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

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

Matthew Blackwell

April 13, 2013
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  1. How to Make Causal Inferences with
    Time-Series Cross-Sectional Data
    Matthew Blackwell
    University of Rochester
    Adam Glynn
    Harvard University

    View Slide

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

    View Slide

  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

    ۼ
    ܎
    ۼƓ
    ܎
    ۦ=ۺ܎

    ۼ
    ܎
    ˞ ۺ܎

    ۼƓ
    ܎
    ?
    Average Treatment
    History Effect

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  15. The effect of history

    ۼ
    ܎
    ۼƓ
    ܎
    ۦ=ۺ܎

    ۼ
    ܎
    ˞ ۺ܎

    ۼƓ
    ܎
    ?
    Average Treatment
    History Effect
    ATHE

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  16. The effect of history

    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

    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 ۢ܎

    ܎
    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܎˞଀
    ঑ଃ
    ۢ܎˞଀

    View Slide

  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

    View Slide

  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܎˞଀

    View Slide

  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
    ঑ଁ
    ঑଀
    ঑ଁ

    View Slide

  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

    View Slide

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

    View Slide

  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?

    View Slide

  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)

    View Slide

  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)

    View Slide

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

    View Slide

  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)

    View Slide

  60. TSCS data under unmeasured confounding

    View Slide

  61. TSCS data under unmeasured confounding
    ۺ܃܎

    ۼ
    ܎
    е
    е ۢ܃܎

    ܃܎
    ۢ
    ܃܎˞଀
    ۼ
    ܎˞଀
    ۶

    View Slide

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

    ۼ
    ܎
    е
    е ۢ܃܎

    ܃܎
    ۢ
    ܃܎˞଀
    ۼ
    ܎˞଀
    ۶

    View Slide

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

    ۼ
    ܎
    е
    е ۢ܃܎

    ܃܎
    ۢ
    ܃܎˞଀
    ۼ
    ܎˞଀
    ۶

    View Slide

  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.

    View Slide

  65. How unit-specific weighting can help

    View Slide

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

    View Slide

  67. How unit-specific weighting can help
    ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    ۶
    ۸܃܎

    ܎
    ಿ
    ܍଀

    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ۶?

    View Slide

  68. How unit-specific weighting can help
    ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    ۶
    ۸܃܎

    ܎
    ಿ
    ܍଀

    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ۶?
    Weighting balances the
    treatment groups.

    View Slide

  69. ۺ܎˞଀
    ۢ܎˞଀
    ۢ܎
    ۹܎
    ۺ܎
    How unit-specific weighting can help
    ۶
    ۸܃܎

    ܎
    ಿ
    ܍଀

    2T=ۢ܃܍
    ^ۢ܎˞଀
    ۹܎
    ۺ܎˞଀
    ۶?

    View Slide

  70. A weighting approach to fixed effects

    View Slide

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

    View Slide

  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

    View Slide

  73. k-order sequential ignorability

    View Slide

  74. k-order sequential ignorability
    ۺ܃܎

    ۼ
    ܎
    е
    е ۢ܃܎

    ܃܎̂܎˞܅
    ۢ
    ܃܎˞଀̂܎˞܅
    ۼ
    ܎˞଀̂܎˞܅
    ۶

    View Slide

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

    ۼ
    ܎
    е
    е ۢ܃܎

    ܃܎̂܎˞܅
    ۢ
    ܃܎˞଀̂܎˞܅
    ۼ
    ܎˞଀̂܎˞܅
    ۶

    View Slide

  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

    View Slide

  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

    View Slide

  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
    ● ● ● ● ● ●

    View Slide

  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
    ● ● ● ● ● ●

    View Slide

  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
    ● ● ● ● ● ●

    View Slide

  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
    ● ● ● ● ● ●

    View Slide

  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.

    View Slide