How to Make Causal Inferences with Time-Series Cross-Sectional Data

Transcript

1. How to Make Causal Inferences with Time-Series Cross-Sectional Data Matthew Blackwell University of Rochester Adam Glynn Harvard University
2. How to Make Causal Inferences with Time-Series Cross-Sectional Data
3. How to Make Causal Inferences with Time-Series Cross-Sectional Data Very Carefully.
4. How to Make Causal Inferences with Time-Series Cross-Sectional Data Using weights.
5. ۺ܎˞଀ ۢ܎˞଀ ۹܎˞଀ ۢ܎ ۹܎ ۺ܎
6. What is the effect of A on Y? ۺ܎˞଀ ۢ܎˞଀ ۹܎˞଀ ۢ܎ ۹܎ ۺ܎
7. What is the effect of A on Y? contemporaneous ۺ܎˞଀ ۢ܎˞଀ ۹܎˞଀ ۢ܎ ۹܎ ۺ܎
8. What is the effect of A on Y? treatment history ۺ܎˞଀ ۢ܎˞଀ ۹܎˞଀ ۢ܎ ۹܎ ۺ܎
9. Shouldn't we have more notation?
10. ۢ ܎  ۢ଀  Ɛ  ۢ܎ Treatment history Shouldn't we have more notation?
11. ۢ ܎  ۢ଀  Ɛ  ۢ܎ Treatment history Shouldn't we have more notation? Specific instance of a treatment history ۼ ܎  ۼ଀  Ɛ  ۼ܎
12. ۢ ܎  ۢ଀  Ɛ  ۢ܎ Treatment history ۺ܎ ۼ ܎ Potential outcomes Shouldn't we have more notation? Specific instance of a treatment history ۼ ܎  ۼ଀  Ɛ  ۼ܎
13. The effect of history
14. The effect of history ণ ۼ ܎  ۼƓ ܎  ۦ=ۺ܎ ۼ ܎ ˞ ۺ܎ ۼƓ ܎ ? Average Treatment History Effect
15. The effect of history ণ ۼ ܎  ۼƓ ܎  ۦ=ۺ܎ ۼ ܎ ˞ ۺ܎ ۼƓ ܎ ? Average Treatment History Effect ATHE
16. The effect of history ণ ۼ ܎  ۼƓ ܎  ۦ=ۺ܎ ۼ ܎ ˞ ۺ܎ ۼƓ ܎ ? Average Treatment History Effect 1 1 1 1 1 1 1 ATHE
17. The effect of history ণ ۼ ܎  ۼƓ ܎  ۦ=ۺ܎ ۼ ܎ ˞ ۺ܎ ۼƓ ܎ ? Average Treatment History Effect 1 1 1 1 1 1 1 0 0 0 0 0 0 0 vs ATHE
