ScPoEconometrics
Differences-in-Differences
Bluebery Planterose
SciencesPo Paris 
 2023-02-07
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Recap from last session

Applied inference tools to regression analysis
Standard error of regression coefficients
Statistical significance of regression coefficients

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Recap from last session

Applied inference tools to regression analysis
Standard error of regression coefficients
Statistical significance of regression coefficients

Today: Differences-in-differences

Exploits changes in policy over time that don't affect everyone
Need to find (or construct) appropriate control group(s)
Key assumption: parallel trends
Empirical application: impact of minimum wage on employment

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Evaluation methods

Multiple regression often does not provide causal estimates because of selection on unobservables.

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Evaluation methods

Multiple regression often does not provide causal estimates because of selection on unobservables.
RCTs are one way to solve this problem but they are often impossible to do.

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Evaluation methods

Multiple regression often does not provide causal estimates because of selection on unobservables.
RCTs are one way to solve this problem but they are often impossible to do.
Four main causal evaluation methods used in economics:
- instrumental variables (IV),
- propensity-score matching,
- differences-in-differences (DiD), and
- regression discontinuity designs (RDD).

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Evaluation methods

Multiple regression often does not provide causal estimates because of selection on unobservables.
RCTs are one way to solve this problem but they are often impossible to do.
Four main causal evaluation methods used in economics:
- instrumental variables (IV),
- propensity-score matching,
- differences-in-differences (DiD), and
- regression discontinuity designs (RDD).
These methods are used to identify causal relationships between treatments and outcomes.

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Evaluation methods

Multiple regression often does not provide causal estimates because of selection on unobservables.
RCTs are one way to solve this problem but they are often impossible to do.
Four main causal evaluation methods used in economics:
- instrumental variables (IV),
- propensity-score matching,
- differences-in-differences (DiD), and
- regression discontinuity designs (RDD).
These methods are used to identify causal relationships between treatments and outcomes.
In this lecture, we will cover a popular and rigorous program evaluation method: differences-in-differences.

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Differences-in-Differences (DiD)

Usual starting point: subjects are not randomly allocated to treatment ⚠️

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Differences-in-Differences (DiD)

Usual starting point: subjects are not randomly allocated to treatment ⚠️

DiD Requirements:

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Differences-in-Differences (DiD)

Usual starting point: subjects are not randomly allocated to treatment ⚠️

DiD Requirements:

2 time periods: before and after treatment.

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Differences-in-Differences (DiD)

Usual starting point: subjects are not randomly allocated to treatment ⚠️

DiD Requirements:

2 time periods: before and after treatment.
2 groups:

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Differences-in-Differences (DiD)

Usual starting point: subjects are not randomly allocated to treatment ⚠️

DiD Requirements:

2 time periods: before and after treatment.
2 groups:
- control group: never receives treatment,

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Differences-in-Differences (DiD)

Usual starting point: subjects are not randomly allocated to treatment ⚠️

DiD Requirements:

2 time periods: before and after treatment.
2 groups:
- control group: never receives treatment,
- treatment group: initially untreated and then fully treated.

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Differences-in-Differences (DiD)

Usual starting point: subjects are not randomly allocated to treatment ⚠️

DiD Requirements:

2 time periods: before and after treatment.
2 groups:
- control group: never receives treatment,
- treatment group: initially untreated and then fully treated.
Under certain assumptions, control group can be used as the counterfactual for treatment group

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An Example: Minimum Wage and Employment

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An Example: Minimum Wage and Employment

Imagine you are interested in assessing the causal impact of increasing the minimum wage on (un)employment.

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An Example: Minimum Wage and Employment

Imagine you are interested in assessing the causal impact of increasing the minimum wage on (un)employment.
Why is this not that straightforward? What should the control group be?

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An Example: Minimum Wage and Employment

Imagine you are interested in assessing the causal impact of increasing the minimum wage on (un)employment.
Why is this not that straightforward? What should the control group be?
Seminal 1994 paper by prominent labor economists David Card and Alan Krueger entitled "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania"

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An Example: Minimum Wage and Employment

Imagine you are interested in assessing the causal impact of increasing the minimum wage on (un)employment.
Why is this not that straightforward? What should the control group be?
Seminal 1994 paper by prominent labor economists David Card and Alan Krueger entitled "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania"
Estimates the effect of an increase in the minimum wage on the employment rate in the fast-food industry. Why this industry?

