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 AandN) and Hyndman and Billah ([2003. Unmasking the theta method. Int. J. Forecast., 19, 287–290], hereafter HandB) 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 AandN and HandB, 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 AandN 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 AandN in M3 competition—rather than an optimal value for the drift as in HandB. The paper concludes with a series of simulations as well as empirical investigations on the M3 yearly data.