  Providing forecasts for ultra-long time series plays a vital role in various
activities, such as investment decisions, industrial production arrangements,
and farm management. This paper develops a novel distributed forecasting
framework to tackle challenges associated with forecasting ultra-long time
series by using the industry-standard MapReduce framework. The proposed model
combination approach facilitates distributed time series forecasting by
combining the local estimators of time series models delivered from worker
nodes and minimizing a global loss function. In this way, instead of
unrealistically assuming the data generating process (DGP) of an ultra-long
time series stays invariant, we make assumptions only on the DGP of subseries
spanning shorter time periods. We investigate the performance of the proposed
approach with AutoRegressive Integrated Moving Average (ARIMA) models using the
real data application as well as numerical simulations. Compared to directly
fitting the whole data with ARIMA models, our approach results in improved
forecasting accuracy and computational efficiency both in point forecasts and
prediction intervals, especially for longer forecast horizons. Moreover, we
explore some potential factors that may affect the forecasting performance of
our approach.
