Estimating Trend and Growth Rates in Seasonal Time Series

Abstract

Abstract We consider the problem of trend estimation from an ARIMA-model-based perspective. Given that the time series of interest is well represented by a multiplicative seasonal ARIMA model (Box and Jenkins 1970), a method is presented for estimating the trend of this series as a component of the model's forecast function. This method is applied to the “airline model,” a commonly occurring model form for economic and social time series. For those numerous series that are rendered stationary by logging and first-differencing, the resulting quantity supplies an estimate of the rate of change, or growth rate, of the series. As this trend estimate is given in terms of the underlying series' forecast function, we analyze forecast functions of ARIMA models in general and of the airline model in particular, extending the development in Box and Jenkins (1970, chaps. 5 and 9). Representing the forecast function as the solution of a difference equation, the trend estimate is the component of the forecast function...