Dierence-in-Dierences for Ordinal Outcomes Soichiro Yamauchi - - PowerPoint PPT Presentation

Dierence-in-Dierences for Ordinal Outcomes

Soichiro Yamauchi Harvard University Applied Stascs Workshop, IQSS April 1, 2020

1 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies

Adjust for the me-invariant confounders by ulizing the past outcome Key idencaon assumpon: parallel trends assumpon Idencal trends across the treated & the control without the treatment Relies on the dierences between two potenal outcomes: Linearity In social science, many outcomes are measured on an ordinal scale (e.g., survey quesons) “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome

Key idencaon assumpon: parallel trends assumpon Idencal trends across the treated & the control without the treatment Relies on the dierences between two potenal outcomes: Linearity In social science, many outcomes are measured on an ordinal scale (e.g., survey quesons) “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

Idencal trends across the treated & the control without the treatment Relies on the dierences between two potenal outcomes: Linearity In social science, many outcomes are measured on an ordinal scale (e.g., survey quesons) “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment Relies on the dierences between two potenal outcomes: Linearity In social science, many outcomes are measured on an ordinal scale (e.g., survey quesons) “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment

pre−treatment post−treatment

• Treatment Group
• Control Group

Relies on the dierences between two potenal outcomes: Linearity In social science, many outcomes are measured on an ordinal scale (e.g., survey quesons) “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment

pre−treatment post−treatment

• Treatment Group
• Control Group
• Treatment Group w/o Treatment

Relies on the dierences between two potenal outcomes: Linearity In social science, many outcomes are measured on an ordinal scale (e.g., survey quesons) “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity In social science, many outcomes are measured on an ordinal scale (e.g., survey quesons) “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity

• In social science, many outcomes are measured on an ordinal scale (e.g.,

survey quesons) ❀ “dierences” are not well dened Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity

• In social science, many outcomes are measured on an ordinal scale (e.g.,

survey quesons) ❀ “dierences” are not well dened

• Problems in common pracces:

Treat as a connuous variable Dicult to interpret + linearity Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity

• In social science, many outcomes are measured on an ordinal scale (e.g.,

survey quesons) ❀ “dierences” are not well dened

• Problems in common pracces:
• Treat as a connuous variable ❀ Dicult to interpret + linearity

Dichotomize the outcome Mulple disnct parallel trends assumpons Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity

• In social science, many outcomes are measured on an ordinal scale (e.g.,

survey quesons) ❀ “dierences” are not well dened

• Problems in common pracces:
• Treat as a connuous variable ❀ Dicult to interpret + linearity
• Dichotomize the outcome ❀ Mulple disnct parallel trends assumpons

Ordered probit/logit Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity

• In social science, many outcomes are measured on an ordinal scale (e.g.,

survey quesons) ❀ “dierences” are not well dened

• Problems in common pracces:
• Treat as a connuous variable ❀ Dicult to interpret + linearity
• Dichotomize the outcome ❀ Mulple disnct parallel trends assumpons
• Ordered probit/logit ❀ Idencaon assumpons are not explicitly stated

Propose: A latent variable framework for DiD for the ordinal outcomes Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity

• In social science, many outcomes are measured on an ordinal scale (e.g.,

survey quesons) ❀ “dierences” are not well dened

• Problems in common pracces:
• Treat as a connuous variable ❀ Dicult to interpret + linearity
• Dichotomize the outcome ❀ Mulple disnct parallel trends assumpons
• Ordered probit/logit ❀ Idencaon assumpons are not explicitly stated
• Propose: A latent variable framework for DiD for the ordinal outcomes

Applicaon: Revisit a recent debate on the relaonship between the mass shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences Design in Observaonal Studies

• Dierence-in-dierences for causal inference in observaonal studies
• Adjust for the me-invariant confounders by ulizing the past outcome
• Key idencaon assumpon: parallel trends assumpon

❀ Idencal trends across the treated & the control without the treatment ❀ Relies on the dierences between two potenal outcomes: Linearity

• In social science, many outcomes are measured on an ordinal scale (e.g.,

survey quesons) ❀ “dierences” are not well dened

• Problems in common pracces:
• Treat as a connuous variable ❀ Dicult to interpret + linearity
• Dichotomize the outcome ❀ Mulple disnct parallel trends assumpons
• Ordered probit/logit ❀ Idencaon assumpons are not explicitly stated
• Propose: A latent variable framework for DiD for the ordinal outcomes
• Applicaon: Revisit a recent debate on the relaonship between the mass

shoongs and the atude toward gun control

2 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool Introduce a latent variable framework Extend the latent ulity representaon of the standard probit/logit Apply the assumpon by Athey & Imbens (2006) on the latent variable scale Assumes temporal changes in quanles are idencal across two groups Avoid imposing the linearity assumpon in the standard DiD Derive a diagnosc with one addional pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework

Extend the latent ulity representaon of the standard probit/logit Apply the assumpon by Athey & Imbens (2006) on the latent variable scale Assumes temporal changes in quanles are idencal across two groups Avoid imposing the linearity assumpon in the standard DiD Derive a diagnosc with one addional pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework
• Extend the latent ulity representaon of the standard probit/logit

Apply the assumpon by Athey & Imbens (2006) on the latent variable scale Assumes temporal changes in quanles are idencal across two groups Avoid imposing the linearity assumpon in the standard DiD Derive a diagnosc with one addional pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework
• Extend the latent ulity representaon of the standard probit/logit
• Apply the assumpon by Athey & Imbens (2006) on the latent variable scale

Assumes temporal changes in quanles are idencal across two groups Avoid imposing the linearity assumpon in the standard DiD Derive a diagnosc with one addional pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework
• Extend the latent ulity representaon of the standard probit/logit
• Apply the assumpon by Athey & Imbens (2006) on the latent variable scale
• Assumes temporal changes in quanles are idencal across two groups

Avoid imposing the linearity assumpon in the standard DiD Derive a diagnosc with one addional pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework
• Extend the latent ulity representaon of the standard probit/logit
• Apply the assumpon by Athey & Imbens (2006) on the latent variable scale
• Assumes temporal changes in quanles are idencal across two groups

❀ Avoid imposing the linearity assumpon in the standard DiD Derive a diagnosc with one addional pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework
• Extend the latent ulity representaon of the standard probit/logit
• Apply the assumpon by Athey & Imbens (2006) on the latent variable scale
• Assumes temporal changes in quanles are idencal across two groups

