格蘭傑因果關係是一個透過結合向量自迴歸模型中所有變數的資訊於衡量兩組時間序列間可預測性的經典統計分析工具，傳統分析格蘭傑因果關係的推論方法為 Wald 類型的檢定方法，然而這些檢定方法可能會面臨以下問題: 一、需要挑選微調參數，二、當預估測之共變異數矩陣為奇異矩陣時，用於推論的臨界值會失效。在這篇論文中，我們發展了一個基於非樞紐統計量的格蘭傑因果關係檢定，此方法不僅避免了以上兩個問題，相較於 Wald 類型的檢定，我們的方法有更佳的檢定力，最後我們也通過幾個模擬例子和實際資料分析驗證此方法的有效性。

Granger causality is a classical tool for measuring predictability from one group of time series to another by incorporating information of variables described by a vector autoregressive (VAR) model. Traditional methods for validating Granger causality are based on the Wald type tests, which may encounter a problem with (i) tuning parameter selection or (ii) test-statistic inflation when the true covariance matrix is singular or near-singular. In this study, we propose an alternative procedure for testing Granger causality based on non-pivotal statistics. The proposed hypothesis testing method is valuable in that (i) it does not require any calibration of tuning parameters (thus saving huge computational cost); and (ii) it yields very competitive power values as compared with the Wald type tests. Finally, a number of simulation examples and a real data set are used to illustrate and evaluate the proposed method.

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