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 and . 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.

中文翻译：

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衡量同时聚合在预测中的优势

假设在已知的聚合机制下对底层多元时间序列进行同时聚合，得到较低维的多元聚合时间序列。为了预测聚合时间序列，可以考虑两种通用策略：第一，聚合基础时间序列的预测；其次，直接预测聚合时间序列。直观上，第一个策略应该更准确，因为底层时间序列比聚合时间序列包含更全面的信息。然而，与低维聚合时间序列相比，高维基础多元时间序列的模型构建过程和估计过程更加复杂，这可能会增加模型错误指定的机会并导致更大的估计误差。因此，最好直接预测聚合时间序列。因此，在实践中衡量两种预测策略之间的相对精度至关重要。为此，我们引入了一种预测方法来量化在均方误差意义上使用同期聚合进行预测的优势。预测措施是在以下假设下构建的：基础时间序列遵循向量自回归移动平均 (VARMA) 过程。估计过程不需要指定 VARMA 的任何特定形式，即滞后阶数和。建立了估计过程的渐近性质，并通过蒙特卡洛模拟和真实数据示例评估了所提出方法的有限样本性能。