Dichotomize the outcome Mul�ple dis�nct parallel trends assump�ons Ordered probit/logit Iden��ca�on assump�ons are not explicitly stated Propose: A latent variable framework for DiD for the ordinal outcomes Applica�on: Revisit a recent debate on the rela�onship between the mass shoo�ngs and the a�tude toward gun control Di�erence-in-Di�erences Design in Observa�onal Studies • Di�erence-in-di�erences for causal inference in observa�onal studies • Adjust for the �me-invariant confounders by u�lizing the past outcome • Key iden��ca�on assump�on: parallel trends assump�on ❀ Iden�cal trends across the treated & the control without the treatment ❀ Relies on the di�erences between two poten�al outcomes: Linearity • In social science, many outcomes are measured on an ordinal scale (e.g., survey ques�ons) ❀ “di�erences” are not well de�ned • Problems in common prac�ces: • Treat as a con�nuous variable ❀ Di�cult to interpret + linearity 2 | 15
Ordered probit/logit Iden��ca�on assump�ons are not explicitly stated Propose: A latent variable framework for DiD for the ordinal outcomes Applica�on: Revisit a recent debate on the rela�onship between the mass shoo�ngs and the a�tude toward gun control Di�erence-in-Di�erences Design in Observa�onal Studies • Di�erence-in-di�erences for causal inference in observa�onal studies • Adjust for the �me-invariant confounders by u�lizing the past outcome • Key iden��ca�on assump�on: parallel trends assump�on ❀ Iden�cal trends across the treated & the control without the treatment ❀ Relies on the di�erences between two poten�al outcomes: Linearity • In social science, many outcomes are measured on an ordinal scale (e.g., survey ques�ons) ❀ “di�erences” are not well de�ned • Problems in common prac�ces: • Treat as a con�nuous variable ❀ Di�cult to interpret + linearity • Dichotomize the outcome ❀ Mul�ple dis�nct parallel trends assump�ons 2 | 15
Propose: A latent variable framework for DiD for the ordinal outcomes Applica�on: Revisit a recent debate on the rela�onship between the mass shoo�ngs and the a�tude toward gun control Di�erence-in-Di�erences Design in Observa�onal Studies • Di�erence-in-di�erences for causal inference in observa�onal studies • Adjust for the �me-invariant confounders by u�lizing the past outcome • Key iden��ca�on assump�on: parallel trends assump�on ❀ Iden�cal trends across the treated & the control without the treatment ❀ Relies on the di�erences between two poten�al outcomes: Linearity • In social science, many outcomes are measured on an ordinal scale (e.g., survey ques�ons) ❀ “di�erences” are not well de�ned • Problems in common prac�ces: • Treat as a con�nuous variable ❀ Di�cult to interpret + linearity • Dichotomize the outcome ❀ Mul�ple dis�nct parallel trends assump�ons • Ordered probit/logit ❀ Iden��ca�on assump�ons are not explicitly stated 2 | 15
Applica�on: Revisit a recent debate on the rela�onship between the mass shoo�ngs and the a�tude toward gun control Di�erence-in-Di�erences Design in Observa�onal Studies • Di�erence-in-di�erences for causal inference in observa�onal studies • Adjust for the �me-invariant confounders by u�lizing the past outcome • Key iden��ca�on assump�on: parallel trends assump�on ❀ Iden�cal trends across the treated & the control without the treatment ❀ Relies on the di�erences between two poten�al outcomes: Linearity • In social science, many outcomes are measured on an ordinal scale (e.g., survey ques�ons) ❀ “di�erences” are not well de�ned • Problems in common prac�ces: • Treat as a con�nuous variable ❀ Di�cult to interpret + linearity • Dichotomize the outcome ❀ Mul�ple dis�nct parallel trends assump�ons • Ordered probit/logit ❀ Iden��ca�on assump�ons are not explicitly stated • Propose: A latent variable framework for DiD for the ordinal outcomes 2 | 15
Di�erence-in-Di�erences Design in Observa�onal Studies • Di�erence-in-di�erences for causal inference in observa�onal studies • Adjust for the �me-invariant confounders by u�lizing the past outcome • Key iden��ca�on assump�on: parallel trends assump�on ❀ Iden�cal trends across the treated & the control without the treatment ❀ Relies on the di�erences between two poten�al outcomes: Linearity • In social science, many outcomes are measured on an ordinal scale (e.g., survey ques�ons) ❀ “di�erences” are not well de�ned • Problems in common prac�ces: • Treat as a con�nuous variable ❀ Di�cult to interpret + linearity • Dichotomize the outcome ❀ Mul�ple dis�nct parallel trends assump�ons • Ordered probit/logit ❀ Iden��ca�on assump�ons are not explicitly stated • Propose: A latent variable framework for DiD for the ordinal outcomes • Applica�on: Revisit a recent debate on the rela�onship between the mass shoo�ngs and the a�tude toward gun control 2 | 15
Introduce a latent variable framework Extend the latent u�lity representa�on of the standard probit/logit Apply the assump�on by Athey & Imbens (2006) on the latent variable scale Assumes temporal changes in quan�les are iden�cal across two groups Avoid imposing the linearity assump�on in the standard DiD Derive a diagnos�c with one addi�onal pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool 3 | 15
