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Wayne A. Woodward and H. L. Gray

Abstract

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).

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Wayne A. Woodward and H. L. Gray

Abstract

In recent years a number of statistical tests have been proposed for testing the hypothesis that global warming is occurring. The standard approach is to examine one or two of the more prominent global temperature datasets by letting Yt = a + bt + Et, where Yt represents the temperature at time t and Et represents error from the trend line, and to test the hypothesis that b = 0. Several authors have applied these tests for trend to determine whether or not a significant long-term or deterministic trend exists, and have generally concluded that there is a significant deterministic trend in the data. However, we show that certain autoregressive-moving average (ARMA) models may also be very reasonable models for these data due to the random trends present in their realizations. In this paper, we provide simulation evidence to show that the tests for trend detect a deterministic trend in a relatively high percentage of realizations from a wide range of ARMA models, including those obtained for the temperature series, for which it is improper to forecast a trend to continue over more than a very short time period. Thus, we demonstrate that trend tests based on models such as Yt = a + bt + Et, where Yt for the purpose of prediction or inference concerning future behavior should be used with caution.

Of course, the projections that the warming trend will extend into the future are largely based on such factors as the buildup of atmospheric greenhouse gases. We have shown here, however, that based solely on the available temperature anomaly series, it is difficult to conclude that the trend will continue over any extended length of time.

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