❀ Avoid imposing the linearity assumpon in the standard DiD

• Derive a diagnosc with one addional pre-treatment period

Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework
• Extend the latent ulity representaon of the standard probit/logit
• Apply the assumpon by Athey & Imbens (2006) on the latent variable scale
• Assumes temporal changes in quanles are idencal across two groups

❀ Avoid imposing the linearity assumpon in the standard DiD

• Derive a diagnosc with one addional pre-treatment period
• Analogous to the pre-treatment trend check in the standard DiD

Equivalence based test to assess the plausibility of the assumpon

3 | 15

Dierence-in-Dierences for Ordinal Outcomes

Contribuons: New idencaon strategy & diagnosc tool

• Introduce a latent variable framework
• Extend the latent ulity representaon of the standard probit/logit
• Apply the assumpon by Athey & Imbens (2006) on the latent variable scale
• Assumes temporal changes in quanles are idencal across two groups

❀ Avoid imposing the linearity assumpon in the standard DiD

• Derive a diagnosc with one addional pre-treatment period
• Analogous to the pre-treatment trend check in the standard DiD

❀ Equivalence based test to assess the plausibility of the assumpon

3 | 15

Mass Shoongs and Atudes toward Gun Control

• Recent debate on the topic (Barney & Schaner, 2019; Hartman& Newman,

2019; Newman & Hartman, 2019) Proximity to the shoongs as a treatment (dichotomized by 100 miles) Ordinal survey outcome: less-strict, keep-the-same and more-strict

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

4 | 15

Mass Shoongs and Atudes toward Gun Control

• Recent debate on the topic (Barney & Schaner, 2019; Hartman& Newman,

2019; Newman & Hartman, 2019)

• Proximity to the shoongs as a treatment (dichotomized by 100 miles)

Ordinal survey outcome: less-strict, keep-the-same and more-strict

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

4 | 15

Mass Shoongs and Atudes toward Gun Control

• Recent debate on the topic (Barney & Schaner, 2019; Hartman& Newman,

2019; Newman & Hartman, 2019)

• Proximity to the shoongs as a treatment (dichotomized by 100 miles)
• Ordinal survey outcome: less-strict, keep-the-same and more-strict

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

4 | 15

Mass Shoongs and Atudes toward Gun Control

• Recent debate on the topic (Barney & Schaner, 2019; Hartman& Newman,

2019; Newman & Hartman, 2019)

• Proximity to the shoongs as a treatment (dichotomized by 100 miles)
• Ordinal survey outcome: less-strict, keep-the-same and more-strict

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• 0.35

0.40 0.45 0.50

'more−strict' as 1

Pr(Y = 1 | D = d) 2010 2012 2014

• Treatment Group

Control Group

4 | 15

Mass Shoongs and Atudes toward Gun Control

• Recent debate on the topic (Barney & Schaner, 2019; Hartman& Newman,

2019; Newman & Hartman, 2019)

• Proximity to the shoongs as a treatment (dichotomized by 100 miles)
• Ordinal survey outcome: less-strict, keep-the-same and more-strict

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• 0.35

0.40 0.45 0.50

'more−strict' as 1

Pr(Y = 1 | D = d) 2010 2012 2014

• Treatment Group

Control Group

• 0.75

0.80 0.85 0.90

'keep−the−same' & 'more−strict' as 1

Pr(Y = 1 | D = d) 2010 2012 2014

• connuous

4 | 15

DiD for Ordinal Outcomes: Setup

• Observed outcome: Yit ∈ {0, . . . , J − 1} for i = 1, . . . , n and t ∈ {0, 1}

Binary treatment: Di 0 1 Potenal outcome: Yit d for d 0 1 Esmand: Dierences in choice probabilies for the treated

j

Example: Dierence in prob. of choosing more-strict under two condions Pr Yi1 1 j Di 1 is observed from the data: Pr Yi1 j Di 1 Need to idenfy Pr Yi1 0 j Di 1 with addional assumpons

5 | 15

DiD for Ordinal Outcomes: Setup

• Observed outcome: Yit ∈ {0, . . . , J − 1} for i = 1, . . . , n and t ∈ {0, 1}
• Binary treatment: Di ∈ {0, 1}

Potenal outcome: Yit d for d 0 1 Esmand: Dierences in choice probabilies for the treated

j

Example: Dierence in prob. of choosing more-strict under two condions Pr Yi1 1 j Di 1 is observed from the data: Pr Yi1 j Di 1 Need to idenfy Pr Yi1 0 j Di 1 with addional assumpons

5 | 15

DiD for Ordinal Outcomes: Setup

• Observed outcome: Yit ∈ {0, . . . , J − 1} for i = 1, . . . , n and t ∈ {0, 1}
• Binary treatment: Di ∈ {0, 1}
• Potenal outcome: Yit(d) for d ∈ {0, 1}

Esmand: Dierences in choice probabilies for the treated

j

Example: Dierence in prob. of choosing more-strict under two condions Pr Yi1 1 j Di 1 is observed from the data: Pr Yi1 j Di 1 Need to idenfy Pr Yi1 0 j Di 1 with addional assumpons

5 | 15

DiD for Ordinal Outcomes: Setup

• Observed outcome: Yit ∈ {0, . . . , J − 1} for i = 1, . . . , n and t ∈ {0, 1}
• Binary treatment: Di ∈ {0, 1}
• Potenal outcome: Yit(d) for d ∈ {0, 1}
• Esmand: Dierences in choice probabilies for the treated

ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1) Example: Dierence in prob. of choosing more-strict under two condions Pr Yi1 1 j Di 1 is observed from the data: Pr Yi1 j Di 1 Need to idenfy Pr Yi1 0 j Di 1 with addional assumpons

5 | 15

DiD for Ordinal Outcomes: Setup

• Observed outcome: Yit ∈ {0, . . . , J − 1} for i = 1, . . . , n and t ∈ {0, 1}
• Binary treatment: Di ∈ {0, 1}
• Potenal outcome: Yit(d) for d ∈ {0, 1}
• Esmand: Dierences in choice probabilies for the treated

ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)

• Example: Dierence in prob. of choosing more-strict under two condions

Pr Yi1 1 j Di 1 is observed from the data: Pr Yi1 j Di 1 Need to idenfy Pr Yi1 0 j Di 1 with addional assumpons