Extend the latent u�lity representa�on of the standard probit/logit Apply the assump�on by Athey & Imbens (2006) on the latent variable scale Assumes temporal changes in quan�les are iden�cal across two groups Avoid imposing the linearity assump�on in the standard DiD Derive a diagnos�c with one addi�onal pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework 3 | 15
Apply the assump�on by Athey & Imbens (2006) on the latent variable scale Assumes temporal changes in quan�les are iden�cal across two groups Avoid imposing the linearity assump�on in the standard DiD Derive a diagnos�c with one addi�onal pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework • Extend the latent u�lity representa�on of the standard probit/logit 3 | 15
Assumes temporal changes in quan�les are iden�cal across two groups Avoid imposing the linearity assump�on in the standard DiD Derive a diagnos�c with one addi�onal pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework • Extend the latent u�lity representa�on of the standard probit/logit • Apply the assump�on by Athey & Imbens (2006) on the latent variable scale 3 | 15
Avoid imposing the linearity assump�on in the standard DiD Derive a diagnos�c with one addi�onal pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework • Extend the latent u�lity representa�on of the standard probit/logit • Apply the assump�on by Athey & Imbens (2006) on the latent variable scale • Assumes temporal changes in quan�les are iden�cal across two groups 3 | 15
Derive a diagnos�c with one addi�onal pre-treatment period Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework • Extend the latent u�lity representa�on of the standard probit/logit • Apply the assump�on by Athey & Imbens (2006) on the latent variable scale • Assumes temporal changes in quan�les are iden�cal across two groups ❀ Avoid imposing the linearity assump�on in the standard DiD 3 | 15
Analogous to the pre-treatment trend check in the standard DiD Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework • Extend the latent u�lity representa�on of the standard probit/logit • Apply the assump�on by Athey & Imbens (2006) on the latent variable scale • Assumes temporal changes in quan�les are iden�cal across two groups ❀ Avoid imposing the linearity assump�on in the standard DiD • Derive a diagnos�c with one addi�onal pre-treatment period 3 | 15
Equivalence based test to assess the plausibility of the assump�on Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework • Extend the latent u�lity representa�on of the standard probit/logit • Apply the assump�on by Athey & Imbens (2006) on the latent variable scale • Assumes temporal changes in quan�les are iden�cal across two groups ❀ Avoid imposing the linearity assump�on in the standard DiD • Derive a diagnos�c with one addi�onal pre-treatment period • Analogous to the pre-treatment trend check in the standard DiD 3 | 15
Di�erence-in-Di�erences for Ordinal Outcomes Contribu�ons: New iden��ca�on strategy & diagnos�c tool • Introduce a latent variable framework • Extend the latent u�lity representa�on of the standard probit/logit • Apply the assump�on by Athey & Imbens (2006) on the latent variable scale • Assumes temporal changes in quan�les are iden�cal across two groups ❀ Avoid imposing the linearity assump�on in the standard DiD • Derive a diagnos�c with one addi�onal pre-treatment period • Analogous to the pre-treatment trend check in the standard DiD ❀ Equivalence based test to assess the plausibility of the assump�on 3 | 15
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. Proximity to the shoo�ngs as a treatment (dichotomized by 100 miles) Ordinal survey outcome: less-strict , keep-the-same and more-strict Mass Shoo�ngs and A�tudes toward Gun Control • Recent debate on the topic (Barney & Scha�ner, 2019; Hartman& Newman, 2019; Newman & Hartman, 2019) 4 | 15
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. Ordinal survey outcome: less-strict , keep-the-same and more-strict Mass Shoo�ngs and A�tudes toward Gun Control • Recent debate on the topic (Barney & Scha�ner, 2019; Hartman& Newman, 2019; Newman & Hartman, 2019) • Proximity to the shoo�ngs as a treatment (dichotomized by 100 miles) 4 | 15
Mass Shoo�ngs and A�tudes toward Gun Control • Recent debate on the topic (Barney & Scha�ner, 2019; Hartman& Newman, 2019; Newman & Hartman, 2019) • Proximity to the shoo�ngs 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
(0) Less Strict; (1) Kept As They Are; (2) More Strict. Mass Shoo�ngs and A�tudes toward Gun Control • Recent debate on the topic (Barney & Scha�ner, 2019; Hartman& Newman, 2019; Newman & Hartman, 2019) • Proximity to the shoo�ngs 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? 'more−strict' as 1 0.50 Treatment Group ● ● Pr(Y = 1 | D = d) 0.45 ● ● 0.40 Control Group ● ● 0.35 2010 2012 2014 4 | 15
(0) Less Strict; (1) Kept As They Are; (2) More Strict. Mass Shoo�ngs and A�tudes toward Gun Control • Recent debate on the topic (Barney & Scha�ner, 2019; Hartman& Newman, 2019; Newman & Hartman, 2019) • Proximity to the shoo�ngs 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? 'keep−the−same' & 'more−strict' as 1 'more−strict' as 1 0.50 0.90 Treatment Group ● ● Pr(Y = 1 | D = d) Pr(Y = 1 | D = d) 0.45 0.85 ● ● ● ● 0.40 0.80 ● Control Group ● ● ● ● ● 0.35 0.75 2010 2012 2014 2010 2012 2014 con�nuous 4 | 15
