual approach determines the possibility of solving the problem of "Common" cycle estimation with the methodology of principle component analysis.

The factor loadings to "common" cycle are presented in Table 3. This and the following tables are the copies of the reports on the performed calculations in the software package of statistical analysis "STATISTICA," which gives an opportunity to imagine the practical side of the research (Tables 3–5).

The general cycle corresponds to the first principle component, since it is characterized by significant statistical relationships<sup>4</sup> with most types of activity, i.e., manifests its effect of the aggregated cycle by Mitchell's definition of business cycle. In both cases the first principle component will explain about 40% of the variance of the initial feature space.

The difference of the application of HP and BK filters is opened by Baxter and King [1]. As they note, the BK filter is easier in design and gives more exact results for data sampled at other-thanquarterly frequencies. However, the HP filter provides a longer cycle curve, because it is not based on moving average calculation (we can see it on Figure 3: the black line is shorter than the

Because we use the quarterly data by country, the HP filter is more suitable for business cycle

At stage 2 of our algorithm, we will evaluate the delays and correlations between the "total" business cycle reflected by the GDP curve and the business cycles of various economic activities (sectors) presented by added value curves, and on this basis, we will estimate the "common" cycle as the result of aggregating specific cycles by sectors. This approach corresponds to

According to our approach, the difference between "total" and "common" cycle consists in the

"Total" cycle is a cyclical curve of any macroeconomic statistical indicator chosen for the purposes of measurement and analysis of business cycle (usually GDP or in our case the gross value added as a sum of added value by sectors (analogue of GDP). "Common" business cycle is a result of aggregation of different phases of sector's business cycles. The modern scientific problem is measurement of leading or lagging sector's business cycles relatively "common" cycle.

Coincidence of "total" and "common" cycles confirms the correctness of choosing the statistical

The results are presented by two countries (Germany and Russia) because of the limitation of

The results presented in Table 1 make it possible to conclude that in Germany the sectors enter the general cycle not simultaneously. Analyzed types of activities can be divided into three groups:

1. Industry production, information, and communications are the "leading" activities. Their phases are two quarters earlier as the same phases of the "total" cycle, reflected by fluctuations of the gross value added indicator. We can name them as "pro-cyclical," because they have the same (parallel) phases and shift peaks, relatively the peaks of "total" cycle in time

2. Agriculture, forestry and fishing, real estate activities, public administration, defense, education, health and social work are the economic activities which have lagging and "counter-

3. Construction, wholesale and retail, transport, accommodation and food service activities, financial and insurance activities, professional, scientific and technical activities and arts, entertainment and recreation, and other types of activities show simultaneous entry into the same phases with a "total" cycle. As the first group, these activities are "pro-cyclical."

Similar analysis for Russia (Table 2) led to the conclusion that the majority of sectors (D, F, G, H, I, J, K, N, O) show simultaneous and "pro-cyclical" oscillations with the "total" cycle. Such

Mitchell's definition of business cycle presented above in this paper.

indicator for measurement of business cycle on the macroeconomic level.

gray line).

following:

an article volume.

(also as a shift of trough).

cyclical" phases relatively the "total" cycle.

extraction as a basis of further analysis.

36 Statistics - Growing Data Sets and Growing Demand for Statistics

<sup>4</sup> The statistically significant correlations are marked in red.


Table 2. Spearmen correlation coefficients of "total" business cycle estimated by gross value added and business cycles on different types of activity (sectors) with different lags of influence (in quarters of year) (Russia).

In Figures 5 and 6, we can see the difference in the adequacy of the reflection of the business cycle in Germany and in Russia with the statistical indicator "gross value added" (accounting for almost 100% of GDP produced) (but in both cases, there are high correlation coefficients: 0.954 and 0.837).

Modeling the cyclic component as a harmonic regression on stage 3 of our algorithm is necessary for quantifying the duration and amplitude both the "common" and specific cycles of different types of activity (sectors). For these purposes, it is expedient to use a Fourier series —the periodic function with a finite number of elements.