5 | 15

DiD for Ordinal Outcomes: Setup

• Observed outcome: Yit ∈ {0, . . . , J − 1} for i = 1, . . . , n and t ∈ {0, 1}
• Binary treatment: Di ∈ {0, 1}
• Potenal outcome: Yit(d) for d ∈ {0, 1}
• Esmand: Dierences in choice probabilies for the treated

ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)

• Example: Dierence in prob. of choosing more-strict under two condions
• Pr(Yi1(1) = j | Di = 1) is observed from the data: Pr(Yi1 = j | Di = 1)

Need to idenfy Pr Yi1 0 j Di 1 with addional assumpons

5 | 15

DiD for Ordinal Outcomes: Setup

• Observed outcome: Yit ∈ {0, . . . , J − 1} for i = 1, . . . , n and t ∈ {0, 1}
• Binary treatment: Di ∈ {0, 1}
• Potenal outcome: Yit(d) for d ∈ {0, 1}
• Esmand: Dierences in choice probabilies for the treated

ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)

• Example: Dierence in prob. of choosing more-strict under two condions
• Pr(Yi1(1) = j | Di = 1) is observed from the data: Pr(Yi1 = j | Di = 1)
• Need to idenfy Pr(Yi1(0) = j | Di = 1) with addional assumpons

5 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

Y11 is the counterfactual outcome Latent “ulity” generang the ordinal outcome: Ydt Index model: Mapping Ydt to Ydt Ydt if

1

Ydt j if

j 1

Ydt

j

J 1 if

J

Ydt

J 1

Locaon-scale family: Imposing distribuon on Ydt Ydt

dt locaon dt scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

❀ Y11 is the counterfactual outcome Latent “ulity” generang the ordinal outcome: Ydt Index model: Mapping Ydt to Ydt Ydt if

1

Ydt j if

j 1

Ydt

j

J 1 if

J

Ydt

J 1

Locaon-scale family: Imposing distribuon on Ydt Ydt

dt locaon dt scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

❀ Y11 is the counterfactual outcome

• Latent “ulity” generang the ordinal outcome: Y∗

dt ∈ R

Index model: Mapping Ydt to Ydt Ydt if

1

Ydt j if

j 1

Ydt

j

J 1 if

J

Ydt

J 1

Locaon-scale family: Imposing distribuon on Ydt Ydt

dt locaon dt scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

❀ Y11 is the counterfactual outcome

• Latent “ulity” generang the ordinal outcome: Y∗

dt ∈ R

• Index model: Mapping Y∗

dt to Ydt

Ydt =      if κ1 ≥ Y∗

dt ≥ κ0

j if κj+1 ≥ Y∗

dt ≥ κj

J − 1 if κJ ≥ Y∗

dt ≥ κJ−1

Locaon-scale family: Imposing distribuon on Ydt Ydt

dt locaon dt scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

❀ Y11 is the counterfactual outcome

• Latent “ulity” generang the ordinal outcome: Y∗

dt ∈ R

• Index model: Mapping Y∗

dt to Ydt

Ydt =      if κ1 ≥ Y∗

dt ≥ κ0

j if κj+1 ≥ Y∗

dt ≥ κj

J − 1 if κJ ≥ Y∗

dt ≥ κJ−1

Latent Utility Y *

κ0 κ3 κ1 κ2 Ydt = less−strict

Locaon-scale family: Imposing distribuon on Ydt Ydt

dt locaon dt scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

❀ Y11 is the counterfactual outcome

• Latent “ulity” generang the ordinal outcome: Y∗

dt ∈ R

• Index model: Mapping Y∗

dt to Ydt

Ydt =      if κ1 ≥ Y∗

dt ≥ κ0

j if κj+1 ≥ Y∗

dt ≥ κj

J − 1 if κJ ≥ Y∗

dt ≥ κJ−1

Latent Utility Y *

κ0 κ3 κ1 κ2 Ydt = keep−the−same

Locaon-scale family: Imposing distribuon on Ydt Ydt

dt locaon dt scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

❀ Y11 is the counterfactual outcome

• Latent “ulity” generang the ordinal outcome: Y∗

dt ∈ R

• Index model: Mapping Y∗

dt to Ydt

Ydt =      if κ1 ≥ Y∗

dt ≥ κ0

j if κj+1 ≥ Y∗

dt ≥ κj

J − 1 if κJ ≥ Y∗

dt ≥ κJ−1

Latent Utility Y *

κ0 κ3 κ1 κ2 Ydt = more−strict

Locaon-scale family: Imposing distribuon on Ydt Ydt

dt locaon dt scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Latent Variable Formulaon

• Ordinal outcome: Ydt ∼ Yit(0) | Di = d

❀ Y11 is the counterfactual outcome

• Latent “ulity” generang the ordinal outcome: Y∗

dt ∈ R

• Index model: Mapping Y∗

dt to Ydt

Ydt =      if κ1 ≥ Y∗

dt ≥ κ0

j if κj+1 ≥ Y∗

dt ≥ κj

J − 1 if κJ ≥ Y∗

dt ≥ κJ−1

• Locaon-scale family: Imposing distribuon on Y∗

dt

Y∗

dt ∼ µdt

• locaon

+ σdt

• scale

U where U belongs to a parametric family (e.g., normal, logisc, t-dist.)

6 | 15

Main Result

• Distribuonal parallel-trends assumpon (Athey & Imbens 2006)

FY∗

00(F−1

Y∗

01 (v)) = FY∗ 10(F−1

Y∗

11 (v))

∀v ∈ [0, 1] Proposion: the distribuon of the counterfactual latent variable given by

11 10 01 00 00 10

and

11 10 01 00

When variances are constant

dt

, recovers the usual parallel trends form

11 10 01 00

7 | 15

Main Result

• Distribuonal parallel-trends assumpon (Athey & Imbens 2006)

FY∗

00(F−1

Y∗

01 (v))

• = q0(v)

= FY∗

10(F−1

Y∗

11 (v))

• = q1(v)

∀v ∈ [0, 1]

−4 −2 2 4

Control

Latent Utility 1

Y00

*

−4 −2 2 4

Treated

Latent Utility 1

Y10

*

Proposion: the distribuon of the counterfactual latent variable given by

11 10 01 00 00 10

and

11 10 01 00

When variances are constant

dt

, recovers the usual parallel trends form

11 10 01 00

7 | 15

Main Result

• Distribuonal parallel-trends assumpon (Athey & Imbens 2006)