Binary treatment: D i 0 1 Poten�al outcome: Y it d for d 0 1 Es�mand: Di�erences in choice probabili�es for the treated j Example: Di�erence in prob. of choosing more-strict under two condi�ons Pr Y i 1 1 j D i 1 is observed from the data: Pr Y i 1 j D i 1 Need to iden�fy Pr Y i 1 0 j D i 1 with addi�onal assump�ons DiD for Ordinal Outcomes: Setup • Observed outcome: Y it ∈ { 0 , . . . , J − 1 } for i = 1 , . . . , n and t ∈ { 0 , 1 } 5 | 15
Poten�al outcome: Y it d for d 0 1 Es�mand: Di�erences in choice probabili�es for the treated j Example: Di�erence in prob. of choosing more-strict under two condi�ons Pr Y i 1 1 j D i 1 is observed from the data: Pr Y i 1 j D i 1 Need to iden�fy Pr Y i 1 0 j D i 1 with addi�onal assump�ons DiD for Ordinal Outcomes: Setup • Observed outcome: Y it ∈ { 0 , . . . , J − 1 } for i = 1 , . . . , n and t ∈ { 0 , 1 } • Binary treatment: D i ∈ { 0 , 1 } 5 | 15
Es�mand: Di�erences in choice probabili�es for the treated j Example: Di�erence in prob. of choosing more-strict under two condi�ons Pr Y i 1 1 j D i 1 is observed from the data: Pr Y i 1 j D i 1 Need to iden�fy Pr Y i 1 0 j D i 1 with addi�onal assump�ons DiD for Ordinal Outcomes: Setup • Observed outcome: Y it ∈ { 0 , . . . , J − 1 } for i = 1 , . . . , n and t ∈ { 0 , 1 } • Binary treatment: D i ∈ { 0 , 1 } • Poten�al outcome: Y it ( d ) for d ∈ { 0 , 1 } 5 | 15
Example: Di�erence in prob. of choosing more-strict under two condi�ons Pr Y i 1 1 j D i 1 is observed from the data: Pr Y i 1 j D i 1 Need to iden�fy Pr Y i 1 0 j D i 1 with addi�onal assump�ons DiD for Ordinal Outcomes: Setup • Observed outcome: Y it ∈ { 0 , . . . , J − 1 } for i = 1 , . . . , n and t ∈ { 0 , 1 } • Binary treatment: D i ∈ { 0 , 1 } • Poten�al outcome: Y it ( d ) for d ∈ { 0 , 1 } • Es�mand: Di�erences in choice probabili�es for the treated ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) 5 | 15
Pr Y i 1 1 j D i 1 is observed from the data: Pr Y i 1 j D i 1 Need to iden�fy Pr Y i 1 0 j D i 1 with addi�onal assump�ons DiD for Ordinal Outcomes: Setup • Observed outcome: Y it ∈ { 0 , . . . , J − 1 } for i = 1 , . . . , n and t ∈ { 0 , 1 } • Binary treatment: D i ∈ { 0 , 1 } • Poten�al outcome: Y it ( d ) for d ∈ { 0 , 1 } • Es�mand: Di�erences in choice probabili�es for the treated ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Example: Di�erence in prob. of choosing more-strict under two condi�ons 5 | 15
Need to iden�fy Pr Y i 1 0 j D i 1 with addi�onal assump�ons DiD for Ordinal Outcomes: Setup • Observed outcome: Y it ∈ { 0 , . . . , J − 1 } for i = 1 , . . . , n and t ∈ { 0 , 1 } • Binary treatment: D i ∈ { 0 , 1 } • Poten�al outcome: Y it ( d ) for d ∈ { 0 , 1 } • Es�mand: Di�erences in choice probabili�es for the treated ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Example: Di�erence in prob. of choosing more-strict under two condi�ons • Pr ( Y i 1 ( 1 ) = j | D i = 1 ) is observed from the data: Pr ( Y i 1 = j | D i = 1 ) 5 | 15
DiD for Ordinal Outcomes: Setup • Observed outcome: Y it ∈ { 0 , . . . , J − 1 } for i = 1 , . . . , n and t ∈ { 0 , 1 } • Binary treatment: D i ∈ { 0 , 1 } • Poten�al outcome: Y it ( d ) for d ∈ { 0 , 1 } • Es�mand: Di�erences in choice probabili�es for the treated ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Example: Di�erence in prob. of choosing more-strict under two condi�ons • Pr ( Y i 1 ( 1 ) = j | D i = 1 ) is observed from the data: Pr ( Y i 1 = j | D i = 1 ) • Need to iden�fy Pr ( Y i 1 ( 0 ) = j | D i = 1 ) with addi�onal assump�ons 5 | 15
Y 11 is the counterfactual outcome Latent “u�lity” genera�ng the ordinal outcome: Y dt Index model: Mapping Y dt to Y dt 0 if Y dt 1 0 Y dt j if Y dt j 1 j J 1 if Y dt J J 1 Loca�on-scale family: Imposing distribu�on on Y dt Y dt U dt dt loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d 6 | 15
Latent “u�lity” genera�ng the ordinal outcome: Y dt Index model: Mapping Y dt to Y dt 0 if Y dt 1 0 Y dt j if Y dt j 1 j J 1 if Y dt J J 1 Loca�on-scale family: Imposing distribu�on on Y dt Y dt U dt dt loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d ❀ Y 11 is the counterfactual outcome 6 | 15
Index model: Mapping Y dt to Y dt 0 if Y dt 1 0 Y dt j if Y dt j 1 j J 1 if Y dt J J 1 Loca�on-scale family: Imposing distribu�on on Y dt Y dt U dt dt loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d ❀ Y 11 is the counterfactual outcome • Latent “u�lity” genera�ng the ordinal outcome: Y ∗ dt ∈ R 6 | 15
Loca�on-scale family: Imposing distribu�on on Y dt Y dt U dt dt loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d ❀ Y 11 is the counterfactual outcome • Latent “u�lity” genera�ng the ordinal outcome: Y ∗ dt ∈ R • Index model: Mapping Y ∗ dt to Y dt κ 1 ≥ Y ∗ dt ≥ κ 0 0 if κ j + 1 ≥ Y ∗ Y dt = dt ≥ κ j j if κ J ≥ Y ∗ J − 1 if dt ≥ κ J − 1 6 | 15
Loca�on-scale family: Imposing distribu�on on Y dt Y dt U dt dt loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d ❀ Y 11 is the counterfactual outcome • Latent “u�lity” genera�ng the ordinal outcome: Y ∗ dt ∈ R • Index model: Mapping Y ∗ dt to Y dt κ 1 ≥ Y ∗ 0 if dt ≥ κ 0 κ j + 1 ≥ Y ∗ Y dt = j if dt ≥ κ j κ J ≥ Y ∗ J − 1 dt ≥ κ J − 1 if Y dt = less−strict κ 0 κ 1 κ 2 κ 3 Latent Utility Y * 6 | 15
