Unmasking the Theta method

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Abstract

The ‘Theta method’ of forecasting performed particularly well in the M3-competition and is therefore of interest to forecast practitioners. The original description of the method given by Assimakopoulos and Nikolopoulos [International Journal of Forecasting 16 (2000) 521] involves several pages of algebraic manipulation. We show that the method can be expressed much more simply and that the forecasts obtained are equivalent to simple exponential smoothing with drift.

Introduction

The ‘Theta method’ of forecasting was introduced by Assimakopoulos and Nikolopoulos (2000), hereafter referred to as A&N. Their description of the method is complicated, potentially confusing, and involves several pages of algebra. However, the method performed particularly well in the M3-competition (Makridakis & Hibon, 2000) and is, therefore, of interest to forecast practitioners.

We examine the Theta method and show that it can be expressed much more simply than in A&N; furthermore we show that the forecasts obtained are equivalent to simple exponential smoothing (SES) with drift. Using this equivalence, we derive appropriate prediction intervals for the method based on a state space model underlying SES with drift. Finally, we show that SES with drift can produce better forecasts than the Theta method if the parameters are optimized using a maximum likelihood approach.

Section 2 reproduces the main results from A&N using a different (and much simpler) notation. We obtain an explicit expression for point forecasts in Section 3 and show that these are equivalent to the point forecasts from SES with drift. In Section 4, we describe a state space model with equivalent forecasts, thus enabling the computation of prediction intervals and likelihood estimates. Finally, in Section 5 we compare the Theta method with fully optimized SES with drift by applying both methods to the annual data from the M3-competition. Appendix A contains a list of equivalencies in the notation in A&N and in this paper.

Section snippets

Theta method

Let {X1,…,Xn} denote the observed univariate time series. From this series A&N construct a new series {Y1,θ,…,Yn,θ} such that for t=3,…,n:Yt,θ″=θXtwhere Xt″ denotes the second difference of Xt and Yt,θ″, denotes the second difference of Yt,θ. We note that (1) is a second-order difference equation and has the solution (see Kelley, 2000):Yt,θ=aθ+bθ(t−1)+θXtwhere aθ and bθ are constants and t=1,…,n. Thus, Yt,θ is equivalent to a linear function of Xt with a linear trend added. A&N call Yt,θ a

Point forecasts

The above results can be combined to obtain a simple expression for the forecasts X̂n(h). From (4) we obtain:Ŷn,2(h)=αi=0n−1(1−α)iâ2+b̂2(n−i−1)+2Xn−i+(1−α)n(â2+2X1)=â2+b̂2n−1α+(1−α)nα+2X̃n(h)where X̃n(h) is the SES forecast of the series {Xt}. Noting that â2=−â0 and b̂2=−b̂0, we obtain:X̂n(h)=X̃n(h)+12b̂0h−1+1α(1−α)nαFor large n, this can be written as:X̂n(h)=X̃n(h)+12b̂0(h−1+1/α)Thus it is SES with an added trend plus a constant, where the slope of the trend is half that of the fitted

Underlying stochastic models

A&N do not give an underlying stochastic model for their forecasting method. However, it is possible to find such a model using a state space approach. We initialize the model by setting X1=l1 and then for t=2, 3,…, let:Xt=lt−1+b+εtandlt=lt−1+b+αεtwhere {εt} is Gaussian white noise with mean zero and variance σ2.

Then Xt follows a state space model which gives forecasts equivalent to SES with drift. This is a special case of Holt’s method with the smoothing parameter for the slope set to zero.

Application to annual M3 competition data

The preceding analysis suggests we may be able to obtain better forecasts if we optimize the value of b rather than setting it equal to b̂0/2. To evaluate this idea, we apply the model to the 645 annual series from the M3 competition (Makridakis & Hibon, 2000).

We computed forecasts up to six steps ahead and then we computed the symmetric mean absolute percentage error (SMAPE) as in Makridakis and Hibon (2000). The results are presented in Table 1.

Table 1 shows the average SMAPE for: (a) the

Conclusion

We have demonstrated that the Theta method proposed by A&N is simply a special case of SES with drift where the drift parameter is half the slope of the linear trend fitted to the data. We have also demonstrated that prediction intervals and likelihood-based estimation of the parameters can be obtained using a state space model.

Biographies: Rob HYNDMAN holds a PhD in statistics from the University of Melbourne and is currently Associate Professor and Director of the Business and Economic Forecasting Unit at Monash University. He is co-author of the textbook Forecasting: methods and applications with Spyros Makridakis and Steven Wheelwright. His research papers have appeared in Journal of the Royal Society, International Journal of Forecasting, Journal of Forecasting, American Statistician, Journal of Computational and

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Biographies: Rob HYNDMAN holds a PhD in statistics from the University of Melbourne and is currently Associate Professor and Director of the Business and Economic Forecasting Unit at Monash University. He is co-author of the textbook Forecasting: methods and applications with Spyros Makridakis and Steven Wheelwright. His research papers have appeared in Journal of the Royal Society, International Journal of Forecasting, Journal of Forecasting, American Statistician, Journal of Computational and Graphical Statistics, Computational Statistics and Data Analysis, and elsewhere. He is Editor of the Australian and New Zealand Journal of Statistics and Associate Editor of the International Journal of Forecasting. His research interests include time series analysis, forecasting, and nonparametric smoothing.

Baki BILLAH is Research Fellow in the Business and Economics Forecasting Unit at Monash University, Australia. He received a Masters in Applied Statistics from Memorial University of Newfoundland, Canada and his PhD in Econometrics and Business Statistics from Monash University. His research interests include business forecasting, model selection and shrinkage estimation

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