On Autoregressive Time Series

Publisher Website
Google Scholar
Abstract
Let [chi]i, [chi]i . . . , [chi]n be values measured about their mean, trend-free and defined at equal intervals of time. The kth serial correlation coefficient rk, is defined as the correlation between members of the series, k intervals apart. Correlograms are obtained if rk is plotted against k. If the series consists of a simple harmonic the correlogram reproduces this harmonic, hence the correlogram provides a criterion for distinguishing various kinds of oscillatory times series. This paper treats of correlograms of series defined by the difference equation, [image] called the generating equation, where et+2 is a random variable. The solution of this difference equation gives a series. The author discusses the relation of the period given by the generating equation and that of the generated series and the relation of the variances of U and [epsilon]. Two series are constructed from particular difference equations and discussed in details, in relation to damping, covariances and periods. Superposed variations are also discussed. The author concludes that (a) the correlogram will be damped according to the damping factor of the equation; but if the series is short the damping may be considerably less than the theoretical value; (b) the correlogram will show a period equal within limits of error, to that of the fundamental period of the system; (c) it does not appear that in short series the periodic movement is substantially affected, at any rate not to such an extent as the damping; (d) when superposed random variation is present, the fundamental period calculated from the observed regressions will be too short and may be very considerably so; (e) the period of the generated series, defined as the mean distance from upcross to upcross, is not the same as the fundamental period; but the difference is not likely to be important in practice. The appendix contains a development of the relation between variate differences and serial correlations.

This publication has 1 reference indexed in Scilit: