本文提出在一階移動平均模型﹙moving average model of order one, MA(1)﹚中，四種拔靴法﹙bootstrap﹚單根﹙unit root﹚檢定。這四個檢定分別如下：以最小平方法﹙假設誤差項服從常態分配﹚來估計參數且沒有虛無假設限制﹙先估計模型參數再建立殘差的經驗函數﹚的拔靴單根檢定，以最小平方法來估計參數且有虛無假設限制﹙令模型參數為1來建立殘差的經驗函數﹚的拔靴單根檢定，以最小絕對值﹙least absolute deviation (LAD)﹚法﹙假設誤差項服從雙重指數﹙double exponential﹚分配﹚來估計參數且沒有虛無假設限制的拔靴單根檢定，與以最小絕對值法來估計參數且有虛無假設限制的拔靴單根檢定。此外，藉由模擬，我們比較在不同誤差項的分配下，這些檢定的顯著水準與檢定力。模擬結果顯示，若以最小平方法來估計參數的拔靴單根檢定，不論是否有虛無假設的限制，其檢定力並無差別；若以最小絕對值法來估計參數的拔靴單根檢定，無虛無假設限制的檢定檢定力較高。而在不同分配的誤差項中，若誤差項為厚尾﹙heavy tail﹚分怖，則以最小絕對值法來估計參數的拔靴單根檢定的檢定力高於以最小平方法來估計參數的拔靴單根檢定。

Four new bootstrap tests for testing unit root in the moving average model of order one (MA(1)) are proposed in this thesis. The four bootstrap tests are the test without restriction under Gaussian (i.e., the test which is based on the least square estimator and is not imposed the null hypothesis), the test with restriction under Gaussian (i.e., which is the test based on the least square estimator and is imposed the null hypothesis), the test without restriction under Laplace, (i.e., the test which is based on the least absolute deviation (LAD) estimator and is not imposed the null hypothesis), and last, the test with restriction under Laplace (i.e., the test which is based on the LAD estimator and is imposed the null hypothesis). Besides, we compare the performance among these tests in terms of the power and size under four different error distributions by simulation. It shows that under Gaussian estimation, the performance is no difference whether we impose the null hypothesis or not. However, under Laplace estimation, the test without restriction is more powerful than the test with restriction. It also shows that when the distribution of the error term has a heavy tail, the tests under Laplace outperform the tests under Gaussian.

Breidt, F.J., Davis, R.A., Hsu, N.J., and Rosenblatt, M. (2007). Pile-up probabilities for the Laplace likelihood estimator of a non-invertible first order moving average. IMS Lecture Notes-Monograph Series, forthcoming.
Breidt, F.J., and Hsu, N.J. (2005). Best mean square prediction for moving average. Statistica Sinica, 15, 427-446.
Chen, M., Davis, R.A., and Dunsmuir, W.T.M. (1995). Inference for MA(1) processes with a root on or near the unit circle. Probability and Mathematical Statistics 15, 227-242.
Dickey, D.A., and Fuller, W.A. (1979). Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427-431.
Herce, M.A. (1996). Asymptotic theory of LAD estimation in a unit root process with finite variance errors. Econometric Theory 12, 129-153.
Knight, K. (1989). Limit theory for autoregressive parameter estimates in an infinite variance random walk. Canadian Journal of Statistics 17, 261-278.
Knight, K. (1991). Limit theory for M-estimates in an integrated infinite variance process. Econometric Theory 7,200-212.
Moreno, M., and Romo, J. (2000). Bootstrap tests for unit roots based on LAD estimation. Journal of Statistical Planning and Inference 83, 347-367.
Paparoditis, E., and Politis, D.N. (2005). Bootstrapping unit root tests for autoregressive time series. Journal of the American Statistical Association 100, 545-553.
Pollard, D. (1991). Asymptotic for least absolute deviation regression estimators. Econometric Theory 7, 186-199.