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
‫଀˞܎ۢ‬
‫଀˞܎ۺ‬
‫଀˞܎‪۹‬‬
What is the effect of A on Y?
ۢ܎˞଀
‫଀˞܎ۺ‬
۹܎˞଀
ۢ܎
‫܎ۺ‬
۹܎
What is the effect of A on Y?
contemporaneous
ۢ܎˞଀
‫଀˞܎ۺ‬
۹܎˞଀
ۢ܎
‫܎ۺ‬
۹܎
What is the effect of A on Y?
treatment history
ۢ܎˞଀
‫଀˞܎ۺ‬
۹܎˞଀
ۢ܎
‫܎ۺ‬
۹܎
Shouldn't we have more notation?
Shouldn't we have more notation?
ۢ ۢ଀ Ɛ ۢ܎ ܎
Treatment history
Shouldn't we have more notation?
ۢ ۢ଀ Ɛ ۢ܎ ܎
Treatment history
‫ ଀ۼ
ۼ‬Ɛ ‫ ܎ۼ‬
܎
Specific instance of a treatment history
Shouldn't we have more notation?
ۢ ۢ଀ Ɛ ۢ܎ ܎
Treatment history
‫ ଀ۼ
ۼ‬Ɛ ‫ ܎ۼ‬
܎
Specific instance of a treatment history
‫ ۼ
܎ۺ‬
܎
Potential outcomes
The effect of history
The effect of history
Average Treatment
History Effect
Ɠ
ণ
‫ ۼ ۼ‬
܎ ܎
‫˞ ۼ
܎ۺ=ۦ‬
܎
Ɠ
‫? ۼ
܎ۺ‬
܎
The effect of history
Average Treatment
History Effect
ATHE
Ɠ
ণ
‫ ۼ ۼ‬
܎ ܎
‫˞ ۼ
܎ۺ=ۦ‬
܎
Ɠ
‫? ۼ
܎ۺ‬
܎
The effect of history
Average Treatment
History Effect
Ɠ
ণ
‫ ۼ ۼ‬
܎ ܎
‫˞ ۼ
܎ۺ=ۦ‬
܎
Ɠ
‫? ۼ
܎ۺ‬
܎
ATHE
1
1
1
1
1
1
1
The effect of history
Average Treatment
History Effect
Ɠ
ণ
‫ ۼ ۼ‬
܎ ܎
‫˞ ۼ
܎ۺ=ۦ‬
܎
Ɠ
‫? ۼ
܎ۺ‬
܎
ATHE
1
1
1
1
1
1
1
0
0
0
vs
0
0
0
0
The effect of history
The effect of history
Blip Effect
ণ‫ۼ
۽‬
܎˞଀
‫ۼ
܎ۺ=ۦ‬
܎˞଀
˞ ‫ۼ
܎ۺ‬
܎˞଀
?
The effect of history
ণ‫ۼ
۽‬
Blip Effect
0
0
܎˞଀
0
0
‫ۼ
܎ۺ=ۦ‬
܎˞଀
0
˞ ‫ۼ
܎ۺ‬
܎˞଀
0
?
1
vs
0
0
0
0
0
0
0
The effect of history
ণ‫ۼ
۽‬
Blip Effect
0
0
܎˞଀
0
0
‫ۼ
܎ۺ=ۦ‬
܎˞଀
0
˞ ‫ۼ
܎ۺ‬
܎˞଀
0
?
1
vs
0
0
0
0
0
0
0
The effect of history
ণ‫ۼ
۽‬
Blip Effect
0
0
܎˞଀
0
1
‫ۼ
܎ۺ=ۦ‬
܎˞଀
1
˞ ‫ۼ
܎ۺ‬
܎˞଀
1
?
1
vs
0
0
0
1
1
1
0
The effect of history
ণ‫ۼ
۽‬
Blip Effect
1
1
܎˞଀
1
1
‫ۼ
܎ۺ=ۦ‬
܎˞଀
1
˞ ‫ۼ
܎ۺ‬
܎˞଀
1
?
1
vs
1
1
1
1
1
1
0
The effect of history
The effect of history
Contemporaneous
Effect of Treatment
ণ܎ ‫=ۦ‬ণ‫ۼ
۽‬
܎˞଀
?
The effect of history
Contemporaneous
Effect of Treatment
CET
ণ܎ ‫=ۦ‬ণ‫ۼ
۽‬
܎˞଀
?