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Institutional Details

In the US, there is a national minimum wage, but states can depart from it.

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Institutional Details

In the US, there is a national minimum wage, but states can depart from it.
April 1, 1992: New Jersey minimum wage increases from $4.25 to $5.05 per hour.

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Institutional Details

In the US, there is a national minimum wage, but states can depart from it.
April 1, 1992: New Jersey minimum wage increases from $4.25 to $5.05 per hour.
Neighboring Pennsylvania did not change its minimum wage level.

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Institutional Details

In the US, there is a national minimum wage, but states can depart from it.
April 1, 1992: New Jersey minimum wage increases from $4.25 to $5.05 per hour.
Neighboring Pennsylvania did not change its minimum wage level.

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Institutional Details

In the US, there is a national minimum wage, but states can depart from it.
April 1, 1992: New Jersey minimum wage increases from $4.25 to $5.05 per hour.
Neighboring Pennsylvania did not change its minimum wage level.

Pennsylvania and New Jersey are very similar: similar institutions, similar habits, similar consumers, similar incomes, similar weather, etc.

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Card and Krueger (1994): Methodology

Surveyed 410 fast-food establishments in New Jersey (NJ) and eastern Pennsylvania

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Card and Krueger (1994): Methodology

Surveyed 410 fast-food establishments in New Jersey (NJ) and eastern Pennsylvania
Timing:

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Card and Krueger (1994): Methodology

Surveyed 410 fast-food establishments in New Jersey (NJ) and eastern Pennsylvania
Timing:
- Survey before NJ MW increase: Feb/March 1992

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Card and Krueger (1994): Methodology

Surveyed 410 fast-food establishments in New Jersey (NJ) and eastern Pennsylvania
Timing:
- Survey before NJ MW increase: Feb/March 1992
- Survey after NJ MW increase: Nov/Dec 1992

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Card and Krueger (1994): Methodology

Surveyed 410 fast-food establishments in New Jersey (NJ) and eastern Pennsylvania
Timing:
- Survey before NJ MW increase: Feb/March 1992
- Survey after NJ MW increase: Nov/Dec 1992
What comparisons do you think they did?

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Card and Krueger (1994): Methodology

Surveyed 410 fast-food establishments in New Jersey (NJ) and eastern Pennsylvania
Timing:
- Survey before NJ MW increase: Feb/March 1992
- Survey after NJ MW increase: Nov/Dec 1992
What comparisons do you think they did?

Let's take a closer at their data

  # install package that contains the cleaned data
remotes::install_github("b-rodrigues/diffindiff")
  # load package
library(diffindiff)
  # load data
ck1994 <- njmin

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Card and Krueger (1994): Methodology

Surveyed 410 fast-food establishments in New Jersey (NJ) and eastern Pennsylvania
Timing:
- Survey before NJ MW increase: Feb/March 1992
- Survey after NJ MW increase: Nov/Dec 1992
What comparisons do you think they did?

Let's take a closer at their data

  # install package that contains the cleaned data
remotes::install_github("b-rodrigues/diffindiff")
  # load package
library(diffindiff)
  # load data
ck1994 <- njmin

ck1994 %>%
  select(sheet,chain,state,observation,empft,emppt) %>%
  head()

## # A tibble: 6 × 6
##   sheet chain  state        observation   empft emppt
##   <chr> <chr>  <chr>        <chr>         <dbl> <dbl>
## 1 46    bk     Pennsylvania February 1992  30    15  
## 2 49    kfc    Pennsylvania February 1992   6.5   6.5
## 3 506   kfc    Pennsylvania February 1992   3     7  
## 4 56    wendys Pennsylvania February 1992  20    20  
## 5 61    wendys Pennsylvania February 1992   6    26  
## 6 62    wendys Pennsylvania February 1992   0    31

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Task 1 (10 minutes)

Take a look at the dataset and list the variables. Check the variable definitions with ?njmin.
Tabulate the number of stores by state and by survey wave (observation). Does it match what's in Table 1 of the paper?
Create a full-time equivalent (FTE) employees variable called empfte equal to empft + 0.5*emppt + nmgrs. empft and emppt correspond respectively to the number of full-time and part-time employees. nmgrs corresponds to the number of managers. This is how Card and Krueger compute their full-time equivalent (FTE) employment variable (p.775 of the paper).
Compute the average number of FTE employment, average percentage of FT employees (out of the number of FTE employees), and average starting wage (wage_st) by state and by survey wave. Compare your results with Table 2 of the paper.
How different are New Jersey and Pennsylvania's fast-food restaurants before the minimum wage increase?