Loca�on-scale family: Imposing distribu�on on Y dt Y dt U dt dt loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d ❀ Y 11 is the counterfactual outcome • Latent “u�lity” genera�ng the ordinal outcome: Y ∗ dt ∈ R • Index model: Mapping Y ∗ dt to Y dt κ 1 ≥ Y ∗ 0 if dt ≥ κ 0 κ j + 1 ≥ Y ∗ Y dt = j if dt ≥ κ j κ J ≥ Y ∗ J − 1 dt ≥ κ J − 1 if Y dt = keep−the−same κ 0 κ 1 κ 2 κ 3 Latent Utility Y * 6 | 15
Loca�on-scale family: Imposing distribu�on on Y dt Y dt U dt dt loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d ❀ Y 11 is the counterfactual outcome • Latent “u�lity” genera�ng the ordinal outcome: Y ∗ dt ∈ R • Index model: Mapping Y ∗ dt to Y dt κ 1 ≥ Y ∗ 0 if dt ≥ κ 0 κ j + 1 ≥ Y ∗ Y dt = j if dt ≥ κ j κ J ≥ Y ∗ J − 1 dt ≥ κ J − 1 if Y dt = more−strict κ 0 κ 1 κ 2 κ 3 Latent Utility Y * 6 | 15
Latent Variable Formula�on • Ordinal outcome: Y dt ∼ Y it ( 0 ) | D i = d ❀ Y 11 is the counterfactual outcome • Latent “u�lity” genera�ng the ordinal outcome: Y ∗ dt ∈ R • Index model: Mapping Y ∗ dt to Y dt κ 1 ≥ Y ∗ dt ≥ κ 0 0 if κ j + 1 ≥ Y ∗ Y dt = dt ≥ κ j j if κ J ≥ Y ∗ J − 1 if dt ≥ κ J − 1 • Loca�on-scale family: Imposing distribu�on on Y ∗ dt Y ∗ dt ∼ µ dt + σ dt U ���� ���� loca�on scale where U belongs to a parametric family (e.g., normal, logis�c, t -dist.) 6 | 15
Proposi�on: the distribu�on of the counterfactual latent variable given by 01 00 10 01 and 11 10 11 00 10 00 When variances are constant , recovers the usual parallel trends form dt 11 10 01 00 Main Result • Distribu�onal parallel-trends assump�on (Athey & Imbens 2006) 00 ( F − 1 10 ( F − 1 F Y ∗ 01 ( v )) = F Y ∗ 11 ( v )) ∀ v ∈ [ 0 , 1 ] Y ∗ Y ∗ 7 | 15
Proposi�on: the distribu�on of the counterfactual latent variable given by 01 00 10 01 and 11 10 11 00 10 00 When variances are constant , recovers the usual parallel trends form dt 11 10 01 00 Main Result • Distribu�onal parallel-trends assump�on (Athey & Imbens 2006) 00 ( F − 1 10 ( F − 1 F Y ∗ 01 ( v )) = F Y ∗ 11 ( v )) ∀ v ∈ [ 0 , 1 ] Y ∗ Y ∗ � �� � � �� � = q 0 ( v ) = q 1 ( v ) Control Treated 1 1 * Y 00 * Y 10 0 0 −4 −2 0 2 4 −4 −2 0 2 4 Latent Utility Latent Utility 7 | 15
Proposi�on: the distribu�on of the counterfactual latent variable given by 01 00 10 01 and 11 10 11 00 10 00 When variances are constant , recovers the usual parallel trends form dt 11 10 01 00 Main Result • Distribu�onal parallel-trends assump�on (Athey & Imbens 2006) 00 ( F − 1 10 ( F − 1 F Y ∗ 01 ( v )) = F Y ∗ 11 ( v )) ∀ v ∈ [ 0 , 1 ] Y ∗ Y ∗ � �� � � �� � = q 0 ( v ) = q 1 ( v ) Control Treated 1 1 * Y 00 * * * Y 01 Y 10 Y 11 0 0 −4 −2 0 2 4 −4 −2 0 2 4 Latent Utility Latent Utility 7 | 15
Proposi�on: the distribu�on of the counterfactual latent variable given by 01 00 10 01 and 11 10 11 00 10 00 When variances are constant , recovers the usual parallel trends form dt 11 10 01 00 Main Result • Distribu�onal parallel-trends assump�on (Athey & Imbens 2006) 00 ( F − 1 10 ( F − 1 F Y ∗ 01 ( v )) = F Y ∗ 11 ( v )) ∀ v ∈ [ 0 , 1 ] Y ∗ Y ∗ � �� � � �� � = q 0 ( v ) = q 1 ( v ) Control Treated 1 1 q 0 (v) * Y 00 v * * * Y 01 Y 10 Y 11 0 0 −4 −2 0 2 4 −4 −2 0 2 4 Latent Utility Latent Utility 7 | 15
Proposi�on: the distribu�on of the counterfactual latent variable given by 01 00 10 01 and 11 10 11 00 10 00 When variances are constant , recovers the usual parallel trends form dt 11 10 01 00 Main Result • Distribu�onal parallel-trends assump�on (Athey & Imbens 2006) 00 ( F − 1 10 ( F − 1 F Y ∗ 01 ( v )) = F Y ∗ 11 ( v )) ∀ v ∈ [ 0 , 1 ] Y ∗ Y ∗ � �� � � �� � = q 0 ( v ) = q 1 ( v ) Control Treated 1 1 q 0 (v) q 1 (v) * Y 00 v v * * * Y 01 Y 10 Y 11 0 0 −4 −2 0 2 4 −4 −2 0 2 4 Latent Utility Latent Utility 7 | 15
When variances are constant , recovers the usual parallel trends form dt 11 10 01 00 Main Result • Distribu�onal parallel-trends assump�on (Athey & Imbens 2006) 00 ( F − 1 10 ( F − 1 F Y ∗ 01 ( v )) = F Y ∗ 11 ( v )) ∀ v ∈ [ 0 , 1 ] Y ∗ Y ∗ � �� � � �� � = q 0 ( v ) = q 1 ( v ) Control Treated 1 q 1 (v) 1 q 0 (v) * Y 00 v v * * * Y 01 Y 10 Y 11 0 0 −4 −2 0 2 4 −4 −2 0 2 4 Latent Utility Latent Utility • Proposi�on: the distribu�on of the counterfactual latent variable given by µ 11 = µ 10 + µ 01 − µ 00 σ 11 = σ 10 σ 01 , and σ 00 /σ 10 σ 00 7 | 15
Main Result • Distribu�onal parallel-trends assump�on (Athey & Imbens 2006) 00 ( F − 1 10 ( F − 1 F Y ∗ 01 ( v )) = F Y ∗ 11 ( v )) ∀ v ∈ [ 0 , 1 ] Y ∗ Y ∗ � �� � � �� � = q 0 ( v ) = q 1 ( v ) Control Treated 1 q 1 (v) 1 q 0 (v) * Y 00 v v * * * Y 01 Y 10 Y 11 0 0 −4 −2 0 2 4 −4 −2 0 2 4 Latent Utility Latent Utility • Proposi�on: the distribu�on of the counterfactual latent variable given by µ 11 = µ 10 + µ 01 − µ 00 σ 11 = σ 10 σ 01 , and σ 00 /σ 10 σ 00 • When variances are constant σ dt = σ , recovers the usual parallel trends form µ 11 − µ 10 = µ 01 − µ 00 7 | 15
Obtain variance es�mates by the block-bootstrap Es�mate parameters by MLE (e.g., variant of the ordered probit) Plug-in es�mator for the counter-factual distribu�on 01 00 10 01 and 11 10 11 00 10 00 Obtain causal es�mates: 1 J 1 j Cuto� Mean-Variance Es�ma�on • Impose a distribu�on on U (base distribu�on): U ∼ N ( 0 , 1 ) 8 | 15
