本論文重新檢視當檢定兩整合階次不相等之時間序列(y_(t,i) and y_(t,j))間相關性時，傳統皮爾森樣本橫斷面相關係數(ρ ˆ_ij)之極限漸進分配。
我們首先證明當兩時間序列之整合階次不相等時，√T ρ ˆ_ij→N(0,1)之假設將不再成立，以及，由Breusch and Pagan (1980)提出，以此統計推論為基石，所衍生建構之拉氏乘數(LM)相關性檢定統計量亦因而不再適用。而後，本文提出以經過自我相關漸進法濾過之殘差項，重新建構適用於檢定兩整合階次不相等時間序列間相關性之 ρ ˆ_ij以及衍生之LM檢定量。經數理推論證實，重新建構之皮爾森樣本相關係數將服從標準常態分配，亦即，√T ρ ˆ_(ij,AR)→N(0,1)，並且，與其相對應之LM統計量亦將服從自由度為N(N-1)/2之卡方分配。此外，本文考慮Hong(1996)之橫斷面相關性檢定，並利用自我相關漸進法，提出兩個實行上相當簡易之檢定統計量，以用於檢測兩整合階次不相等時間序列，其落後期間之相關性。
我們的蒙地卡羅模擬結果顯示，若所考慮之兩個時間序列出現整合階次不相等的情形，與過去所熟知ρ ˆ_ij之表現相異，傳統ρ ˆ_ij之極限分配將不再服從標準常態分配。此外，根據模擬實驗結果更顯示，在有限樣本下，應用新建構之皮爾森樣本相關係數所進行之相關性檢定，其容忍度(size)可被有效控制，且其檢定力(power)亦相當可信，較傳統方法有十分顯著的進步，
更甚者，利用本文所提出之新統計量進行相關性檢定，亦可避免掉過去文獻中被分析討論的，因整合階次不等所導致之假性相關，以及結果出現偏誤的問題。
最後，以此新建構之估計方法，本文除提供一得準確捕捉金融危機之偵測指標，亦應用至風險報酬抵換議題上，以玆證明此方法之實用性。

This paper re-examines the limiting distribution of the conventional Pearson sample cross- correlation coefficients (𝜌ˆ𝑖𝑗) when testing for the uncorrelatedness between two time series (𝑦𝑡,𝑖 and 𝑦𝑡,𝑗), of which the integrated orders are not equal to each other. We first demonstrate the invalidation of √𝑇𝜌ˆ𝑖𝑗 → 𝑁(0,1) and of conventional liming distribution for its resulting Lagrangian multiplier (LM) correlation test statistics of Breusch and Pagan (1980) when the integrated orders of two time series are imbalanced. We then suggest to reconstruct this 𝜌ˆ𝑖𝑗 by using the AR-filtered residuals from two integrated order-imbalanced time series as well as the AR-filtering version of LM test. The mathematical theorems justify the standard normal distribution followed by the new built Pearson sample correlation coefficient, i.e., √𝑇𝜌ˆ𝑖𝑗,𝐴𝑅 → 𝑁(0,1) and then show a chi-squared distribution with 𝑁(𝑁 − 1)/2 degrees of freedom of its corresponding LM test. Extending the Hong(1996) cross-correlation tests, we further propose two easy-to-implement lagged correlation tests for two integrated order-imbalanced processes being uncorrelated via AR approximations. Our simulations confirm the limiting distribution of the conventional 𝜌ˆ𝑖𝑗 fails to follow the standard normal distribution as two time series displaying different integrated orders in contrast to the traditional understanding of 𝜌ˆ𝑖𝑗, as well as demonstrate that in finite samples, the significant improvement of the newly built Pearson sample correlation coefficient in terms of the size and promising power performances for the new proposed correlation tests. More importantly, our new methodology could be treated as a simple correlation test which avoids the spurious correlation and inaccurate results caused by the two order-imbalanced series analyzed in previous literature. Finally, a newly constructed crises-detecting index and a revisiting study of risk-return trade-offs provide the usefulness of our methodology.

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