The business cycles extracted with the HP filter in accordance with the expansion of the Fourier series can be represented as a periodic function of time (byt ) in order to decrease the number of allocated harmonics, (i):

$$\hat{y}\_t = a\_0 + \sum\_{1}^{i} \left\{ a\_{i\bar{\jmath}} \sin \left( b\_{i\bar{\jmath}} t + k\_{i\bar{\jmath}} \right) + a\_{i\bar{\jmath}} \cos \left( b\_{i\bar{\jmath}} t + k\_{i\bar{\jmath}} \right) \right\},\tag{2}$$

Factor loadings (Germany) Table 3. Factor loadings of specific cycles of the types of activity to

"common" cycle latent variable (with lagging effect of influence).

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Factor loadings (Russia)

where <sup>y</sup><sup>b</sup> is the harmonic model of the business cycle

a<sup>0</sup> , aij, bij, and kij are the parameters of the harmonic model of the business cycle

j is the number of parameters

Factor loadings (Germany)

Factor loadings (Russia)

In Figures 5 and 6, we can see the difference in the adequacy of the reflection of the business cycle in Germany and in Russia with the statistical indicator "gross value added" (accounting for almost 100% of GDP produced) (but in both cases, there are high correlation coefficients:

Table 2. Spearmen correlation coefficients of "total" business cycle estimated by gross value added and business cycles

on different types of activity (sectors) with different lags of influence (in quarters of year) (Russia).

Types of activity (sectors) according to NACE2 Code Lags of correlation with the "total" cycle in

Agriculture, hunting, and forestry A – – �0.381 �0.378 �0.295 Fishing and fish farming B – 0.281 ––– Mining C –– – 0.437 0.598 Manufacturing D 0.791 0.754 0.421 – – Production and distribution of electricity, gas, and water E –– – – 0.313 Development F 0.853 0.589 – – �0.423

Hotels and restaurants H 0.864 0.598 – – -0.493 Transport and communications I 0.732 0.644 0.391 – – Finance J 0.683 0.415 – –0.337 –0.656 Real estate, renting, and business activities K 0.531 –––0.457 –0.744 Public administration and defense and social security L – �0.309 �0.508 �0.563 �0.551 Education M – – �0.307 –0.336 – Health and social services N 0.663 0.349 –––0.304 Other community and social and personal services O 0.635 0.382 –––0.512

quarters

Lag = 0 Lag = �2 Lag = �4 Lag = �6 Lag = �8

G 0.901 0.670 0.231 – �0.293

Modeling the cyclic component as a harmonic regression on stage 3 of our algorithm is necessary for quantifying the duration and amplitude both the "common" and specific cycles of different types of activity (sectors). For these purposes, it is expedient to use a Fourier series

The business cycles extracted with the HP filter in accordance with the expansion of the

� � <sup>þ</sup> aij cos bijt <sup>þ</sup> kij

� � � � , (2)

aijsin bijt þ kij

a<sup>0</sup> , aij, bij, and kij are the parameters of the harmonic model of the business cycle

) in order to decrease the

—the periodic function with a finite number of elements.

<sup>b</sup>yt <sup>¼</sup> <sup>a</sup><sup>0</sup> <sup>þ</sup><sup>X</sup>

where <sup>y</sup><sup>b</sup> is the harmonic model of the business cycle

number of allocated harmonics, (i):

Wholesale and retail trade, repair of motor vehicles, motorcycles, household goods, and personal items

38 Statistics - Growing Data Sets and Growing Demand for Statistics

j is the number of parameters

Fourier series can be represented as a periodic function of time (byt

i

1

0.954 and 0.837).


Table 3. Factor loadings of specific cycles of the types of activity to "common" cycle latent variable (with lagging effect of influence).



The following tables (Tables 4 and 5) and figures (Figures 7–10) present the results of the assessment of harmonic models of "common" business cycles and the most closely related sectoral business cycles for Germany and Russia. We can see that the models as a whole and

Table 5. Parameters of the harmonic models: (A) of the "common" business cycle and (B) of the manufacturing type of activity (sector "D") business cycle with estimations of their significance level (p-value) and confident interval; Russia.

