配對交易是一種市場中立高槓桿的投資策略。如何選擇配對的股票，來獲得利潤是一個重要的議題。本研究利用 S&P 500 的股票進行實證分析，探討影響配對交易利潤的股票變數。希望藉由這些變數的資訊，選取可獲利的配對交易組合。本文所考慮的股票變數包含共整合係數、相關係數、每股盈餘、貝它係數等。

我們利用主成分分析找出能夠解釋資料主要變異的變數。此外我們也利用切片逆回歸法將利潤視為應變數選取主要的解釋變數。最後我們利用廣義相關圖，對於利潤以及解釋變數進行視覺化的集群分析。

Pairs trading is a market neutral high leverage trading strategy. How to choose profitable trading pairs is an important issue. In this work, we perform an empirical study for S&P 500 stocks and investigate the effects of selected variables on trading profits. The objective is to use the information of the variables to help select profitable trading pairs. The considered variables include : cointegration coefficient, correlation, earnings per share (EPS) and β-coefficient etc.

We use principal component analysis to find the variables which explain the major variability of the data.
In addition, we treat profit as the dependent variable and select the major explanatory variables by sliced inverse regression method. Finally, we use generalized association plots for visualization clustering analysis of profit and explanatory variables.

論文審定書 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i
誌謝 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ii
摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .iv
1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
2 Pairs Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
2.1 Cointegration Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Trading Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .5
3.1 Principal Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Number of Principal Components . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Sliced Inverse Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 The SIR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Generalized Association Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 Calculation of proximity matrices . . . . . . . . . . . . . . . . . . . . . . . 12
5.3 Colors Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.4 Tree Seriation and and Flipping Mechanism . . . . . . . . . . . . . . . . . 13
6 Empirical Clustering Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15
6.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.1.1 Strategies 1, 2 & 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6.2 Sliced Inverse Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2.1 Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.2.2 Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2.3 Strategy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3 Generalized Association Plots . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.3.1 Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.3.2 Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.3.3 Strategy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.Chen, C. H. (2002). Generalized association plots: Information visualization via iteratively generated correlation matrices. Statistica Sinica 12, 7-29.
2. Chen, N. H. (2010). Time Series: Applications to Finance with R and S-Plus (2nd ed.). John Wiley & Sons, New Jersey.
3. Duan, N. and Li, K.-C. (1991). Slicing regression: A link-free regression method. Annals of Statistics. 19(2): 505-530.
4. Engle, R. and Granger, C. (1987). Cointegration and error correction: Representation, estimation, and testing. Econometrica, 55, pp. 251-276.
5.Huber, P. (1985). Projection pursuit. Annals of Statistic}. 13(2): 435-475.
6. Johnson, R. D. and Wichern, D. W (2007). Applied Multivariate Statistical Analysis. 6th ed. Pearson, Essex.
7. Wu, H. M. (2006). Introduction to generalized association plots (PCA) for dimension-free data visualization, Lecture notes, Department of Mathematics, Tamkang University,. Taipei, Taiwan.
8. Wu, H. M., Tien, Y. J. and Chen, C. H. (2006). GAP: A graphical environment for matrix visualization and information mining. User Manual. Elsevier.
9. Wu, P. S. (2015). Trading Strategy Based on Cointegration Pairs. Master thesis, Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung.
10. 陳君厚 (2014). 互動式統計繪圖與矩陣視覺化。 中央研究院暑期研習營講義，台北。
11. 吳漢銘 (2014). R 統計圖形及資料視覺化。 中央研究院暑期研習營講義，台北。