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Card and Krueger DiD: Tabular Results

Average Employment Per Store Before and After the Rise in NJ Minimum Wage

Variables	Pennsylvania	New Jersey
FTE employment before	23.33	20.44
FTE employment after	21.17	21.03
Change in mean FTE employment	-2.17	0.59

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Card and Krueger DiD: Tabular Results

Average Employment Per Store Before and After the Rise in NJ Minimum Wage

Variables	Pennsylvania	New Jersey
FTE employment before	23.33	20.44
FTE employment after	21.17	21.03
Change in mean FTE employment	-2.17	0.59

DiD Estimate

Differences-in-differences causal estimate: $0.59 - (- 2.17) = 2.76$

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Card and Krueger DiD: Tabular Results

Average Employment Per Store Before and After the Rise in NJ Minimum Wage

Variables	Pennsylvania	New Jersey
FTE employment before	23.33	20.44
FTE employment after	21.17	21.03
Change in mean FTE employment	-2.17	0.59

DiD Estimate

Differences-in-differences causal estimate: $0.59 - (- 2.17) = 2.76$

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Card and Krueger DiD: Tabular Results

Average Employment Per Store Before and After the Rise in NJ Minimum Wage

Variables	Pennsylvania	New Jersey
FTE employment before	23.33	20.44
FTE employment after	21.17	21.03
Change in mean FTE employment	-2.17	0.59

DiD Estimate

Differences-in-differences causal estimate: $0.59 - (- 2.17) = 2.76$

Yes the essence of differences-in-differences is that simple! 😀

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Card and Krueger DiD: Tabular Results

Average Employment Per Store Before and After the Rise in NJ Minimum Wage

Variables	Pennsylvania	New Jersey
FTE employment before	23.33	20.44
FTE employment after	21.17	21.03
Change in mean FTE employment	-2.17	0.59

DiD Estimate

Differences-in-differences causal estimate: $0.59 - (- 2.17) = 2.76$

Yes the essence of differences-in-differences is that simple! 😀

Let's look at these results graphically.

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DiD Graphically

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DiD Graphically

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DiD Graphically

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DiD Graphically

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DiD Graphically

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DiD Graphically

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What if we had done a naive after/before comparison?

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What if we had done a naive after/before comparison?

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What if we had done a naive after NJ/PA comparison?

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What if we had done a naive after NJ/PA comparison?

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Estimation20 / 42

DiD in Regression Form

In practice, DiD is usually estimated on more than 2 periods (4 observations)
There are more data points before and after the policy change

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DiD in Regression Form

In practice, DiD is usually estimated on more than 2 periods (4 observations)
There are more data points before and after the policy change

3 ingredients:

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DiD in Regression Form

In practice, DiD is usually estimated on more than 2 periods (4 observations)
There are more data points before and after the policy change

3 ingredients:

Treatment dummy variable: $T R E A T_{s}$ where the $s$ subscript reminds us that the treatment is at the state level

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DiD in Regression Form

In practice, DiD is usually estimated on more than 2 periods (4 observations)
There are more data points before and after the policy change

3 ingredients:

Treatment dummy variable: $T R E A T_{s}$ where the $s$ subscript reminds us that the treatment is at the state level
Post-treatment periods dummy variables: $P O S T_{t}$ where the $t$ subscript reminds us that this variable varies over time

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DiD in Regression Form

In practice, DiD is usually estimated on more than 2 periods (4 observations)
There are more data points before and after the policy change

3 ingredients:

Treatment dummy variable: $T R E A T_{s}$ where the $s$ subscript reminds us that the treatment is at the state level
Post-treatment periods dummy variables: $P O S T_{t}$ where the $t$ subscript reminds us that this variable varies over time
Interaction term between the two: $T R E A T_{s} \times P O S T_{t}$ 👉 the coefficient on this term is the DiD causal effect!

