Stationary Time Series

Abstract

The concept of a stationary time series was, apparently, formalized by Khintchine in 1932. An infinite sequence y(t), t = 0, ±1,…, of random variables is called stationary if the joint probability law of y(t₁), y(t₂),…, y(t_n), is the same as that of y(t₁ + t),…, y(t_n + t), for any integers, t₁, t₂,…, t_n, t and any n. Thus the stochastic mechanism generating the sequence is not changing. In the natural sciences approximately stationary phenomena abound, but the continuing social evolution of man makes such phenomena rarer in social science. Nevertheless, stationary time series models have been widely used in econometrics since they may fit the data well over periods of time that are not too long and thus may provide a basis for short term predictions. The notions of trend, cycle, seasonal are closely related to a frequency decomposition of a series, with the trend corresponding to very low frequencies, and the spectral decomposition of a stationary series [see (2) below] is therefore of interest to economists. Finally, models have also been used where the observed series is regarded as the output of an evolving mechanism whose input is stationary.