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Jiyong Park

Bryan School of Business and Economics

University of North Carolina at Greensboro

jiyong.park@uncg.edu

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Korea Summer Workshop on Causal Inference 2022

Session Website: https://sites.google.com/view/causal-inference2022

Boot Camp for Beginners

Quasi-Experiments

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Korea Summer Workshop on Causal Inference 2022

Quasi-Experimental Designs

: Research Designs without Random Assignment

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Causal Hierarchy from the Perspective of Potential Outcomes

Level of Causal Inference

Meta-Analysis

Randomized Controlled Trial

Quasi-Experiment

Instrumental Variable

“Designed” Regression/Matching

(based on causal knowledge or theory)

Model-Free Descriptive Statistics (no causal inference)

Exploiting Research Design

Regression/Matching (little causal inference)

Selection on Unobservables Strategies

Selection on Observables Strategies

Exploiting

Random Assignment

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Random Assignment is not Always Feasible

  • Quasi-experiment = RCT without random assignment
    • The only difference between RCT and quasi-experiment lies at the treatment assignment mechanism.

  • Quasi-experiment is a research design where a control group is comparable to the treatment, albeit not randomly assigned, except the fact that they didn’t receive the treatment (i.e., Ceteris Paribus).

Key Identification Assumption

Is the treatment assignment without random assignment as good as random?

How similar is the control group to the treatment group in the absence of the treatment?

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Research Design (1) Institution and Economic Growth

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Twin cities

Comparable people

Comparable geography

Comparable culture

Research Design (1) Institution and Economic Growth

Nogales, Mexico

Nogales, Arizona

GDP per capita (2019)

USA: $65,297

Mexico: $9,946

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Research Design (1) Institution and Economic Growth

Acemoglu, D., Johnson, S. and Robinson, J.A., 2001. The colonial origins of comparative development: An empirical investigation. American Economic Review91(5), pp.1369-1401.

Acemoglu, D. and Johnson, S., 2005. Unbundling institutions. Journal of Political Economy113(5), pp.949-995.

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Research Design (2) Minimum Wage and Employment

    • Ideal causal effect of the increase in minimum wage in New Jersey

Causal effect

= (Employment after the increase in minimum wage in New Jersey) – (Potential employment if the minimum wage were not increased in New Jersey)

Counterfactual

Causal effect estimation

= (Employment after the increase in minimum wage in New Jersey) – (Employment in Pennsylvania after the increase in minimum wage in New Jersey)

    • A natural experiment in which Pennsylvania serves as a comparable control group

Card, D. and Krueger, A.B., 1994. Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania. American Economic Review84(4), pp.772-93.

Comparable control group

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Research Design (3) 1995 Chicago Heat Wave

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Research Design (3) 1995 Chicago Heat Wave

1995년 여름 미국 시카고에 섭씨 41도까지 올라가는 폭염이 찾아왔고 무려 700여 명이 사망했다. 그중에는 노인과 빈곤층, 1인 가구 거주자가 많았다. 이들은 이웃이나 국가의 외면으로 혼자 더위를 견디다 사망한 것으로 드러났다. 정부가 대비책을 아예 마련하지 않았던 것은 아니었지만, 취약계층이 능동적으로 대비책을 활용할 수 있다고 판단한 것이 문제였다.

1999년 시카고에 또다시 폭염이 찾아왔지만, 이번에는 여름철 기상이변에 대한 대비가 잘 돼 있었다. 경찰은 집집마다 방문하여 취약계층의 안전을 확인했다. 시카고시는 냉방센터를 여럿 만들고 냉방센터에 오려는 사람들이 무료로 버스를 이용할 수 있게 했다. 과거의 실수를 반복하지 않으려는 시카고시의 노력으로 사망자 수가 110명으로 줄었다. 여전히 높은 수치지만 99년도의 더위가 95년도와 크게 다르지 않았다는 점을 생각하면 큰 성과라 할 수 있다.