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DiD in Regression Form

Treatment dummy variable $T R E A T_{s} = {\begin{cases} \begin{array}{lcl} 0 if s = Pennsylvania \\ 1 if s = New Jersey \end{array} \end{cases}$

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DiD in Regression Form

Treatment dummy variable $T R E A T_{s} = {\begin{cases} \begin{array}{lcl} 0 if s = Pennsylvania \\ 1 if s = New Jersey \end{array} \end{cases}$

Post-treatment periods dummy variable $P O S T_{t} = {\begin{cases} \begin{array}{lcl} 0 if t < April 1, 1992 \\ 1 if t \geq April 1, 1992 \end{array} \end{cases}$

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DiD in Regression Form

Treatment dummy variable $T R E A T_{s} = {\begin{cases} \begin{array}{lcl} 0 if s = Pennsylvania \\ 1 if s = New Jersey \end{array} \end{cases}$

Post-treatment periods dummy variable $P O S T_{t} = {\begin{cases} \begin{array}{lcl} 0 if t < April 1, 1992 \\ 1 if t \geq April 1, 1992 \end{array} \end{cases}$

Which observations correspond to $T R E A T_{s} \times P O S T_{t} = 1$ ?

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DiD in Regression Form

Treatment dummy variable $T R E A T_{s} = {\begin{cases} \begin{array}{lcl} 0 if s = Pennsylvania \\ 1 if s = New Jersey \end{array} \end{cases}$

Post-treatment periods dummy variable $P O S T_{t} = {\begin{cases} \begin{array}{lcl} 0 if t < April 1, 1992 \\ 1 if t \geq April 1, 1992 \end{array} \end{cases}$

Which observations correspond to $T R E A T_{s} \times P O S T_{t} = 1$ ?

Let's put all these ingredients together: $E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$
$δ$ : causal effect of the minimum wage increase on employment

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

We have the following:

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

We have the following:

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 0) = α$

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

We have the following:

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 0) = α$

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 1) = α + γ$

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

We have the following:

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 0) = α$

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 1) = α + γ$

$E (E M P_{s t} | T R E A T_{s} = 1, P O S T_{t} = 0) = α + β$

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

We have the following:

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 0) = α$

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 1) = α + γ$

$E (E M P_{s t} | T R E A T_{s} = 1, P O S T_{t} = 0) = α + β$

$E (E M P_{s t} | T R E A T_{s} = 1, P O S T_{t} = 1) = α + β + γ + δ$

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

We have the following:

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 0) = α$

$E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 1) = α + γ$

$E (E M P_{s t} | T R E A T_{s} = 1, P O S T_{t} = 0) = α + β$

$E (E M P_{s t} | T R E A T_{s} = 1, P O S T_{t} = 1) = α + β + γ + δ$

$[E (E M P_{s t} | T R E A T_{s} = 1, P O S T_{t} = 1) - E (E M P_{s t} | T R E A T_{s} = 1, P O S T_{t} = 0)] - [E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 1) - E (E M P_{s t} | T R E A T_{s} = 0, P O S T_{t} = 0)] = δ$

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

In table form:

	Pre mean	Post mean	$Δ$ (post - pre)
Pennsylvania (PA)	$α$	$α + γ$	$γ$
New Jersey (NJ)	$α + β$	$α + β + γ + δ$	$γ + δ$
$Δ$ (NJ - PA)	$β$	$β + δ$	$δ$

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Understanding the Regression

$E M P_{s t} = α + β T R E A T_{s} + γ P O S T_{t} + δ (T R E A T_{s} \times P O S T_{t}) + ε_{s t}$

In table form:

	Pre mean	Post mean	$Δ$ (post - pre)
Pennsylvania (PA)	$α$	$α + γ$	$γ$
New Jersey (NJ)	$α + β$	$α + β + γ + δ$	$γ + δ$
$Δ$ (NJ - PA)	$β$	$β + δ$	$δ$

This table generalizes to other settings by substituting Pennsylvania with Control and New Jersey with Treatment

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Task 2 (10 minutes)

Create a dummy variable, treat, equal to FALSE if state is Pennsylvania and TRUE if New Jersey.
Create a dummy variable, post, equal to FALSE if observation is February 1992 and TRUE otherwise.
Estimate the following regression model. Do you obtain the same results as in slide 9?

