The Wold decomposition gives a moving average (MA) representation of a purely non‐deterministic stationary process. In this article, we derive estimates of the Wold matrices for a d‐dimensional process by using a Cholesky decomposition of a banded and tapered version of the sample autocovariance matrix, and we derive convergence rates for the estimation error of the (possibly infinite) sequence of Wold matrices. By analogy to lag‐window estimates of the spectral density, this method can be used to obtain finite vector MA models with an adaptive lag‐order. We additionally show how these results can be applied to impulse response analysis and to derive a bootstrap procedure. Our theoretical results are complemented by simulations which investigate the finite sample performance of the estimator.

中文翻译：

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估计沃尔特矩阵和矢量移动平均过程

Wold分解给出了纯非确定性平稳过程的移动平均值（MA）表示。在本文中，我们使用样本自协方差矩阵的带状和锥形版本的Cholesky分解，得出了d维过程的Wold矩阵的估计，并且得出了（可能是无限的）序列的估计误差的收敛速度。 Wold矩阵。类似于频谱密度的滞后窗口估计，该方法可用于获得具有自适应滞后阶的有限矢量MA模型。我们还显示了如何将这些结果应用于脉冲响应分析和导出引导程序。我们的理论结果得到了模拟的补充，这些模拟研究了估计器的有限样本性能。