時間數列之同步性研究

TESTS OF SYNCHRONIZATION AND THEIR APPLICATION

邱惠玉(Huei-Yu Chiu)

指導教授 ： 管中閔

國立臺灣大學/社會科學院/經濟學研究所/博士(2006年)

https://doi.org/10.6342/NTU.2006.02304

摘要

由於科技進步、貿易和財務整合與全球化經濟政策, 許多經濟變數跨國之間呈現互相關聯或是同步變動的現象。 同步(synchronization) 主要是用來描述經濟變數的共同變動現象。 一旦我們關心的經濟變數為同步, 這個訊息不僅使我們了解其變數之間的關係, 同時也讓我們能夠利用此同步關係制定相關的經濟政策。 文獻上, Granger (1988) 具體的提出了兩個同步性概念: 完全同步 (complete synchronization) 和 長期同步 (long-run synchronization)。 本篇論文即是針對 Granger 的同步性概念做探討、並且提出檢定完全同步性的統計檢定量、最後將新檢定方法應用於分析各國景氣循環的關係。

Granger (1988) 利用頻率域 (frequency domain) 的交叉相位頻譜 (cross phase spectrum) 來定義同步性概念。 故在第一章中, 我們介紹頻率域上常用的概念和探討同步的定義。 當兩數列的交叉相位頻譜對所有頻率而言為零時, 這兩數列則被稱為完全同步。 亦即表示這兩數列有相同的變動行為。 當兩數列之交叉相位頻譜的一階微分為零時, 這兩數列則被稱為長期同步。 長期同步的定義放鬆了完全同步的條件, 只要求兩數列有相似的變動行為即可, 而不要求其有相同的變動。

經由簡單的計算, 完全同步的交叉相位頻譜為零之條件可改寫為要求交叉共變異函數對原點對稱。 在第二章中, 我們即是針對這個完全同步條件提出三種檢定方法。 由於樣本交叉共變異函數的極限變異數與共變異數矩陣 (asymptotic variance-covariance matrix) 很複雜, 包括了自我共變異函數與交叉共變異函數。 故我們首先採用 Melard, Paesmans and Roy (1991) 的方法直接估計, 主要使用核函數 (kernel function) 平滑樣本自我共變異函數與交叉共變異函數。 在虛無假設下, 可推導得到這個統計檢定量服從卡方分配。 為了避免直接估計變異數與共變異數矩陣, 第二種檢定方法是利用疊代的移動區間拔靴法 (overlapping moving-block bootstrap) 得到經驗分配 (empirical distribution)。 經由對樣本區間抽取產生的拔靴樣本, 可保留原樣本中的依存性 (dependence)。 以上兩種檢定方法都需要對特定的參數做設定: 第一種方法需要選取適當的核函數與頻寬 (bandwidth); 第二種方法則是需要選取適當的區間長度 (block length)。 為了避免這類任意選取的問題, 第三種方法採用 Kuan and Lee (2006) 利用隨機標準化矩陣 (random normalizing matrix) 的方法來估計極限變異數與共變異數矩陣。 在虛無假設下, 這個統計檢定量的極限性質可推導得到為布朗橋的函數 (functional of Brownian bridge)。
蒙地卡羅模擬結果顯示, 第一種方法的統計檢定量有較多的誤差扭曲 (size distortion), 但卻有較佳的檢定力 (power); 第二種方法的統計檢定量的誤差與檢定力表現取決於區間長度的選取; 而第三種方法的統計檢定量有較佳的誤差表現, 但卻在某些情況下有檢定力損失。

在第三章中, 我們檢驗 7 個歐洲地區與亞洲地區的國家是否分別對德國和日本存在景氣循環完全同步關係 (business cycle complete synchronization), 並且也檢驗 OECD (Organization for Economic Cooperation and Development) 7 個國家對美國是否有同樣的現象。 有別於一般文獻上利用同期交叉相關係數來描述跨國關係, 我們則是採用第二章所提出的三種完全同步檢定方法來進行檢驗。 結果顯示, 除了英國, 其他歐洲地區國家對德國有景氣循環完全同步現象。 亞洲地區國家皆對日本存在景氣循環完全同步關係。 OECD 國家中, 除了加拿大、法國與德國對美國沒有景氣循環完全同步現象外, 其他國家和美國皆具景氣循環完全同步現象。

關鍵字

景氣循環完全同步 ； 完全同步 ； 完全同步檢定 ； 交叉頻譜 ； 長期同步 ； 相位頻譜

並列摘要

Due to technology advancement, trade and financial integration and globalist economic policy, many economic variables are related and may be synchronized across countries. The concept of synchronization characterizes the comovement property of oscillations among economic variables. Once we learn two economic series of interest are synchronized, we not only acquire the inter-relationship information but also may use such information to design an adjusting mechanism. Two notions of synchronization, "complete synchronization" and "long-run synchronization", were formally proposed by Granger (1988).

Granger (1988) defined synchronization in terms of the
cross phase spectrum in frequency domain. We introduce
some commonly used notions in frequency domain and explore
the concepts of synchronization in Chapter 1. When the cross phase spectrum of two series is zero for all frequencies, they are said to be "completely
synchronized". It means that the two series have the same oscillating behavior. When the first derivative of cross phase spectrum of two series is zero at zero frequency, they are said to be "long-run synchronized". It implies that the two series may not have the same oscillating
behavior but have similar vibrating paths.

In Chapter 2, we propose three diagnostic tests for complete synchronization. The condition of complete synchronization can be examined by testing the symmetry of the cross covariance function at the origin for all lags.
Since the asymptotic variance-covariance matrix of the sample cross covariance function is complicated, we first adopt the method of Melard, Paesmans and Roy (1991) and employ a kernel function to smooth sample auto- and cross-covariances. Under the null hypothesis, this test is shown to follow a chi-squared distribution. To avoid direct estimation of the variance-covariance matrix, we also adopt overlapping moving-block bootstrap and compute the empirical distribution of the test statistic. Both methods have user-chosen parameters, such as the kernel function, bandwidth and block length. In order to avoid such a problem, we follow Kuan and Lee (2006) which employed a random normalizing matrix to eliminate the nuisance parameters. Under the null hypothesis, this test is shown to follow a functional of Brownian bridge. The Monte Carlo simulations show that the test constructed by Melard et. al. (1991) has more size distortion but better power performance. The sizes and powers of the moving-block bootstrap test depend on the choice of block length.
The test constructed by Kuan and Lee (2006) method has better size performance but suffers from power loss in some cases.

In Chapter 3, we examine the phenomenon of business cycle complete synchronization between different countries in the Europe and Asia. Unlike the commonly used measurement of contemporary cross correlation, we employ three proposed tests in Chapter 2. Germany and Japan are supposed to be the leading countries in these two areas so that we analyze the complete synchronization relation between each country with them. We also investigate the synchronization relation between the OECD countries with the US. The results show that most European countries are
completely synchronized with Germany, except that the UK is synchronized with Germany for a small number of lags.
In Asia, the countries all exhibit complete synchronization
with Japan. Canada, France and Germany are also found not to have complete synchronization with the US for a small number of lags.

並列關鍵字

business cycle synchronization ； complete synchronization ； complete synchronization tests ； cross spectrum ； long-run synchronization ； phase spectrum

參考文獻

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