$e m p f t e_{s t} = α + β t r e a t_{s} + γ p o s t_{t} + δ (t r e a t_{s} \times p o s t_{t}) + ε_{s t}$

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Identifying Assumptions26 / 42

DiD Crucial Assumption: Parallel Trends

Common or parallel trends assumption: absent any minimum wage increase, Pennsylvania's fast-food employment trend would have been what we should have expected to see in New Jersey.

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DiD Crucial Assumption: Parallel Trends

Common or parallel trends assumption: absent any minimum wage increase, Pennsylvania's fast-food employment trend would have been what we should have expected to see in New Jersey.

This assumption states that Pennsylvania's fast-food employment trend between February and November 1992 provides a reliable counterfactual employment trend New Jersey's fast-food industry would have experienced had New Jersey not increased its minimum wage.

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DiD Crucial Assumption: Parallel Trends

Common or parallel trends assumption: absent any minimum wage increase, Pennsylvania's fast-food employment trend would have been what we should have expected to see in New Jersey.

This assumption states that Pennsylvania's fast-food employment trend between February and November 1992 provides a reliable counterfactual employment trend New Jersey's fast-food industry would have experienced had New Jersey not increased its minimum wage.
Impossible to completely validate or invalidate this assumption.
Intuitive check: compare trends before policy change (and after policy change if no expected medium-term effects)

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Parallel Trends: Graphically

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Checking the parallel trends assumption

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Checking the parallel trends assumption

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Parallel trends assumption $\to$ Verified ✅

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Parallel trends assumption $\to$ Verified ✅

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Parallel trends assumption $\to$ Not verified ❌

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Parallel trends assumption $\to$ Not verified ❌

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Parallel Trends Assumption: Card and Krueger (2000)

Here is the actual trends for Pennsylvania and New Jersey

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Parallel Trends Assumption: Card and Krueger (2000)

Here is the actual trends for Pennsylvania and New Jersey

Is the common trend assumption likely to be verified?

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Parallel Trends Assumption: Formally

Let:

$Y_{i s t}^{1}$ : fast food employment at restaurant $i$ in state $s$ at time $t$ if there is a high state MW;

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Parallel Trends Assumption: Formally

Let:

$Y_{i s t}^{1}$ : fast food employment at restaurant $i$ in state $s$ at time $t$ if there is a high state MW;
$Y_{i s t}^{0}$ : fast food employment at restaurant $i$ in state $s$ at time $t$ if there is a low state MW;

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Parallel Trends Assumption: Formally

Let:

$Y_{i s t}^{1}$ : fast food employment at restaurant $i$ in state $s$ at time $t$ if there is a high state MW;
$Y_{i s t}^{0}$ : fast food employment at restaurant $i$ in state $s$ at time $t$ if there is a low state MW;

These are potential outcomes, you can only observe one of the two.

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Parallel Trends Assumption: Formally

Let:

$Y_{i s t}^{1}$ : fast food employment at restaurant $i$ in state $s$ at time $t$ if there is a high state MW;
$Y_{i s t}^{0}$ : fast food employment at restaurant $i$ in state $s$ at time $t$ if there is a low state MW;

These are potential outcomes, you can only observe one of the two.

The key assumption underlying DiD estimation is that, in the no-treatment state, restaurant $i$ 's outcome in state $s$ at time $t$ is given by:

$E [Y_{i s t}^{0} | s, t] = γ_{s} + λ_{t}$

2 implicit assumptions:

Selection bias: relates to fixed state characteristics $(γ)$
Time trend: same time trend for treatment and control group $(λ)$

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Parallel Trends Assumption: Formally

Outcomes in the comparison group:

$E [Y_{i s t} | s = Pennsylvania, t = Feb] = γ_{P A} + λ_{F e b}$

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Parallel Trends Assumption: Formally

Outcomes in the comparison group:

$E [Y_{i s t} | s = Pennsylvania, t = Feb] = γ_{P A} + λ_{F e b}$ $E [Y_{i s t} | s = Pennsylvania, t = Nov] = γ_{P A} + λ_{N o v}$

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Parallel Trends Assumption: Formally

Outcomes in the comparison group:

$E [Y_{i s t} | s = Pennsylvania, t = Feb] = γ_{P A} + λ_{F e b}$ $E [Y_{i s t} | s = Pennsylvania, t = Nov] = γ_{P A} + λ_{N o v}$

