2. A literature review of the definition of a business cycle and methods for extracting cyclic components

indicators. In most cases, the authors of scientific publications use the real GDP (gross domestic product), as an indicator for investigation of macroeconomic business cycle. The development of this direction is the study of the interplay of the "total" business cycles identified in the GDP dynamics by countries. The second direction is the definition and quantitative description of the cyclic components in the dynamics of indicators characterizing the processes in individual sectors or spheres of the economy. The "specific cycles" thus estimated are the basis for the identification and quantification of the so-called common cycle using cluster analysis methods. In this case, the "common" (or as it is also called the "reference") cycle is represented as a multivariate value of the actual reaction of individual industries or economic subsystems, e.g., financial, investment, labor market, etc., observed with the help of statistical indicators. This chapter proposes a method for identifying and quantifying the common business cycle as a directly unmeasurable phenomenon, which manifests itself in fluctuations in the dynamics of the specific indicators of industries and economic systems, but has an objective and independent economic nature. In the author's opinion, specific cycles, even if they have a leading character with respect to the general cycle, are its economic consequence. The chapter suggests methods and their applications for identifying and quantifying the macroeconomic business cycle as a latent system-wide phenomenon, as well as methods for estimating intercountry synchronization and harmonization of common cycles. At the same time, the author gives her own definition of these forms of the interaction of business cycles. This approach is different to the view expressed in many publications about the identity of the concepts of synchronization, harmonization,

The chapter in addition to this introduction includes two main sections and a conclusion. The first section is devoted to the review of scientific publications that disclose the concept of the "business cycle," and also this section presents the author's systematization of the methods outlined in a number of publications for identifying cyclic components in the dynamics of macroeconomic indicators. In the second section of the chapter, an algorithm for quantifying the overall business cycle based on the principal component method is proposed, and methods for estimating synchronization and harmonizing business cycles are substantiated. The conclusion of the chapter contains a concentrated expression of scientific novelty of the author's methodological proposals of the business cycle quantification and the features of the algorithms for evaluating their synchronization and harmonization at

The proof of the concept proposed in this chapter and the approbation of the corresponding algorithm were realized on the basis of statistics of Eurostat and Rosstat. Most of the examples in this chapter are compiled from the results of calculations for Germany and Russia, which allows a comparative analysis provided that there is a significant difference in the duration and stages of the history of the market economy that have a significant impact on the forma-

Approbation of the proposed algorithm can be based on R-packages or the "STATISTICA"

concordance, and correlation of business cycles.

28 Statistics - Growing Data Sets and Growing Demand for Statistics

tion of sustainable multi-year business cycles.

the macroeconomic level.

program.

The number of scientific papers devoted to the theory and methods of quantifying business cycles is measured in hundreds. Nevertheless, there is a basis for their distribution into two large parts.

This basis is the definition of the concept of fluctuations in economic activity. This determines the choice of mathematical and statistical methods used to determine business cycles (Kijek [20]).

The first part includes the papers which are based on Burns and Mitchell's definition of a cycle [3]. According to them, a business cycle represents the four distinct phases of "aggregate economic activity" development that evolve from one into another: expansion, recession, depression, and revival.

The second part is the works based on a different view of business cycles, which was presented by Lucas [23], who does not interpret cycles as inevitable transitions between different phases of the cycle. He sees the business cycle as a process of oscillation of GNP around a long-term trend.

Some authors define the business cycle in accordance with Burns and Mitchell and then continue to measure the fluctuations of the macroeconomic aggregate values (GNP, GDP, industrial production, investment, and so on) relatively to the long-term trend.

This approach cannot be adopted without a preliminary study confirming that the observed cyclicality of the national economy is a violation of macroeconomic equilibrium rather than synchronous periodic fluctuations of various economic activities without changing of their balance.

The two basic approaches to the definition of the concept of a business cycle, mentioned above, were the basis for developing analytical tools for recognizing cycles.

Harding and Pagan [17] define three directions ("traditions") in the deviation of approaches presented in the literature on the development of cycle indicators from information available in a time-continuous random variable (yt).

The main idea of the first direction is changing of the initial time series (yt) by a series of binary random variable St, which includes turning points. In this case, peaks (troughs) are considered as local maxima (minima) in the series yt, and they are taking the value unity and zero otherwise:

$$\begin{aligned} \mathsf{V}\mathsf{t} &= \mathsf{1}\left(y\_{t} < y\_{t\pm 1}\leq \mathsf{j}\leq \mathsf{k}\right); \\ \mathsf{h}\mathsf{t} &= \mathsf{1}\left(y\_{t} < y\_{t\pm 1}\leq \mathsf{j}\leq \mathsf{k}\right). \end{aligned} \tag{1}$$

where ∨t (∧t) are binary variables and k is the length of period for local maxima (minima) estimation.

