Estimating Trend and Growth Rates in Seasonal Time Series

原文摘要

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 that is a linear combination of those roots of the observable series' autoregressive operator that are associated with trend (an allocation that is usually standard and unambigous in practice). Thus it is important to determine the coefficients in this linear combination, which are adaptive in the time origin of the forecast, and we describe three general methods for doing this: in terms of initial forecasts, directly from the present and recent values of the observed series, and recursively. For the airline model the trend estimate is a straight line, with both intercept and slope changing in each time period, adapting to the new information becoming available. We also connect this trend estimate with trend forecasts in unobserved-components ARIMA models, which underlie much seasonal adjustment research of the last decade. Trend components in such models are not unique, even given the association of certain of the autoregressive roots with the trend. We show, however, that all admissible trend-seasonal-irregular decompositions have identical trend-component forecasts and that this common forecast is supplied by our trend estimate. Thus the “true trend” (that which our trend estimate estimates) is given only with reference to a particular decomposition. One such decomposition, commonly used in model-based seasonal adjustment, is the one (whose existence and uniqueness are known under fairly general conditions) that maximizes the irregular-component variance or, equivalently, minimizes the trend- and seasonal-component innovation variances. We show that, among all admissible decompositions, this “canonical decomposition” minimizes the mean squared error of our trend estimate.

摘要来源：OpenAlex 索引摘要（稿件版本未注明）