에릭 클라이넨버그는 “1995년도와 1999년도의 차이는 사회적 연결망에 있었다”고 말한다. 비슷한 폭염 상황이라 해도 취약계층의 안부를 수시로 묻고 그들을 챙기려는 노력이 피해를 크게 줄였다.

http://www.snunews.com/news/articleView.html?idxno=18455

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Research Design (3) 1995 Chicago Heat Wave

1995년 여름 미국 시카고에 섭씨 41도까지 올라가는 폭염이 찾아왔고 무려 700여 명이 사망했다. 그중에는 노인과 빈곤층, 1인 가구 거주자가 많았다. 이들은 이웃이나 국가의 외면으로 혼자 더위를 견디다 사망한 것으로 드러났다. 정부가 대비책을 아예 마련하지 않았던 것은 아니었지만, 취약계층이 능동적으로 대비책을 활용할 수 있다고 판단한 것이 문제였다.

1999년 시카고에 또다시 폭염이 찾아왔지만, 이번에는 여름철 기상이변에 대한 대비가 잘 돼 있었다. 경찰은 집집마다 방문하여 취약계층의 안전을 확인했다. 시카고시는 냉방센터를 여럿 만들고 냉방센터에 오려는 사람들이 무료로 버스를 이용할 수 있게 했다. 과거의 실수를 반복하지 않으려는 시카고시의 노력으로 사망자 수가 110명으로 줄었다. 여전히 높은 수치지만 99년도의 더위가 95년도와 크게 다르지 않았다는 점을 생각하면 큰 성과라 할 수 있다.

에릭 클라이넨버그는 “1995년도와 1999년도의 차이는 사회적 연결망에 있었다”고 말한다. 비슷한 폭염 상황이라 해도 취약계층의 안부를 수시로 묻고 그들을 챙기려는 노력이 피해를 크게 줄였다.

http://www.snunews.com/news/articleView.html?idxno=18455

Causal inference aims at designing intervention strategies of inputs (causes)

to improve an output (outcome).

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Research Design (3) 1995 Chicago Heat Wave

노스론데일과 사우스론데일의 사례가 대표적이다. 두 지역은 시카고 서쪽에 인접해 있는 마을이다. 그해 여름 두 지역의 기후는 유사했을 게 불문가지다. 독거노인 비율이나 빈곤율도 비슷했다.

하지만 두 지역의 폭염 피해 수준은 크게 달랐다. 노스론데일에서 폭염으로 숨진 사람은 19명(10만명 당 40명)이었다. 반면 사우스론데일에서는 3명(10만명 당 4명)이 사망했다. 희생자 비율이 10배나 차이가 났던 셈이다. 어떻게 이런 차이가 나타났던 걸까.

노스론데일은 공동화(空洞化)된 마을이었다. 60년대 이후 지역경제가 휘청거리자 인구는 급감했고, 약국/식료품점/음식점 같은 상점도 하나둘 사라졌다. 그러면서 범죄율이 치솟았다. 마약상이 활개를 쳤다. 하지만 사우스론데일은 달랐다. 상점가에는 여전히 사람이 넘쳐났다. 범죄율도 낮았다.

저자는 “위험한 지역의 주민들이 고독사로 죽어간 것에는 그들이 좀처럼 집을 떠나지 못하도록 만드는 사회적 환경에서 살았기 때문”이라고 적었다.

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Research Design (3) 1995 Chicago Heat Wave

Klinenberg E (2003) Heat wave: A social autopsy of disaster in Chicago: University of Chicago Press

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Research Design (3) 1995 Chicago Heat Wave

Klinenberg E (2003) Heat wave: A social autopsy of disaster in Chicago: University of Chicago Press

Quasi-experiment is to take advantage of research designs in which the control groups are comparable to the treatment groups in all aspects, on average,

but the fact that they are treated or not.

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Research Design in Marketing Research

Goldfarb, A., Tucker, C. and Wang, Y., 2022. Conducting research in marketing with quasi-experiments. Journal of Marketing86(3), pp.1-20.

  • Research designs are very diverse by context.

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Research Design Matters

Angrist, J.D. and Pischke, J.S., 2017. Undergraduate econometrics instruction: through our classes, darkly. Journal of Economic Perspectives, 31(2), pp.125-144.

Keele, L., 2015. The Statistics of Causal Inference: A View from Political Methodology. Political Analysis, 23(3), pp.313-335.