Harding and Pagan [18] show that for Burns and Mitchell's specific cycle dating procedures it is necessary to set k = 5 for monthly data or k = 2 for quarterly data, but it is not the single variant. Mathematics and IT providing of extracting and estimation of turning points in structure of economic time series have been actively developing.

Other two directions, or "traditions," by Harding and Pagan [17] are combined because they both are based on the prior transformation of yt so as to remove a permanent component, leaving only a transitory one (zt). The first direction of them provides for the isolation of the cycle on the basis of an analysis of the presence or absence of peaks in the spectral density. The second direction combines the methods of extracting a cycle by the results of the analysis of serial correlations in zt series.

The abovementioned approach to the systematization of methods for extracting and quantifying cycles in long-term economic dynamics, set forth in papers of Harding and Pagan, is accepted by many scientists, but is not the only one. An example of a somewhat different classification of methods for isolating stable fluctuations reflecting business cycles is presented in the paper of Kijek [20].

According to this author's view, three dominant approaches are used to distinguish cycles. Two of them coincide with the groups of methods singled out by Harding and Pagan (the last two "traditions"). They are (1) the presentation of the time series as a difference-stationary process and the application of autoregressive integrated moving average (ARIMA) models and (2) the presentation of economic dynamics as a trendstationary process and treat it as a sum (but also it can be multiple interconnection) of polynomial as deterministic trend and stochastic deviation around it. The last one is considered as a residual cyclical component, which includes business cycle's pattern and random deviation.

The third concept of time series decomposition according to Kijek [20] is the use of frequency filters, methodology, and software which are widely represented in the statistical literature.

The above methods are detailed and combined in different documents, but we must take into account that "The crucial question is not which method is more appropriate but whether different concepts of cycle are likely to produce alternative information which can be used to get a better perspective into economic phenomena and to validate theories" (Canova [4]).

Figure 1. System of methods for extraction of cyclic components from the seasonal adjusted time series of macroeconomic

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

http://dx.doi.org/10.5772/intechopen.75580

31

indicators.

The purpose of this paper is to create and demonstrate the results of the application of the algorithm of business cycle synchronization and harmonization proceed from economic content definition of business cycle by Burns and Mitchell and methods of estimation of economic fluctuations presented in publications. The main idea of it is based on the two key features in Burns and Mitchell's definition of business cycles. They are determined in many articles in the same variant. For example, in the paper of Diebold and Rudebusch ([8], p. 1), we can read "The first (key feature ) is the comovement among individual economic variables…In their analysis, Burns and Mitchell considered the historical concordance of hundreds of series, including those measuring commodity output, income, prices, interest rates, banking transactions, and transportation services…The second prominent element of Burns and Mitchell's definition of

business cycles is their division of business cycles into separate phases or regimes."

We believe that the first point should be based on another focus of "aggregate economic activity" in determining the business cycles of Burns and Mitchell: not only as a combination of the abovementioned indicators characterizing aggregate economic activity but above all the multiple effect of the entry of various economic activities into one and the same phase of the business cycle (the second key features in Burns and Mitchell's definition of business cycles).

Some authors include in the classification of methods for extracting from the time series of macroeconomic indicators of cyclic components adequate to real business cycles and groups of methods based on the evaluation of regression model parameters (such as the Fourier series models and multifactor models of the dependence of the cyclicality of GDP on the dynamics of factor variables).

We consider that it is not the methods of extraction of business cycles, but the methods for the next step of business cycle analysis—the quantification of cycle's characteristics (such as its phases, amplitude, duration of the period, and others).

The results of grouping of methods for extraction of cyclic components from the time series of macroeconomic indicators in conditions of its primly seasonal adjusting are presented in Figure 1. They include the group of "turning point" methods: methods for spectral density analysis and their realization with band-pass filtering (the statistical tools that pass frequencies within a certain range and reject frequencies outside that range) and methods for determining and subsequently evaluating cyclic components. We will not dwell on detrending methods in detail, since they are detailed in the articles, including in connection with the algorithms for isolating and quantifying the cycles (e.g., Canova [4]).

