Infemotiond StafLstical Review, (1986), 54, 1, pp. 51-66. Printed in Great Britain 0 International Statistical Institute Forecast Functions Implied by Autoregressive Integrated Moving Average Models and Other Related Forecast Procedures Bovas Abraham1 and Johannes Ledolte? 'Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Department of Statistics and Actuarial Science and Department of Management Sciences, University of Iowa, Iowa City, Iowa 52242, USA Summary The simplifying operators in ARIMA (autrogressive integrated moving average) models deter- mine the form of the corresponding forecast functions. For example, regular differences imply polynomial trends and seasonal differences certain periodic functions. The same functions also arise in the context of many other forecast procedures, such as regressions on time, exponential smoothing and Kalman filtering. In this paper we describe how the various methods update the coefficients in these forecast functions and discuss their similarities and differences. In addition, we compare the forecasts from seasonal ARIMA models and the forecasts from Winters' additive and multiplicative smoothing methods. Key words: ARIMA models; Exponential smoothing; Forecasting; Kalman filter; Simplifying operators; Time series; Winters' forecast procedures. 1 Introduction Suppose that Z,, for t = 0, f1, f 2. . . , are time series observations on some measurable phenomenon. It is common to model such observations with stochastic difference equations of the form where B is the backward shift operator such that BmZf = Z,-,, is the generalized autoregressive operator of order p, 8(B) = 1- BIB - . . . - 8,Bq is a moving average operator of order q, and {a,, t = 0, f 1, . . .) is a sequence of independent identically distributed normal random variables with mean zero and variance a2, subsequently referred to as a white noise sequence. The generalized autoregressive operator q(B) consists of a simplifying operator U(B) and nonseasonal and seasonal autoregressive operators. The simplifying operator U(B) has all its roots on the unit circle. Special cases are the ordinary differences (1 - B)d, the seasonal differences (1 - ByD, where s is the period of seasonality, and their products and factors such as (1 - V~B + B~), (1 - B + B~), etc. The autoregressive components (B) = 1- B - . . - B (nonseasonal) and @(By = 1- Q1BS - . . . - @,B," (seasonal) have their roots outside the unit circle. For seasonal time series the moving