本研究提出一個非平穩半參數空間模型(non-stationary semi-parametric spatial model)來描述所得分配不均於空間上的相依性。該模型為數個基底函數(basis function)及平穩過程(stationary process)的線性組合，由於此模型有大量參數需要估計，我們使用Tibshirani (1996)提出的最小絕對壓縮與篩選運算法(least absolute shrinkage and selection operator, lasso)進行參數估計，該方法可以同時估計參數及作變數選取。接著使用Efron et al. (2001)提出的最小角迴歸法(least angle regression, lars)求lasso估計值，並利用交叉驗證法(cross-validation, cv)選擇lasso模型裡最適合的調整參數(tuning parameter)。本研究將估計結果繪製成空間分佈圖，透過空間分佈圖來描述歐洲地區所得分配不均資料在空間上的分佈情形。根據研究結果顯示，波羅地海三小國：愛沙尼亞(Estonia)、拉脫維亞(Latvia)及立陶宛(Lithuania)的變異程度較大；所得分配不均於愛沙尼亞(Estonia)和瑞典(Sweden)附近有較高的相依性，在德國(Germany)、英國(UK)及西班牙(Spain)附近相依性較低。

In this paper, we propose a non-stationary semi-parametric spatial model to describe the spatial dependence for the income inequality data in Europe. The model is presented by a linear combination of some basis functions and some stationary porcesses. We use least absolute shrinkage and selection operator (lasso, Tibshirani. 2006)  to estimate the parameters. Lasso is very efficient because it can select and estimate parameters simultaneously. The least angle regression (lars, Efron et al. 2001) is used to solve the lasso estimates. The tuning parameter of lasso is selected by cross-validation (cv). In this research, the spatial dependence of the income inequality data in Europe is demonstrated by plots. According to our results, Baltic states: Estonia, Latvia and Lithuania have higher variances in income inequality. The correlation of income inequality in Estonia and Sweden to other countries is higher. Moreover, the correlation of income inequality in Germany, UK and Spain to other countries is lower. The results present a non-stationary structure of the income inequality data.

目錄
第一章導論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
第二章半參數空間模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
第三章估計方法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
第一節最小絕對壓縮與篩選運算法(Least Absolute Shrinkage and
Selection Operator, lasso) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
第二節交叉驗證法(Cross-Validation, CV) . . . . . . . . . . . . . . . . . . . 11
第三節本研究之估計方法(Estimation Method) . . . . . . . . . . . . . . . . 13
第四章實例分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
第一節資料介紹(Data Description) . . . . . . . . . . . . . . . . . . . . . . . 16
第二節迴歸模型(Regression Model) . . . . . . . . . . . . . . . . . . . . . . . 23
第三節殘差檢定(Residual Diagnostic) . . . . . . . . . . . . . . . . . . . . . 25
第四節分析結果(Result of Analysis) . . . . . . . . . . . . . . . . . . . . . . 26
第五章結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

圖目錄
3.1 lasso 及 ridge 差異圖, Tibshirani (1996) 9 
3.2 lars 計算步驟示意圖, Chen et al. (2017) 10 
3.3 5-fold cv 流程圖 11 
3.4 lasso 示意圖 12 
4.1 歐洲 31 個國家地圖 17 
4.2 歐洲 31 個國家中心分布圖 18 
4.3 訓練樣本、測試樣本國家分佈圖 19 
4.4 變數散佈圖矩陣 21 
4.5 歐洲國家吉尼係數樣本平均值 27 
4.6 歐洲國家吉尼係數樣本標準差 27 
4.7 半參數空間模型隨機效應之標準差估計 29 
4.8 愛沙尼亞、德國、西班牙、瑞典及英國吉尼係數相關係數圖 31


表目錄
4.1 歐洲 31 個國家列表 17 
4.2 測試樣本國家 19 
4.3 變數相關係數矩陣 22 
4.4 模型一與模型二之參數估計值及修正 R 平方表 23 
4.5 模型一與模型二之 VIF 共線性診斷 24 
4.6 殘差檢定 25 
4.7 半參數模型之參數估計 28 
4.8 迴歸模型與半參數空間模型之 MSE 32

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