“We call this framework the design-based approach to econometrics because the skills and strategies required to use it successfully are related to research design.” (Angrist and Pischke 2017, p. 126)

“The analyst is successful at identifying the causal effect not because of the complex statistical methods that are applied to the data, but due to the effort in developing a design before data are collected.” (Keele 2015, p. 331)

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“Taken together, therefore, the Laureates’ contributions have played a central role in establishing the so-called design-based approach in economics. This approach – aimed at emulating a randomized experiment to answer a causal question using observational data – has transformed applied work and improved researchers’ ability to answer causal questions of great importance for economic and social policy using observational data.” (P. 2)

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Korea Summer Workshop on Causal Inference 2022

Overview of Quasi-Experimental Methods

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Counterfactual Revisited

  • Causation is defined as the difference in potential outcomes after the treatment.
    • “What if the treatment was not applied?”
    • Causal effect = (Actual outcome for treated if treated) – (Potential outcome for treated if not treated)

Counterfactual

Treatment

Potential Outcomes

Causal Effect

Subject i

1

1

3

counterfactual

ATE on the Treated (ATET)

2

1

1

counterfactual

3

0

1

ATE on the Untreated (ATEU)

4

0

1

  • All causal inference methods basically aim to approximate the counterfactual.
  • All quasi-experiments estimate ATET.

Counterfactual

ATE

For ATET = ATE, the treatment and control groups should be comparable.

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Data Structure from the Perspective of Counterfactual

Time

Treated Unit A

Outcome

Outcome

Outcome

  • Counterfactual can be classified into time-invariant outcome and time-varying outcome even in the absence of the treatment.

Treatment

Counterfactual

Outcome

Counterfactual

Outcome

Counterfactual

  • Time-invariant outcome
  • Time-varying outcome

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Data Structure from the Perspective of Counterfactual

Time

Treated Unit A

Treated Unit B

Untreated Unit C

Untreated Unit D

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

  • Longitudinal data before and after the treatment has an advantage over cross-sectional data.
    • For longitudinal data, the counterfactual of the treatment group is estimated using its own prior patterns. Time-invariant outcome can be easily accounted for (i.e., by using fixed-effects).

Treatment

Counterfactual

Counterfactual

Counterfactual

Counterfactual

Counterfactual

Counterfactual

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Data Structure from the Perspective of Counterfactual

Time

Treated Unit A

Treated Unit B

Untreated Unit C

Untreated Unit D

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

Outcome

  • Panel data with the control group has an advantage over pure time-series data of the treatment group.
    • For pure time-series data of the treatment group, it is challenging to account for time-varying outcome in the absence of the treatment (due to outside confounders). It can be estimated using time trends in the control group.

Treatment

Counterfactual

Counterfactual

Counterfactual

Counterfactual

Counterfactual

Counterfactual

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What’s Your Research Design and Data Structure?

Quasi-experimental designs are available.

Treatment and control groups are observed.

Parallel trend assumption is valid.

Treatment is assigned by arbitrary threshold.

There is an exogenous variable that can induce the treatment.

No

Local Average Treatment Effect

Yes

No

DID + Matching

Synthetic Control

Regression Discontinuity

Quasi-Experiments

The goal is causal inference, and random assignment is feasible.

Matching/

Weighting

There is a single treatment group and no information about functional forms.

Regression

Selection on Observables

No

Yes

No

Yes

No

Note that the flowchart may depend on the context.

Yes

Randomized Controlled Trial

No

Longitudinal data is observed.

Interrupted Time-Series Analysis

Yes

No

Longitudinal data is observed.

Not feasible

No

Yes

Yes

Difference-in-Differences (DID)

Yes

or

No

Yes

Control Function & Selection Model

or

Instrumental Variable

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What’s Your Research Design and Data Structure?

Treatment and control groups are observed.

Parallel trend assumption is valid.

Treatment is assigned by arbitrary threshold.

There is an exogenous variable that can induce the treatment.

No

Local Average Treatment Effect

Yes

No

DID + Matching

Synthetic Control

Regression Discontinuity

Quasi-Experiments

The goal is causal inference, and random assignment is feasible.

Matching/

Weighting

There is a single treatment group and no information about functional forms.

Regression

Selection on Observables

No

Yes

No

Yes

No

Yes

Randomized Controlled Trial

No

Longitudinal data is observed.

Interrupted Time-Series Analysis

Yes

No

Longitudinal data is observed.

Not feasible

No

Yes

Yes

Difference-in-Differences (DID)

Yes

or

No

Yes

Control Function & Selection Model

or

Instrumental Variable

Quasi-experimental designs are available.

Note that the flowchart may depend on the context.

