Spectral Analysis: Time Series Analysis in Frequency Domain

This paper describes how the frequency domain analysis provides an alternative approach to time domain analysis of a given time series. Spectral and periodogram analyses of a given time series are performed to detect trends and seasonalities in the data. A cross-spectral analysis is done to find causality and comovements in two different time series. Univariate frequency domain analysis is done using time series of varying nature including simulated white noise process, random walk process, AR(1) process, Wolfer's Sunspot data and Box-Jenkins Airlines data; while bivariate (cross-spectral) analysis is done for macroeconomic variables such as money in circulation and inflation. (ProQuest: ... denotes formulae omitted.) Introduction Most of the macroeconomic and financial data are time series data, which can be expressed in the form: X(t) = {x^sub 1[,sub], x^sub 2^,..., X^sub T^} ...(1) The behavior of a time series data may be disintegrated into three main parts: long-, mediumand short-run behavior. These three parts are respectively associated with slowly evolving secular movements (the trend), a faster oscillating part (the business cycles) and a rapidly varying, often irregular, component (the seasonality) (Iacobucci, 2003). Generally, the properties of time series are enquired in time domain by analyzing the projections of interesting functions of time onto the phase space (Sella, 2008). The economic time series data, which refer to important indicators like exchange rate, sales, prices, export, etc., are characterized by trend. This trend is further used to produce forecasts for the series. At the same time, the general trend is important in measuring the accuracy of the estimated parameters (Manole, 2007). Most of the time series analysis is done in the time domain, but frequency domain analysis can also be used as a complementary tool. The information obtained by the time domain could be effectively supplemented by a frequency domain approach, i.e., spectral analysis. The main oscillatory components of the time series are described quantitatively by spectral decomposition; thus, it is possible to formally identify trends, low-frequency components, business cycles, seasonalities, etc. (Sella, 2008). Taking cue from concepts of mathematics and physics, financial and economic time series can also be analyzed in the frequency domain to determine trends and seasonalities. This paper has tried to show how well frequency domain analysis can be used to analyze time series of varying nature. It demonstrates the complementary nature of frequency domain analysis with reference to economic and financial data. Spectral analysis is a tool used in the frequency domain to determine seasonality in the time series. Cross-spectral analysis is basically used to find the direction and magnitude of comovements between two time series. A brief explanation of the same is provided in the next section. Spectral analysis has been used in this paper to analyze time series like some basic simulated white noise series, random walk process and AR(1) series. Also the Wolfer's Sunspot data is used, which is a long dataset and difficult to model and forecast in the time domain. The well-known Box Jenkins Airlines seasonality data has been used to show the efficacy of spectral analysis. Macroeconomic indicators like money in circulation in the economy and the inflation rate of India have been used to determine the magnitude and direction of comovement using cross-spectral analysis. Frequency Domain Analysis The main assumption underlying this use of time and frequency domain analysis techniques to inquire global system dynamics is that the series X (t) we actually observe is an effective realization of the underlying stochastic data generating process X (Sella, 2008). Fourier Analysis Take any series of numbers {x^sub t^}. We define the Fourier Transform as: . …