Obtain variance es�mates by the block-bootstrap Plug-in es�mator for the counter-factual distribu�on 01 00 10 01 and 11 10 11 00 10 00 Obtain causal es�mates: 1 J 1 j Cuto� Mean-Variance Es�ma�on • Impose a distribu�on on U (base distribu�on): U ∼ N ( 0 , 1 ) ❀ Es�mate parameters by MLE (e.g., variant of the ordered probit) 8 | 15
Obtain variance es�mates by the block-bootstrap Obtain causal es�mates: 1 J 1 j Cuto� Mean-Variance Es�ma�on • Impose a distribu�on on U (base distribu�on): U ∼ N ( 0 , 1 ) ❀ Es�mate parameters by MLE (e.g., variant of the ordered probit) • Plug-in es�mator for the counter-factual distribu�on µ 10 + � µ 01 − � µ 00 σ 11 ← � σ 10 � σ 01 µ 11 ← � � , and � σ 00 / � � σ 10 σ 00 � 8 | 15
Obtain variance es�mates by the block-bootstrap Cuto� Mean-Variance Es�ma�on • Impose a distribu�on on U (base distribu�on): U ∼ N ( 0 , 1 ) ❀ Es�mate parameters by MLE (e.g., variant of the ordered probit) • Plug-in es�mator for the counter-factual distribu�on µ 10 + � µ 01 − � µ 00 σ 11 ← � σ 10 � σ 01 µ 11 ← � � , and � � σ 00 / � σ 10 � σ 00 • Obtain causal es�mates: � ζ = ( � ζ 1 , . . . , � ζ J − 1 ) ⊤ n ∑ { ( ) ( )} ζ j = 1 � D i 1 { Y i 1 = j } − Φ ( � κ j + 1 − � µ 11 ) / � σ 11 − Φ ( � κ j − � µ 11 ) / � σ 11 n 1 � �� � i = 1 � �� � = � Pr ( Y i 1 ( 0 )= j | D i = 1 ) = � Pr ( Y i 1 ( 1 )= j | D i = 1 ) 8 | 15
Obtain variance es�mates by the block-bootstrap Cuto� Mean-Variance Es�ma�on • Impose a distribu�on on U (base distribu�on): U ∼ N ( 0 , 1 ) ❀ Es�mate parameters by MLE (e.g., variant of the ordered probit) • Plug-in es�mator for the counter-factual distribu�on µ 10 + � µ 01 − � µ 00 σ 11 ← � σ 10 � σ 01 µ 11 ← � � , and � � σ 00 / � σ 10 � σ 00 • Obtain causal es�mates: � ζ = ( � ζ 1 , . . . , � ζ J − 1 ) ⊤ n ∑ { ( ) ( )} ζ j = 1 � D i 1 { Y i 1 = j } − Φ ( � κ j + 1 − � µ 11 ) / � σ 11 − Φ ( � κ j − � µ 11 ) / � σ 11 n 1 � �� � i = 1 � �� � = � Pr ( Y i 1 ( 0 )= j | D i = 1 ) = � Pr ( Y i 1 ( 1 )= j | D i = 1 ) 8 | 15
Obtain variance es�mates by the block-bootstrap Cuto� Mean-Variance Es�ma�on • Impose a distribu�on on U (base distribu�on): U ∼ N ( 0 , 1 ) ❀ Es�mate parameters by MLE (e.g., variant of the ordered probit) • Plug-in es�mator for the counter-factual distribu�on µ 10 + � µ 01 − � µ 00 σ 11 ← � σ 10 � σ 01 µ 11 ← � � , and � � σ 00 / � σ 10 � σ 00 • Obtain causal es�mates: � ζ = ( � ζ 1 , . . . , � ζ J − 1 ) ⊤ n ∑ { ( ) ( )} ζ j = 1 � D i 1 { Y i 1 = j } − Φ ( � κ j + 1 − � µ 11 ) / � σ 11 − Φ ( � κ j − � µ 11 ) / � σ 11 n 1 � �� � i = 1 � �� � = � Pr ( Y i 1 ( 0 )= j | D i = 1 ) = � Pr ( Y i 1 ( 1 )= j | D i = 1 ) 8 | 15
Es�ma�on • Impose a distribu�on on U (base distribu�on): U ∼ N ( 0 , 1 ) ❀ Es�mate parameters by MLE (e.g., variant of the ordered probit) • Plug-in es�mator for the counter-factual distribu�on µ 10 + � µ 01 − � µ 00 σ 11 ← � σ 10 � σ 01 µ 11 ← � � , and � � σ 00 / � σ 10 � σ 00 • Obtain causal es�mates: � ζ = ( � ζ 1 , . . . , � ζ J − 1 ) ⊤ n ∑ { ( ) ( )} ζ j = 1 � D i 1 { Y i 1 = j } − Φ ( � κ j + 1 − � µ 11 ) / � σ 11 − Φ ( � κ j − � µ 11 ) / � σ 11 n 1 � �� � i = 1 � �� � = � Pr ( Y i 1 ( 0 )= j | D i = 1 ) = � Pr ( Y i 1 ( 1 )= j | D i = 1 ) • Obtain variance es�mates by the block-bootstrap Cuto� Mean-Variance 8 | 15
16 mass-shoo�ngs coded at the zip-code level Approx. 30% of respondents (out of 16620) are treated Partisanship : 3-point scale party self-iden��ca�on Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Respondents are “treated” if living within 100 miles from the shoo�ngs Subgroups Previous studies used: 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. De�n�on of Mass Shoo�ngs Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on Study: 9 | 15
16 mass-shoo�ngs coded at the zip-code level Approx. 30% of respondents (out of 16620) are treated Partisanship : 3-point scale party self-iden��ca�on Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Respondents are “treated” if living within 100 miles from the shoo�ngs Subgroups Previous studies used: De�n�on of Mass Shoo�ngs Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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. 9 | 15
Partisanship : 3-point scale party self-iden��ca�on Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) 16 mass-shoo�ngs coded at the zip-code level Approx. 30% of respondents (out of 16620) are treated Subgroups Previous studies used: Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs De�n�on of Mass Shoo�ngs 9 | 15
Partisanship : 3-point scale party self-iden��ca�on Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Approx. 30% of respondents (out of 16620) are treated Subgroups Previous studies used: Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level De�n�on of Mass Shoo�ngs 9 | 15
Partisanship : 3-point scale party self-iden��ca�on Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Subgroups Previous studies used: Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level • Approx. 30% of respondents (out of 16620) are treated De�n�on of Mass Shoo�ngs 9 | 15
Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Partisanship : 3-point scale party self-iden��ca�on Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years Previous studies used: Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level • Approx. 30% of respondents (out of 16620) are treated • Subgroups De�n�on of Mass Shoo�ngs 9 | 15
Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years Previous studies used: Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level • Approx. 30% of respondents (out of 16620) are treated • Subgroups • Partisanship : 3-point scale party self-iden��ca�on De�n�on of Mass Shoo�ngs 9 | 15
Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Previous studies used: Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level • Approx. 30% of respondents (out of 16620) are treated • Subgroups • Partisanship : 3-point scale party self-iden��ca�on • Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years De�n�on of Mass Shoo�ngs 9 | 15
Ordinal logit with RE (NH19 and HN19) Linear two-way FE (BS19) Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level • Approx. 30% of respondents (out of 16620) are treated • Subgroups • Partisanship : 3-point scale party self-iden��ca�on • Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years • Previous studies used: De�n�on of Mass Shoo�ngs 9 | 15
Linear two-way FE (BS19) Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level • Approx. 30% of respondents (out of 16620) are treated • Subgroups • Partisanship : 3-point scale party self-iden��ca�on • Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years • Previous studies used: • Ordinal logit with RE (NH19 and HN19) De�n�on of Mass Shoo�ngs 9 | 15
Revisi�ng the Empirical Applica�on • Two-wave (2010-12) panel from Coopera�ve Congressional Elec�on 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 shoo�ngs • 16 mass-shoo�ngs coded at the zip-code level • Approx. 30% of respondents (out of 16620) are treated • Subgroups • Partisanship : 3-point scale party self-iden��ca�on • Prior-exposure : Living in areas with mass shoo�ngs in the past 10 years • Previous studies used: • Ordinal logit with RE (NH19 and HN19) • Linear two-way FE (BS19) De�n�on of Mass Shoo�ngs 9 | 15
Distribu�on of Outcome in 2010 & 2012 0: less-strict ; 1: keep-the-same ; 2: more-strict 0.8 0.7 Treated Control 0.6 2010 0.5 0.4 Freq 0.3 0.2 0.1 0 0 1 2 10 | 15 Detail
Distribu�on of Outcome in 2010 & 2012 0: less-strict ; 1: keep-the-same ; 2: more-strict 0.8 0.7 Treated Control 0.6 2010 2012 0.5 0.4 Freq 0.3 0.2 0.1 0 0 1 2 0 1 2 10 | 15 Detail
Results • Es�mate ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Block-bootstrap at the zip-code level 0.10 Effect in 2012 (CCES 2010−12) Full Sample n = 16553 0.05 Difference in probabilities 0.00 ● −0.05 −0.10 ● Effect on 'Less Strict' Effect on 'Keep the Same' Effect on 'More Strict' 25-miles PID-7 Cumula�ve e�ect Bounds 11 | 15
Results • Es�mate ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Block-bootstrap at the zip-code level 0.10 Effect in 2012 (CCES 2010−12) Full Sample No Prior Exposure Prior Exposure n = 16553 n = 7123 n = 9430 0.05 Difference in probabilities ● 0.00 ● ● −0.05 −0.10 ● Effect on 'Less Strict' Effect on 'Keep the Same' Effect on 'More Strict' 25-miles PID-7 Cumula�ve e�ect Bounds 11 | 15
Results • Es�mate ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Block-bootstrap at the zip-code level 0.10 Effect in 2012 (CCES 2010−12) Full Sample No Prior Exposure Prior Exposure Democrat Republican Independent n = 16553 n = 7123 n = 9430 n = 5526 n = 5126 n = 4996 0.05 Difference in probabilities ● ● 0.00 ● ● ● ● −0.05 −0.10 ● Effect on 'Less Strict' Effect on 'Keep the Same' Effect on 'More Strict' 25-miles PID-7 Cumula�ve e�ect Bounds 11 | 15
Results • Es�mate ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Block-bootstrap at the zip-code level No Prior Exposure 0.15 Democrat Republican Independent n = 2118 n = 2359 n = 2222 0.10 Difference in probabilities 0.05 ● ● 0.00 ● −0.05 −0.10 −0.15 ● Effect on 'Less Strict' Effect on 'Keep the Same' Effect on 'More Strict' 25-miles PID-7 Cumula�ve e�ect Bounds 11 | 15
Results • Es�mate ζ j = Pr ( Y i 1 ( 1 ) = j | D i = 1 ) − Pr ( Y i 1 ( 0 ) = j | D i = 1 ) • Block-bootstrap at the zip-code level No Prior Exposure Prior Exposure 0.15 Democrat Republican Independent Democrat Republican Independent n = 2118 n = 2359 n = 2222 n = 3408 n = 2767 n = 2774 0.10 Difference in probabilities 0.05 ● ● ● 0.00 ● ● −0.05 ● −0.10 −0.15 ● Effect on 'Less Strict' Effect on 'Keep the Same' Effect on 'More Strict' 25-miles PID-7 Cumula�ve e�ect Bounds 11 | 15
If the assump�on holds for the pre-treatment, we have q 1 v q 0 v 0 1 q d v F Y d 0 F Y d 1 v Test the following non-equivalence null (i.e., H 0 : Assump�on does not hold) H 0 0 1 q 1 v q 0 v vs H 1 0 1 q 1 v q 0 v v v Two one-sided tests (TOST) H 0 0 1 q 1 v q 0 v and H 0 0 1 q 1 v q 0 v v v Construct one-sided point-wise CIs: U 1 v and L 1 v reject H 0 at level 0 1 U 1 v v We reject H 0 if we reject both H 0 and H 0 Choose delta Assessing the Distribu�onal Parallel Trends Assump�on • Addi�onal pre-treatment �me-periods ❀ Assessment of the distribu�onal PT 12 | 15