$\begin{aligned} E [Y_{i s t} & | s = Pennsylvania, t = Nov] - E [Y_{i s t} | s = Pennsylvania, t = Feb] \\ = γ_{P A} + λ_{N o v} - (γ_{P A} + λ_{F e b}) \\ = λ_{N o v} - λ_{F e b} \end{aligned}$

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Parallel Trends Assumption: Formally

Outcomes in the comparison group:

$E [Y_{i s t} | s = Pennsylvania, t = Feb] = γ_{P A} + λ_{F e b}$

$E [Y_{i s t} | s = Pennsylvania, t = Nov] = γ_{P A} + λ_{N o v}$

$\begin{aligned} E [Y_{i s t} & | s = Pennsylvania, t = Nov] - E [Y_{i s t} | s = Pennsylvania, t = Feb] \\ = γ_{P A} + λ_{N o v} - (γ_{P A} + λ_{F e b}) \\ = \underset{time trend}{\underset{⏟}{λ_{N o v} - λ_{F e b}}} \end{aligned}$

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Parallel Trends Assumption: Formally

Outcomes in the comparison group:

$E [Y_{i s t} | s = Pennsylvania, t = Feb] = γ_{P A} + λ_{F e b}$

$E [Y_{i s t} | s = Pennsylvania, t = Nov] = γ_{P A} + λ_{N o v}$

$\to$ the comparison group allows to estimate the time trend.

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Parallel Trends Assumption: Formally

Let $δ$ denote the true impact of the minimum wage increase:

$E [Y_{i s t}^{1} - Y_{i s t}^{0} | s, t] = δ$

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Parallel Trends Assumption: Formally

Let $δ$ denote the true impact of the minimum wage increase:

$E [Y_{i s t}^{1} - Y_{i s t}^{0} | s, t] = δ$

Outcomes in the treatment group:

$E [Y_{i s t} | s = New Jersey, t = Feb] = γ_{N J} + λ_{F e b}$

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Parallel Trends Assumption: Formally

Let $δ$ denote the true impact of the minimum wage increase:

$E [Y_{i s t}^{1} - Y_{i s t}^{0} | s, t] = δ$

Outcomes in the treatment group:

$E [Y_{i s t} | s = New Jersey, t = Feb] = γ_{N J} + λ_{F e b}$ $E [Y_{i s t} | s = New Jersey, t = Nov] = γ_{N J} + δ + λ_{N o v}$

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Parallel Trends Assumption: Formally

Let $δ$ denote the true impact of the minimum wage increase:

$E [Y_{i s t}^{1} - Y_{i s t}^{0} | s, t] = δ$

Outcomes in the treatment group:

$E [Y_{i s t} | s = New Jersey, t = Feb] = γ_{N J} + λ_{F e b}$ $E [Y_{i s t} | s = New Jersey, t = Nov] = γ_{N J} + δ + λ_{N o v}$ $\begin{aligned} E [Y_{i s t} & | s = New Jersey, t = Nov] - E [Y_{i s t} | s = New Jersey, t = Feb] \\ = γ_{N J} + δ + λ_{N o v} - (γ_{N J} + λ_{F e b}) \\ = δ + λ_{N o v} - λ_{F e b} \end{aligned}$

39 / 42

Parallel Trends Assumption: Formally

Let $δ$ denote the true impact of the minimum wage increase:

$E [Y_{i s t}^{1} - Y_{i s t}^{0} | s, t] = δ$

Outcomes in the treatment group:

$E [Y_{i s t} | s = New Jersey, t = Feb] = γ_{N J} + λ_{F e b}$

$E [Y_{i s t} | s = New Jersey, t = Nov] = γ_{N J} + δ + λ_{N o v}$

$\begin{aligned} E [Y_{i s t} & | s = New Jersey, t = Nov] - E [Y_{i s t} | s = New Jersey, t = Feb] \\ = γ_{N J} + δ + λ_{N o v} - (γ_{N J} + λ_{F e b}) \\ = δ + \underset{time trend}{\underset{⏟}{λ_{N o v} - λ_{F e b}}} \end{aligned}$

40 / 42

Parallel Trends Assumption: Formally

Therefore we have:

$\begin{aligned} E [Y_{i s t} & | s = PA, t = Nov] - E [Y_{i s t} | s = PA, t = Feb] = \underset{time trend}{\underset{⏟}{λ_{N o v} - λ_{F e b}}} \end{aligned}$