FY∗

00(F−1

Y∗

01 (v))

• = q0(v)

= FY∗

10(F−1

Y∗

11 (v))

• = q1(v)

∀v ∈ [0, 1]

−4 −2 2 4

Control

Latent Utility 1

Y00

*

Y01

*

−4 −2 2 4

Treated

Latent Utility 1

Y10

*

Y11

*

Proposion: the distribuon of the counterfactual latent variable given by

11 10 01 00 00 10

and

11 10 01 00

When variances are constant

dt

, recovers the usual parallel trends form

11 10 01 00

7 | 15

Main Result

• Distribuonal parallel-trends assumpon (Athey & Imbens 2006)

FY∗

00(F−1

Y∗

01 (v))

• = q0(v)

= FY∗

10(F−1

Y∗

11 (v))

• = q1(v)

∀v ∈ [0, 1]

−4 −2 2 4

Control

Latent Utility 1

Y00

*

Y01

*

v q0 (v) −4 −2 2 4

Treated

Latent Utility 1

Y10

*

Y11

*

Proposion: the distribuon of the counterfactual latent variable given by

11 10 01 00 00 10

and

11 10 01 00

When variances are constant

dt

, recovers the usual parallel trends form

11 10 01 00

7 | 15

Main Result

• Distribuonal parallel-trends assumpon (Athey & Imbens 2006)

FY∗

00(F−1

Y∗

01 (v))

• = q0(v)

= FY∗

10(F−1

Y∗

11 (v))

• = q1(v)

∀v ∈ [0, 1]

−4 −2 2 4

Control

Latent Utility 1

Y00

*

Y01

*

v q0 (v) −4 −2 2 4

Treated

Latent Utility 1

Y10

*

Y11

*

v q1 (v)

Proposion: the distribuon of the counterfactual latent variable given by

11 10 01 00 00 10

and

11 10 01 00

When variances are constant

dt

, recovers the usual parallel trends form

11 10 01 00

7 | 15

Main Result

• Distribuonal parallel-trends assumpon (Athey & Imbens 2006)

FY∗

00(F−1

Y∗

01 (v))

• = q0(v)

= FY∗

10(F−1

Y∗

11 (v))

• = q1(v)

∀v ∈ [0, 1]

−4 −2 2 4

Control

Latent Utility v 1 q0 (v)

Y00

*

Y01

*

−4 −2 2 4

Treated

Latent Utility v q1 (v) 1

Y10

*

Y11

*

• Proposion: the distribuon of the counterfactual latent variable given by

µ11 = µ10 + µ01 − µ00 σ00/σ10 , and σ11 = σ10σ01 σ00 When variances are constant

dt

, recovers the usual parallel trends form

11 10 01 00

7 | 15

Main Result

• Distribuonal parallel-trends assumpon (Athey & Imbens 2006)

FY∗

00(F−1

Y∗

01 (v))

• = q0(v)

= FY∗

10(F−1

Y∗

11 (v))

• = q1(v)

∀v ∈ [0, 1]

−4 −2 2 4

Control

Latent Utility v 1 q0 (v)

Y00

*

Y01

*

−4 −2 2 4

Treated

Latent Utility v q1 (v) 1

Y10

*

Y11

*

• Proposion: the distribuon of the counterfactual latent variable given by

µ11 = µ10 + µ01 − µ00 σ00/σ10 , and σ11 = σ10σ01 σ00

• When variances are constant σdt = σ, recovers the usual parallel trends form

µ11 − µ10 = µ01 − µ00

7 | 15

Esmaon

• Impose a distribuon on U (base distribuon): U ∼ N(0, 1)

Esmate parameters by MLE (e.g., variant of the ordered probit) Plug-in esmator for the counter-factual distribuon

11 10 01 00 00 10

and

11 10 01 00

Obtain causal esmates:

1 J 1 j

Obtain variance esmates by the block-bootstrap

Cuto Mean-Variance 8 | 15

Esmaon

• Impose a distribuon on U (base distribuon): U ∼ N(0, 1)

❀ Esmate parameters by MLE (e.g., variant of the ordered probit) Plug-in esmator for the counter-factual distribuon

11 10 01 00 00 10

and

11 10 01 00

Obtain causal esmates:

1 J 1 j

Obtain variance esmates by the block-bootstrap

Cuto Mean-Variance 8 | 15

Esmaon

• Impose a distribuon on U (base distribuon): U ∼ N(0, 1)

❀ Esmate parameters by MLE (e.g., variant of the ordered probit)

• Plug-in esmator for the counter-factual distribuon
• µ11 ←

µ10 + µ01 − µ00

• σ00/

σ10 , and

• σ11 ←

σ10 σ01

• σ00

Obtain causal esmates:

1 J 1 j

Obtain variance esmates by the block-bootstrap

Cuto Mean-Variance 8 | 15

Esmaon

• Impose a distribuon on U (base distribuon): U ∼ N(0, 1)

❀ Esmate parameters by MLE (e.g., variant of the ordered probit)

• Plug-in esmator for the counter-factual distribuon
• µ11 ←

µ10 + µ01 − µ00

• σ00/

σ10 , and

• σ11 ←

σ10 σ01

• σ00
• Obtain causal esmates:

ζ = ( ζ1, . . . , ζJ−1)⊤

• ζj = 1

n1

n

∑

i=1

Di1{Yi1 = j}

• =

Pr(Yi1(1)=j|Di=1)

− { Φ ( ( κj+1 − µ11)/ σ11 ) − Φ ( ( κj − µ11)/ σ11 )}

• =

Pr(Yi1(0)=j|Di=1)

Obtain variance esmates by the block-bootstrap

Cuto Mean-Variance 8 | 15

Esmaon

• Impose a distribuon on U (base distribuon): U ∼ N(0, 1)

❀ Esmate parameters by MLE (e.g., variant of the ordered probit)

• Plug-in esmator for the counter-factual distribuon
• µ11 ←

µ10 + µ01 − µ00

• σ00/

σ10 , and

• σ11 ←

σ10 σ01

• σ00
• Obtain causal esmates:

ζ = ( ζ1, . . . , ζJ−1)⊤

• ζj = 1

n1

n

∑

i=1

Di1{Yi1 = j}

• =

Pr(Yi1(1)=j|Di=1)