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The theory and methodology of measurement of the business cycle synchronization and harmonization can be divided into two groups: an approach based on the clustering of turning points

For the implementation of stages 3–5 of our algorithm (Figure 3), we suggest the next definitions of the synchronization and harmonization of business cycles in the general approach:

their parameters are significant.

and the evaluation of the concordance of business cycles.

Table 4. Parameters of the harmonic models: (A) of the "common" business cycle and (B) of the manufacturing type of activity (sector "C") business cycle with estimations of their significance level (p-value) and confident intervals; Germany.



Table 5. Parameters of the harmonic models: (A) of the "common" business cycle and (B) of the manufacturing type of activity (sector "D") business cycle with estimations of their significance level (p-value) and confident interval; Russia.

The following tables (Tables 4 and 5) and figures (Figures 7–10) present the results of the assessment of harmonic models of "common" business cycles and the most closely related sectoral business cycles for Germany and Russia. We can see that the models as a whole and their parameters are significant.

The theory and methodology of measurement of the business cycle synchronization and harmonization can be divided into two groups: an approach based on the clustering of turning points and the evaluation of the concordance of business cycles.

For the implementation of stages 3–5 of our algorithm (Figure 3), we suggest the next definitions of the synchronization and harmonization of business cycles in the general approach:

Table 4. Parameters of the harmonic models: (A) of the "common" business cycle and (B) of the manufacturing type of activity (sector "C") business cycle with estimations of their significance level (p-value) and confident intervals;

Germany.

40 Statistics - Growing Data Sets and Growing Demand for Statistics

Figure 5. Business cycle graphs in Germany in "total" and "common" presentations.

Figure 6. Business cycle graphs in Russia in "total" and "common" presentations.

• Business cycles are synchronous if their curves are parallel, with a possible shift in time.

For the comparison of two cycles, it can be described by the following mathematical conditions for any i-harmonics:

$$\begin{cases} \mathcal{\mathcal{Y}}\_{\text{lt(1)}} \cong \mathcal{\mathcal{Y}}\_{\text{lt(2)}}\\ \mathcal{T}\_{\text{1,l}} \approx T\_{2,l}, \text{ for any } L\_{2,l}, \text{ where } L\_{2,l} \text{ is any constant time shift } \mathcal{Y}\_{\text{lt(2)}} \text{ relatively } \mathcal{Y}\_{\text{lt(1)}} \end{cases} (3)$$

classical cycles as Kondratieff, Kitchin, Juglar, Kuznets, and others is the main cause of global economic crises, such as the 2007–2009 global financial and economic crisis or Great Recession [29]. Using the parameters of harmonic models of "common" and specific (by types of activity) cycles, we can estimate in-country and intercountry synchronization and

Figure 8. The manufacturing type of activity (sector "C") business cycle in Germany estimated above with principle

Figure 7. The "common" business cycle in Germany estimated above with principle components method (observed) and

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Using the parameters of harmonic models of "common" and specific (by types of activity) cycles, we can estimate in-country and intercountry synchronization and harmonization of

harmonization of business cycles.

components method (observed) and its harmonic model (predicted).

business cycles.

its harmonic model (predicted).

According to the theory of the Fourier analysis, for any business cycle harmonic model, the length of the period of the i-order harmonic is T(i) = 2π/ (i):

• Business cycles are harmonized if the peaks (or troughs) of these cycles fall on the same time point. The coincidence of the minima of the harmonic waves of such superstrong

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization… http://dx.doi.org/10.5772/intechopen.75580 43

Figure 7. The "common" business cycle in Germany estimated above with principle components method (observed) and its harmonic model (predicted).

Figure 8. The manufacturing type of activity (sector "C") business cycle in Germany estimated above with principle components method (observed) and its harmonic model (predicted).

• Business cycles are synchronous if their curves are parallel, with a possible shift in time. For the comparison of two cycles, it can be described by the following mathematical conditions

According to the theory of the Fourier analysis, for any business cycle harmonic model, the

• Business cycles are harmonized if the peaks (or troughs) of these cycles fall on the same time point. The coincidence of the minima of the harmonic waves of such superstrong

length of the period of the i-order harmonic is T(i) = 2π/ (i):

Figure 6. Business cycle graphs in Russia in "total" and "common" presentations.

