本文係研究結構改變發生於一般自我迴歸型式的條件變異數不齊一性 (gen- eralized autoregressive conditional heteroscadasticity; GARCH) 模型的情況。 採用 的方法是 Lin 和 Terasvirta (1994) 的平滑轉換模型及 LM 檢驗統計量。文中首先探討平 滑轉換的 GARCH 模型。其次，介紹模型正確設定下三種 GARCH 模型結構性改變的 LM 檢定 ，一是由一階微分的記分 (score) 函數出發的測驗; 二是由 Breusch 和 Pagan (1979) 及 Koenker (1981) 的推導， 簡化的 nR2 (n 為樣本點個數 ) 檢定統計量， 三則是 Bollerslev (1986) 提出用 BHHH 演算法以斜率外積 (outer product of the gradient; OPG) 作為訊息矩陣的基礎， 經由第一次 0LS 輔迴歸造出的 nR2 的統計量。 而因為以 GARCH 模型出發點的條件分配一般為常態，當模型的條件分配不是真正常態的情況下，訊息 矩陣等式不成立， 我們提出針對誤差項仍為鞅差序列 (martingale diffrence sequence) 時修正的 LM 統計量。另外，如果條件分配不是常態且模型也設定錯誤，以上四種 LM 測驗 的方法都無法使用。 針對這種情況本文根據 White (1982) 以 Newey 和 West (1987) 及 Andrews (1991) 的文章為基礎， 經由變換訊息矩陣找到一個可以在模型設定錯誤且不知道 條件分配情況下的頑強 LM 檢驗統計量。實證的結果發現臺灣的股票加權平均指數報酬率的 波動性，在經歷 12495 的歷史性最高點下，仍沒有結構性改變。 而美國的三個股票指數報 酬率都有顯著的結構性改變。 而其滾動 (rolling) 的檢驗，發現其改變點也符合歷史事實 。

　　　　　This paper proposes tests for a structural change in the generalized autoregressive conditional heteroscadasticity (GARCH) model. We use the smooth transition method and Lagrange multiplier (LM) test discussed in Lin and Terasvirta (1994). First, we investigate the smooth transition form in the GARCH model, and then derive three LM tests under the correct specification of the model. These three tests include the original form from the score function; the nR2 version (n is the sample size) in Breusch and Pagan (1979) and Koenker (1981); the nR2 version from the first OLS auxiliary regression in the outer product of the gradient (OPG) in the BHHH algorithm proposed by Bollerslev (1986). However, the GARCH model assumes the underline distribution with normality. When the conditional distribution is no longer Gaussian, the information matrix equality does not hold but the errors are still a martingale difference sequence. For this case, we suggest a modified LM test which can be applied when the error terms have the martingal difference property. Moreover, when the conditional distribution is wrongly assumed and the model is wrongly specified, the errors are no longer a martingale difference sequence. The LM tests discussed above cannot be applied, and we have proposed another robust LM test discussed in White (1982), using Newey and West (1987) and Andrews (1991) to calculate the heteroskedasticity and autocorrelation consistent estimator. The empirical result shows that the index return in the Taiwan Stock Market did not present structural change, even though the index fluctuated a lot and reached a historical high of 12495 points. The three stock indices i the US market do show the structural change and the rolling test in the sample finds the change points can match the historical events.