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Difference-in-Differences

Treatment Group

After Period

Actual Treatment

Potential Outcomes

Causal Effect

Subject i

1

1

0

0

1

ATET = 1

1

1

1

3

1 + 0.5

2

1

0

0

0

1

1

1

1

0 + 0.5

3

0

0

0

1

0

1

0

1

4

0

0

0

0

0

1

0

1

Average increase by 0.5

Parallel trend assumption is required.

If it is not satisfied, the control group can be recomposed using matching.

  • With panel data, DID approximates the counterfactual using its own prior and time trends of the control group.

Difference-in-Differences (DID)

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What’s Your Research Design and Data Structure?

Treatment and control groups are observed.

Parallel trend assumption is valid.

Treatment is assigned by arbitrary threshold.

There is an exogenous variable that can induce the treatment.

No

Local Average Treatment Effect

Yes

No

DID + Matching

Synthetic Control

Regression Discontinuity

Quasi-Experiments

The goal is causal inference, and random assignment is feasible.

Matching/

Weighting

There is a single treatment group and no information about functional forms.

Regression

Selection on Observables

No

Yes

No

Yes

No

Yes

Randomized Controlled Trial

No

Longitudinal data is observed.

Interrupted Time-Series Analysis

Yes

No

Longitudinal data is observed.

Not feasible

No

Yes

Yes

Difference-in-Differences (DID)

Yes

or

No

Yes

Control Function & Selection Model

or

Instrumental Variable

Quasi-experimental designs are available.

Note that the flowchart may depend on the context.

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Synthetic Control

Treatment Group

After Period

Actual Treatment

Potential Outcomes

Causal Effect

Subject i

1

1

0

0

1

1

1

1

3

2

1

0

0

0

1

1

1

1

3

0

0

0

1

0

1

0

1

4

0

0

0

0

0

1

0

1

(2) Predicting the counterfactual (a.k.a. synthetic control)

Synthetic Control (SC)

 

 

(1) Training a prediction model

Even parallel trends assumption may not be necessary.

  • With panel data, SC approximates the counterfactual using a combination of the control group.

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Synthetic Control vs. Difference-in-Differences

Difference-in-Differences

Difference-in-Differences + Matching

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Synthetic Control vs. Difference-in-Differences

Synthetic Control

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What’s Your Research Design and Data Structure?

Treatment and control groups are observed.

Parallel trend assumption is valid.

Treatment is assigned by arbitrary threshold.

There is an exogenous variable that can induce the treatment.

No

Local Average Treatment Effect

Yes

No

DID + Matching

Synthetic Control

Regression Discontinuity

Quasi-Experiments

The goal is causal inference, and random assignment is feasible.

Matching/

Weighting

There is a single treatment group and no information about functional forms.

Regression

Selection on Observables

No

Yes

No

Yes

No

Yes

Randomized Controlled Trial

No

Longitudinal data is observed.

Interrupted Time-Series Analysis

Yes

No

Longitudinal data is observed.

Not feasible

No

Yes

Yes

Difference-in-Differences (DID)

Yes

or

No

Yes

Control Function & Selection Model

or

Instrumental Variable

Quasi-experimental designs are available.

Note that the flowchart may depend on the context.

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Interrupted Time-Series Analysis

Treatment Group

After Period

Actual Treatment

Potential Outcomes

Causal Effect

Subject i

1

1

-1

0

1

ATET = 0

1

0

0

2

1

1

1

3

2 + 1

Interrupted Time-Series Analysis

+1

  • With pure time-series data, interrupted time-series analysis approximates the counterfactual using its own prior and time trends.

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Difference-in-Differences

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Difference-in-Differences (DID)

  • Rethinking the DID model from the perspective of potential outcomes

Treatment Group

Control Group

Before Treatment

After Treatment

 

 

 

 

 

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Difference-in-Differences (DID)

  • Rethinking the DID model from the perspective of potential outcomes

Treatment Group

Control Group

Before Treatment

After Treatment

 

 

 

 

 

Counterfactual outcome in the absence of treatment

 

 

Counterfactual outcome inferred from the control group

 

 

 

Change for the treated in the absence of treatment (not observed)

Inferred from the control group unaffected by the treatment

Inferred counterfactual

 

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Difference-in-Differences (DID)

  • Rethinking the DID model from the perspective of potential outcomes

Treatment Group

Control Group

Before Treatment

After Treatment

 

 

 

 

 

Counterfactual outcome in the absence of treatment

 

 

Counterfactual outcome inferred from the control group

 