Statistical Methodology for Evaluating Business Cycles with the Conditions of Their Synchronization… http://dx.doi.org/10.5772/intechopen.75580 31

Other two directions, or "traditions," by Harding and Pagan [17] are combined because they both are based on the prior transformation of yt so as to remove a permanent component, leaving only a transitory one (zt). The first direction of them provides for the isolation of the cycle on the basis of an analysis of the presence or absence of peaks in the spectral density. The second direction combines the methods of extracting a cycle by the results of the analysis of

The abovementioned approach to the systematization of methods for extracting and quantifying cycles in long-term economic dynamics, set forth in papers of Harding and Pagan, is accepted by many scientists, but is not the only one. An example of a somewhat different classification of methods for isolating stable fluctuations reflecting business cycles is presented

According to this author's view, three dominant approaches are used to distinguish cycles. Two of them coincide with the groups of methods singled out by Harding and Pagan (the last two "traditions"). They are (1) the presentation of the time series as a difference-stationary process and the application of autoregressive integrated moving average (ARIMA) models and (2) the presentation of economic dynamics as a trendstationary process and treat it as a sum (but also it can be multiple interconnection) of polynomial as deterministic trend and stochastic deviation around it. The last one is considered as a residual cyclical component, which includes business cycle's pattern and

The third concept of time series decomposition according to Kijek [20] is the use of frequency filters, methodology, and software which are widely represented in the statistical

Some authors include in the classification of methods for extracting from the time series of macroeconomic indicators of cyclic components adequate to real business cycles and groups of methods based on the evaluation of regression model parameters (such as the Fourier series models and multifactor models of the dependence of the cyclicality of GDP on the dynamics of

We consider that it is not the methods of extraction of business cycles, but the methods for the next step of business cycle analysis—the quantification of cycle's characteristics (such as its

The results of grouping of methods for extraction of cyclic components from the time series of macroeconomic indicators in conditions of its primly seasonal adjusting are presented in Figure 1. They include the group of "turning point" methods: methods for spectral density analysis and their realization with band-pass filtering (the statistical tools that pass frequencies within a certain range and reject frequencies outside that range) and methods for determining and subsequently evaluating cyclic components. We will not dwell on detrending methods in detail, since they are detailed in the articles, including in connection with the algorithms for

phases, amplitude, duration of the period, and others).

isolating and quantifying the cycles (e.g., Canova [4]).

serial correlations in zt series.

30 Statistics - Growing Data Sets and Growing Demand for Statistics

in the paper of Kijek [20].

random deviation.

literature.

factor variables).

Figure 1. System of methods for extraction of cyclic components from the seasonal adjusted time series of macroeconomic indicators.

The above methods are detailed and combined in different documents, but we must take into account that "The crucial question is not which method is more appropriate but whether different concepts of cycle are likely to produce alternative information which can be used to get a better perspective into economic phenomena and to validate theories" (Canova [4]).

The purpose of this paper is to create and demonstrate the results of the application of the algorithm of business cycle synchronization and harmonization proceed from economic content definition of business cycle by Burns and Mitchell and methods of estimation of economic fluctuations presented in publications. The main idea of it is based on the two key features in Burns and Mitchell's definition of business cycles. They are determined in many articles in the same variant. For example, in the paper of Diebold and Rudebusch ([8], p. 1), we can read "The first (key feature ) is the comovement among individual economic variables…In their analysis, Burns and Mitchell considered the historical concordance of hundreds of series, including those measuring commodity output, income, prices, interest rates, banking transactions, and transportation services…The second prominent element of Burns and Mitchell's definition of business cycles is their division of business cycles into separate phases or regimes."

We believe that the first point should be based on another focus of "aggregate economic activity" in determining the business cycles of Burns and Mitchell: not only as a combination of the abovementioned indicators characterizing aggregate economic activity but above all the multiple effect of the entry of various economic activities into one and the same phase of the business cycle (the second key features in Burns and Mitchell's definition of business cycles). 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) phases of the "common"<sup>1</sup> cycle.

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

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

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

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

tors 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

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

1. The filter must extract the specified range of periodicity, which means that it passes through time series components with periodic oscillations between low and high frequencies, defining a specific cycle. Baxter and King recommend a filter that approximates periodic oscillations between 6 and 32 quarters (according to the definition of the Mitchell

2. An ideal band-pass filter should not introduce a phase shift, i.e., do not change the time

3. The expression of the discrepancy between the exact and approximate filters should be

4. The application of the band-pass filter must extract of a deterministic trend from a time series and result in a stationary time series, even when applied to trending data.

5. The filter should allocate the same components of the business cycle, regardless of the length of the observation period. Baxter and King notes that "Technically, this means that

stamps of the turning points at any frequency.

expressed mathematically.

Eurostat: http://ec.europa.eu/eurostat

the moving averages we construct." 6. Method of filtering must be operational.

Russian Federal State Statistics Service (Rosstat): http://www.gks.ru

. The indica-

33

http://dx.doi.org/10.5772/intechopen.75580

added by the types of activity (sector) represented by Eurostat<sup>2</sup> and Rosstat<sup>3</sup>

trough.

product (GDP).

six requirements:

cycle).

2

3

algorithm, presented in Figure 3.

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