The effect of history
Contemporaneous
Effect of Treatment
ণ܎ ‫=ۦ‬ণ‫ۼ
۽‬
܎˞଀
?
CET
1
vs
0
The effect of history
Contemporaneous
Effect of Treatment
ণ܎ ‫=ۦ‬ণ‫ۼ
۽‬
܎˞଀
?
CET
1
Marginalize over the past
vs
0
TSCS data under sequential ignorability
‫ ۼ
܎ۺ‬ее ۢ܎ ^۹ ‫ۺ‬
܎
Treatment is unrelated
to the potential outcomes
܎
܎˞଀
ۢ
‫଀˞܎܃‬
‫ۼ‬
܎˞଀
...conditional on the
covariate history.
How conditioning leads you astray
How conditioning leads
you
astray
...for some questions.
How conditioning leads
you
astray
...for some questions.
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
We “fix” these
‫܎ۺ‬
۹܎
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
We “fix” these
‫܎ۺ‬
۹܎
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
We “fix” these
‫܎ۺ‬
۹܎
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
এ૾
‫܎ۺ‬
۹܎
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
?
? ? ??
ۢ܎˞଀
?
‫଀˞܎ۺ‬
ۢ܎
এ૾
‫܎ۺ‬
۹܎
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How conditioning leads
you
astray
...for some questions.
?
? ? ??
ۢ܎˞଀
?
‫଀˞܎ۺ‬
CET: (1,0) vs (0,0) এ૾
ATHE: (0,1) vs (0,0) ̪ এଁ
ۢ܎
এ૾
‫܎ۺ‬
۹܎
ATHE: (1,1) vs (0,0) ̪ এ૾ এଁ
‫ ܎ۺ‬঑૿ ঑଀ ۢ܎ ঑ଁ ۹܎ ঑ଂ ‫ ଀˞܎ۺ‬঑ଃ ۢ܎˞଀
How weighting can help
How weighting can help
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
How weighting can help
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
܎
۸‫ ܎܃‬ಿ
‫଀܍‬
2T=ۢ‫ ଀˞܎ۢ^ ܍܃‬۹܎ ‫? ଀˞܎ۺ‬
How weighting can help
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
We weight to create balance
܎
۸‫ ܎܃‬ಿ
‫଀܍‬
2T=ۢ‫ ଀˞܎ۢ^ ܍܃‬۹܎ ‫? ଀˞܎ۺ‬
How weighting can help
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
We weight to create balance
܎
۸‫ ܎܃‬ಿ
‫଀܍‬
2T=ۢ‫ ଀˞܎ۢ^ ܍܃‬۹܎ ‫? ଀˞܎ۺ‬
How weighting can help
Unconfounded
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
No posttreatment bias
܎
۸‫ ܎܃‬ಿ
‫଀܍‬
‫܎ۺ‬
۹܎
2T=ۢ‫ ଀˞܎ۢ^ ܍܃‬۹܎ ‫? ଀˞܎ۺ‬
How weighting can help
How weighting can help
‫ۦ ? ଀˞܎ۼ ܎ۼ
܎ۺ=ۦ‬۸ =‫? ଀˞܎ۼ ଀˞܎ۢ ܎ۼ ܎ۢ^ ܎ۺ‬
঑૿ ঑଀ ‫ ܎ۼ‬঑ଁ ‫଀˞܎ۼ‬
How weighting can help
WLS
‫ۦ ? ଀˞܎ۼ ܎ۼ
܎ۺ=ۦ‬۸ =‫? ଀˞܎ۼ ଀˞܎ۢ ܎ۼ ܎ۢ^ ܎ۺ‬
঑૿ ঑଀ ‫ ܎ۼ‬঑ଁ ‫଀˞܎ۼ‬
How weighting can help
WLS
‫ۦ ? ଀˞܎ۼ ܎ۼ
܎ۺ=ۦ‬۸ =‫? ଀˞܎ۼ ଀˞܎ۢ ܎ۼ ܎ۢ^ ܎ۺ‬
঑૿ ঑଀ ‫ ܎ۼ‬঑ଁ ‫଀˞܎ۼ‬
CET: (1,0) vs (0,0) এ૾
ATHE: (0,1) vs (0,0) ঑ଁ
ATHE: (1,1) vs (0,0) ঑଀ ঑ଁ
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
Democratic Peace
Literature
War in
year t
The Long Arm of the Democratic Peace?
Democracy
in year t
Democratic Peace
Literature
History of
Democracy
War in
year t
The Long Arm of the Democratic Peace?
