本篇文章主要在探討，AR模型在小樣本的情況下，會因為該資料在估計時使用了最小平方法來做估計，而產生了AR係數的估計向下偏誤，在使用逐步檢定方法的結構性變化檢定時，會因為使用了估計具向下偏誤的AR係數，來計算各資料間的殘差平方和，所以逐步檢定法的檢定統計量會同時出現估計偏誤，導致這個檢定方法會出現型一誤差。
文中主要是使用蒙地卡羅法來隨機抽樣資料，之後使用Rudebusch(1993) 和Andrews-Chen(1994)三位學者提出的兩種中位數修正法，修正該資料具向下偏誤的估計值，使用修正後的估計值回去計算殘差平方和與檢定統計量，試看中位數修正法帶給結構性變化的型一誤差是不是具有良好的改善。
由本篇論文模擬的結果顯示出，因為結構性變化多是使用資料間的殘差平方和來互相比較，所以的確會因為估計值的向下偏誤，使得在計算各資料間的殘差平方和時也會出現了些許誤差，導致檢定出現了型一誤差，經由兩種中位數修正參數後，雖不及信心水準，但修正後型一誤差皆得到大幅的改善，使得小樣本資料在結構性變化檢定中會比修正前更較為精準。

This paper is exploring AR (Autoregression) model which estimated by the ordinary least squares estimator in a small sample , it will produce the downward bias in the estimate of the AR coefficient. Because of the use of estimated AR coefficients with downward bias to calculate the residual sum of squares between the data when the structural changes test of the sequential method is used, the statistic value of the sequential method will also have an estimation error, causing this method to appear a Type I Error.
We resample by using Monte Carlo simulation, and then uses two kinds of median-unbiased estimations proposed by Rudebusch (1993) and Andrews-Chen (1994) to correct the estimated value of the data with downward bias by using the corrected estimated value to calculate the residual sum of squares and the statistic value to see if the median-unbiased estimations will bring a good improvement for Type I Error of the structural changes test.
The simulation results of this paper show that the structural changes test mostly uses the residual sum of squares between the data to compare with each other, so it is true that due to the downward bias of the estimated value there will be some errors in the calculation of the residual sum of squares between the data, causing the Type I Error to appear in the test. After correcting the parameters by two median-unbiased estimations, though it doesn’t reach the confidence level, the Type I Error after the correction get greatly improved, making the small sample data more accurate in the structural changes test than before the bias estimated value.

論文審定書 i
致謝 ii
摘要 iii
Abstract iv
目錄 v
表目錄 vii
第一章 緒論 1
第一節 研究背景 1
第二節 研究動機 2
第三節 研究目的 3
第四節 論文架構 3
第二章 文獻回顧 5
第一節 AR(1)係數估計的向下偏誤 5
第二節 偏誤修正法 6
第三章 模型設定 7
第一節 AR(1)係數估計的向下偏誤 7
第二節 結構性變化檢定 8
第一小節 Chow Test 9
第二小節 Bai-Perron檢定方法 10
第四章 模擬結果 15
第一節 資料的生成 15
第二節 估計的向下偏誤 15
第三節 檢視SupF_T (m+1|m)逐步檢定法的型一誤差 16
第四節 中位數偏誤修正法 19
第五節 SupF_T (m+1|m)逐步檢定法的型一誤差修正 21
第六節 Andrews-Chen Method修正結果 24
第七節 Rudebusch Method修正結果 27
第五章 結論 31
第一節 研究結論 31
第二節 未來展望 32
參考文獻 33

李慶男, (2006). “時間序列分析.” 國立中山大學經濟學研究所.
陳旭昇, (2009). “時間序列分析,總體經濟與財務金融之應用. ”東華書局出版.
張銘仁、姜宗廷,(2009). “購買力平價說的實證檢定：考慮不偏估計與整合偏誤. ”應用經濟論叢.
Andrews, D. W. K. and Peter, P. C. B., (1987)“Best median-unbiased estimation in linear regression with bounded asymmetric loss functions.” Journal of the American Statistical Association, 886-893
Andrews, D. W. K., (1993)“Exactly median-unbiased estimation of first order autoregressive/unit root models.”Econometrica,61(1),139-165.
Andrews, D. W. K., (1993)“Tests for parameter instability and structural change with unknown change point. ” Econometrica,61(4),821-856.
Andrews, D. W. K. and Chen, H. Y., (1994)“Approximately median-unbiased estimation of autoregressive models.”Journal of Business & Economic Statistics,12(2),187-204.
Bartlett, M. S., (1946)“On the theoretical specification and sampling properties of autocorrelated time-series.” Journal of the Royal Statistical Society,27-41.
Bai,J., (1997)“Estimating multiple breaks One at a time.” Econometric Theory,315-352.
Bai,J., and Perron, P., (1998)“Estimating and testing linear models with multiple structural changes.” Econometrica,66(1),47-78.
Bai,J., (1999)“Likelihood ratio tests for multiple structural changes.” Journal of Econometrics, 91(2),299-323.
Bai,J.,and Perron, P., (2003a)“Computation and analysis of multiple structural change model.”Journal of applied econometrics, 18(1),1-22.
Bai,J.,and Perron, P., (2003b)“Critical values for multiple structural change tests.”Econometrics Journal,6(1),72-78.
Chow C. G., (1960)“Tests of equality between sets of coefficients in two linear regressions.” Econometrica：Journal of the Econometric Society,28(3),591-605.
Hurwicz, L., (1950)“Least squares bias in time series”T. Koopmans, ed., Statistical inference in dynamic economic models,365-383.
Kendall, M. G., (1954)“Note on bias in the estimation of autocorrelation.” Biometrika,403-404.
Kim, J. H., (2003)“Forecasting autoregressive time series with bias-corrected parameter estimators.” International Journal of Forecasting,493-502.
Li, H., and Maddala, G. S., (1996)“Bootstrapping time series models” Econometric Reviews,115-158.
Marriott, F. H. C., & Pope, J. A., (1954). “Bias in the estimation of autocorrelations. ” Biometrika, 393–403.
Orcutt, G. H. and Irwin, J. O., (1948)“A study of the autoregressive nature of the time series used for tinbergen's model of the economic system of the united states, 1919-1932.” Journal of the Royal Statistical Society,1-53.
Orcutt, G. H., and H.S. Winokur, (1969)“First order autoregression: inference, estimation, and prediction.” Econometrica：Journal of the Econometric Society,37(1),1-14.
Peter, P. C. B., (1977)“Approximations to some finite sample distributions associated with a first-order stochastic difference equation.” Econometrica,55(290),463-485.
Quandt, R. E., (1958)“The estimation of the parameters of a linear regression system obeying two separate regimes.” Journal of the American Statistical Association,873-880.
Quandt, R. E., (1960)“Tests of the hypothesis that a linear regression system obeys two separate regimes.” Journal of the American Statistical Association,55(290),324-330.
Rudebusch, G. D., (1992)“Trends and random walks in macroeconomic time series: A re-examinatio.” International Economic Review, 661-680.
Rudebusch, G. D., (1993)“The uncertain unit root in real GNP.” American Economic Review, 264-272.
Sawa, T., (1978)“The exact moments of the least squares estimator for the autoregressive model.” Journal of Econometrics,159-172.
Stock, J. H., (1991)“Confidence intervals for the largest autoregressive root in U.S. macroeconomic time series.” Journal of Monetary Economics,435-459.
White, J. S., (1961)“Asymptotic expansions for the mean and variance of the serial correlation coefficien.” Biometrika,85-94.