Time Series Analysis

T000032 Any series of observations ordered along a single dimension, such as time, may be thought of as a time series. The emphasis in time series analysis is on studying the dependence among observations at different points in time. What distinguishes time series analysis from general multivariate analysis is precisely the temporal order imposed on the observations. Many economic variables, such as GNP and its components, price indices, sales, and stock returns are observed over time. In addition to being interested in the contemporaneous relationships among such variables, we are often concerned with relationships between their current and past values, i.e., relationships over time. The study of time series of, for example, astronomical observations predates recorded history. Early writers on economic subjects occasionally made explicit reference to astronomy as the source of their ideas. For example, in 1838 Cournot said, 'As in astronomy, it is necessary to recognize the secular variations which are independent of the periodic variations' (Cournot, 1838, translation 1927). Jevons (1884) remarks that his study of short-term fluctuations uses the methods of astronomy and meteorology. During the 19th century interest in, and analysis of, social and economic time series evolved into a new field of study independent of developments in astronomy and meteorology. See Nerlove et al., 1979, pp. 1–21, for a historical survey. Harmonic analysis is one of the earliest methods of analyzing time series thought to exhibit some form of periodicity. In this type of analysis, the time series, or some simple transformation of it, is assumed to be the result of the superposition of sine and cosine waves of different frequencies. However, since summing a finite number of such strictly periodic functions always results in a perfectly periodic series, which is seldom observed in practice, one usually allows for an additive stochastic component, sometimes called 'noise'. Thus, an observer must confront the problem of searching for 'hidden periodicities' in the data, i.e., the unknown frequencies and amplitudes of sinusoidal fluctuations hidden amidst noise. An early method for this purpose is periodogram analysis, suggested by Stokes (1879) and used by Schuster (1898) to analyze sunspot data and later by others, principally William Beveridge (1921, 1922), to analyze economic time series. Spectral analysis is a modernized version of periodogram analysis modified to take account of the stochastic nature of the entire time series, not just the noise component. If it is assumed that economic time series …