Selecting a Model for Detecting the Presence of a Trend

The authors consider the problem of determining whether the upward trending behavior in the global temperature anomaly series should be forecast to continue. To address this question, the generic problem of determining whether an observed trend in a time series realization is a random (i.e., short-term) trend or a deterministic (i.e., permanent) trend is considered. The importance of making this determination is that forecasts based on these two scenarios are dramatically different. Forecasts based on a series with random trends will not predict the observed trend to continue, while forecasts based on a model with deterministic trend will forecast the trend to continue into the future. In this paper, the authors consider an autoregressive integrated moving average (ARIMA) model and a “deterministic forcing function + autoregressive (AR) noise” model as possible random trend and deterministic trend models, respectively, for realizations displaying trending behavior. A bootstrap-based classification procedure for classifying an observed time series realization as ARIMA or “function + AR” using linear and quadratic forcing functions is introduced. A simulation study demonstrates that the procedure is useful in distinguishing between realizations from these two models. A unit-root test is also examined in an effort to distinguish between these two types of models. Using the techniques developed here, the temperature anomaly series are classified as ARIMA (i.e., having random trends).