The Theta Model in the Presence of a Unit Root Some new results on “optimal” theta forecasts

We significantly extend earlier work by Assimakopoulos and Nikopoloulos (2000) and Hyndman and Billah (2003) on the properties and performance of the Theta model, and potentially explain its very good performance in the M3 forecasting competition. We derive a number of new theoretical results for theta forecasts when the data generating process contains both deterministic and stochastic trends. In particular (a) we show that using the standard theta forecasts coincides with the minimum mean-squared error forecast when the innovations are uncorrelated; (b) we provide, for the first time, an optimal value for the theta parameter, which coincides with the first order autocorrelation of the innovations, and thus provide a single optimal theta line; (c) we show that the optimal linear combination of two standard theta lines coincides with the single optimal theta line of (b). Under (b) and (c) we show that the optimal theta forecast function is identical with that of an ARIMA(1,1,0) model. Furthermore, we illustrate how the Theta model can be generalized to include local behavior in two different ways.