2.1 Approach 1

Calculate the difference in New Jersey’s employment level before and after the change in minimum wage

\[employment_t = \beta_0 + \beta_1Time_{t} + \mu_t\]

What are the problems with this approach?

There may be variables in the error term that impact employment and also changed from \(Time_0\) to \(Time_1\). This would result in a biased \(\beta_1\). For instance, if a recession took place from \(Time_0\) to \(Time_1\), this would be related to the variable \(Time\) and also to \(employment\), preventing us from learning the true impact of the minimum wage increase on employment. The coefficient for \(Time\) just shows the difference in employment between the periods after and before, but does not necessarily reveal the effect of the new policy.

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2.2 Approach 2

Calculate the difference in New Jersey’s employment level and another state’s employment level (not affected by the minimum wage policy) after the change in minimum wage

\[employment_i = \beta_0 + \beta_1TreatedState_i + \mu_i\]

What are the problems with this approach?

There may be variables in the error term that impact employment and that are state specific. This would result in a biased \(\beta_1\). For instance, if the two states have different political parties in power, the party’s policies may impact both the employment level and the decision to raise minimum wage. This would result in a violation of MLR.4 (zero conditional mean assumption), and our estimate of the impact of the minimum wage increase on employment would suffer from omitted variable bias. Similarly, even if we observe a difference between states, this difference could have already existed in the period before the new law was passed. The coefficient for \(TreatedState\) just shows the difference in employment between the treated State and the control State, but does not necessarily reveal the effect of the new policy.

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2.3 Approach 3

Difference-in-differences

\[employment_{it} = \beta_0 + \beta_1TreatedState_i + \beta_2Time_{t} + \beta_3TreatedState_i\times Time_{t} + \mu_{it}\]

To proceed with a Difference-in-differences analysis, the authors collected data for New Jersey (NJ) and Pennsylvania (PA). Pennsylvania shares a border with the state of New Jersey but the minimum wage was unchanged there. It serves as a control group.

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3.1 Data Structure

How is this data set different from the data sets we have seen so far?

What is our unit of analysis? The fast-food store.

# Print the first rows of the dataset
head(tb.fastfood)

This is a panel data set. Each fast-food store’s id shows up twice in the data, once for time period 0 and once for time period 1.

Is this a balanced panel? Meaning, do we observe each store in each of the time periods?

# Number of observations per time period
tb.fastfood %>% 
  group_by(time) %>%
  summarize(n = n()) %>%
  ungroup()

# Number of time periods per ID
tb.fastfood %>% 
  group_by(id) %>%
  summarize(n = n()) %>%
  filter(n!=2) %>%
  ungroup()

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3.2 Data Pre-processing

We will now look at the summary statistics of our numerical variables. Note that some of the variables are affected by missing data.

# Summary statistics
stargazer(  data.frame(tb.fastfood %>% select(-id))
          , type = "text"
          , header = FALSE
          , font.size = "small"
          , title = "Summary Statistics")

Summary Statistics
==========================================
Statistic  N   Mean  St. Dev.  Min   Max  
------------------------------------------
time      714 0.500   0.500     0     1   
emptot    714 20.979  9.508   0.000 85.000
state     714 0.812   0.391     0     1   
chain     714 2.090   1.079     1     4   
co_owned  714 0.356   0.479     0     1   
atmin     714 0.322   0.468     0     1   
wage      708 4.805   0.359   4.250 6.250 
hrsopen   707 14.452  2.821   7.000 24.000
pmeal     672 3.341   0.658   2.140 5.860 
fracft    708 0.344   0.239   0.000 0.904 
------------------------------------------

To make this example simpler for today’s lab we will clean the data set by eliminating stores with missing information:

# Create indicator for missing data by ID
tb.fastfood <- tb.fastfood %>% 
  mutate(missing = as.double(complete.cases(tb.fastfood)==FALSE)) %>%
  group_by(id) %>%
  mutate(missing = max(missing)) %>%
  ungroup()

# Check data
# View(tb.fastfood %>% filter(missing>0))

# Check how many stores have missing data
tb.fastfood %>% 
  filter(missing>0) %>% 
  summarize(n = n_distinct(id))

# Keep only IDs with complete information
tb.fastfood <- tb.fastfood %>% filter(missing==0)
tb.fastfood <- tb.fastfood %>% select(-missing)

# Alternatively you could drop the stores with NAs using:
tb.fastfood <- tb.fastfood %>%
  drop_na %>%
  group_by(id) %>%
  mutate(n = n()) %>%
  filter(n==2) %>%
  select(-n) %>% 
  ungroup()

We will also format the state and time variables as factors:

# Create state and time factors
tb.fastfood <- tb.fastfood %>%
  mutate(  state = factor(state, levels = c(0,1), labels = c("PA", "NJ"))
         , time  = factor(time,  levels = c(0,1), labels = c("Before", "After")))

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3.3 Descriptive Statistics

We can now produce a summary statistics table for the clean data set.

