Measuring the advantages of contemporaneous aggregation in forecasting

Suppose an underlying multivariate time series is contemporaneously aggregated under a known aggregation mechanism, and a lower dimensional multivariate aggregated time series is obtained. To forecast the aggregated time series, one could consider two general strategies: first, aggregate the forecasts of the underlying time series; second, forecast the aggregated time series directly. Intuitively, the first strategy should be more accurate, as the underlying time series contains more comprehensive information than the aggregated time series. However, the model-building process and estimation procedure for the higher dimensional underlying multivariate time series are more complex compared with that for the lower dimensional aggregated time series, which may increase the chances of model misspecification and result in larger estimation errors. Therefore, it may be preferable to forecast the aggregated time series directly. It is then crucial to measure the relative precision between the two forecasting strategies in practice. To this end, we introduce a forecasting measure to quantify the advantages of using contemporaneous aggregation in forecasting in the sense of the mean-squared error. The forecasting measure is constructed under the assumption that the underlying time series follows the vector autoregressive moving average (VARMA) process. The estimation procedure does not require specifying any particular form of the VARMA, namely, the lag order p$$ p $$ and q$$ q $$. Asymptotic properties of the estimation procedure are established, and we evaluate the finite-sample performance of the proposed method through Monte Carlo simulations and a real data example.