Test the following non-equivalence null (i.e., H 0 : Assump�on does not hold) H 0 0 1 q 1 v q 0 v vs H 1 0 1 q 1 v q 0 v v v Two one-sided tests (TOST) H 0 0 1 q 1 v q 0 v and H 0 0 1 q 1 v q 0 v v v Construct one-sided point-wise CIs: U 1 v and L 1 v reject H 0 at level 0 1 U 1 v v We reject H 0 if we reject both H 0 and H 0 Choose delta Assessing the Distribu�onal Parallel Trends Assump�on • Addi�onal pre-treatment �me-periods ❀ Assessment of the distribu�onal PT • If the assump�on holds for the pre-treatment, we have ˜ q 1 ( v ) − ˜ q 0 ( v ) = 0 d 0 ( F − 1 q d ( v ) = F Y ∗ ˜ d 1 ( v )) Y ∗ 12 | 15
Two one-sided tests (TOST) H 0 0 1 q 1 v q 0 v and H 0 0 1 q 1 v q 0 v v v Construct one-sided point-wise CIs: U 1 v and L 1 v reject H 0 at level 0 1 U 1 v v We reject H 0 if we reject both H 0 and H 0 Choose delta Assessing the Distribu�onal Parallel Trends Assump�on • Addi�onal pre-treatment �me-periods ❀ Assessment of the distribu�onal PT • If the assump�on holds for the pre-treatment, we have ˜ q 1 ( v ) − ˜ q 0 ( v ) = 0 d 0 ( F − 1 ˜ q d ( v ) = F Y ∗ d 1 ( v )) Y ∗ • Test the following non-equivalence null (i.e., H 0 : Assump�on does not hold) H 0 : max v ∈ [ 0 , 1 ] | ˜ q 1 ( v ) − ˜ q 0 ( v ) | > δ vs H 1 : max v ∈ [ 0 , 1 ] | ˜ q 1 ( v ) − ˜ q 0 ( v ) | ≤ δ 12 | 15
Construct one-sided point-wise CIs: U 1 v and L 1 v reject H 0 at level 0 1 U 1 v v We reject H 0 if we reject both H 0 and H 0 Choose delta Assessing the Distribu�onal Parallel Trends Assump�on • Addi�onal pre-treatment �me-periods ❀ Assessment of the distribu�onal PT • If the assump�on holds for the pre-treatment, we have ˜ q 1 ( v ) − ˜ q 0 ( v ) = 0 d 0 ( F − 1 ˜ q d ( v ) = F Y ∗ d 1 ( v )) Y ∗ • Test the following non-equivalence null (i.e., H 0 : Assump�on does not hold) H 0 : max v ∈ [ 0 , 1 ] | ˜ q 1 ( v ) − ˜ q 0 ( v ) | > δ vs H 1 : max v ∈ [ 0 , 1 ] | ˜ q 1 ( v ) − ˜ q 0 ( v ) | ≤ δ ❀ Two one-sided tests (TOST) H + H − 0 : max v ∈ [ 0 , 1 ] { ˜ q 1 ( v ) − ˜ q 0 ( v ) } > δ and 0 : max v ∈ [ 0 , 1 ] { ˜ q 1 ( v ) − ˜ q 0 ( v ) } < − δ 12 | 15
Assessing the Distribu�onal Parallel Trends Assump�on • Addi�onal pre-treatment �me-periods ❀ Assessment of the distribu�onal PT • If the assump�on holds for the pre-treatment, we have ˜ q 1 ( v ) − ˜ q 0 ( v ) = 0 d 0 ( F − 1 ˜ q d ( v ) = F Y ∗ d 1 ( v )) Y ∗ • Test the following non-equivalence null (i.e., H 0 : Assump�on does not hold) H 0 : max v ∈ [ 0 , 1 ] | ˜ q 1 ( v ) − ˜ q 0 ( v ) | > δ vs H 1 : max v ∈ [ 0 , 1 ] | ˜ q 1 ( v ) − ˜ q 0 ( v ) | ≤ δ ❀ Two one-sided tests (TOST) H + H − 0 : max v ∈ [ 0 , 1 ] { ˜ q 1 ( v ) − ˜ q 0 ( v ) } > δ and 0 : max v ∈ [ 0 , 1 ] { ˜ q 1 ( v ) − ˜ q 0 ( v ) } < − δ • Construct one-sided point-wise CIs: � U 1 − α ( v ) and � L 1 − α ( v ) reject : H + � 0 at α level ⇐ ⇒ max U 1 − α ( v ) < δ v ∈ [ 0 , 1 ] ❀ We reject H 0 if we reject both H + 0 and H − CI construc�on Choose delta 0 12 | 15
Did not have the “prior exposure” as of 2010 Were not treated between 2010 and 2012 28 shoo�ngs Focus on 2817 respondents who Approx. 25% of them (667) are newly treated between 2012 and 2014 Assess the distribu�onal parallel trends assump�on using 2010–12 Using Three-wave Panel to Assess the Assump�on • Some respondents of CCES 2010–12 panel are reinterviewed in 2014 13 | 15
28 shoo�ngs Approx. 25% of them (667) are newly treated between 2012 and 2014 Assess the distribu�onal parallel trends assump�on using 2010–12 Using Three-wave Panel to Assess the Assump�on • 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 13 | 15
Assess the distribu�onal parallel trends assump�on using 2010–12 Using Three-wave Panel to Assess the Assump�on • 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 shoo�ngs 13 | 15
Using Three-wave Panel to Assess the Assump�on • 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 shoo�ngs • Assess the distribu�onal parallel trends assump�on using 2010–12 13 | 15
Using Three-wave Panel to Assess the Assump�on • 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 shoo�ngs • Assess the distribu�onal parallel trends assump�on using 2010–12 Test Statistic (Pre−Treatment Outcome) t max = 0.021 equivalence threshold = 0.054 0.05 0 (v) ~ 1 (v) − q 0.00 ~ ^ (v) = q t −0.05 minimum threshold = 0.039 0.0 0.2 0.4 0.6 0.8 1.0 13 | 15 Quantile (v)
Using Three-wave Panel to Assess the Assump�on • 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 shoo�ngs • Assess the distribu�onal parallel trends assump�on using 2010–12 Test Statistic Effect in 2014 (Pre−Treatment Outcome) (CCES 10−12−14 Subsamples) 0.10 t max = 0.021 equivalence threshold = 0.054 No prior exposure & Untreated in 2012 0.05 n = 2812 Difference in probabilities Less strict Keep the same More strict 0 (v) 0.05 ~ 1 (v) − q 0.00 ~ ^ (v) = q 0.00 ● t −0.05 minimum threshold = 0.039 −0.05 0.0 0.2 0.4 0.6 0.8 1.0 13 | 15 Quantile (v)
Find that e�ects are concentrated among Democrats who do not have “prior exposure” to shoo�ngs and among Independents Linearity assump�on in DID is inappropriate for ordinal outcomes Propose a latent variable framework to address the issue Revisit the recent debate on the rela�onship between the mass shoo�ngs and the a�tudes toward gun control: Concluding Remarks • Di�erence-in-di�erences is widely used in social science research 14 | 15
