In this section I am replicate a study by David Card and Alan B. Krueger about the effect of a raise in minimum wages on employment.

Conventional economic theory suggests that in a labour market with perfect competition an increase in the minimum wage leads to an increase in unemployment.

In April 1992, the U.S. state of New Jersey (NJ) raised the minimum wage from $4.25 to $5.05. Card and Krueger (1994) use a DiD approach and show that this increase in minimum wages led to an increase in employment in the sector of fast food restaurants.

The control group in their setting is the neighbouring U.S. state of Pennsylvania (PA), which was not subject to this policy change The authors conducted a survey before and after the raise of the minimum wage with a representative sample of fast food restaurants in NJ and PA. This setting can be regarded as quasi experimental, as both states are not identical in many aspects and the legislative procedure, in order the raise the minimum wage, was not initiated at random.

• \(emp^{PA,feb} = \beta_{0} + u\)

• \(emp^{PA,nov} = \beta_{0} + \beta_{2} + u\)

• \(emp^{NJ,feb} = \beta_{0} + \beta_{1} + u\)

• \(emp^{NJ,nov} = \beta_{0} + \beta_{1} + \beta_{2} + \beta_{3} + u\)

Diffence in Difference Estimator:

\(counterfactual = emp^{NJ,feb} - (emp^{PA,feb} - emp^{PA,nov})\)

\(counterfactual = (\beta_{0} + \beta_{1}) - ((\beta_{0}) - (\beta_{0} + \beta_{2}))\)

\(counterfactual = \beta_{0} + \beta_{1} + \beta_{2}\)

• Raw Data

• Cleaned Data

• Transposed Data (February 1992 vs November 1992)

• Final Data

Generate a variable that measures employment. According to the paper, the full-time equivalents (FTE/emptot) consist of full-time employees (empft), managers (nmgrs) and part-time employees (emppt). The latter are multiplied by factor 0.5 before entering the calculation. Also, I am generating the share of full-time employees of all FTE (pct_ftw).

card_krueger_1994_mod <- card_krueger_1994 %>%
  mutate(emptot = empft + nmgrs + 0.5 * emppt,
         pct_fte = empft / emptot * 100)

Before calculating the DiD estimator with OLS, I want to deduce it by means of differencing the mean values of employment (emptot) between each group. This is easily done with functions group_by() and summarise(). We obtain four groups with distinct mean values!

differences <- card_krueger_1994_mod %>%
  group_by(observation, state) %>%
  summarise(emptot = mean(emptot, na.rm = TRUE))

# Treatment group (NJ) before treatment
njfeb <- differences[1,3]

# Control group (PA) before treatment
pafeb <- differences[2,3]

# Treatment group (NJ) after treatment
njnov <- differences[3,3]

# Control group (PA) after treatment
panov <- differences[4,3]

The Average Treatment Effect (ATE) in this setting can be determined in two ways:

• Calculate the difference between the difference of November and February within NJ and PA

(njnov-njfeb)-(panov-pafeb) 

##     emptot
## 1 2.753606

• Calculate the difference between the difference of NJ and PA within November and February

(njnov-panov)-(njfeb-pafeb)

##     emptot
## 1 2.753606

First, I use the differences of variable emptotcalculated in the previous step for NJ and PJ in February and November. Additionally, we require the outcome of NJ if the treatment (raise of the minimum wage) did not happen. This is called the counterfactual outcome (nj_counterfactual). The DiD assumption states that the trends of treatment and control group are identical until the treatment takes place. Hence, without the treatment the employment (emptot) of NJ would decline from February to November by the same amount as PA.

# Calculate counterfactual outcome
nj_counterfactual <- tibble(
  observation = c("February 1992","November 1992"), 
  state = c("New Jersey (Counterfactual)","New Jersey (Counterfactual)"),
  emptot = as.numeric(c(njfeb, njfeb-(pafeb-panov)))) 
# Data points for treatment event
intervention <- tibble(
    observation = c("Intervention", "Intervention", "Intervention"),
    state = c("New Jersey", "Pennsylvania", "New Jersey (Counterfactual)"),
    emptot = c(19.35, 22.3, 19.35)) 
# Combine data
did_plotdata <- bind_rows(differences, nj_counterfactual, intervention)

In November 1992, the distance between the actual and counterfactual employment (emptot) of NJ identifies the causal effect of an increase in minimum wages on employment.

The coefficient of time:treated is the difference-in-differences estimator. The treatment (raise of the minimum wage) leads on average to an increase of employment (emptot) in NJ by 2.75 FTE.

# Declare as panel data
panel <- pdata.frame(card_krueger_1994_mod, "sheet")

# Within model
did.reg <- plm(emptot ~ time + treated + time:treated, 
               data = panel, model = "within")

# obtain clustered standard errors
coeftest(did.reg, vcov = function(x) 
  vcovHC(x, cluster = "group", type = "HC1"))

## 
## t test of coefficients:
## 
##               Estimate Std. Error t value Pr(>|t|)  
## time2          -2.2833     1.2465 -1.8319  0.06775 .
## time2:treated   2.7500     1.3359  2.0585  0.04022 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1