 

 

Change for the treated in the absence of treatment (not observed)

Inferred from the control group unaffected by the treatment

Inferred counterfactual

 

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Difference-in-Differences (DID)

  • Rethinking the DID model from the perspective of potential outcomes

Treatment Group

Control Group

Before Treatment

After Treatment

 

 

 

 

 

Counterfactual outcome in the absence of treatment

 

 

Counterfactual outcome inferred from the control group

 

 

 

Change for the treated in the absence of treatment (not observed)

Inferred from the control group unaffected by the treatment

Inferred counterfactual

 

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Difference-in-Differences (DID)

  • Rethinking the DID model from the perspective of potential outcomes

Treatment Group

Control Group

Before Treatment

After Treatment

 

 

 

 

 

Counterfactual outcome in the absence of treatment

 

 

Counterfactual outcome inferred from the control group

 

 

 

Change for the treated in the absence of treatment (not observed)

Inferred from the control group unaffected by the treatment

Inferred counterfactual

 

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Difference-in-Differences (DID)

  • Rethinking the DID model from the perspective of potential outcomes

Treatment Group

Control Group

Before Treatment

After Treatment

 

 

 

 

 

Counterfactual outcome in the absence of treatment

 

 

Counterfactual outcome inferred from the control group

 

 

 

Change for the treated in the absence of treatment (not observed)

Inferred from the control group unaffected by the treatment

Inferred counterfactual

 

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Identification Assumption for DID

  • For DID analysis, the parallel trend assumption (in the absence of treatment) must be hold.
    • Basically, the parallel trend assumption is not verifiable, and what we can show is parallel pretrends.
    • Matching methods might help to achieve the parallel trend assumption.

No parallel pretrends

Parallel pretrends

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Relative Time Model (Leads-and-Lags Model)

  • Relative time model is useful when the treatment timing is heterogenous across treated units.
  • Example: Impact of Uber entry on alcohol-related motor vehicle fatalities (Greenwood and Wattal 2017)

“This is done by creating a second series of time dummies, in addition to the chronological time dummies, which indicate the relative chronological distance between time t and the time Uber is implemented in city j.”

“Econometrically, the primary benefit of this model is that it can determine if a pretreatment trend exists (i.e., a significant difference between treated and untreated counties before treatment) in order to determine if the untreated counties are an acceptable control group. If such a trend exists, it would violate one of the primary assumptions of the model.” (p. 170)

Greenwood, B.N. and Wattal, S., 2017. Show me the way to go home: an empirical investigation of ride-sharing and alcohol related motor vehicle fatalities. MIS Quarterly41(1), pp.163-187.

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Various Cases of Difference-in-Differences

  • Case 1. Same treatment timing + Treatment/control groups are assigned by the treatment
  • Example: NBC's decision to remove its own content from iTunes (Danaher et al. 2010)

Danaher, B., Dhanasobhon, S., Smith, M.D. and Telang, R., 2010. Converting Pirates without Cannibalizing Purchasers: The Impact of Digital Distribution on Physical Sales and Internet Piracy. Marketing Science, 29(6), pp.1138-1151.

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Various Cases of Difference-in-Differences

  • Case 1. Same treatment timing + Treatment/control groups are assigned by the treatment
  • Example: NBC's decision to remove its own content from iTunes (Danaher et al. 2010)

Danaher, B., Dhanasobhon, S., Smith, M.D. and Telang, R., 2010. Converting Pirates without Cannibalizing Purchasers: The Impact of Digital Distribution on Physical Sales and Internet Piracy. Marketing Science, 29(6), pp.1138-1151.

 

Absorbed by unit fixed effects

Absorbed by time fixed effects

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Various Cases of Difference-in-Differences

  • Case 2. Different treatment timing + Treatment/control groups are assigned by the treatment
  • Example: Effect of offline showrooms on demand generation and operational efficiency (Bell et al. 2018)

Bell, D.R., Gallino, S. and Moreno, A., 2018. Offline showrooms in omnichannel retail: Demand and operational benefits. Management Science, 64(4), pp.1629-1651.

Offline showroom (no purchase)

30 miles

Control Group

Treatment Group

Warby Parker’s website

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Various Cases of Difference-in-Differences

  • Case 2. Different treatment timing + Treatment/control groups are assigned by the treatment
  • Example: Effect of offline showrooms on demand generation and operational efficiency (Bell et al. 2018)

Bell, D.R., Gallino, S. and Moreno, A., 2018. Offline showrooms in omnichannel retail: Demand and operational benefits. Management Science, 64(4), pp.1629-1651.