Figure 5. Business cycle graphs in Germany in "total" and "common" presentations.

42 Statistics - Growing Data Sets and Growing Demand for Statistics

ð3Þ

for any i-harmonics:

classical cycles as Kondratieff, Kitchin, Juglar, Kuznets, and others is the main cause of global economic crises, such as the 2007–2009 global financial and economic crisis or Great Recession [29]. Using the parameters of harmonic models of "common" and specific (by types of activity) cycles, we can estimate in-country and intercountry synchronization and harmonization of business cycles.

Using the parameters of harmonic models of "common" and specific (by types of activity) cycles, we can estimate in-country and intercountry synchronization and harmonization of business cycles.

emphasis on its key position that the business cycle is an integrated multifactorial phenomenon. The effect of integration is understood as the interaction of business cycles of different

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization…

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The novelty of the approach is to assess the statistical relationship between the indicators of business cycles of sectors of the economy with a certain macroeconomic indicator characterizing the cyclical nature of the economy as a whole and representing a "total" cycle. There is an

At the same time, the proposed algorithm takes into account the lagging effect of the mutual

Measurement of the effect of business cycle synchronization and harmonization is presented

The above examples of calculations for Germany and Russia show the general and distinctive

The results presented in the chapter can serve as a basis for further research in both theoretical and applied aspects. The main areas should be the development of methods for forecasting the entry of the economy into different phases of cyclicality and the expansion of groups of

[1] Baxter M, King RG. Measuring Business Cycles Approximate Band-Pass Filters for Eco-

[2] Blinder AS, Fischer S. Inventories, Rational Expectation, and the Business Cycle. Cam-

[3] Burns AF, Mitchell WC. Measuring Business Cycles. New York, New York: National

[4] Canova F. Detrending and business cycle facts. Journal of Monetary Economics. 1998;41:

Refs. 5, 6, 7, 9–15, 19, 21, 22, 24, 25, 27, 28 are recommended for reading for deep understanding of the topic of the article.

nomic Time Series. Cambridge: National Bureau of Economic Research; 1995

types of activities (sectors), which are simultaneously at different phases of cyclicality.

evaluation of some objective, but not directly measured, "common" cycle.

countries for analysis in-country and intercountry interaction of cycles.

on the basis of the construction of harmonic models.

characteristics of the business cycles of these countries.

Address all correspondence to: zarova.ru@gmail.com

Plekhanov Russian University of Economics, Moscow, Russia

bridge: National Bureau of Economic Research; 1979

Bureau of Economic Research; 1946

Author details

Elena Zarova

References\*

475-512

\*

influence of business cycles, as well as their synchronization and harmonization.

Figure 9. The "common" business cycle in Russia estimated above with principle components method (observed) and its harmonic model (predicted).

Figure 10. The manufacturing type of activity (sector "D") business cycle in Russia estimated above with principle components method (observed) and its harmonic model (predicted).

### 4. Conclusions

The approach proposed and tested in this chapter has a theoretical basis and originality. The theoretical basis of this approach is the classic definition of Mitchell's business cycle with an emphasis on its key position that the business cycle is an integrated multifactorial phenomenon. The effect of integration is understood as the interaction of business cycles of different types of activities (sectors), which are simultaneously at different phases of cyclicality.

The novelty of the approach is to assess the statistical relationship between the indicators of business cycles of sectors of the economy with a certain macroeconomic indicator characterizing the cyclical nature of the economy as a whole and representing a "total" cycle. There is an evaluation of some objective, but not directly measured, "common" cycle.

At the same time, the proposed algorithm takes into account the lagging effect of the mutual influence of business cycles, as well as their synchronization and harmonization.

Measurement of the effect of business cycle synchronization and harmonization is presented on the basis of the construction of harmonic models.

The above examples of calculations for Germany and Russia show the general and distinctive characteristics of the business cycles of these countries.

The results presented in the chapter can serve as a basis for further research in both theoretical and applied aspects. The main areas should be the development of methods for forecasting the entry of the economy into different phases of cyclicality and the expansion of groups of countries for analysis in-country and intercountry interaction of cycles.
