Forecasting aggregates of independent Arima processes

In this paper I derive the properties of linear aggregates of independent ARIMA processes, including those with seasonality. I show that such aggregates are ARIMA processes, but that multiplicative seasonal structure on the moving average side will generally break down in the aggregation. The forecasting efficiency of a direct model of an aggregate is contrasted with that of an optimal predictor which uses the structural information. The former is shown to suffer from a kind of aggregation bias, a bias which can be expressed as a mean zero stochastic process.