儘管已有許多文獻在探討當非定態時間序列變數被受限時，應該如何建立模型去描述此類變數。但在如何解釋與分析此類變數，當其應為 I ( 1 ) 的隨機過程卻又被固定的界限所限制時卻鮮少有相關文獻探討，以下我們將建議兩篇論文對此變數的相關研究。

Lee. et. al. ( 2005 ) 建議利用Gibbs sampling 的模擬抽樣方式去處理受到截堵的資料，也證明出當利用 Gibbs sampling 所還原的資料後採用 ADF 檢定時，其檢定統計式仍與 Said and Dickey ( 1984 ) 所導出的分配相同。而 Cavaliere ( 2005 ) 則在當時間序列變數服從近單根的隨機過程但卻因某些外在因素而被限制住時，發展出另一種漸進分配去檢定單根的存在。而本論文則以 Lee. et. al. ( 2005 ) 及 Cavaliere ( 2005 ) 為主軸，分別討論兩論文所建議的方法，並於最後將模型設定在相同的情況下，比較及分析兩者在有限樣本下型一誤差 ( size ) 的表現。

Despite the extensive literature on modeling nonstationary economice time series and on limited-dependent variables, a controversial and rarely discussed topic is how to interpret and analyze time series whose behaviors can be well approximated by means of integrated processes, I ( 1 ), but are “ limited ” in the sense that their range is constrained by fixed bounds.

Lee. et. al. ( 2005 ) proposes to deal with censored data via Gibbs sampling, and show an Augmented Dickey-Fuller test on the augmented data that has the same asymptotic distribution as that of Said and Dickey ( 1984 ) is derived. Cavaliere ( 2005 ) develops an asymptotic theory for integrated and near-integrated time series whose range is constrained in some ways. We hereby compare the performances of the size of these two different papers which are assumed in the same conditions.

Chapter 1 緒論………………………………………………1

1.1 研究背景與動機…………………………………………1
1.2 研究目的…………………………………………………2
1.3 論文研究架構與流程……………………………………3

Chapter 2 相關模型回顧……………………………………5

2.1 時間序列模型……………………………………………5
2.1.1 時間序列模型的一般化ARMA ( p, q ) 模型………5
2.1.2 隨機漫步………………………………………………7
2.1.3 單根模型………………………………………………10
2.2 單根檢定…………………………………………………12
2.2.1 Dickey-Fuller 檢定…………………………………12
2.2.2 擴充的Dickey-Fuller檢定 …………………………14
2.2.3 Phillips-Perron 檢定………………………………16
2.3 截堵迴歸模型……………………………………………18
2.4 Gibbs Sampling…………………………………………21

Chapter 3 理論模型…………………………………………24

3.1 Gibbs Sampling 在截堵單根模型下的應用 …………24
3.1.1 截堵單根模型…………………………………………24
3.1.2 截堵 AR ( K ) 模型條件分配之推導………………27
3.1.2.1 的推導說明………………………………………29
3.1.2.2 的推導說明………………………………………30
3.1.2.3 的推導說明………………………………………32
3.1.3 Gibbs Sampling 模擬抽樣程序 ……………………34
3.1.4 統計式的極限分配……………………………………37
3.1.5 截堵模型下單根檢定之探討…………………………38
3.2 受限制時間序列模型下的單根分配……………………39
3.2.1 受限制的自我迴歸模型………………………………40
3.2.2 受限的布朗運動………………………………………41
3.2.3 受限制單根分配………………………………………42
3.2.4 受限模型下單根檢定之探討…………………………44

Chapter 4 截堵模型在有限樣本下型一誤差之比較………46

Chapter 5 結論與建議………………………………………51
參考文獻………………………………………………………53
附錄……………………………………………………………56

英文文獻：
Albert, J.H. and Chib, S. ( 1993 ), “Bayesian Analysis of Binary and Polychotomous Response Data”, Journal of the American Statistical Association, 88, 669-679.

Cavaliere, G. ( 2005 ), “Limited Time Series with a Unit Root”, Econometric Theory, 21, 907-945.

Cavaliere, G. ( 2001 ), “Testing the Unit Root Hypothesis Using Generalizied Rescaled Range Statistics”, Econometrica, 69, 1555-1596.

Chib, S. and Greenberg, E. ( 1996 ), “Markov Chain Monte Carlo Simulation Methods in Econometrics”, Econometric Theory, 12 ( 3 ), 409-431.

Dickey, D.A. and Fuller, W.A. ( 1979 ), “Distribution of the Estimator for Autoregressive Time Series with a Unit Root”, Journal of the American Statistical Association, 74, 427-431.

Dueker, M. ( 2005 ), “Dynamic Forecasts of Qualitative Variables : A Qual VAR Model of U.S. Recessions”, Journal of Business and Economic Statistic, 23, 96-104.

Gelfand, A. and Smith, A. ( 1990 ), “Sampling-Based Approaches to Calculating Marginal Densities”, Journal of the American Statistical Association, 85,398-409.

Gelman, A. and Rubin, D.B. ( 1992 ), “Inference from Iterative Simulation Using Multiple Sequence”, Statistical Science, 7, 457-511.

Geman, S. and Geman, D. ( 1984 ), “Stochastic Relaxation, Gibbs Distribution, and the Bayesain Restoration of Image”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-741.

Karatzas, I. and Shreve S.E. ( 1988 ), “Brownian Motion and Stochastic Calculus”, Springer-Verlag.

Kosuke, I. and David, A.V.D. ( 2005 ), “A Bayesian Analysis of the Multinomial Probit Model Using Marginal Sata Augmentation”, Journal of Econometrics, 124 ( 2 ), 311-334.

Lee, C., Chen, P. and Wu, W. ( 2005 ), “Testing for a Unit Root in Censored Variables”, Working paper, Institute of Economics, National Sun Yat-Sen University, Kaohsing, Taiwan.

Ng, S. and Perron P. ( 2001 ), “Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power”, Econometrica, 69, 1519-1554.

Phillips, P.C.B. ( 1987a ), “Time Series Regression with a Unit Root”, Econometrica, 55, 277-301.

Phillips, P.C.B. ( 1987b ), “Toward a Unified Asymptotic Theory for Autoregression”, Biometrika, 74, 535-547.
Said, S. and Dickey, D. ( 1984 ), “Testing for Unit Roots in Autoregressive-Moving Average Model of Unknown Order”, Biometrica, 71, 599-607.

Tanner, T.A. and Wong, W.H. ( 1987 ), “The Calculation of Posterior Distribution by Data Augmentation”, Journal of the American Statistical Association, 82, 528-549.

中文文獻：
吳瑋倫 ( 2005 )，「Gibbs Sampling 在截堵模型下的應用及其單根檢定」，碩士論文，國立中山大學經濟學研究所。