Can we
estimate this?
History of
Democracy
Democracy
in year t
Democratic Peace
Literature
War in
year t
Revisiting Beck, Katz, and Tucker (1998)
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%FNPDSBDZ #MJQ
$VNVMBUJWF %FNPDSBDZ
(SPXUI
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$VNVMBUJWF .PEFM
*158
.4.
˞૿૿଀૿
૿૿଀ଁ
˞૿૿ଃ଄ˣˣˣ
૿૿଀ଂ
ଁ૿ ଃଃଇ
ଁ૿ ଃଃଇ
˞૿ଅ଄଀ˣˣˣ
૿଀ଅ૿
˞ଂଇଂଆˣˣˣ
ଁ૿ ଃଃଇ
˞ଃଂଅ૿ˣˣˣ
ˣ Q ˣˣ Q ˣˣˣ Q
Revisiting Beck, Katz, and Tucker (1998)
%FQFOEFOU WBSJBCMF %JTQVUF
%FNPDSBDZ #MJQ
$VNVMBUJWF %FNPDSBDZ
(SPXUI
0CTFSWBUJPOT
/PUF
#,5
.PEFM
.JTTQFDJĕFE
$VNVMBUJWF .PEFM
*158
.4.
˞૿૿଀૿
૿૿଀ଁ
˞૿૿ଃ଄ˣˣˣ
૿૿଀ଂ
ଁ૿ ଃଃଇ
ଁ૿ ଃଃଇ
˞૿ଅ଄଀ˣˣˣ
૿଀ଅ૿
˞ଂଇଂଆˣˣˣ
ଁ૿ ଃଃଇ
˞ଃଂଅ૿ˣˣˣ
ˣ Q ˣˣ Q ˣˣˣ Q
Misspecification of an ATHE
Economic
Growth in
year t
Democracy
in year t
History of
Democracy
War in
year t
Time-Varying
Confounder
Revisiting Beck, Katz, and Tucker (1998)
%FQFOEFOU WBSJBCMF %JTQVUF
%FNPDSBDZ #MJQ
$VNVMBUJWF %FNPDSBDZ
(SPXUI
0CTFSWBUJPOT
/PUF
#,5
.PEFM
.JTTQFDJĕFE
$VNVMBUJWF .PEFM
*158
.4.
˞૿૿଀૿
૿૿଀ଁ
˞૿૿ଃ଄ˣˣˣ
૿૿଀ଂ
ଁ૿ ଃଃଇ
ଁ૿ ଃଃଇ
˞૿ଅ଄଀ˣˣˣ
૿଀ଅ૿
˞ଂଇଂଆˣˣˣ
ଁ૿ ଃଃଇ
˞ଃଂଅ૿ˣˣˣ
ˣ Q ˣˣ Q ˣˣˣ Q
TSCS data under unmeasured confounding
TSCS data under unmeasured confounding
‫ ۼ
܎܃ۺ‬ее ۢ‫^ ܎܃‬۹ ۢ
܎
‫܎܃‬
‫଀˞܎܃‬
‫ۼ‬
܎˞଀
۶
TSCS data under unmeasured confounding
‫ ۼ
܎܃ۺ‬ее ۢ‫^ ܎܃‬۹ ۢ
܎
Treatment is unrelated
to the potential outcomes
‫܎܃‬
‫଀˞܎܃‬
‫ۼ‬
܎˞଀
۶
TSCS data under unmeasured confounding
‫ ۼ
܎܃ۺ‬ее ۢ‫^ ܎܃‬۹ ۢ
܎
Treatment is unrelated
to the potential outcomes
‫܎܃‬
‫଀˞܎܃‬
‫ۼ‬
܎˞଀
۶
...conditional on the
covariate history
TSCS data under unmeasured confounding
...and a time-fixed
unmeasured confounder.
‫ ۼ
܎܃ۺ‬ее ۢ‫^ ܎܃‬۹ ۢ
܎
Treatment is unrelated
to the potential outcomes
‫܎܃‬
‫଀˞܎܃‬
‫ۼ‬
܎˞଀
۶
...conditional on the
covariate history
How unit-specific weighting can help
How unit-specific weighting can help
۶
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
How unit-specific weighting can help
۶
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
܎
۸‫ ܎܃‬ಿ
2T=ۢ‫ ଀˞܎ۢ^ ܍܃‬۹܎ ‫ ଀˞܎ۺ‬۶?