# Summary statistics
stargazer(data.frame(tb.fastfood %>% select(-id))
          , type = "text"
          , header = FALSE
          , font.size = "small"
          , title = "Summary Statistics")

Summary Statistics
==========================================
Statistic  N   Mean  St. Dev.  Min   Max  
------------------------------------------
emptot    632 20.691  8.731   5.000 70.500
chain     632 2.092   1.075     1     4   
co_owned  632 0.348   0.477     0     1   
atmin     632 0.326   0.469     0     1   
wage      632 4.804   0.358   4.250 6.250 
hrsopen   632 14.295  2.679   7.000 24.000
pmeal     632 3.355   0.662   2.390 5.860 
fracft    632 0.343   0.240   0.000 0.904 
------------------------------------------

# Plot the distribution of wages per state before and after the policy
ggplot(data = tb.fastfood, aes(x = wage)) + 
  geom_histogram(binwidth=0.2) + 
  facet_wrap(~ state + time) + theme_bw()

# Boxplot of full time employment per state before and after
ggplot(data = tb.fastfood, aes(x = time, y = emptot, fill = state)) + 
  geom_boxplot() +
  theme_bw() +
  labs(x = "Time", y = "FTE Employment", fill="State") +
  scale_fill_fivethirtyeight()

# Create a table with the variable means for treatment and control groups before and after the policy
tb.bf.aft <- tb.fastfood %>%
  group_by(state, time) %>%
  summarise(
    mean_emptot     = mean(emptot),
    mean_wage       = mean(wage),
    mean_pmeal      = mean(pmeal),
    mean_hrsopen    = mean(hrsopen),
    mean_fracft     = mean(fracft)) %>%
  ungroup()

tb.bf.aft

# Diff-in-diff plot
ggplot(  data = tb.bf.aft
       , aes(x = time, y = mean_emptot, group = state, color = state)) + 
  geom_point() + geom_line() + theme_bw() + 
  scale_color_fivethirtyeight() + ylim(15, 25)

4.1 Manual Diff-in-Diff

The differences-in-differences strategy amounts to comparing the change in mean FTE in New Jersey to the change in mean FTE in Pennsylvania.

\[\text{Treatment Effect} = (\bar{Y}_{NJ,T1}-\bar{Y}_{NJ,T0})-(\bar{Y}_{PA,T1}-\bar{Y}_{PA,T0})\]

# Difference in differences (manual)
(20.47745 - 20.00588) - (21.54098-23.59836)

[1] 2.5289

Surprisingly, employment rose in New Jersey relative to Pennsylvania after the minimum wage change. New Jersey stores were initially smaller than their Pennsylvania counterparts but grew relative to Pennsylvania stores after the rise in the minimum wage. The relative gain (the “difference in differences” of the changes in employment) is 2.52 FTE employees.

We can also write the treatment effect as:

\[\text{Treatment Effect} = (\bar{Y}_{NJ,T1}-\bar{Y}_{PA,T1})-(\bar{Y}_{NJ,T0}-\bar{Y}_{PA,T0})\]

# Difference in differences (manual)
(20.47745 - 21.54098) - (20.00588-23.59836)

[1] 2.5289

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4.2 Regression

We can estimate the difference-in-differences (DiD) estimator in a regression framework using the lm() function. This approach has several advantages:

• It is easy to calculate standard errors.

• We can control for other covariates, which may reduce residual variance and lead to more precise estimates.

• It naturally accommodates extensions to more than two periods.

• It can be adapted to varying treatment intensities (e.g., different levels of a policy across groups).

Although we are working with panel data, for a two-period DiD setting, the use of lm() is sufficient and preferred for its simplicity and transparency.

However, since the same unit (e.g., store, individual, firm) appears more than once, the assumption of independent errors is violated. To address this, we use robust (heteroskedasticity-consistent) standard errors, which can be computed with the sandwich and lmtest packages.

Note: While lm() is appropriate and convenient for estimating difference-in-differences with two-period panel data, for panels with more than two time periods, it is recommended to use the plm() function from the plm package.

