本論文探討受限因子模型(Tsai and Tsay, 2010)之建模議題。此模型最初提出利用事先決定之群結構來達到降維之成效。首先，透過允許共同因子存在時間之相關性，將靜態受限因子模型延伸至動態受限因子模型。第二，我們採用EM演算法來求得參數之最大概似估計值。此外，我們提出概似比檢定以選取因子個數之最適值。最後，我們利用奇異值分解和集群分析等方法尋找受限矩陣之群結構，並且再次利用概似比檢定選取群個數之最適值。模擬研究顯示所提出之方法在簡化模式結構上頗具成效，此方法已成功地運用在財務相關之實例分析上。

This thesis concerns about some modeling issues for a constrained factor model (Tsai and Tsay, 2010). This model was initially proposed for summarizing high-dimensional variables in a low-dimensional form with a pre-specified constrained structure. First, the static constrained factor model is extended to the dynamic one by incorporating temporal dependence in the common factors. Second, we employ the Expectation and Maximization algorithm to solve the maximum likelihood estimate for a dynamic constrained factor model. In addition, a sequential testing procedure based on likelihood ratio is proposed to determine a suitable dimension for common factors. Finally, we apply the singular value decomposition coupled with several clustering methods to determine the grouping structure among variables in the constraint matrix. Again, the number of clusters is determined via a sequential testing procedure based on likelihood ratio. The performance of the proposed methodology is demonstrated by a simulation study and an application with real data.

Contents
1 Introduction 1
2 The Model 4
2.1 Dynamic Factor Model 4
2.2 Constrained Factor Model 5
3 Estimation 7
3.1 Likelihood Function 7
3.2 Solve MLE via EM Algorithm 8
4 Structure Exploration for Constraint Matrix H 12
4.1 PCA and SVD 12
4.2 Variable Clustering Analysis 15
5 Likelihood Ratio Test for Determining r and m 18
5.1 Test for the Number of Factors 19
5.2 Test for the Number of Clusters 20
6 Simulation Study 23
6.1 Setup 23
6.2 Performance of the MLE via EM 25
6.3 Performance of the LR Test 33
7 Application 40
8 Conclusion 46
References 47

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