Expanded Summary

On the current stage of world economic development periodic macroeconomic fluctuations and crises are becoming increasingly important and amplitudinous issues. The problem of economic cycles and crises has been researched since 19^th century by leading economists (Schumpeter, Marx, Keynes and others). Nowadays this issue is thoroughly studied in western economies, whereas in Georgia the challenge of cyclical economic development is relatively new.

In modern surveys keen interest is expressed to the issue of interrelations between cycles of different length (Sanvi Avouyi-Dovi, Julien Matheron (2003); Acie S. Forrer, Donald A. Forrer (2014); Modelski G. (2001); Korotayev A., Tsirel S. (2010) and others), which proves actuality of this stream of economic theory.

The purpose of this article is to reviews theoretical concepts of spectral analysis-one of the most widespread modern methods used for research of economic cycles, as well as its practical use for the economy of Georgia.

Research of economic cyclicality is performed using different methods. Along with visual methods, which enable to reveal long-term cycles based on the graphical analysis of time series dynamics, nowadays more and more popular are exact methods, based on mathematic approach. They enable researchers to make accurate conclusions about dynamics of concrete macroeconomic indicators, make forecast and implement anti-crisis regulation.

One such method is spectrum analysis. Initially it was used in other branches of science, such as physics and mathematics. In economics it is usually used on micro-level for analysis of seasonality of sales, stock market dynamics and change in production stocks. Nevertheless, it is successfully applied on macroeconomic level for analysis of cyclicality. Spectrum analysis is particularly interesting in economics, since it enables separation of such important components of economic time series as trend, cyclicality, seasonal fluctuations and others. Economic analysis performed from this point of view makes it possible to create exact definition of an economic cycle based on reliable mathematic methods, reveal cycles of various periods and calculate so-called leading indicators to predict economic crises.

In the process of use of spectrum analysis there are often specific requirements to the initial data. For example, sometimes it is necessary to ensure stationary state of a time series, which in turn occurs quite seldom when speaking about economic time series. In order to make such series usable for further analysis specific methods of spectrum analysis were developed, such as Singular Spectrum Analysis (SSA).

The main idea of spectrum analysis is to break down the time series into several components, each of which represent cycles of a different frequency, assess the impact of each component on the initial dynamics of the time series, and reveal cycles by the strongest impact of several components. Spectrum analysis makes it possible not only to indicate concrete cycles, but also to find so-called quasi-periodic components, which should be taken into consideration in analysis of cyclicality.

Classical method of spectrum analysis is based on Fourier transform. Its general idea is to break down the initial time series, or a function, into harmonic fluctuations and their frequencies. The formula of discrete Fourier transform is as follows:

,

Where: X(m)Where: – m-th member of a transformed series;

X(k)-k-th component of the initial data;

N– length of the initial data;

i-imaginary value,   .

According to the classical interpretation of spectrum analysis by Fourier transform, an initial signal (or a time series) can be presented as a sinusoidal model, which is characterized by a concrete frequency, amplitude and phase. Thus, a function f_p(t) with a period p can be represented as follows:

 where .    

a_k coefficient is calculated as follows:

 The results of a spectrum analysis can be shown in a graphical form, which is called periodogram and shows different cycles and their frequencies in a time series.

In the survey described in this article spectral analysis was performed using MATLAB software in order to analyze dynamics of the economy of Georgia. As initial time series, the following indicators were chosen;

ü Real Gross Domestic Product per capita in 1965-2014, in constant 2005 USD;

ü Seasonally adjusted index of industrial production (more precisely, of a manufacturing section of industrial production, according to its share in total industry.

According to the results, the following cycles were revealed in the economy of Georgia:

ü 12.5-year cycle, which was classified as a Juglar medium-term cycle (business cycle);

ü 3.7-year cycle, which was assumed to be a Kitchen short-term cycle.

Besides, cycles with periods of 25 and 14.8 were found, but due to the length of the initial data they were excluded from the further analysis.