Title Page
ABSTRACT
Contents
Chapter 1. Introduction 14
Chapter 2. Review: low-rank matrix completion 18
2.1. Introduction 18
2.2. Setup and notation 20
2.3. Low-rank matrix approximation 22
2.4. Low-rank matrix completion 23
2.4.1. Rank constraint 23
2.4.2. Nuclear norm penalty 24
2.4.3. Nonconvex penalty 26
2.5. Theoretical studies for low-rank matrix completion 28
2.5.1. Assumption 1: Coherence 28
2.5.2. Assumption 2: Spikiness 29
2.5.3. Review of existing studies 31
2.6. Algorithms for low-rank matrix completion 33
2.6.1. SOFT-IMPUTE algorithm 33
2.6.2. PGH algorithm 34
Chapter 3. Nonconvex penalized matrix completion for causal inference in panel data 39
3.1. Introduction 39
3.2. Setup and notation 41
3.3. Review of existing matrix completion methods 43
3.3.1. Nuclear norm penalized estimator for the potential control outcomes 43
3.3.2. De-biased estimator for the average treatment effect 46
3.4. The proposed estimator 48
3.5. Theoretical results 50
3.5.1. Recovery of the potential control outcomes 50
3.5.2. Estimation of the average treatment effect 57
3.6. Numerical studies 60
3.6.1. Recovery of the potential control outcomes 60
3.6.2. Estimation of the average treatment effect 67
Chapter 4. Conclusions 72
Bibliography 74
Appendices 83
Appendix A 83
A.1. Computation of the PGH algorithm 83
Appendix B 85
B.1. Proof of Theorems 85
B.2. Proof of Lemmas and Proposition 101
Appendix C 111
C.1. Comparison with the Synthetic control method 111
국문초록 119
Table 3.1. The average RMSE and the average Rank of the low-rank matrix in the parentheses for the analysis of the Cigarette... 65
Table 3.2. The average RMSE and the average Rank of the low-rank matrix in the parentheses for the analysis of the GDP data... 66
Table 3.3. The average ANE on the Cigarette sales data with the block and the staggered structures 70
Table 3.4. The average ANE on the GDP data with the block and the staggered structures 71
Figure 2.1. Examples of (Left) an incomplete observed matrix and (Right) a result of MC. 22
Figure 2.2. The graph of some penalty functions: lasso, MCP, and SCAD. 28
Figure 3.1. The simulation results for the block structure with the nuclear norm and the SCAD penalty: the average MSE (Left)... 62
Figure 3.2. The simulation results for the staggered structure with the nuclear norm and the SCAD penalty: the average MSE (Left)... 63
Figure 3.3. Empirical distributions of (θ-θ*)/Vθ for our estimator (Left) and of (θd - θ*)/Vθ,d for the de-biased estimator (Right)....[이미지참조] 69
Figure C.1. Time series of the observed cigarette sales for a specific repetition in the block structures. 113
Figure C.2. Time series of the estimated and observed cigarette sales for all treated states in a specific repetition of the block struc-... 116
Figure C.3. Time series of the observed cigarette sales for a specific repetition in the staggered structures 117
Figure C.4. Time series of the estimated and observed cigarette sales for all treated states in a specific repetition of the staggered... 118