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How to Make Causal Inferences with Time-Series Cross-Sectional Data
Matthew Blackwell
April 13, 2013
Science
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How to Make Causal Inferences with Time-Series Cross-Sectional Data
Matthew Blackwell
April 13, 2013
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Transcript
How to Make Causal Inferences with Time-Series Cross-Sectional Data Matthew
Blackwell University of Rochester Adam Glynn Harvard University
How to Make Causal Inferences with Time-Series Cross-Sectional Data
How to Make Causal Inferences with Time-Series Cross-Sectional Data Very
Carefully.
How to Make Causal Inferences with Time-Series Cross-Sectional Data Using
weights.
ۺ˞ ۢ˞ ۹˞ ۢ ۹ ۺ
What is the effect of A on Y? ۺ˞ ۢ˞
۹˞ ۢ ۹ ۺ
What is the effect of A on Y? contemporaneous ۺ˞
ۢ˞ ۹˞ ۢ ۹ ۺ
What is the effect of A on Y? treatment history
ۺ˞ ۢ˞ ۹˞ ۢ ۹ ۺ
Shouldn't we have more notation?
ۢ ۢ Ɛ ۢ Treatment history
Shouldn't we have more notation?
ۢ ۢ Ɛ ۢ Treatment history
Shouldn't we have more notation? Speciﬁc instance of a treatment history ۼ ۼ Ɛ ۼ
ۢ ۢ Ɛ ۢ Treatment history
ۺ ۼ Potential outcomes Shouldn't we have more notation? Speciﬁc instance of a treatment history ۼ ۼ Ɛ ۼ
The effect of history
The effect of history ণ ۼ ۼƓ
ۦ=ۺ ۼ ˞ ۺ ۼƓ ? Average Treatment History Effect
The effect of history ণ ۼ ۼƓ
ۦ=ۺ ۼ ˞ ۺ ۼƓ ? Average Treatment History Effect ATHE
The effect of history ণ ۼ ۼƓ
ۦ=ۺ ۼ ˞ ۺ ۼƓ ? Average Treatment History Effect 1 1 1 1 1 1 1 ATHE
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
The effect of history
The effect of history Blip Effect ণ۽ ۼ ˞
ۦ=ۺ ۼ ˞ ˞ ۺ ۼ ˞ ?
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 ˞ ?
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 ˞ ?
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
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
The effect of history
The effect of history Contemporaneous Effect of Treatment ণ
ۦ=ণ۽ ۼ ˞ ?
The effect of history Contemporaneous Effect of Treatment ণ
ۦ=ণ۽ ۼ ˞ ? CET
The effect of history 1 0 vs Contemporaneous Effect of
Treatment ণ ۦ=ণ۽ ۼ ˞ ? CET
The effect of history 1 0 vs Contemporaneous Effect of
Treatment ণ ۦ=ণ۽ ۼ ˞ ? CET Marginalize over the past
TSCS data under sequential ignorability Treatment is unrelated to the
potential outcomes ...conditional on the covariate history. ۺ ۼ е е ۢ ^۹ ۺ ˞ ۢ ܃˞ ۼ ˞
How conditioning leads you astray
How conditioning leads you astray ...for some questions.
How conditioning leads you astray ...for some questions. ۺ
૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
ۺ˞ ۢ˞ ۢ ۹ ۺ How conditioning leads you astray
...for some questions. ۺ ૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
ۺ˞ ۢ˞ ۢ ۹ ۺ How conditioning leads you astray
...for some questions. We “ﬁx” these ۺ ૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
ۺ˞ ۢ˞ ۢ ۹ ۺ How conditioning leads you astray
...for some questions. We “ﬁx” these ۺ ૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
ۺ˞ ۢ˞ ۢ ۹ ۺ How conditioning leads you astray
We “ﬁx” these ...for some questions. ۺ ૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
ۺ˞ ۢ˞ ۢ ۹ ۺ How conditioning leads you astray
...for some questions. ۺ ૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
ۺ˞ ۢ˞ ۢ ۹ ۺ How conditioning leads you astray
এ૾ ...for some questions. ۺ ૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
ۺ˞ ۢ˞ ۢ ۹ ۺ How conditioning leads you astray
এ૾ ? ? ? ? ? ? ...for some questions. ۺ ૿ ۢ ଁ ۹ ଂ ۺ˞ ଃ ۢ˞
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˞ ଃ ۢ˞
How weighting can help
ۺ˞ ۢ˞ ۢ ۹ ۺ How weighting can help
ۺ˞ ۢ˞ ۢ ۹ ۺ How weighting can help ۸܃
ಿ ܍ 2T=ۢ܃܍ ^ۢ˞ ۹ ۺ˞ ?
ۺ˞ ۢ˞ ۢ ۹ ۺ How weighting can help ۸܃
ಿ ܍ 2T=ۢ܃܍ ^ۢ˞ ۹ ۺ˞ ? We weight to create balance
ۺ˞ ۢ˞ ۢ ۹ ۺ How weighting can help We
weight to create balance ۸܃ ಿ ܍ 2T=ۢ܃܍ ^ۢ˞ ۹ ۺ˞ ?
ۺ˞ ۢ˞ ۢ ۹ ۺ How weighting can help ۸܃
ಿ ܍ 2T=ۢ܃܍ ^ۢ˞ ۹ ۺ˞ ? Unconfounded No posttreatment bias
How weighting can help
How weighting can help ۦ=ۺ ۼ ۼ˞ ?
ۦ۸ =ۺ ^ۢ ۼ ۢ˞ ۼ˞ ? ૿ ۼ ଁ ۼ˞
How weighting can help ۦ=ۺ ۼ ۼ˞ ?