18. The effect of history
19. The effect of history Blip Effect ণ۽ ۼ ܎˞଀  ۦ=ۺ܎ ۼ ܎˞଀   ˞ ۺ܎ ۼ ܎˞଀   ?
20. The effect of history 1 0 0 0 0 0 0 0 vs 0 0 0 0 0 0 Blip Effect ণ۽ ۼ ܎˞଀  ۦ=ۺ܎ ۼ ܎˞଀   ˞ ۺ܎ ۼ ܎˞଀   ?
21. The effect of history 1 0 0 0 0 0 0 0 vs 0 0 0 0 0 0 Blip Effect ণ۽ ۼ ܎˞଀  ۦ=ۺ܎ ۼ ܎˞଀   ˞ ۺ܎ ۼ ܎˞଀   ?
22. The effect of history 1 0 0 0 0 vs 0 0 0 1 1 1 Blip Effect ণ۽ ۼ ܎˞଀  ۦ=ۺ܎ ۼ ܎˞଀   ˞ ۺ܎ ۼ ܎˞଀   ? 1 1 1
23. The effect of history 1 0 vs 1 1 1 Blip Effect ণ۽ ۼ ܎˞଀  ۦ=ۺ܎ ۼ ܎˞଀   ˞ ۺ܎ ۼ ܎˞଀   ? 1 1 1 1 1 1 1 1 1
24. The effect of history
25. The effect of history Contemporaneous Effect of Treatment ণ܎  ۦ=ণ۽ ۼ ܎˞଀ ?
26. The effect of history Contemporaneous Effect of Treatment ণ܎  ۦ=ণ۽ ۼ ܎˞଀ ? CET
27. The effect of history 1 0 vs Contemporaneous Effect of Treatment ণ܎  ۦ=ণ۽ ۼ ܎˞଀ ? CET
28. The effect of history 1 0 vs Contemporaneous Effect of Treatment ণ܎  ۦ=ণ۽ ۼ ܎˞଀ ? CET Marginalize over the past
29. TSCS data under sequential ignorability Treatment is unrelated to the potential outcomes ...conditional on the covariate history. ۺ܎ ۼ ܎ е е ۢ܎ ^۹ ܎  ۺ ܎˞଀  ۢ ܃܎˞଀  ۼ ܎˞଀
30. How conditioning leads you astray
31. How conditioning leads you astray ...for some questions.
32. How conditioning leads you astray ...for some questions. ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
33. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray ...for some questions. ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
34. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray ...for some questions. We “fix” these ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
35. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray ...for some questions. We “fix” these ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
36. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray We “fix” these ...for some questions. ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
37. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray ...for some questions. ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
38. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray এ૾ ...for some questions. ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
39. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray এ૾ ? ? ? ? ? ? ...for some questions. ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
40. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How conditioning leads you astray এ૾ ? ? ? ? ? ? ...for some questions. CET: (1,0) vs (0,0) ATHE: (0,1) vs (0,0) ATHE: (1,1) vs (0,0)  এ૾ ̪ এଁ ̪ এ૾ এଁ ۺ܎  ঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ۺ܎˞଀ ঑ଃ ۢ܎˞଀
41. How weighting can help
42. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How weighting can help
43. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How weighting can help ۸܃܎  ܎ ಿ ܍଀  2T=ۢ܃܍ ^ۢ܎˞଀  ۹܎  ۺ܎˞଀ ? 
44. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How weighting can help ۸܃܎  ܎ ಿ ܍଀  2T=ۢ܃܍ ^ۢ܎˞଀  ۹܎  ۺ܎˞଀ ?  We weight to create balance
45. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How weighting can help We weight to create balance ۸܃܎  ܎ ಿ ܍଀  2T=ۢ܃܍ ^ۢ܎˞଀  ۹܎  ۺ܎˞଀ ? 
46. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How weighting can help ۸܃܎  ܎ ಿ ܍଀  2T=ۢ܃܍ ^ۢ܎˞଀  ۹܎  ۺ܎˞଀ ?  Unconfounded No posttreatment bias
47. How weighting can help
48. How weighting can help ۦ=ۺ܎ ۼ܎  ۼ܎˞଀ ?  ۦ۸ =ۺ܎ ^ۢ܎  ۼ܎  ۢ܎˞଀  ۼ܎˞଀ ?  ঑૿ ঑଀ ۼ܎ ঑ଁ ۼ܎˞଀
49. How weighting can help ۦ=ۺ܎ ۼ܎  ۼ܎˞଀ ?  ۦ۸ =ۺ܎ ^ۢ܎  ۼ܎  ۢ܎˞଀  ۼ܎˞଀ ?  ঑૿ ঑଀ ۼ܎ ঑ଁ ۼ܎˞଀ WLS
50. How weighting can help ۦ=ۺ܎ ۼ܎  ۼ܎˞଀ ?  ۦ۸ =ۺ܎ ^ۢ܎  ۼ܎  ۢ܎˞଀  ۼ܎˞଀ ?  ঑૿ ঑଀ ۼ܎ ঑ଁ ۼ܎˞଀ WLS CET: (1,0) vs (0,0) ATHE: (0,1) vs (0,0) ATHE: (1,1) vs (0,0)  এ૾  ঑ଁ  ঑଀ ঑ଁ
51. The Long Arm of the Democratic Peace?
52. The Long Arm of the Democratic Peace? Democracy in year t War in year t
53. The Long Arm of the Democratic Peace? Democracy in year t War in year t Democratic Peace Literature
54. The Long Arm of the Democratic Peace? Democracy in year t War in year t Democratic Peace Literature History of Democracy
55. The Long Arm of the Democratic Peace? Democracy in year t War in year t Democratic Peace Literature History of Democracy Can we estimate this?
