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
   

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

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3. How to Make Causal Inferences with
   Time-Series Cross-Sectional Data
   Very Carefully.
   

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4. How to Make Causal Inferences with
   Time-Series Cross-Sectional Data
   Using weights.
   

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5. ۺ܎˞଀
   ۢ܎˞଀
   ۹܎˞଀
   ۢ܎
   ۹܎
   ۺ܎
   

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6. What is the effect of A on Y?
   ۺ܎˞଀
   ۢ܎˞଀
   ۹܎˞଀
   ۢ܎
   ۹܎
   ۺ܎
   

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7. What is the effect of A on Y?
   contemporaneous
   ۺ܎˞଀
   ۢ܎˞଀
   ۹܎˞଀
   ۢ܎
   ۹܎
   ۺ܎
   

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8. What is the effect of A on Y?
   treatment history
   ۺ܎˞଀
   ۢ܎˞଀
   ۹܎˞଀
   ۢ܎
   ۹܎
   ۺ܎
   

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9. Shouldn't we have more notation?
   

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10. ۢ
    ܎
    
    ۢ଀
    Ɛ ۢ܎
    
    Treatment history
    Shouldn't we have more notation?
    

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11. ۢ
    ܎
    
    ۢ଀
    Ɛ ۢ܎
    
    Treatment history
    Shouldn't we have more notation?
    Specific instance of a treatment history
    ۼ
    ܎
    
    ۼ଀
    Ɛ ۼ܎
    
    

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12. ۢ
    ܎
    
    ۢ଀
    Ɛ ۢ܎
    
    Treatment history
    ۺ܎
    
    ۼ
    ܎
    
    Potential outcomes
    Shouldn't we have more notation?
    Specific instance of a treatment history
    ۼ
    ܎
    
    ۼ଀
    Ɛ ۼ܎
    
    

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13. The effect of history
    

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14. The effect of history
    ণ
    ۼ
    ܎
    ۼƓ
    ܎
    ۦ=ۺ܎
    
    ۼ
    ܎
    ˞ ۺ܎
    
    ۼƓ
    ܎
    ?
    Average Treatment
    History Effect
    

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15. The effect of history
    ণ
    ۼ
    ܎
    ۼƓ
    ܎
    ۦ=ۺ܎
    
    ۼ
    ܎
    ˞ ۺ܎
    
    ۼƓ
    ܎
    ?
    Average Treatment
    History Effect
    ATHE
    

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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
    

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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
    

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18. The effect of history
    

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19. The effect of history
    Blip Effect ণ۽
    
    ۼ
    ܎˞଀
    ۦ=ۺ܎
    
    ۼ
    ܎˞଀
    ˞ ۺ܎
    
    ۼ
    ܎˞଀
    ?
    

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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
    ܎˞଀
    ?
    

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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
    ܎˞଀
    ?
    

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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
    

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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
    

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24. The effect of history
    

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25. The effect of history
    Contemporaneous
    Effect of Treatment
    ণ܎
    ۦ=ণ۽
    
    ۼ
    ܎˞଀
    ?
    

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26. The effect of history
    Contemporaneous
    Effect of Treatment
    ণ܎
    ۦ=ণ۽
    
    ۼ
    ܎˞଀
    ?
    CET
    

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27. The effect of history
    1
    0
    vs
    Contemporaneous
    Effect of Treatment
    ণ܎
    ۦ=ণ۽
    
    ۼ
    ܎˞଀
    ?
    CET
    

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28. The effect of history
    1
    0
    vs
    Contemporaneous
    Effect of Treatment
    ণ܎
    ۦ=ণ۽
    
    ۼ
    ܎˞଀
    ?
    CET
    Marginalize over the past
    

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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
    ܎˞଀
    

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30. How conditioning leads you astray
    

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31. How conditioning leads you astray
    ...for some questions.
    

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32. How conditioning leads you astray
    ...for some questions.
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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33. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How conditioning leads you astray
    ...for some questions.
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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34. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How conditioning leads you astray
    ...for some questions.
    We “fix” these
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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35. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How conditioning leads you astray
    ...for some questions.
    We “fix” these
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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36. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How conditioning leads you astray
    We “fix” these
    ...for some questions.
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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37. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How conditioning leads you astray
    ...for some questions.
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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38. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How conditioning leads you astray
    এ૾
    ...for some questions.
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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39. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How conditioning leads you astray
    এ૾
    ? ?
    ?
    ?
    ?
    ?
    ...for some questions.
    ۺ܎
    ঑૿
    ঑଀
    ۢ܎
    ঑ଁ
    ۹܎
    ঑ଂ
    ۺ܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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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܎˞଀
    ঑ଃ
    ۢ܎˞଀
    

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41. How weighting can help
    

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42. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How weighting can help
    

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43. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How weighting can help
    ۸܃܎
    
    ܎
    ಿ
    ܍଀
    
    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ?
    
    

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44. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How weighting can help
    ۸܃܎
    
    ܎
    ಿ
    ܍଀
    
    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ?
    
