Fathoming the Theta Method for a Unit Root Process

In this paper, building on earlier work by Assimakopoulos and Nikolopoulos ([2000. The theta model: a decomposition approach to forecasting. Int. J. Forecast., 16, 521�530], hereafter A&N) and Hyndman and Billah ([2003. Unmasking the theta method. Int. J. Forecast., 19, 287�290], hereafter H&B) on the properties and performance of the theta method, we derive new results for a unit root data generating process. In particular, (a) we investigate the theoretical underpinnings of the method when a single �theta line� is used, rather than a combination of two �theta lines� as in A&N and H&B, and we provide an optimal value for the theta parameter that coincides with the first-order autocorrelation of the innovations; (b) we demonstrate that the optimal forecast function for the model examined in A&N is identical with that of ARIMA(1,1,0) and (c) we provide formulae for optimal weights when combining two �theta lines� as in the model used by A&N in M3 competition�rather than an optimal value for the drift as in H&B. The paper concludes with a series of simulations as well as empirical investigations on the M3 yearly data