Directionality and Reversibility in Time Series

Summary This paper gives a review and development of what is currently known about the directionality (irreversibility) of time series models, together with briefer coverage of the still limited statistical methodology. Reversibility is shown to imply stationarity; Weiss's result concerning the reversibility of linear Gaussian processes is stressed, and contrasted to the directional nature of much time series data. Reversed ARMA models are explored, and non-linear examples given; the stationarity and invertibility conditions of ARMA models are shown to be implicitly directional, and a consequence of the future-independent nature of such models. Invertibility is extended to the two-sided futuredependent generalised linear model, and applied to reversible moving average models. The directional and reversible implications of autoregressive roots are covered. Work applying directional-sensitive methods of statistical analysis to reversed data series is mentioned; possible dangers in transforming directional series to Gaussian marginal distributions are noted. The directional nature of most non-linear models is invoked to emphasise the current importance of the area.