− { Φ ( ( κj+1 − µ11)/ σ11 ) − Φ ( ( κj − µ11)/ σ11 )}

• =

Pr(Yi1(0)=j|Di=1)

Obtain variance esmates by the block-bootstrap

Cuto Mean-Variance 8 | 15

Esmaon

• Impose a distribuon on U (base distribuon): U ∼ N(0, 1)

❀ Esmate parameters by MLE (e.g., variant of the ordered probit)

• Plug-in esmator for the counter-factual distribuon
• µ11 ←

µ10 + µ01 − µ00

• σ00/

σ10 , and

• σ11 ←

σ10 σ01

• σ00
• Obtain causal esmates:

ζ = ( ζ1, . . . , ζJ−1)⊤

• ζj = 1

n1

n

∑

i=1

Di1{Yi1 = j}

• =

Pr(Yi1(1)=j|Di=1)

− { Φ ( ( κj+1 − µ11)/ σ11 ) − Φ ( ( κj − µ11)/ σ11 )}

• =

Pr(Yi1(0)=j|Di=1)

Obtain variance esmates by the block-bootstrap

Cuto Mean-Variance 8 | 15

Esmaon

• Impose a distribuon on U (base distribuon): U ∼ N(0, 1)

❀ Esmate parameters by MLE (e.g., variant of the ordered probit)

• Plug-in esmator for the counter-factual distribuon
• µ11 ←

µ10 + µ01 − µ00

• σ00/

σ10 , and

• σ11 ←

σ10 σ01

• σ00
• Obtain causal esmates:

ζ = ( ζ1, . . . , ζJ−1)⊤

• ζj = 1

n1

n

∑

i=1

Di1{Yi1 = j}

• =

Pr(Yi1(1)=j|Di=1)

− { Φ ( ( κj+1 − µ11)/ σ11 ) − Φ ( ( κj − µ11)/ σ11 )}

• =

Pr(Yi1(0)=j|Di=1)

• Obtain variance esmates by the block-bootstrap

Cuto Mean-Variance 8 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

Respondents are “treated” if living within 100 miles from the shoongs

16 mass-shoongs coded at the zip-code level

• Approx. 30% of respondents (out of 16620) are treated

Subgroups

Partisanship: 3-point scale party self-idencaon Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

Respondents are “treated” if living within 100 miles from the shoongs

16 mass-shoongs coded at the zip-code level

• Approx. 30% of respondents (out of 16620) are treated

Subgroups

Partisanship: 3-point scale party self-idencaon Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs

16 mass-shoongs coded at the zip-code level

• Approx. 30% of respondents (out of 16620) are treated

Subgroups

Partisanship: 3-point scale party self-idencaon Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated

Subgroups

Partisanship: 3-point scale party self-idencaon Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated

Subgroups

Partisanship: 3-point scale party self-idencaon Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated
• Subgroups

Partisanship: 3-point scale party self-idencaon Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated
• Subgroups
• Partisanship: 3-point scale party self-idencaon

Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated
• Subgroups
• Partisanship: 3-point scale party self-idencaon
• Prior-exposure: Living in areas with mass shoongs in the past 10 years

Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated
• Subgroups
• Partisanship: 3-point scale party self-idencaon
• Prior-exposure: Living in areas with mass shoongs in the past 10 years
• Previous studies used:

Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated
• Subgroups
• Partisanship: 3-point scale party self-idencaon
• Prior-exposure: Living in areas with mass shoongs in the past 10 years
• Previous studies used:
• Ordinal logit with RE (NH19 and HN19)

Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Revising the Empirical Applicaon

• Two-wave (2010-12) panel from Cooperave Congressional Elecon Study:

In general, do you feel that laws covering the sale of firearms should be made more strict, less strict, or kept as they are? (0) Less Strict; (1) Kept As They Are; (2) More Strict.

• Respondents are “treated” if living within 100 miles from the shoongs
• 16 mass-shoongs coded at the zip-code level
• Approx. 30% of respondents (out of 16620) are treated
• Subgroups
• Partisanship: 3-point scale party self-idencaon
• Prior-exposure: Living in areas with mass shoongs in the past 10 years
• Previous studies used:
• Ordinal logit with RE (NH19 and HN19)
• Linear two-way FE (BS19)

Denon of Mass Shoongs 9 | 15

Distribuon of Outcome in 2010 & 2012

0: less-strict; 1: keep-the-same; 2: more-strict

Freq 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 2

2010 Treated Control

Detail 10 | 15

Distribuon of Outcome in 2010 & 2012

0: less-strict; 1: keep-the-same; 2: more-strict

Freq 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 2

2010 2012

1 2

Treated Control

Detail 10 | 15

Results

• Esmate ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)
• Block-bootstrap at the zip-code level

−0.10 −0.05 0.00 0.05 0.10 Difference in probabilities

• Full Sample

n = 16553

Effect in 2012 (CCES 2010−12)

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

25-miles PID-7 Cumulave eect Bounds 11 | 15

Results

• Esmate ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)
• Block-bootstrap at the zip-code level

−0.10 −0.05 0.00 0.05 0.10 Difference in probabilities

• Full Sample

n = 16553

• No Prior Exposure

n = 7123

• Prior Exposure

n = 9430

Effect in 2012 (CCES 2010−12)

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

25-miles PID-7 Cumulave eect Bounds 11 | 15

Results

• Esmate ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)
• Block-bootstrap at the zip-code level

−0.10 −0.05 0.00 0.05 0.10 Difference in probabilities

• Full Sample

n = 16553

• No Prior Exposure

n = 7123

• Prior Exposure

n = 9430

• Democrat

n = 5526

• Republican

n = 5126

• Independent

n = 4996

Effect in 2012 (CCES 2010−12)

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

25-miles PID-7 Cumulave eect Bounds 11 | 15

Results

• Esmate ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)
• Block-bootstrap at the zip-code level

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 Difference in probabilities

• Democrat

n = 2118

No Prior Exposure

• Republican

n = 2359

• Independent

n = 2222

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

25-miles PID-7 Cumulave eect Bounds 11 | 15

Results

• Esmate ζj = Pr(Yi1(1) = j | Di = 1) − Pr(Yi1(0) = j | Di = 1)
• Block-bootstrap at the zip-code level

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 Difference in probabilities