Find that e�ects are concentrated among Democrats who do not have “prior exposure” to shoo�ngs and among Independents Propose a latent variable framework to address the issue Revisit the recent debate on the rela�onship between the mass shoo�ngs and the a�tudes toward gun control: Concluding Remarks • Di�erence-in-di�erences is widely used in social science research • Linearity assump�on in DID is inappropriate for ordinal outcomes 14 | 15
Find that e�ects are concentrated among Democrats who do not have “prior exposure” to shoo�ngs and among Independents Revisit the recent debate on the rela�onship between the mass shoo�ngs and the a�tudes toward gun control: Concluding Remarks • Di�erence-in-di�erences is widely used in social science research • Linearity assump�on in DID is inappropriate for ordinal outcomes • Propose a latent variable framework to address the issue 14 | 15
Find that e�ects are concentrated among Democrats who do not have “prior exposure” to shoo�ngs and among Independents Concluding Remarks • Di�erence-in-di�erences is widely used in social science research • Linearity assump�on in DID is inappropriate for ordinal outcomes • Propose a latent variable framework to address the issue • Revisit the recent debate on the rela�onship between the mass shoo�ngs and the a�tudes toward gun control: 14 | 15
Concluding Remarks • Di�erence-in-di�erences is widely used in social science research • Linearity assump�on in DID is inappropriate for ordinal outcomes • Propose a latent variable framework to address the issue • Revisit the recent debate on the rela�onship between the mass shoo�ngs and the a�tudes toward gun control: • Find that e�ects are concentrated among Democrats who do not have “prior exposure” to shoo�ngs and among Independents 14 | 15
References • Yamauchi, Soichiro. (2020). “Di�erence-in-Di�erences for Ordinal Outcomes: Applica�on to the E�ect of Mass Shoo�ngs on A�tudes towards Gun Control” Working Paper . • orddid : R package for implemen�ng the di�erence-in-di�erence for the ordinal outcomes. Available at github.com/soichiroy/orddid Send comments and sugges�ons to syamauchi@g.harvard.edu For more informa�on soichiroy.github.io 15 | 15
Addi�onal Results 1 | 16
Trea�ng as Con�nuous Outcome • Consider a cross-sec�onal se�ng: ζ j = Pr ( Y i ( 1 ) = j ) − Pr ( Y i ( 0 ) = j ) • Rescale the outcome: � Y i = Y i / ( J − 1 ) • The di�erence-in-means es�amtor on � Y i can be wri�en as J ∑ ( J − j ) − 1 � τ DiM = � ζ j j = 1 where ∑ n ∑ n τ DiM = 1 D i Y i − 1 ( 1 − D i ) Y i � n 1 n 0 i = 1 i = 1 ❀ Weighted average of � ζ j with weights are 1 / ( J − j ) • This can poten�ally cancel out the e�ects: E.g., � ζ 1 > 0 and � ζ 2 < 0 2 | 16
Invariance of Causal E�ect to Choice of Cuto�s • Proposi�on : Suppose Y it ∈ J ≡ { 0 , 1 , 2 } and U ∼ N ( 0 , 1 ) . Let κ and κ ′ be the di�erent sets of cuto�s. Then, for all j ∈ J . ζ j ( κ ) = � � ζ j ( κ ′ ) • Intui�on : (1) Assump�on is imposed on the quan�le scale (i.e., distribu�onal PT) ❀ counterfactual distribu�on is iden��ed as long as quan�le info. is preserved (2) Changing cuto�s a�ect mean & scale ❀ transform the latent variables (3) But quan�le informa�on is preserved, Pr( Y ∗ ≤ κ 1 ) = Pr(˜ Y ∗ ≤ κ ′ 1 ) ∫ κ 2 ∫ κ ′ 2 φ (( y ∗ − µ 00 ) /σ 00 ) dy ∗ = Pr( Y 00 = 1 ) φ (( y ∗ − µ ′ 00 ) /σ ′ 00 ) dy ∗ = � �� � κ ′ κ 1 1 observed prob. ❀ uniquely recovers the counterfactual distribu�on Y ∗ 11 Back 3 | 16
Iden��ca�on of Latent Variables • Suppose that the cuto�s are �xed at κ 1 and κ 2 for Y dt = j ∈ { 0 , 1 , 2 } . Then, µ dt and σ dt in Y ∗ dt ∼ µ dt + σ dt U are uniquely iden��ed from the observed probability distribu�on. • Proof: Suppose that U has the density f U ( u ) . Then, we can form a non-linear system of equa�ons ∫ κ 1 f U (( y ∗ − µ dt ) /σ dt ) dy ∗ Pr( Y dt = 0 ) = −∞ ∫ ∞ f U (( y ∗ − µ dt ) /σ dt ) dy ∗ Pr( Y dt = 2 ) = κ 2 which are su�cient for es�ma�ng µ and σ . Back 4 | 16
Alterna�ve Formula of Iden��ca�on • Suppose Y dt = j ∈ { 0 , 1 , 2 } . Let v 1 = F 01 ( κ 1 ) and v 2 = F 01 ( κ 2 ) where κ is a set of �xed cuto�s. • Under the assump�ons, we iden�fy µ 11 and σ 11 by the following system of non-linear equa�ons: ∫ F − 1 10 ( v 1 ) f U (( y ∗ − µ 11 ) /σ 11 ) dy ∗ q 0 ( v 1 ) = −∞ ∫ F − 1 10 ( v 2 ) f U (( y ∗ − µ 11 ) /σ 11 ) dy ∗ . q 0 ( v 2 ) = −∞ 5 | 16
Construc�ng Con�dence Intervals for Tes�ng • Let t ( v ) = ˜ q 1 ( v ) − ˜ q 0 ( v ) • Point-wise upper and lower ( 1 − α ) level con�dence intervals: √ U 1 − α ( v ) = ˆ � t ( v ) + Φ − 1 ( 1 − α ) Var (ˆ t ( v )) / n √ � L 1 − α ( v ) = ˆ t ( v ) − Φ − 1 ( 1 − α ) Var (ˆ t ( v )) / n • Proposi�on ( ) � U 1 − α ( v ′ ) Pr v ∈ [ 0 , 1 ] t ( v ) ≤ max max ≥ 1 − α v ′ ∈ [ 0 , 1 ] ( ) � L 1 − α ( v ′ ) Pr v ∈ [ 0 , 1 ] t ( v ) ≥ min min ≥ 1 − α v ′ ∈ [ 0 , 1 ] Back 6 | 16
Choosing Delta • Value of δ re�ect the admissible level of “non equivalence” ❀ Larger values of δ correspond to lenient thresholds • Calibrate δ based on the rejec�on threshold for the KS test {√ √ } n 1 + n 0 δ n = min − log( α ) / 2 , 1 n 1 n 0 and take α = 0 . 05 • Can report the equivalence CI: minimum possible value of δ at α level { } | � U 1 − α ( v ) | , | � δ min , n = max L 1 − α ( v ) | v ∈ [ 0 , 1 ] ❀ Equivalence CI is given by [ − δ min , n , δ min , n ] Back 7 | 16
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