高維度時間序列資料分析廣泛地應用在財務金融及環境科學等領域，而在多變量時間序列的模型建構上，向量自迴歸模型 (VAR) 是最廣泛常用的模型。然而當序列的變數維度大時，VAR 模型的參數個數便會大幅度增加，致使估計推論與建模的困難度增加。因此，如何能在不損失序列關聯資訊的前提下適當地精簡模型結構，有助於降低建模的計算成本與提高估計推論的準確性。動態因子模型 (dynamic factor model) 是一種有效率的時序資料降維的方法，本篇論文將在動態因子模型的架構下，研討高維度時間序列間相關函數(CCF)的交互對稱結構，並據此建議高維度動態因子模型建構時的簡化流程。本文提出了兩種CCF 對稱性的假設檢定：第一種檢定是利用資料樣本相關函數 (sample CCF) 的漸進分佈 (asymptotic distribution) 所建構出的Wald 檢定統計量，適用於較低維度的時序資料；第二種檢定是考量前述漸進分佈的邊際分佈(marginal distribution)，並利用Benjamini-Hochberg (Benjamini and Hochberg, 1995; Benjamini and Yekutieli, 2001) 的方法所建構的多重檢定程序，適用於中高維度的時序資料。本論文經由數值模擬驗證了兩種統計檢定在CCF 不對稱的多種情境下都有良好的檢定力，並將所提出的檢定方法應用在兩個經濟指標的實證分析上，藉由驗證CCF 的對稱性，確實能有效地輔助高維度動態因子模型的結構簡化。

Dynamic factor models are commonly used in modeling high-dimensional time series and have wide applications in many fields, including economics, finance, and environmental science. Due to numerous parameters in a high-dimensional setting, properly simplifying the model is essential to reduce the computational cost of modeling and making inferences. For this purpose, this thesis explores the cross-correlation structures among multiple time series, particularly concerning the symmetry structure of the cross-correlation functions (CCF). This thesis proposed two hypothesis tests on the symmetry of CCF. One is based on the asymptotic distribution of the sample estimator of the auto-correlation matrices; the other is the multiple tests based on Benjamini-Hochberg (Benjamini and Hochberg, 1995; Benjamini and Yekutieli, 2001) procedures. Rejection of both tests provides a guideline for factor model simplifications, suggesting that an exact factor model (a simplified model form) would be sufficient for the data under study. The performance of the proposed tests is examined via a simulation study, which shows a fairly accurate error control and good testing powers against a variety of asymmetric CCF scenarios. For illustration, the proposed methodology is applied to two economic datasets. According to the test results on the symmetry of CCF, we properly simplified the factor model specifications for both data resulting in satisfactory fitting performance.

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