‫଀܍‬
How unit-specific weighting can help
۶
ۢ܎˞଀
‫଀˞܎ۺ‬
Weighting balances the
treatment groups.
܎
ۢ܎
‫܎ۺ‬
۹܎
۸‫ ܎܃‬ಿ
2T=ۢ‫ ଀˞܎ۢ^ ܍܃‬۹܎ ‫ ଀˞܎ۺ‬۶?
‫଀܍‬
How unit-specific weighting can help
۶
ۢ܎˞଀
‫଀˞܎ۺ‬
ۢ܎
‫܎ۺ‬
۹܎
܎
۸‫ ܎܃‬ಿ
2T=ۢ‫ ଀˞܎ۢ^ ܍܃‬۹܎ ‫ ଀˞܎ۺ‬۶?
‫଀܍‬
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
‫ ۼ
܎܃ۺ‬ее ۢ‫^ ܎܃‬۹
܎
‫܅˞܎̂܎܃‬
ۢ
‫܅˞܎̂଀˞܎܃‬
‫ۼ‬
܎˞଀̂܎˞‫܅‬
۶
k-order sequential ignorability
‫ ۼ
܎܃ۺ‬ее ۢ‫^ ܎܃‬۹
܎
‫܅˞܎̂܎܃‬
ۢ
‫܅˞܎̂଀˞܎܃‬
‫ۼ‬
܎˞଀̂܎˞‫܅‬
۶
Only the last k
periods matter.
Blip effect: (1,0) vs (0,0)
0.7
Blip effect
0.6
0.5
0.4
0.3
0.2
10
25
50
75
Time periods
100
125
Blip effect: (1,0) vs (0,0)
0.7
Blip effect
0.6
0.5
0.4
0.3
0.2
10
25
50
75
Time periods
100
125
Blip effect: (1,0) vs (0,0)
0.7
Blip effect
0.6
0.5
0.4
0.3
●
●
●
●
●
●
Pooled
0.2
10
25
50
75
Time periods
100
125
Blip effect: (1,0) vs (0,0)
Outcome fixed effects
0.7
Blip effect
0.6
0.5
0.4
0.3
●
●
●
●
●
●
Pooled
0.2
10
25
50
75
Time periods
100
125
Blip effect: (1,0) vs (0,0)
Outcome fixed effects
IPTW true weights
0.7
Blip effect
0.6
0.5
0.4
0.3
●
●
●
●
●
●
Pooled
0.2
10
25
50
75
Time periods
100
125
Blip effect: (1,0) vs (0,0)
IPTW fixed effects
Outcome fixed effects
IPTW true weights
0.7
Blip effect
0.6
0.5
0.4
0.3
●
●
●
●
●
●
Pooled
0.2
10
25
50
75
Time periods
100
125
Treatment History Effect: (1,1) vs (0,0)
1.4
1.2
ATHE
1.0
0.8
0.6
0.4
0.2
0.0
10
25
50
75
Time periods
100
125
Treatment History Effect: (1,1) vs (0,0)
1.4
1.2
ATHE
1.0
0.8
0.6
0.4
0.2
0.0
10
25
50
75
Time periods
100
125
Treatment History Effect: (1,1) vs (0,0)
1.4
1.2
ATHE
1.0
0.8
0.6
0.4
Pooled
0.2
●
●
●
●
●
●
0.0
10
25
50
75
Time periods
100
125
Treatment History Effect: (1,1) vs (0,0)
1.4
1.2
ATHE
1.0
0.8
Outcome fixed effects
0.6
0.4
Pooled
0.2
●
●
●
●
●
●
0.0
10
25
50
75
Time periods
100
125
Treatment History Effect: (1,1) vs (0,0)
IPTW true weights
1.4
1.2
ATHE
1.0
0.8
Outcome fixed effects
0.6
0.4
Pooled
0.2
●
●
●
●
●
●
0.0
10
25
50
75
Time periods
100
125
Treatment History Effect: (1,1) vs (0,0)
IPTW fixed effects
IPTW true weights
1.4
1.2
ATHE
1.0
0.8
Outcome fixed effects
0.6
0.4
Pooled
0.2
●
●
●
●
●
●
0.0
10
25
50
75
Time periods
100
125
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
Even under strong assumptions, conditional estimators
cannot recover ATHEs.
Using weights
How to make causal inferences with TSCS data
Very carefully
Even under strong assumptions, conditional estimators
cannot recover ATHEs.
Using weights
A fixed effects weighting approach can recover ATHEs
and CETs even with unmeasured confounding.