Minimum Wages around the World

The Minimum Wage Debate

Minimum wages are a hotly debated topic in economics and politics.

• Proponents argue that minimum wages are necessary to protect workers from exploitation and poverty.

• Opponents argue that minimum wages lead to unemployment and reduce the competitiveness of firms.

• Labour economists have been studying the effects of minimum wages for decades.

• Their answer:

The New Jersey Minimum Wage Experiment

The idea

Why not just compare the employment in New Jersey before and after the minimum wage increase?

Difference in Differences: The Idea

Difference in Differences: The Idea

• Find a control group that did not experience the treatment.
• Compare the difference in the outcome variable between the treatment and control group before the treatment.
• Compare the difference in the outcome variable between the treatment and control group after the treatment.
• The difference in the differences is the treatment effect.
• The treatment effect is estimated as the difference in trends.

Strengths and Requirements

Card and Krueger’s DiD Estimates

• Outcome Variable: \(Y\) = FTE (Full-Time Equivalent employment).
• FTE decreased in Pennsylvania (−2.16) and increased slightly in New Jersey (0.59).
• The difference in differences is 2.75.
• This result speaks against the conventional wisdom that minimum wages reduce employment.

The DiD Estimator

Difference in Differences can be estimated in two ways:

1. Simple DiD: The difference in the average outcome variable between the treatment and control group before and after the treatment.

2. Regression DiD: A linear regression model that includes a treatment indicator and a time indicator.

What Does Simple DiD Identify?

Rearranging reveals that the estimator equals:

\[\begin{aligned} \hat{\delta}^{2\times 2}_{kU} &= \underbrace{E[Y^1_{k}|Post]-E[Y^0_{k}|Post]}_{\text{ATT}} \\[6pt] &+ \underbrace{\bigl(E[Y^0_{k}|Post]-E[Y^0_{k}|Pre]\bigr) - \bigl(E[Y^0_{U}|Post]-E[Y^0_{U}|Pre]\bigr)}_{\text{Bias} = 0 \iff \text{Parallel Trends holds}} \end{aligned}\]

Regression DiD

Difference in Differences can also be estimated using a linear regression model.

Regression DiD Visually

OLS Specification of the DiD Equation

In the MW example the regression DiD model would be:

• \(Y_{it} = \alpha + \gamma NJ_s + \lambda d_t + \delta(NJ \times d)_{st} + \varepsilon_{it}\).

  • PA pre-treatment: \(\alpha\).
  • PA post-treatment: \(\alpha + \lambda\).
  • NJ pre-treatment: \(\alpha + \gamma\).
  • NJ post-treatment: \(\alpha + \gamma + \lambda + \delta\).

• DiD equation = (NJ Post - NJ Pre) - (PA Post - PA Pre) = \(\delta\)

The Two-Way Fixed Effects (TWFE) in R

library(tidyverse); library(modelsummary); library(fixest)
od <- causaldata::organ_donations

# Treatment variable
od <- od %>%
     mutate(Treated = State == 'California' & 
            Quarter %in% c('Q32011','Q42011','Q12012'))

# feols clusters by the first
# fixed effect by default, no adjustment necessary
clfe <- feols(Rate ~ Treated | State + Quarter,
           data = od)
msummary(clfe, stars = c('*' = .1, '**' = .05, '***' = .01),
         coef = "Treated", nobs = TRUE)

Effect of the “Active Choice” Policy on Organ Donations

Standard errors clustered at the state level. * p<0.1, ** p<0.05, *** p<0.01

DiD with Multiple Time Periods and Groups

• The DiD estimator can be extended to multiple time periods and groups.

• Advantages:

  • More variation in the treatment and control groups.
  • More variation over time.

\[Y_{it} = \alpha + \sum^I_{k=i-1} \gamma_k D_{ki} + \sum^T_{j=t-1}\lambda_j P_{jt} + \delta_{DD} (D_i \times P_t) + \varepsilon_{it}\]

• We are adding fixed effects for the treatment and control groups and for the time periods.