41 / 42

Parallel Trends Assumption: Formally

Therefore we have:

$\begin{aligned} E [Y_{i s t} & | s = PA, t = Nov] - E [Y_{i s t} | s = PA, t = Feb] = \underset{time trend}{\underset{⏟}{λ_{N o v} - λ_{F e b}}} \end{aligned}$

$\begin{aligned} E [Y_{i s t} & | s = NJ, t = Nov] - E [Y_{i s t} | s = NJ, t = Feb] = δ + \underset{time trend}{\underset{⏟}{λ_{N o v} - λ_{F e b}}} \end{aligned}$

41 / 42

Parallel Trends Assumption: Formally

Therefore we have:

$\begin{aligned} E [Y_{i s t} & | s = PA, t = Nov] - E [Y_{i s t} | s = PA, t = Feb] = \underset{time trend}{\underset{⏟}{λ_{N o v} - λ_{F e b}}} \end{aligned}$

$\begin{aligned} E [Y_{i s t} & | s = NJ, t = Nov] - E [Y_{i s t} | s = NJ, t = Feb] = δ + \underset{time trend}{\underset{⏟}{λ_{N o v} - λ_{F e b}}} \end{aligned}$

$\begin{aligned} D D & = E [Y_{i s t} | s = NJ, t = Nov] - E [Y_{i s t} | s = NJ, t = Feb] \\ - (E [Y_{i s t} | s = PA, t = Nov] - E [Y_{i s t} | s = PA, t = Feb]) \\ = δ + λ_{N o v} - λ_{F e b} - (λ_{N o v} - λ_{F e b}) \\ = δ \end{aligned}$

41 / 42

END


	bluebery.planterose@sciencespo.fr
	Original Slides from Florian Oswald
	Book
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ScPoEconometrics

Differences-in-Differences

Bluebery Planterose

SciencesPo Paris 2023-02-07

Recap from last session

Recap from last session

Today: Differences-in-differences

Evaluation methods

Evaluation methods

Evaluation methods

Evaluation methods

Evaluation methods

Differences-in-Differences (DiD)

Differences-in-Differences (DiD)

DiD Requirements:

Differences-in-Differences (DiD)

DiD Requirements:

Differences-in-Differences (DiD)

DiD Requirements:

Differences-in-Differences (DiD)

DiD Requirements:

Differences-in-Differences (DiD)

DiD Requirements:

Differences-in-Differences (DiD)

DiD Requirements:

An Example: Minimum Wage and Employment

An Example: Minimum Wage and Employment

An Example: Minimum Wage and Employment

An Example: Minimum Wage and Employment

An Example: Minimum Wage and Employment

Institutional Details

Institutional Details

Institutional Details

Institutional Details

Institutional Details

Card and Krueger (1994): Methodology

Card and Krueger (1994): Methodology

Card and Krueger (1994): Methodology

Card and Krueger (1994): Methodology

Card and Krueger (1994): Methodology

Card and Krueger (1994): Methodology

Card and Krueger (1994): Methodology

Task 1 (10 minutes)

Card and Krueger DiD: Tabular Results

Card and Krueger DiD: Tabular Results

DiD Estimate

Card and Krueger DiD: Tabular Results

DiD Estimate

Card and Krueger DiD: Tabular Results

DiD Estimate

Card and Krueger DiD: Tabular Results

DiD Estimate

DiD Graphically

DiD Graphically

DiD Graphically

DiD Graphically

DiD Graphically

DiD Graphically

What if we had done a naive after/before comparison?

What if we had done a naive after/before comparison?

What if we had done a naive after NJ/PA comparison?

What if we had done a naive after NJ/PA comparison?

Estimation

DiD in Regression Form

DiD in Regression Form

DiD in Regression Form

DiD in Regression Form

DiD in Regression Form

DiD in Regression Form

DiD in Regression Form

DiD in Regression Form

DiD in Regression Form

Understanding the Regression

Understanding the Regression

Understanding the Regression

Understanding the Regression

Understanding the Regression

Understanding the Regression

Understanding the Regression

Understanding the Regression

SciencesPo Paris
2023-02-07

Parallel trends assumption $\to$ Verified ✅

Parallel trends assumption $\to$ Verified ✅

Parallel trends assumption $\to$ Not verified ❌

Parallel trends assumption $\to$ Not verified ❌