• Democrat

n = 2118

No Prior Exposure

• Democrat

n = 3408

Prior Exposure

• Republican

n = 2359

• Republican

n = 2767

• Independent

n = 2222

• Independent

n = 2774

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

25-miles PID-7 Cumulave eect Bounds 11 | 15

Assessing the Distribuonal Parallel Trends Assumpon

• Addional pre-treatment me-periods ❀ Assessment of the distribuonal PT

If the assumpon holds for the pre-treatment, we have q1 v q0 v qd v FYd0 F

1 Yd1 v

Test the following non-equivalence null (i.e., H0: Assumpon does not hold) H0

v 0 1 q1 v

q0 v vs H1

v 0 1 q1 v

q0 v Two one-sided tests (TOST) H0

v 0 1 q1 v

q0 v and H0

v 0 1 q1 v

q0 v Construct one-sided point-wise CIs: U1 v and L1 v reject H0 at level

v 0 1 U1

v We reject H0 if we reject both H0 and H0

Choose delta 12 | 15

Assessing the Distribuonal Parallel Trends Assumpon

• Addional pre-treatment me-periods ❀ Assessment of the distribuonal PT
• If the assumpon holds for the pre-treatment, we have ˜

q1(v) − ˜ q0(v) = 0 ˜ qd(v) = FY∗

d0(F−1

Y∗

d1 (v))

Test the following non-equivalence null (i.e., H0: Assumpon does not hold) H0

v 0 1 q1 v

q0 v vs H1

v 0 1 q1 v

q0 v Two one-sided tests (TOST) H0

v 0 1 q1 v

q0 v and H0

v 0 1 q1 v

q0 v Construct one-sided point-wise CIs: U1 v and L1 v reject H0 at level

v 0 1 U1

v We reject H0 if we reject both H0 and H0

Choose delta 12 | 15

Assessing the Distribuonal Parallel Trends Assumpon

• Addional pre-treatment me-periods ❀ Assessment of the distribuonal PT
• If the assumpon holds for the pre-treatment, we have ˜

q1(v) − ˜ q0(v) = 0 ˜ qd(v) = FY∗

d0(F−1

Y∗

d1 (v))

• Test the following non-equivalence null (i.e., H0: Assumpon does not hold)

H0 : max

v∈[0,1] |˜

q1(v) − ˜ q0(v)| > δ vs H1 : max

v∈[0,1] |˜

q1(v) − ˜ q0(v)| ≤ δ Two one-sided tests (TOST) H0

v 0 1 q1 v

q0 v and H0

v 0 1 q1 v

q0 v Construct one-sided point-wise CIs: U1 v and L1 v reject H0 at level

v 0 1 U1

v We reject H0 if we reject both H0 and H0

Choose delta 12 | 15

Assessing the Distribuonal Parallel Trends Assumpon

• Addional pre-treatment me-periods ❀ Assessment of the distribuonal PT
• If the assumpon holds for the pre-treatment, we have ˜

q1(v) − ˜ q0(v) = 0 ˜ qd(v) = FY∗

d0(F−1

Y∗

d1 (v))

• Test the following non-equivalence null (i.e., H0: Assumpon does not hold)

H0 : max

v∈[0,1] |˜

q1(v) − ˜ q0(v)| > δ vs H1 : max

v∈[0,1] |˜

q1(v) − ˜ q0(v)| ≤ δ ❀ Two one-sided tests (TOST) H+

0 : max v∈[0,1]{˜

q1(v) − ˜ q0(v)} > δ and H−

0 : max v∈[0,1]{˜

q1(v) − ˜ q0(v)} < −δ Construct one-sided point-wise CIs: U1 v and L1 v reject H0 at level

v 0 1 U1

v We reject H0 if we reject both H0 and H0

Choose delta 12 | 15

Assessing the Distribuonal Parallel Trends Assumpon

• Addional pre-treatment me-periods ❀ Assessment of the distribuonal PT
• If the assumpon holds for the pre-treatment, we have ˜

q1(v) − ˜ q0(v) = 0 ˜ qd(v) = FY∗

d0(F−1

Y∗

d1 (v))

• Test the following non-equivalence null (i.e., H0: Assumpon does not hold)

H0 : max

v∈[0,1] |˜

q1(v) − ˜ q0(v)| > δ vs H1 : max

v∈[0,1] |˜

q1(v) − ˜ q0(v)| ≤ δ ❀ Two one-sided tests (TOST) H+

0 : max v∈[0,1]{˜

q1(v) − ˜ q0(v)} > δ and H−

0 : max v∈[0,1]{˜

q1(v) − ˜ q0(v)} < −δ

• Construct one-sided point-wise CIs:

U1−α(v) and L1−α(v) reject: H+

0 at α level ⇐

⇒ max

v∈[0,1]

• U1−α(v) < δ

❀ We reject H0 if we reject both H+

0 and H−

CI construcon Choose delta 12 | 15

Using Three-wave Panel to Assess the Assumpon

• Some respondents of CCES 2010–12 panel are reinterviewed in 2014

Focus on 2817 respondents who

Did not have the “prior exposure” as of 2010 Were not treated between 2010 and 2012

• Approx. 25% of them (667) are newly treated between 2012 and 2014

28 shoongs

Assess the distribuonal parallel trends assumpon using 2010–12

13 | 15

Using Three-wave Panel to Assess the Assumpon

• Some respondents of CCES 2010–12 panel are reinterviewed in 2014
• Focus on 2817 respondents who
• Did not have the “prior exposure” as of 2010
• Were not treated between 2010 and 2012
• Approx. 25% of them (667) are newly treated between 2012 and 2014

28 shoongs

Assess the distribuonal parallel trends assumpon using 2010–12

13 | 15

Using Three-wave Panel to Assess the Assumpon

• Some respondents of CCES 2010–12 panel are reinterviewed in 2014
• Focus on 2817 respondents who
• Did not have the “prior exposure” as of 2010
• Were not treated between 2010 and 2012
• Approx. 25% of them (667) are newly treated between 2012 and 2014
• 28 shoongs

Assess the distribuonal parallel trends assumpon using 2010–12

13 | 15

Using Three-wave Panel to Assess the Assumpon

• Some respondents of CCES 2010–12 panel are reinterviewed in 2014
• Focus on 2817 respondents who
• Did not have the “prior exposure” as of 2010
• Were not treated between 2010 and 2012
• Approx. 25% of them (667) are newly treated between 2012 and 2014
• 28 shoongs
• Assess the distribuonal parallel trends assumpon using 2010–12