Offline showroom (no purchase)

30 miles

Control Group

Treatment Group

Warby Parker’s website

 

Absorbed by unit fixed effects

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Various Cases of Difference-in-Differences

  • Case 2. Different treatment timing + Treatment/control groups are assigned by the treatment
  • Example: Impact of anti-discrimination laws on firm performance (Greene and Shenoy 2021)

Greene, D. and Shenoy, J., 2021. How do anti-discrimination laws affect firm performance and financial policies? Evidence from the post-World War II period. Management Science.

 

Absorbed by unit fixed effects

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Various Cases of Difference-in-Differences

  • Case 3. Same treatment timing + Treatment/control groups are affected disproportionately by the treatment
  • Example: Impact of CSR on firm performance during the financial crisis (Lins et al. 2017)

Lins, K.V., Servaes, H. and Tamayo, A., 2017. Social capital, trust, and firm performance: The value of corporate social responsibility during the financial crisis.  Journal of Finance72(4), pp.1785-1824.

Firms with Higher CSR

Firms with Lower CSR

Before Financial Crisis

After Financial Crisis

 

 

 

 

Although all firms were under the reach of the financial crisis, this study examines whether firms with higher CSR performed better during the financial crisis.

“We estimate a difference-in-differences model with continuous treatment levels.”

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Various Cases of Difference-in-Differences

  • Case 3. Same treatment timing + Treatment/control groups are affected disproportionately by the treatment
  • Example: Impact of machine learning-powered translation on international trade (Brynjolfsson et al. 2019)

Brynjolfsson, E., Hui, X. and Liu, M., 2019. Does machine translation affect international trade? Evidence from a large digital platform. Management Science65(12), pp.5449-5460.

“We aim to create treatment and control groups using variations in title lengths across different listings… The figure shows variation in title lengths, which we use as the treatment intensity in our continuous DiD estimation”

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Various Cases of Difference-in-Differences

  • Case 4. Different treatment timing + Treatment/control groups are affected disproportionately by the treatment
  • Example: Impact of access to local finance on firm formation (Gilje 2019)

Gilje, E.P., 2019. Does local access to finance matter? Evidence from US oil and natural gas shale booms. Management Science65(1), pp.1-18.

Access to Local Finance

Shale Boom

(Treatment)

(Natural Shock)

More deposit in local banks by oil firms

Industries with high external financing requirements

Industries with low external financing requirements

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Various Cases of Difference-in-Differences

  • Case 4. Different treatment timing + Treatment/control groups are affected disproportionately by the treatment
  • Example: Impact of access to local finance on firm formation (Gilje 2019)

Gilje, E.P., 2019. Does local access to finance matter? Evidence from US oil and natural gas shale booms. Management Science65(1), pp.1-18.

Access to Local Finance

Shale Boom

(Treatment)

(Natural Shock)

More deposit in local banks by oil firms

Industries with high external financing requirements

Industries with low external financing requirements

“To achieve this aim, I use a regression form of difference-in-differences, where the first difference (β1) can be thought of as the difference in economic outcomes between boom county-years and nonboom county-years. To identify the effect of the credit component of a boom, I incorporate a second difference (β3), the difference in economic outcomes for industries with high external finance requirements and industries with low external finance requirements.”

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Synthetic Control

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Synthetic Control vs. Difference-in-Differences

  • Synthetic control method is in line with difference-in-differences in that they approximate the counterfactual using untreated units in the control group.

“This method builds on difference-in-differences estimation, but uses systematically more attractive comparisons.”

“The synthetic control approach developed by Abadie, Diamond, and Hainmueller (2010, 2014) and Abadie and Gardeazabal (2003) is arguably the most important innovation in the policy evaluation literature in the last 15 years.” (Athey and Imbens 2017, p. 9)

Athey, S. and Imbens, G.W., 2017. The state of applied econometrics: Causality and policy evaluation. Journal of Economic Perspectives31(2), pp.3-32.

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Synthetic Control vs. Difference-in-Differences

  • A combination of untreated units often provides a more appropriate counterfactual for the treated units.

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Synthetic Control vs. Difference-in-Differences

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Example (1) Impact of California Anti-Tobacco Legislation

  • California vs. All other states

The treated unit (California) is not comparable to untreated units (other states).

