Methodological Background for Spectral Analysis, Illustrated by Studying the Number of Ukrainian Population

Abstract

A data series usually has three components: trend, periodic (seasonal or cyclic) and random. Cyclic recurrence is periodic swings going beyond the limits of one year. The time span between the two next peaks within one year is considered as the length of cycle. Cyclic recurrence reflects repetitive character of economic processes during long periods. Two methods are used to analyze cycles, harmonic (based on periods) and spectral (based on frequencies). Phases of spectral analysis are: (i) testing of a time series for stationary state; (ii) for non-stationary series, transformation of the first difference, to bring it to the stationary form; (iii) construction of the graph of periods and computation of periods, frequencies and coefficients at sinuses and cosines; (iv) construction of the spectral density graph and final computation of frequencies, periods and coefficients at sinuses and cosines, to derive the Fourier series; (v) computation of the cyclic components values on the basis of the derived Fourier series, to build the graph for identifying the extreme points and crisis periods. Cyclic recurrence of social and economic processes in Ukraine can be most explicitly shown by the population data. These data are, therefore, taken for studying the Ukrainian population in 1850-2012 by use of spectral analysis. Results of the spectral analysis of the number of Ukrainian population in 1850-2012 allow for identifying three periods: 12.5 years (Juglar cycle), 27 years (Kuznets cycle) and 81 years. Below zero value of the cyclic component is an evidence of crisis condition and instability. In Ukraine, the crisis condition (instability) started in 1991 and will be on, according to the prognosis, till 2032. The lowest value of the cyclic component occurs in 2018. The crisis condition is aggravated due to random factors, which strongest influence began in 2013, and which will be declining in 2032 and on. These conditions of uncertainty and crisis enhance the significance of forecasting and strategy elaboration at country, industry, enterprise and individual level.