13 | 15

Using Three-wave Panel to Assess the Assumpon

• Some respondents of CCES 2010–12 panel are reinterviewed in 2014
• Focus on 2817 respondents who
• Did not have the “prior exposure” as of 2010
• Were not treated between 2010 and 2012
• Approx. 25% of them (667) are newly treated between 2012 and 2014
• 28 shoongs
• Assess the distribuonal parallel trends assumpon using 2010–12

0.0 0.2 0.4 0.6 0.8 1.0 −0.05 0.00 0.05

Test Statistic (Pre−Treatment Outcome)

Quantile (v) t ^ (v) = q ~

1 (v) − q

~

0 (v)

equivalence threshold = 0.054 minimum threshold = 0.039 tmax = 0.021 13 | 15

Using Three-wave Panel to Assess the Assumpon

• Some respondents of CCES 2010–12 panel are reinterviewed in 2014
• Focus on 2817 respondents who
• Did not have the “prior exposure” as of 2010
• Were not treated between 2010 and 2012
• Approx. 25% of them (667) are newly treated between 2012 and 2014
• 28 shoongs
• Assess the distribuonal parallel trends assumpon using 2010–12

0.0 0.2 0.4 0.6 0.8 1.0 −0.05 0.00 0.05

Test Statistic (Pre−Treatment Outcome)

Quantile (v) t ^ (v) = q ~

1 (v) − q

~

0 (v)

equivalence threshold = 0.054 minimum threshold = 0.039 tmax = 0.021

−0.05 0.00 0.05 0.10

Effect in 2014 (CCES 10−12−14 Subsamples)

Difference in probabilities

• No prior exposure & Untreated in 2012

n = 2812 Less strict Keep the same More strict 13 | 15

Concluding Remarks

• Dierence-in-dierences is widely used in social science research

Linearity assumpon in DID is inappropriate for ordinal outcomes Propose a latent variable framework to address the issue Revisit the recent debate on the relaonship between the mass shoongs and the atudes toward gun control:

Find that eects are concentrated among Democrats who do not have “prior exposure” to shoongs and among Independents

14 | 15

Concluding Remarks

• Dierence-in-dierences is widely used in social science research
• Linearity assumpon in DID is inappropriate for ordinal outcomes

Propose a latent variable framework to address the issue Revisit the recent debate on the relaonship between the mass shoongs and the atudes toward gun control:

Find that eects are concentrated among Democrats who do not have “prior exposure” to shoongs and among Independents

14 | 15

Concluding Remarks

• Dierence-in-dierences is widely used in social science research
• Linearity assumpon in DID is inappropriate for ordinal outcomes
• Propose a latent variable framework to address the issue

Revisit the recent debate on the relaonship between the mass shoongs and the atudes toward gun control:

Find that eects are concentrated among Democrats who do not have “prior exposure” to shoongs and among Independents

14 | 15

Concluding Remarks

• Dierence-in-dierences is widely used in social science research
• Linearity assumpon in DID is inappropriate for ordinal outcomes
• Propose a latent variable framework to address the issue
• Revisit the recent debate on the relaonship between the mass shoongs

and the atudes toward gun control:

Find that eects are concentrated among Democrats who do not have “prior exposure” to shoongs and among Independents

14 | 15

Concluding Remarks

• Dierence-in-dierences is widely used in social science research
• Linearity assumpon in DID is inappropriate for ordinal outcomes
• Propose a latent variable framework to address the issue
• Revisit the recent debate on the relaonship between the mass shoongs

and the atudes toward gun control:

• Find that eects are concentrated among Democrats who do not have “prior

exposure” to shoongs and among Independents

14 | 15

References

• Yamauchi, Soichiro. (2020). “Dierence-in-Dierences for Ordinal Outcomes:

Applicaon to the Eect of Mass Shoongs on Atudes towards Gun Control” Working Paper.

• orddid: R package for implemenng the dierence-in-dierence for the
• rdinal outcomes. Available at github.com/soichiroy/orddid

Send comments and suggesons to syamauchi@g.harvard.edu For more informaon soichiroy.github.io

15 | 15

Addional Results

1 | 16

Treang as Connuous Outcome

• Consider a cross-seconal seng:

ζj = Pr(Yi(1) = j) − Pr(Yi(0) = j)

• Rescale the outcome:

Yi = Yi/(J − 1)

• The dierence-in-means esamtor on

Yi can be wrien as

• τDiM =

J

∑

j=1

(J − j)−1 ζj where

• τDiM = 1

n1

n

∑

i=1

DiYi − 1 n0

n

∑

i=1

(1 − Di)Yi ❀ Weighted average of ζj with weights are 1/(J − j)

• This can potenally cancel out the eects: E.g.,

ζ1 > 0 and ζ2 < 0

2 | 16

Invariance of Causal Eect to Choice of Cutos

• Proposion: Suppose Yit ∈ J ≡ {0, 1, 2} and U ∼ N(0, 1). Let κ and κ′ be

the dierent sets of cutos. Then, for all j ∈ J .

• ζj(κ) =

ζj(κ′)

• Intuion:

(1) Assumpon is imposed on the quanle scale (i.e., distribuonal PT) ❀ counterfactual distribuon is idened as long as quanle info. is preserved (2) Changing cutos aect mean & scale ❀ transform the latent variables (3) But quanle informaon is preserved, Pr(Y∗ ≤ κ1) = Pr(˜ Y∗ ≤ κ′

1)

∫ κ2

κ1

φ((y∗ − µ00)/σ00)dy∗ = Pr(Y00 = 1)

• bserved prob.

= ∫ κ′

2

κ′

1

φ((y∗ − µ′

00)/σ′ 00)dy∗

❀ uniquely recovers the counterfactual distribuon Y∗

11

Back 3 | 16

Idencaon of Latent Variables

• Suppose that the cutos are xed at κ1 and κ2 for Ydt = j ∈ {0, 1, 2}. Then,

µdt and σdt in Y∗

dt ∼ µdt + σdtU are uniquely idened from the observed

probability distribuon.

• Proof: Suppose that U has the density fU(u). Then, we can form a non-linear

system of equaons Pr(Ydt = 0) = ∫ κ1

−∞

fU((y∗ − µdt)/σdt)dy∗ Pr(Ydt = 2) = ∫ ∞

κ2

fU((y∗ − µdt)/σdt)dy∗ which are sucient for esmang µ and σ.