Abadie, A., Diamond, A. and Hainmueller, J., 2010. Synthetic control methods for comparative case studies: Estimating the effect of California’s tobacco control program. Journal of the American Statistical Association105(490), pp.493-505.

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Example (1) Impact of California Anti-Tobacco Legislation

  • California vs. Synthetic California

Abadie, A., Diamond, A. and Hainmueller, J., 2010. Synthetic control methods for comparative case studies: Estimating the effect of California’s tobacco control program. Journal of the American Statistical Association105(490), pp.493-505.

Donor Pool

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Example (2) Impact of Reunification on West Germany

  • We are never able to observe the counterfactual on the reunification, but we can constitute a synthetic control using similar countries that best resembles the economic trajectory of West Germany.

Abadie, A., 2021. Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature59(2), pp.391-425.

Transparency of the counterfactual is one of the most attractive features of the synthetic control.

Donor Pool

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How to Construct the Synthetic Control

  • Original method (Abadie and Gardeazabal 2003; Abadie et al. 2010)
    • It aims to choose weights of control units to minimize the difference in the pre-intervention values of outcomes and predictors (with a constraint that weights are nonnegative and sum is one).

Abadie, A. and Gardeazabal, J., 2003. The economic costs of conflict: A case study of the Basque Country. American Economic Review, 93(1), pp.113-132.

Abadie, A., Diamond, A. and Hainmueller, J., 2010. Synthetic control methods for comparative case studies: Estimating the effect of California’s tobacco control program. Journal of the American Statistical Association105(490), pp.493-505.

Predictors for the treated unit

Weighted predictors for the untreated units

There could be multiple predictors that contribute differently to the synthetic control.

Treatment effect for the treated unit after intervention

Counterfactual computed using the synthetic control (weighted control units that best resemble the treated unit)

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How to Construct the Synthetic Control

  • Original method (Abadie and Gardeazabal 2003; Abadie et al. 2010)
    • It aims to choose weights of control units to minimize the difference in the pre-intervention values of outcomes and predictors (with a constraint that weights are nonnegative and sum is one).

control units

treated unit

pre-intervention outcomes and predictors

(variables in different times are considered separate ones)

 

 

 

 

 

 

 

 

 

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How to Construct the Synthetic Control

  • Original method (Abadie and Gardeazabal 2003; Abadie et al. 2010)
    • It aims to choose weights of control units to minimize the difference in the pre-intervention values of outcomes and predictors (with a constraint that weights are nonnegative and sum is one).

control units

treated unit

pre-intervention outcomes and predictors

(variables in different times are considered separate ones)

 

 

 

 

 

 

 

 

 

    • Synthetic control can be estimated for each treated unit.

    • If there are too many control units in the donor pool, optimization solution is often infeasible.
      • Restrict the donor pool to a relevant subset (e.g., products in the same category)
      • Cluster the data and consider control units in the same cluster

“Computational complexity can be substantially reduced by estimating the composition

of the clusters in a first step (e.g., using K-means)” (p. 11)

Abadie, A. and Zhao, J., 2021. Synthetic controls for experimental design. arXiv preprint arXiv:2108.02196.

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How to Construct the Synthetic Control

  • There are an increasing number of studies that propose a way to construct the synthetic control with different assumptions and different methods.

Doudchenko, N. and Imbens, G.W., 2016. Balancing, regression, difference-in-differences and synthetic control methods: A synthesis (No. w22791). National Bureau of Economic Research.

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How to Construct the Synthetic Control

  • Synthetic difference-in-differences (Arkhangelsky et al. 2021)

Arkhangelsky, D., Athey, S., Hirshberg, D.A., Imbens, G.W. and Wager, S., 2021. Synthetic difference-in-differences. American Economic Review111(12), pp.4088-4118.

  • Unit Fixed Effects
  • Time Fixed Effects
  • Unit Fixed Effects
  • Time Fixed Effects
  • Unit weights
  • Time weights
  • Unit weights

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How to Construct the Synthetic Control

  • Bayesian synthetic control (Kim et al. 2020)

Kim, S., Lee, C. and Gupta, S., 2020. Bayesian synthetic control methods. Journal of Marketing Research57(5), pp.831-852.

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How to Construct the Synthetic Control

Basically, the synthetic control approach is the prediction problem.