Standard Errors in DiD

• Potential serial correlation of error terms within units over time

  • Lead to biased standard errors

• Cluster standard errors at the level of units (e.g., states, firms, individuals)

• Influential paper by Bertrand, Duflo & Mullainathan (2004, QJE)

  • They have shown that many papers were too optimistic (too low standard errors) by neglecting serial correlation

Treatment Effects in DiD

For whom are the treatment effects estimated?

The Parallel Trends Assumption

• Parallel trends assumption is crucial for the validity of the DiD estimator.

• The assumption is that the treatment and control groups would have followed the same trend in the absence of the treatment.

• The assumption is untestable.

• The assumption can be supported by:

  • Graphical analysis.
  • Event study analysis.
  • Placebo tests.

Graphical Analysis

• A typical way to support the parallel trends assumption is to plot the outcome variable for the treatment and control groups before the treatment.

• Parallel pre-trends can mitigate concerns about violations of the parallel trends assumption.

Event Study Analysis

• With only one treatment group and one never-treated group the event study analysis is simply this OLS:

\[Y_{its} = \alpha + \sum_{\ell \leq -1}\mu_{\ell} (D_s \times \mathbf{1}_\ell) + \sum_{\ell \geq 0}\delta_{\ell} (D_s \times \mathbf{1}_\ell) + \lambda_t + D_s + \varepsilon_{ist}\]

• \(D_s\) is a treatment-group indicator for group \(s\); \(\mathbf{1}_\ell\) is an indicator for event-time \(\ell\) (periods relative to treatment onset).
• \(q\) leads (\(\ell < 0\), anticipatory effects) and \(m\) lags (\(\ell \geq 0\), post-treatment effects); \(\lambda_t\) are time fixed effects.

Event Study: How it’s done

• You should plot the coefficients of the interaction terms and their 95% confidence intervals.

• Under no anticipation you should expect the coefficients on the leads to be 0.

Event Study Application

• Miller et al. (2019) study the effect of the introduction of the Affordable Care Act (ACA) on mortality.

• They focus on the near-elderly adults in states with and without the Affordable Care Act Medicaid expansions.

• They find a 0.13-percentage-point decline in annual mortality (9.3% reduction over the sample mean) as a result of the ACA expansion.

• This effect is driven by a reduction in disease-related deaths and grows over time.

Event Study Visualization

Event Study Visualization (cont.)

Event Study in R

library(tidyverse); library(fixest)
od <- causaldata::organ_donations

# Treatment variable
od <- od %>% mutate(California = State == 'California')

# Interact quarter with being in the treated group using
# the fixest i() function, which also lets us specify
# a reference period (using the numeric version of Quarter)
clfe <- feols(Rate ~ i(Quarter_Num, California, ref = 3) | 
            State + Quarter_Num, data = od)

# And use coefplot() for a graph of effects
coefplot(clfe)

Placebo Tests

• Placebo tests look for the effect of the treatment in periods or units where the treatment should not be present.
  • Same outcome in groups that should not be affected by the treatment.
  • Effect on outcomes that should not be affected by the treatment.

Identifying Assumptions in DiD

• Parallel Trends Assumption: The treatment and control groups would have followed the same trend in the absence of the treatment.

• No selection into treatment: treatment assignment does not depend on past or future changes in the outcome variable (otherwise common trends is implausible)

• No spillover effects: The treatment does not affect the control group (or vice versa — the Stable Unit Treatment Value Assumption, SUTVA)

• No concurrent shocks: no other events at the same time affect both the outcome and treatment status

External Validity of DiD

• DiD shows the effects for units with a change in treatment status relative to those without (control)
• Can we generalize these effects to those units without a change in the treatment status?
• Can we extrapolate the results to larger treatments?
• How informative are the results for other situations (countries, etc.)?