在局部線性外插 (local linear extrapolation)的問題上 Li and Heckman (2003)考慮利用局部線性外插來預測，但沒有處理資料有自相關的問題。另外，Gijbels, Pope and Wand (1999)提出無母數迴歸跟指數移動平均(exponential moving average)存在某種關係，但沒有考慮外插。在本文中，我們在有自相關的資料上，考慮加上斜率項的局部線性外插並探討雙指數平滑法 (double exponential smoothing)與無母數迴歸之間的等價關係，也在預測方面推導漸近性質。我們將使用局部線性迴歸轉換成雙指數平滑法來進行預測，在一般傳統的指數平滑法進行預測，需要估計平滑因子(smoothing factor)使平均均方誤差 (average square residuals)在預測表現上最小，而平滑因子是用來調解權重的大小，這對應傳統無母數迴歸選取帶寬 (bandwidth)的問題，針對預測來說，平滑因子或帶寬大小會給不同的資料的權重有所不同。在模擬資料中，我們考慮的預測模型增加斜率項，在某些情況下預測得更加準確。我們也對選取平滑因子和帶寬問題在模擬資料中進行討論和比較。我們發現選取平滑因子比選取帶寬的不穩定性增高，因此建議使用雙指數平滑法時，在選取平滑因子的策略上要相對小心謹慎。

In the literature, Li and Heckman (2003) considered forecasting by local linear extrapolation； however, it does not deal with correlated data. In addition, Gijbels, Pope and Wand (GPW) (1999) investigated some relationships between nonparametric regression and exponential moving average, but extrapolation was not considered. This thesis considers local linear extrapolation for correlated data and re-examines equivalence relationships between double exponential smoothing and nonparametric regression. We give explicit expressions that show the equivalence between double exponential smoothing and local linear regression. Further, we derive the asymptotic bias and variance from the perspectives of double exponential smoothing and the asymptotically optimal smoothing is calculated. Stimulation studies are conducted to examine prediction performance of local linear extrapolation and double exponential smoothing and to compare them with the estimator in GPW (1999). We also investigate selecting the smoothing factor and bandwidth by forward cross validation in stimulations. We find that when data has long-term trends, local linear extrapolation tends to have smaller mean square error than the other two approaches； when data exhibits sudden change of trends, the estimator by GPW performs better; and the double exponential smoothing has the best performance in the state-space model.

第一章 前言 1
第二章 文獻回顧 3
2.1 局部線性估計 3
2.2 雙指數平滑法 5
2.3 局部線性迴歸和雙指數平滑法 9
第三章 預測模型之架構 13
3.1 研究目的及推導 13
3.2 預測模型 16
3.3 預測模型之漸近性質 19
第四章 模擬研究 23
4.1 模擬設定 23
4.2 誤差項為獨立 27
4.2.1 1步預測 27
4.2.2 10步預測 28
4.2.3 20步預測 30
4.3 誤差項有自相關 31
4.3.1 1步預測 31
4.3.2 10步預測 32
4.3.3 20步預測 32
4.4 雙指數平滑法和局部線性估計之比較 33
4.4.1 1步預測 33
4.4.2 10步預測 35
4.4.3 20步預測 35
4.5 狀態空間模型預測 36
第五章 結論 39
表格 40
圖 56
附錄 76
參考文獻 91

[1] Fan, J. and Gijbels, I. (1996), "Local polynomial modeling and its applications", Chapman and Hall, London.

[2] Gardner, E. S. (2006), "Exponential smoothing: The state of the art—Part II'', International journal of forecasting, 22(4), 637-666.

[3] Gijbels, I., Pope, A. and Wand, M.P. (1999), "Understanding exponential smoothing via kernel regression", Journal of the Royal Statistical Society, 61, 39-50.

[4] Harvey, A.C. (1989), "Forecasting, structural time series models and the Kalman filter", Cambridge, New York.

[5] Hart, J.D. (1991), "Kernel regression estimation with time series error", Journal of the Royal Statistical Society, 53, 173-187.

[6] Li, X. and Heckman, N.E. (2003), "Local linear extrapolation", Journal of Nonparametric Statistics, 15, 565-578.

[7] Wand, M.P. (1995), "Kernel smoothing", Chapman and Hall, London.