Spectral Analysis and Linear Filters

Abstract

Up to now we have viewed a time series as a time indexed sequence of random variables. The class of ARMA process was seen as an adequate class of models for the analysis of stationary time series. This approach is usually termed as time series analysis in the time domain. There is, however, an equivalent perspective which views a time series as overlayed waves of different frequencies. This view point is termed in time series analysis as the analysis in the frequency domain. The decomposition of a time series into sinusoids of different frequencies is called the spectral representation. The estimation of the importance of the waves at particular frequencies is referred to as spectral or spectrum estimation. Priestley (1981) provides an excellent account of these methods. The use of frequency domain methods, in particular spectrum estimation, which originated in the natural sciences was introduced to economics by Granger (1964).