Back 4 | 16

Alternave Formula of Idencaon

• Suppose Ydt = j ∈ {0, 1, 2}. Let v1 = F01(κ1) and v2 = F01(κ2) where κ is a

set of xed cutos.

• Under the assumpons, we idenfy µ11 and σ11 by the following system of

non-linear equaons: q0(v1) = ∫ F−1

10 (v1)

−∞

fU((y∗ − µ11)/σ11)dy∗ q0(v2) = ∫ F−1

10 (v2)

−∞

fU((y∗ − µ11)/σ11)dy∗.

5 | 16

Construcng Condence Intervals for Tesng

• Let t(v) = ˜

q1(v) − ˜ q0(v)

• Point-wise upper and lower (1 − α) level condence intervals:
• U1−α(v) = ˆ

t(v) + Φ−1(1 − α) √ Var(ˆ t(v))/n

• L1−α(v) = ˆ

t(v) − Φ−1(1 − α) √ Var(ˆ t(v))/n

• Proposion

Pr ( max

v∈[0,1] t(v) ≤ max v′∈[0,1]

• U1−α(v′)

) ≥ 1 − α Pr ( min

v∈[0,1] t(v) ≥ min v′∈[0,1]

• L1−α(v′)

) ≥ 1 − α

Back 6 | 16

Choosing Delta

• Value of δ reect the admissible level of “non equivalence”

❀ Larger values of δ correspond to lenient thresholds

• Calibrate δ based on the rejecon threshold for the KS test

δn = min {√ − log(α)/2 √ n1 + n0 n1n0 , 1 } and take α = 0.05

• Can report the equivalence CI: minimum possible value of δ at α level

δmin,n = max

v∈[0,1]

{ | U1−α(v)|, | L1−α(v)| } ❀ Equivalence CI is given by [−δmin,n, δmin,n]

Back 7 | 16

Addional Empirical Analysis

8 | 16

Treang as a Connuous Outcome

• 0.55

0.60 0.65 0.70

Treat as Continuous Outcome

Average of 'Normalized' Outcome 2010 2012 2014

• Treatment Group

Control Group Back 9 | 16

Outcome Distribuons by Sub-Group

Distribution of Outcome in 2010

Freq 0.2 0.4 0.6 0.8

Full Sample n = 16553 1 2 No Prior Exposure n = 7123 1 2 Prior Exposure n = 9430 1 2 Democrat n = 5526 1 2 Republican n = 5126 1 2 Independent n = 4996 1 2 Treated Control

Distribution of Outcome in 2012

Freq 0.2 0.4 0.6 0.8

Full Sample n = 16553 1 2 No Prior Exposure n = 7123 1 2 Prior Exposure n = 9430 1 2 Democrat n = 5526 1 2 Republican n = 5126 1 2 Independent n = 4996 1 2 Treated Control 10 | 16

Party ID based on 7-point Scale Measure

−0.10 −0.05 0.00 0.05 0.10 Difference in probabilities

• n = 3914

Strong Dem

• n = 1612

Not Very Strong Dem

• n = 1453

Lean Dem

• n = 1754

Independent

• n = 2526

Lean Rep

• n = 1552

Not Very Strong Rep

• n = 3574

Strong Rep

Effect in 2012 (CCES 2010−12)

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

11 | 16

Dierent Distance Threshold: 25 Miles

−0.10 −0.05 0.00 0.05 0.10 Difference in probabilities

• Full Sample

n = 16553

• No Prior Exposure

n = 13517

• Prior Exposure

n = 3036

• Democrat

n = 5526

• Republican

n = 5126

• Independent

n = 4996

Effect in 2012 (CCES 2010−12) with 25mile Cutoff

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

Back 12 | 16

Dierent Distance Threshold: 25 Miles

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 Difference in probabilities

• Democrat

n = 4404

• Democrat

n = 1122

• Republican

n = 4281

• Republican

n = 845

• Independent

n = 4092

• Independent

n = 904

No Prior Exposure Prior Exposure

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

Back 12 | 16

Three-wave Sub-sample: Eect in 2012

−0.10 −0.05 0.00 0.05 0.10 Difference in probabilities

• Full Sample

n = 8512

• No Prior Exposure

n = 3555

• Prior Exposure

n = 4957

• Democrat

n = 3041

• Independent

n = 2968

• Republican

n = 2503

Effect in 2012 (CCES 2010−12−14)

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

Back 13 | 16

Three-wave Sub-sample: Eect in 2012

−0.20 −0.10 0.00 0.05 0.10 0.15 Difference in probabilities

• Democrat

n = 1133

• Democrat

n = 1908

• Independent

n = 1302

• Independent

n = 1666

• Republican

n = 1120

• Republican

n = 1383

No Prior Exposure Prior Exposure

• Effect on 'Less Strict'

Effect on 'Keep the Same' Effect on 'More Strict'

Back 13 | 16

Cumulave Eect: Two-wave Panel

∆j = Pr(Yi1(1) ≥ j | Di = 1) − Pr(Yi1(0) ≥ j | Di = 1)

−0.06 −0.02 0.00 0.02 0.04 0.06

Effect in 2012 (CCES 2010−12)

Difference in probabilities

• Full Sample

n = 16553 No Prior Exposure n = 7123 Prior Exposure n = 9430 Democrat n = 5526 Republican n = 5126 Independent n = 4996 n = 562

• ∆2 = Pr(Y(1) = more strict | D=1) − Pr(Y(0) = more strict | D=1)

∆1 = Pr(Y(1) >= kept as they are | D=1) − Pr(Y(0) >= kept as they are | D=1)

Back 14 | 16

Bound Results: Two-wave Panel

τ = Pr(Yi1(1) ≥ Yi1(0) | Di = 1), and η = Pr(Yi1(1) > Yi1(0) | Di = 1)

Effect in 2012 (CCES 2010−12)

Probabilities 0.2 0.4 0.6 0.8 1

Full Sample n = 16553 No Prior Exposure n = 7123 Prior Exposure n = 9430 Democrat n = 5526 Republican n = 5126 Independent n = 4996 n = 562 τ = Pr(Y(1) >= Y(0) | D=1) η = Pr(Y(1) > Y(0) | D=1)

Back 15 | 16

Denion of Mass Shoongs

Cases involving the following:

• 1. Firearms as the primary weapon used,
• 2. Aacks on non-family members of the general public
• 3. Aacks in which at least three or more individuals were injured or killed

Back 16 | 16