56. %FQFOEFOU WBSJBCMF %JTQVUF #,5 .JTTQFDJĕFE *158 .PEFM $VNVMBUJWF .PEFM .4.    %FNPDSBDZ #MJQ ˞૿ଅ଄଀ˣˣˣ ૿଀ଅ૿ $VNVMBUJWF %FNPDSBDZ ˞૿૿଀૿ ˞૿૿ଃ଄ˣˣˣ ૿૿଀ଁ ૿૿଀ଂ (SPXUI ˞ଂଇଂଆˣˣˣ ˞ଃଂଅ૿ˣˣˣ 0CTFSWBUJPOT ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ /PUF ˣQ ˣˣQ ˣˣˣQ Revisiting Beck, Katz, and Tucker (1998)
57. %FQFOEFOU WBSJBCMF %JTQVUF #,5 .JTTQFDJĕFE *158 .PEFM $VNVMBUJWF .PEFM .4.    %FNPDSBDZ #MJQ ˞૿ଅ଄଀ˣˣˣ ૿଀ଅ૿ $VNVMBUJWF %FNPDSBDZ ˞૿૿଀૿ ˞૿૿ଃ଄ˣˣˣ ૿૿଀ଁ ૿૿଀ଂ (SPXUI ˞ଂଇଂଆˣˣˣ ˞ଃଂଅ૿ˣˣˣ 0CTFSWBUJPOT ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ /PUF ˣQ ˣˣQ ˣˣˣQ Revisiting Beck, Katz, and Tucker (1998)
58. Democracy in year t War in year t Economic Growth in year t History of Democracy Misspecification of an ATHE Time-Varying Confounder
59. %FQFOEFOU WBSJBCMF %JTQVUF #,5 .JTTQFDJĕFE *158 .PEFM $VNVMBUJWF .PEFM .4.    %FNPDSBDZ #MJQ ˞૿ଅ଄଀ˣˣˣ ૿଀ଅ૿ $VNVMBUJWF %FNPDSBDZ ˞૿૿଀૿ ˞૿૿ଃ଄ˣˣˣ ૿૿଀ଁ ૿૿଀ଂ (SPXUI ˞ଂଇଂଆˣˣˣ ˞ଃଂଅ૿ˣˣˣ 0CTFSWBUJPOT ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ /PUF ˣQ ˣˣQ ˣˣˣQ Revisiting Beck, Katz, and Tucker (1998)
60. TSCS data under unmeasured confounding
61. TSCS data under unmeasured confounding ۺ܃܎ ۼ ܎ е е ۢ܃܎ ^۹ ܃܎  ۢ ܃܎˞଀  ۼ ܎˞଀  ۶
62. TSCS data under unmeasured confounding Treatment is unrelated to the potential outcomes ۺ܃܎ ۼ ܎ е е ۢ܃܎ ^۹ ܃܎  ۢ ܃܎˞଀  ۼ ܎˞଀  ۶
63. TSCS data under unmeasured confounding Treatment is unrelated to the potential outcomes ...conditional on the covariate history ۺ܃܎ ۼ ܎ е е ۢ܃܎ ^۹ ܃܎  ۢ ܃܎˞଀  ۼ ܎˞଀  ۶
64. TSCS data under unmeasured confounding Treatment is unrelated to the potential outcomes ...conditional on the covariate history ۺ܃܎ ۼ ܎ е е ۢ܃܎ ^۹ ܃܎  ۢ ܃܎˞଀  ۼ ܎˞଀  ۶ ...and a time-fixed unmeasured confounder.
65. How unit-specific weighting can help
66. How unit-specific weighting can help ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ ۶
67. How unit-specific weighting can help ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ ۶ ۸܃܎  ܎ ಿ ܍଀  2T=ۢ܃܍ ^ۢ܎˞଀  ۹܎  ۺ܎˞଀  ۶?