ۦ۸ =ۺ ^ۢ ۼ ۢ˞ ۼ˞ ? ૿ ۼ ଁ ۼ˞ WLS
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 ଁ ଁ
The Long Arm of the Democratic Peace?
The Long Arm of the Democratic Peace? Democracy in year
t War in year t
The Long Arm of the Democratic Peace? Democracy in year
t War in year t Democratic Peace Literature
The Long Arm of the Democratic Peace? Democracy in year
t War in year t Democratic Peace Literature History of Democracy
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?
%FQFOEFOU WBSJBCMF %JTQVUF #,5 .JTTQFDJĕFE *158 .PEFM $VNVMBUJWF .PEFM .4.
%FNPDSBDZ #MJQ ˞૿ଅˣˣˣ ૿ଅ૿ $VNVMBUJWF %FNPDSBDZ ˞૿૿૿ ˞૿૿ଃˣˣˣ ૿૿ଁ ૿૿ଂ (SPXUI ˞ଂଇଂଆˣˣˣ ˞ଃଂଅ૿ˣˣˣ 0CTFSWBUJPOT ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ /PUF ˣQ ˣˣQ ˣˣˣQ Revisiting Beck, Katz, and Tucker (1998)
%FQFOEFOU WBSJBCMF %JTQVUF #,5 .JTTQFDJĕFE *158 .PEFM $VNVMBUJWF .PEFM .4.
%FNPDSBDZ #MJQ ˞૿ଅˣˣˣ ૿ଅ૿ $VNVMBUJWF %FNPDSBDZ ˞૿૿૿ ˞૿૿ଃˣˣˣ ૿૿ଁ ૿૿ଂ (SPXUI ˞ଂଇଂଆˣˣˣ ˞ଃଂଅ૿ˣˣˣ 0CTFSWBUJPOT ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ /PUF ˣQ ˣˣQ ˣˣˣQ Revisiting Beck, Katz, and Tucker (1998)
Democracy in year t War in year t Economic Growth
in year t History of Democracy Misspecification of an ATHE Time-Varying Confounder
%FQFOEFOU WBSJBCMF %JTQVUF #,5 .JTTQFDJĕFE *158 .PEFM $VNVMBUJWF .PEFM .4.
%FNPDSBDZ #MJQ ˞૿ଅˣˣˣ ૿ଅ૿ $VNVMBUJWF %FNPDSBDZ ˞૿૿૿ ˞૿૿ଃˣˣˣ ૿૿ଁ ૿૿ଂ (SPXUI ˞ଂଇଂଆˣˣˣ ˞ଃଂଅ૿ˣˣˣ 0CTFSWBUJPOT ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ ଁ૿ ଃଃଇ /PUF ˣQ ˣˣQ ˣˣˣQ Revisiting Beck, Katz, and Tucker (1998)
TSCS data under unmeasured confounding
TSCS data under unmeasured confounding ۺ܃ ۼ е е
ۢ܃ ^۹ ܃ ۢ ܃˞ ۼ ˞ ۶
TSCS data under unmeasured confounding Treatment is unrelated to the
potential outcomes ۺ܃ ۼ е е ۢ܃ ^۹ ܃ ۢ ܃˞ ۼ ˞ ۶
TSCS data under unmeasured confounding Treatment is unrelated to the
potential outcomes ...conditional on the covariate history ۺ܃ ۼ е е ۢ܃ ^۹ ܃ ۢ ܃˞ ۼ ˞ ۶
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.
How unit-specific weighting can help
How unit-specific weighting can help ۺ˞ ۢ˞ ۢ ۹ ۺ
۶
How unit-specific weighting can help ۺ˞ ۢ˞ ۢ ۹ ۺ
۶ ۸܃ ಿ ܍ 2T=ۢ܃܍ ^ۢ˞ ۹ ۺ˞ ۶?
How unit-specific weighting can help ۺ˞ ۢ˞ ۢ ۹ ۺ
۶ ۸܃ ಿ ܍ 2T=ۢ܃܍ ^ۢ˞ ۹ ۺ˞ ۶? Weighting balances the treatment groups.
ۺ˞ ۢ˞ ۢ ۹ ۺ How unit-specific weighting can help
۶ ۸܃ ಿ ܍ 2T=ۢ܃܍ ^ۢ˞ ۹ ۺ˞ ۶?
A weighting approach to fixed effects
A weighting approach to fixed effects 1 Estimate unit-speciﬁc probability
of treatment over time and construct weights.
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
k-order sequential ignorability
k-order sequential ignorability ۺ܃ ۼ е е ۢ܃ ^۹
܃̂˞܅ ۢ ܃˞̂˞܅ ۼ ˞̂˞܅ ۶
k-order sequential ignorability Only the last k periods matter. ۺ܃
ۼ е е ۢ܃ ^۹ ܃̂˞܅ ۢ ܃˞̂˞܅ ۼ ˞̂˞܅ ۶
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
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
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 • • • • • •
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 • • • • • •
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 • • • • • •
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 • • • • • •
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
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
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 • • • • • •
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 • • • • • •
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 • • • • • •
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 • • • • • •
How to make causal inferences with TSCS data
How to make causal inferences with TSCS data Very carefully
How to make causal inferences with TSCS data Very carefully
Even under strong assumptions, conditional estimators cannot recover ATHEs.
How to make causal inferences with TSCS data Very carefully
Using weights Even under strong assumptions, conditional estimators cannot recover ATHEs.
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.