Matthew Blackwell
April 13, 2013
170

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

April 13, 2013

## Transcript

1. How to Make Causal Inferences with
Time-Series Cross-Sectional Data
Matthew Blackwell
University of Rochester
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

ۼ
܎
ۼƓ
܎
ۦ=ۺ܎

ۼ
܎
˞ ۺ܎

ۼƓ
܎
?
Average Treatment
History Effect

15. The effect of history

ۼ
܎
ۼƓ
܎
ۦ=ۺ܎

ۼ
܎
˞ ۺ܎

ۼƓ
܎
?
Average Treatment
History Effect
ATHE

16. The effect of history

ۼ
܎
ۼƓ
܎
ۦ=ۺ܎

ۼ
܎
˞ ۺ܎

ۼƓ
܎
?
Average Treatment
History Effect
1 1 1 1 1 1 1
ATHE

17. The effect of history

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 ۢ܎

܎
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܎
...for some questions.
ۺ܎
঑૿
঑଀
ۢ܎
঑ଁ
۹܎
঑ଂ
ۺ܎˞଀
঑ଃ
ۢ܎˞଀

34. ۺ܎˞଀
ۢ܎˞଀
ۢ܎
۹܎
ۺ܎
...for some questions.
We “ﬁx” these
ۺ܎
঑૿
঑଀
ۢ܎
঑ଁ
۹܎
঑ଂ
ۺ܎˞଀
঑ଃ
ۢ܎˞଀

35. ۺ܎˞଀
ۢ܎˞଀
ۢ܎
۹܎
ۺ܎
...for some questions.
We “ﬁx” these
ۺ܎
঑૿
঑଀
ۢ܎
঑ଁ
۹܎
঑ଂ
ۺ܎˞଀
঑ଃ
ۢ܎˞଀

36. ۺ܎˞଀
ۢ܎˞଀
ۢ܎
۹܎
ۺ܎
We “ﬁx” these
...for some questions.
ۺ܎
঑૿
঑଀
ۢ܎
঑ଁ
۹܎
঑ଂ
ۺ܎˞଀
঑ଃ
ۢ܎˞଀

37. ۺ܎˞଀
ۢ܎˞଀
ۢ܎
۹܎
ۺ܎
...for some questions.
ۺ܎
঑૿
঑଀
ۢ܎
঑ଁ
۹܎
঑ଂ
ۺ܎˞଀
঑ଃ
ۢ܎˞଀

38. ۺ܎˞଀
ۢ܎˞଀
ۢ܎
۹܎
ۺ܎
এ૾
...for some questions.
ۺ܎
঑૿
঑଀
ۢ܎
঑ଁ
۹܎
঑ଂ
ۺ܎˞଀
঑ଃ
ۢ܎˞଀

39. ۺ܎˞଀
ۢ܎˞଀
ۢ܎
۹܎
ۺ܎
এ૾
? ?
?
?
?
?
...for some questions.
ۺ܎
঑૿
঑଀
ۢ܎
঑ଁ
۹܎
঑ଂ
ۺ܎˞଀
঑ଃ
ۢ܎˞଀

40. ۺ܎˞଀
ۢ܎˞଀
ۢ܎
۹܎
ۺ܎
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 ۢ܃܎

܃܎
ۢ
܃܎˞଀
ۼ
܎˞଀
۶

62. TSCS data under unmeasured confounding
Treatment is unrelated
to the potential outcomes
ۺ܃܎

ۼ
܎
е
е ۢ܃܎

܃܎
ۢ
܃܎˞଀
ۼ
܎˞଀
۶

63. TSCS data under unmeasured confounding
Treatment is unrelated
to the potential outcomes
...conditional on the
covariate history
ۺ܃܎

ۼ
܎
е
е ۢ܃܎

܃܎
ۢ
܃܎˞଀
ۼ
܎˞଀
۶

64. TSCS data under unmeasured confounding
Treatment is unrelated
to the potential outcomes
...conditional on the
covariate history
ۺ܃܎

ۼ
܎
е
е ۢ܃܎

܃܎
ۢ
܃܎˞଀
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 ۢ܃܎

܃܎̂܎˞܅
ۢ
܃܎˞଀̂܎˞܅
ۼ
܎˞଀̂܎˞܅
۶

75. k-order sequential ignorability
Only the last k
periods matter.
ۺ܃܎

ۼ
܎
е
е ۢ܃܎

܃܎̂܎˞܅
ۢ
܃܎˞଀̂܎˞܅
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.