# Estimate model
lm1 <- lm(emptot ~ state + time + state:time, data = tb.fastfood)

# Robust standard errors
robust_se1 <- sqrt(diag(vcovHC(lm1, type = "HC3")))

# Stargazer with robust SEs manually added
stargazer(lm1, lm1,
          se = list(NULL, robust_se1),
          column.labels = c("DiD", "DiD Robust SE"),
          type = "text",
          title = "Difference-in-Differences Results",
          omit.stat = c("ser", "f"),
          no.space = TRUE,
          header = FALSE,
          font.size = "small",
          dep.var.caption = "",
          dep.var.labels.include = FALSE,
          model.names = FALSE)

Difference-in-Differences Results
==============================================
                       DiD       DiD Robust SE
                       (1)            (2)     
----------------------------------------------
stateNJ             -3.592***      -3.592**   
                     (1.238)        (1.688)   
timeAfter             -2.057        -2.057    
                     (1.573)        (1.918)   
stateNJ:timeAfter     2.529          2.529    
                     (1.751)        (2.052)   
Constant            23.598***      23.598***  
                     (1.112)        (1.610)   
----------------------------------------------
Observations           632            632     
R2                    0.014          0.014    
Adjusted R2           0.010          0.010    
==============================================
Note:              *p<0.1; **p<0.05; ***p<0.01

Notes:

• state, time, and state:time represent the classic DiD model:

  • state: treated vs control
  • time: before vs after
  • state:time: the DiD interaction (your estimate of interest)

• We use vcovHC() to correct standard errors (HC3 is a good robust default).

Are the coefficients statistically significant? How do we interpret the regression coefficients?

• emptot00: average FTE employment in Pennsylvania at T0 (\(\beta_0\))

• emptot10: average FTE employment in New Jersey at T0 (\(\beta_0\) + \(\beta_1\))

• emptot01: average FTE employment in Pennsylvania at T1 (\(\beta_0\) + \(\beta_2\))

• emptot11: average FTE employment in New Jersey at T1 (\(\beta_0\) + \(\beta_1\) + \(\beta_2\) + \(\beta_3\))

What is the correspondence between the betas and the values from the table?

\(\beta_0 = 23.59836\)

# Store vector with diff-in-diff coefficients
coefs <- coefficients(lm1)

# Intercept
coefs[1]

(Intercept) 
     23.598 

\(\beta_1 = emptot10 - emptot00 =\)

20.00588 - 23.59836

[1] -3.5925

# Coefficient of state
coefs[2]

stateNJ 
-3.5925 

\(\beta_2 = emptot01 - emptot00 =\)

21.54098 - 23.59836

[1] -2.0574

# Coefficient of time
coefs[3]

timeAfter 
  -2.0574 

\(\beta_3 = (emptot11 - emptot10)-(emptot01 - emptot00) =\)

(20.47745 - 20.00588) - (21.54098 - 23.59836)

[1] 2.5289

# Coefficient of interaction term state*time
coefs[4]

stateNJ:timeAfter 
           2.5289 

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4.3 Adding Controls

We can control for other factors to get a better estimate of the treatment effect. Note that the introduction of the co-ownership and chain controls to the model, did not impact the estimated coefficients of timeAfter and stateNJ:timeAfter. This happens when the additional explanatory variables are uncorrelated with the variables that were already included in the model.

# Extended model with controls
lm2 <- update(lm1, . ~ . + factor(co_owned) + factor(chain))

# Robust standard errors (HC3)
robust_se2 <- sqrt(diag(vcovHC(lm2, type = "HC3")))

# Output regression table with robust SEs
stargazer(lm1, lm2,
          se = list(robust_se1, robust_se2),
          type = "text",
          title = "Difference-in-Differences with and without Controls",
          column.labels = c("Simple", "With Controls"),
          dep.var.caption = "",
          dep.var.labels.include = FALSE,
          model.names = FALSE,
          no.space = TRUE,
          omit.stat = c("ser", "f"),
          header = FALSE,
          font.size = "small")

Difference-in-Differences with and without Controls
==============================================
                      Simple     With Controls
                       (1)            (2)     
----------------------------------------------
stateNJ              -3.592**      -2.906**   
                     (1.688)        (1.475)   
timeAfter             -2.057        -2.057    
                     (1.918)        (1.629)   
factor(co_owned)1                   -0.859    
                                    (0.604)   
factor(chain)2                    -10.680***  
                                    (0.671)   
factor(chain)3                     -3.187***  
                                    (0.800)   
factor(chain)4                      -1.823*   
                                    (1.086)   
stateNJ:timeAfter     2.529          2.529    
                     (2.052)        (1.752)   
Constant            23.598***      26.770***  
                     (1.610)        (1.572)   
----------------------------------------------
Observations           632            632     
R2                    0.014          0.251    
Adjusted R2           0.010          0.242    
==============================================
Note:              *p<0.1; **p<0.05; ***p<0.01

After considering other factors, the diff-in-diff estimate stateNJ:timeAfter (treatment effect) is still not statistically significant. There is no significant difference between the relative changes in employment between states.

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