68. How unit-specific weighting can help ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ ۶ ۸܃܎  ܎ ಿ ܍଀  2T=ۢ܃܍ ^ۢ܎˞଀  ۹܎  ۺ܎˞଀  ۶? Weighting balances the treatment groups.
69. ۺ܎˞଀ ۢ܎˞଀ ۢ܎ ۹܎ ۺ܎ How unit-specific weighting can help ۶ ۸܃܎  ܎ ಿ ܍଀  2T=ۢ܃܍ ^ۢ܎˞଀  ۹܎  ۺ܎˞଀  ۶?
70. A weighting approach to fixed effects
71. A weighting approach to fixed effects 1 Estimate unit-specific probability of treatment over time and construct weights.
72. A weighting approach to fixed effects 1 Estimate unit-specific probability of treatment over time and construct weights. 2 Estimate a pooled outcome model with unit-specific weights
73. k-order sequential ignorability
74. k-order sequential ignorability ۺ܃܎ ۼ ܎ е е ۢ܃܎ ^۹ ܃܎̂܎˞܅  ۢ ܃܎˞଀̂܎˞܅  ۼ ܎˞଀̂܎˞܅  ۶
75. k-order sequential ignorability Only the last k periods matter. ۺ܃܎ ۼ ܎ е е ۢ܃܎ ^۹ ܃܎̂܎˞܅  ۢ ܃܎˞଀̂܎˞܅  ۼ ܎˞଀̂܎˞܅  ۶
76. Blip effect: (1,0) vs (0,0) Time periods Blip effect 10 25 50 75 100 125 0.2 0.3 0.4 0.5 0.6 0.7
77. Blip effect: (1,0) vs (0,0) Time periods Blip effect 10 25 50 75 100 125 0.2 0.3 0.4 0.5 0.6 0.7
78. Pooled Blip effect: (1,0) vs (0,0) Time periods Blip effect 10 25 50 75 100 125 0.2 0.3 0.4 0.5 0.6 0.7 • • • • • •
79. Pooled Outcome fixed effects Blip effect: (1,0) vs (0,0) Time periods Blip effect 10 25 50 75 100 125 0.2 0.3 0.4 0.5 0.6 0.7 • • • • • •
80. Pooled Outcome fixed effects Blip effect: (1,0) vs (0,0) IPTW true weights Time periods Blip effect 10 25 50 75 100 125 0.2 0.3 0.4 0.5 0.6 0.7 • • • • • •
81. Pooled IPTW fixed effects Outcome fixed effects IPTW true weights Blip effect: (1,0) vs (0,0) Time periods Blip effect 10 25 50 75 100 125 0.2 0.3 0.4 0.5 0.6 0.7 • • • • • •
82. Treatment History Effect: (1,1) vs (0,0) Time periods ATHE 10 25 50 75 100 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
83. Treatment History Effect: (1,1) vs (0,0) Time periods ATHE 10 25 50 75 100 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time periods ATHE 10 25 50 75 100 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
84. Pooled Treatment History Effect: (1,1) vs (0,0) Time periods ATHE 10 25 50 75 100 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 • • • • • •
85. Pooled Outcome fixed effects Treatment History Effect: (1,1) vs (0,0) Time periods ATHE 10 25 50 75 100 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 • • • • • •
86. Pooled Outcome fixed effects IPTW true weights Treatment History Effect: (1,1) vs (0,0) Time periods ATHE 10 25 50 75 100 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 • • • • • •
87. Pooled IPTW fixed effects Outcome fixed effects IPTW true weights Treatment History Effect: (1,1) vs (0,0) Time periods ATHE 10 25 50 75 100 125 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 • • • • • •
88. How to make causal inferences with TSCS data
89. How to make causal inferences with TSCS data Very carefully
90. How to make causal inferences with TSCS data Very carefully Even under strong assumptions, conditional estimators cannot recover ATHEs.
91. How to make causal inferences with TSCS data Very carefully Using weights Even under strong assumptions, conditional estimators cannot recover ATHEs.
92. How to make causal inferences with TSCS data Very carefully Using weights Even under strong assumptions, conditional estimators cannot recover ATHEs. A fixed effects weighting approach can recover ATHEs and CETs even with unmeasured confounding.