“Like for the lasso, the goal of synthetic controls is out-of-sample prediction” (Abadie 2021, p. 408)

Abadie, A., 2021. Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature59(2), pp.391-425.

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Inference for Synthetic Control

  • Inference is a bit tricky, as we cannot rely on asymptotic theory to get standard errors and p-values.
  • Placebo tests for untreated units are used for inference.

Abadie, A., 2021. Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature, 59(2), pp.391-425.

Example (1) Impact of California Anti-Tobacco Legislation

measure of relative difference post and pre

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Sensitivity Tests for Synthetic Control

  • Sensitivity tests to the choice of predictors / the choice of the donor pool

Amjad, M., Shah, D. and Shen, D., 2018. Robust synthetic control. Journal of Machine Learning Research, 19(1), pp.802-852.

Example (2) Impact of Reunification on West Germany

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Sensitivity Tests for Synthetic Control

  • A train-test split approach can also be applied to the synthetic control.
    • Train – Test – Treat – Compare (TTTC) process

Varian, H.R., 2016. Causal inference in economics and marketing. Proceedings of the National Academy of Sciences113(27), pp.7310-7315.

Pre-treatment period

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Requirements for Synthetic Control

Abadie, A., 2021. Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature59(2), pp.391-425.

  • Contextual Requirements
    • Size of the Effect and Volatility of the Outcome
    • Availability of a Comparison Group
    • No Anticipation
    • No Interference
    • Convex Hull Condition
    • Time Horizon

  • Data Requirements
    • Aggregate Data on Predictors and Outcomes
    • Sufficient Pre-intervention Information
    • Sufficient Post-intervention Information

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Interrupted Time-Series Analysis

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What if There is No Control Group?

  • In the case of no control group, time-series forecasting models can be used to predict the counterfactual.
    • Synthetic control ≅ Predicted counterfactual from a weighted combination of the control group
    • Interrupted time-series ≅ Predicted counterfactual from the prior time trend
    • Time-series models are appropriate to model temporal changes (e.g., seasonality, autocorrelation, long-term time trends, short-term fluctuations).

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Example (1) Effect of Bombing on Fertility Rate

Rodgers, J.L., John, C.A.S. and Coleman, R., 2005. Did fertility go up after the Oklahoma City bombing? An analysis of births in metropolitan counties in Oklahoma, 1990–1999. Demography42(4), pp.675-692.

“Interrupted time series … gain their advantage from the pretreatment series that allows many potential threats [to validity] to be examined.” (p. 679)

“For Oklahoma County, the broad pattern shows an increase beginning exactly at the “bombing effect” month, nine and a half months after the bombing.” (Rodgers et al. 2005, p. 689)

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Example (2) Effect of Online Advertising on Clicks

Brodersen, K.H., Gallusser, F., Koehler, J., Remy, N. and Scott, S.L., 2015. Inferring Causal Impact Using Bayesian Structural Time-Series Models. The Annals of Applied Statistics, 9(1), pp.247-274.

Time series of search-related visits to the advertiser’s website (including both organic and paid clicks).

“Our method generalises the widely used difference-indifferences approach to the time-series setting by explicitly modelling the counterfactual of a time series observed both before and after the intervention… it provides a fully Bayesian time-series estimate for the effect.” (Brodersen et al. 2015)

See CausalImpact R package (http://google.github.io/CausalImpact/)

  • Interrupted time-series can also utilize control time series that were not affected by the intervention.

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Wrap-Up

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What’s Your Research Design and Data Structure?

Treatment and control groups are observed.

Parallel trend assumption is valid.

Treatment is assigned by arbitrary threshold.

There is an exogenous variable that can induce the treatment.

No

Local Average Treatment Effect

Yes

No

DID + Matching

Synthetic Control

Regression Discontinuity

Quasi-Experiments

The goal is causal inference, and random assignment is feasible.

Matching/

Weighting

There is a single treatment group and no information about functional forms.

Regression

Selection on Observables

No

Yes

No

Yes

No

Yes

Randomized Controlled Trial

No

Longitudinal data is observed.

Interrupted Time-Series Analysis

Yes

No

Longitudinal data is observed.

Not feasible

No

Yes

Yes

Difference-in-Differences (DID)

Yes

or

No

Yes

Control Function & Selection Model

or

Instrumental Variable

Quasi-experimental designs are available.

Note that the flowchart may depend on the context.

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