    We weight to create balance
    

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45. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How weighting can help
    We weight to create balance
    ۸܃܎
    
    ܎
    ಿ
    ܍଀
    
    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ?
    
    

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46. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How weighting can help
    ۸܃܎
    
    ܎
    ಿ
    ܍଀
    
    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ?
    
    Unconfounded
    No posttreatment bias
    

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47. How weighting can help
    

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48. How weighting can help
    ۦ=ۺ܎
    
    ۼ܎
    ۼ܎˞଀
    ? ۦ۸
    =ۺ܎
    ^ۢ܎
    ۼ܎
    ۢ܎˞଀
    ۼ܎˞଀
    ?
    ঑૿
    ঑଀
    ۼ܎
    ঑ଁ
    ۼ܎˞଀
    

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49. How weighting can help
    ۦ=ۺ܎
    
    ۼ܎
    ۼ܎˞଀
    ? ۦ۸
    =ۺ܎
    ^ۢ܎
    ۼ܎
    ۢ܎˞଀
    ۼ܎˞଀
    ?
    ঑૿
    ঑଀
    ۼ܎
    ঑ଁ
    ۼ܎˞଀
    WLS
    

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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
    ঑ଁ
    ঑଀
    ঑ଁ
    

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51. The Long Arm of the Democratic Peace?
    

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52. The Long Arm of the Democratic Peace?
    Democracy
    in year t
    War in
    year t
    

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53. The Long Arm of the Democratic Peace?
    Democracy
    in year t
    War in
    year t
    Democratic Peace
    Literature
    

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54. The Long Arm of the Democratic Peace?
    Democracy
    in year t
    War in
    year t
    Democratic Peace
    Literature
    History of
    Democracy
    

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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?
    

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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)
    

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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)
    

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58. Democracy
    in year t
    War in
    year t
    Economic
    Growth in
    year t
    History of
    Democracy
    Misspecification of an ATHE
    Time-Varying
    Confounder
    

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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)
    

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60. TSCS data under unmeasured confounding
    

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61. TSCS data under unmeasured confounding
    ۺ܃܎
    
    ۼ
    ܎
    е
    е ۢ܃܎
    ^۹
    ܃܎
    ۢ
    ܃܎˞଀
    ۼ
    ܎˞଀
    ۶
    

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62. TSCS data under unmeasured confounding
    Treatment is unrelated
    to the potential outcomes
    ۺ܃܎
    
    ۼ
    ܎
    е
    е ۢ܃܎
    ^۹
    ܃܎
    ۢ
    ܃܎˞଀
    ۼ
    ܎˞଀
    ۶
    

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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
    

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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.
    

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65. How unit-specific weighting can help
    

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66. How unit-specific weighting can help
    ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    ۶
    

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67. How unit-specific weighting can help
    ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    ۶
    ۸܃܎
    
    ܎
    ಿ
    ܍଀
    
    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ۶?
    

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68. How unit-specific weighting can help
    ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    ۶
    ۸܃܎
    
    ܎
    ಿ
    ܍଀
    
    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ۶?
    Weighting balances the
    treatment groups.
    

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69. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How unit-specific weighting can help
    ۶
    ۸܃܎
    
    ܎
    ಿ
    ܍଀
    
    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ۶?
    

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70. A weighting approach to fixed effects
    

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71. A weighting approach to fixed effects
    1
    Estimate unit-specific probability of treatment
    over time and construct weights.
    

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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
    

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73. k-order sequential ignorability
    

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74. k-order sequential ignorability
    ۺ܃܎
    
    ۼ
    ܎
    е
    е ۢ܃܎
    ^۹
    ܃܎̂܎˞܅
    ۢ
    ܃܎˞଀̂܎˞܅
    ۼ
    ܎˞଀̂܎˞܅
    ۶
    

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75. k-order sequential ignorability
    Only the last k
    periods matter.
    ۺ܃܎
    
    ۼ
    ܎
    е
    е ۢ܃܎
    ^۹
    ܃܎̂܎˞܅
    ۢ
    ܃܎˞଀̂܎˞܅
    ۼ
    ܎˞଀̂܎˞܅
    ۶
    

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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
    

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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
    

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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
    ● ● ● ● ● ●
    

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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
    ● ● ● ● ● ●
    

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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
    ● ● ● ● ● ●
    

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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
    ● ● ● ● ● ●
    

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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
    

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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
    

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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
    ● ● ● ● ● ●
    

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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
    ● ● ● ● ● ●
    

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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
    ● ● ● ● ● ●
    

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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
    ● ● ● ● ● ●
    

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88. How to make causal inferences with TSCS data
    

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89. How to make causal inferences with TSCS data
    Very carefully
    

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90. How to make causal inferences with TSCS data
    Very carefully
    Even under strong assumptions, conditional estimators
    cannot recover ATHEs.
    

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91. How to make causal inferences with TSCS data
    Very carefully
    Using weights
    Even under strong assumptions, conditional estimators
    cannot recover ATHEs.
    

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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.
    

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