The Dream: Seeing the Road Not Taken

The First Parallel Universe Detective: Semmelweis (1847)

The Setting: Vienna General Hospital

• 1847 pregnant women were routed to either:
  • Physician wing: 13-18% maternal mortality
  • Midwife wing: ~3% mortality

Semmelweis’s Insight:

Physicians came from performing autopsies — could “cadaveric particles” be causing deaths?

Show R Code

semmelweis_data <- tibble(
  period = rep(c("Before\nIntervention", "After\nIntervention"), 2),
  group = rep(c("Physician Wing", "Midwife Wing"), each = 2),
  mortality = c(15.5, 2.5, 3.0, 2.8)
) %>%
  mutate(period = factor(period, levels = c("Before\nIntervention", "After\nIntervention")))

ggplot(semmelweis_data, aes(x = period, y = mortality,
                            color = group, group = group)) +
  geom_line(linewidth = 2) +
  geom_point(size = 5) +
  scale_color_manual(values = c(secondary_teal, accent_coral), name = "") +
  scale_y_continuous(labels = label_percent(scale = 1), limits = c(0, 18)) +
  labs(
    title = "Maternal Mortality by Wing",
    subtitle = "Hand-washing intervention in physician wing",
    x = "", y = "Mortality Rate"
  ) +
  theme_health_econ(base_size = 16) +
  theme(legend.position = "bottom")

Semmelweis’s Intervention Changed Everything

The Intervention:

Instituted hand-washing with chlorine solution for physicians (but not midwives)

The Cost of Being Right Too Early

The Second Detective: John Snow (1854)

The Crisis:

• Three major cholera waves devastated London
• Dominant theory: Miasma (bad air causes disease)

Snow’s Natural Experiment

Snow’s Table: The First DiD Calculation

Snow’s Numbers Tell the Story

The DiD Calculation (with numbers):

\[\delta^{DiD} = (37 - 150) - (147 - 118) = -113 - 29 = \mathbf{-142} \text{ deaths per 10,000}\]

Two Physicians, One Method

From Intuition to Formalism

Semmelweis and Snow saw into the parallel universe before anyone had a name for it. Now let’s formalize what they discovered — turning intuition into a formula we can estimate, test, and defend.

Anatomy of the Bridge: Four Averages, Three Subtractions

# Create DiD visualization data
did_viz <- tibble(
  time = c(0, 1, 0, 1),
  group = c("Control", "Control", "Treated", "Treated"),
  outcome = c(2, 3, 3, 5.5),
  counterfactual = c(NA, NA, NA, 4)
)

# Add counterfactual line for treated
cf_line <- tibble(
  time = c(0, 1),
  outcome = c(3, 4),
  group = "Counterfactual"
)

ggplot() +
  # Control group
  geom_line(data = filter(did_viz, group == "Control"),
            aes(x = time, y = outcome), color = slate_gray, linewidth = 2) +
  geom_point(data = filter(did_viz, group == "Control"),
             aes(x = time, y = outcome), color = slate_gray, size = 6) +
  # Treated group
  geom_line(data = filter(did_viz, group == "Treated"),
            aes(x = time, y = outcome), color = secondary_teal, linewidth = 2) +
  geom_point(data = filter(did_viz, group == "Treated"),
             aes(x = time, y = outcome), color = secondary_teal, size = 6) +
  # Counterfactual
  geom_line(data = cf_line, aes(x = time, y = outcome),
            color = accent_coral, linewidth = 1.5, linetype = "dashed") +
  geom_point(data = filter(cf_line, time == 1),
             aes(x = time, y = outcome), color = accent_coral, size = 5, shape = 1, stroke = 2) +
  # Treatment effect arrow
  annotate("segment", x = 1.05, y = 4, xend = 1.05, yend = 5.5,
           arrow = arrow(length = unit(0.3, "cm"), ends = "both"),
           color = warm_gold, linewidth = 1.5) +
  annotate("text", x = 1.15, y = 4.75, label = "Treatment\nEffect",
           color = warm_gold, fontface = "bold", size = 5, hjust = 0) +
  # Labels
  annotate("text", x = -0.05, y = 2, label = "Control (Pre)", hjust = 1, color = slate_gray, size = 4) +
  annotate("text", x = 1.05, y = 3, label = "Control (Post)", hjust = 0, color = slate_gray, size = 4) +
  annotate("text", x = -0.05, y = 3, label = "Treated (Pre)", hjust = 1, color = secondary_teal, size = 4) +
  annotate("text", x = 1.05, y = 5.5, label = "Treated (Post)", hjust = 0, color = secondary_teal, size = 4) +
  annotate("text", x = 1.05, y = 4, label = "Counterfactual", hjust = 0, color = accent_coral, size = 4) +
  # Vertical line at treatment
  geom_vline(xintercept = 0.5, linetype = "dotted", color = slate_gray, linewidth = 1) +
  annotate("text", x = 0.5, y = 6.2, label = "Treatment\nOccurs", color = slate_gray, size = 4) +
  scale_x_continuous(breaks = c(0, 1), labels = c("Pre", "Post"), limits = c(-0.2, 1.5)) +
  scale_y_continuous(limits = c(1.5, 6.5)) +
  labs(
    title = "The DiD Estimator: Parallel Trends in Action",
    subtitle = "The control group trend estimates what would have happened to treated units",
    x = "", y = "Outcome"
  ) +
  theme_health_econ(base_size = 18) +
  theme(panel.grid = element_blank())

Figure 3: The difference-in-differences estimator

The Blueprint: One Regression Captures the Design

The 2×2 DiD Estimator:

\[\delta^{DiD} = \underbrace{(Y_{Treated,Post} - Y_{Treated,Pre})}_{\text{Change in treated}} - \underbrace{(Y_{Control,Post} - Y_{Control,Pre})}_{\text{Change in control}}\]

The Parallel Universe Test

If the answer is yes, the control group is a valid window into the treated group’s alternate universe.

The Bridge Works Only If the Universes Were Aligned

Express DiD in potential outcomes notation:

\[\delta^{DiD} = \underbrace{E[Y^1_{Treated} - Y^0_{Treated}]}_{\text{ATT (what we want)}} + \underbrace{\Big[E[\Delta Y^0_{Treated}] - E[\Delta Y^0_{Control}]\Big]}_{\text{Bias if trends differ}}\]

From Formula to Assumptions

The formula tells us how to build the bridge between universes. But when does it lead to the right universe? Three rules determine whether the connection is valid.

Three Rules for Parallel Universes

Rule 1: The Universes Must Have Been Aligned

What Parallel Trends IS:

• Control group trend approximates treated counterfactual
• Changes over time would be similar absent treatment

What It’s NOT:

• Does not require groups to be similar at baseline
• Does not require random assignment (though that would guarantee it)

Show R Code

set.seed(123)
time <- -5:5
control <- 10 + 0.5 * time + rnorm(11, 0, 0.2)
treated <- 12 + 0.5 * time + ifelse(time >= 0, 2, 0) + rnorm(11, 0, 0.2)

pt_data <- tibble(
  time = rep(time, 2),
  group = rep(c("Control", "Treated"), each = 11),
  outcome = c(control, treated)
)

ggplot(pt_data, aes(x = time, y = outcome, color = group)) +
  geom_line(linewidth = 1.5) +
  geom_point(size = 3) +
  geom_vline(xintercept = -0.5, linetype = "dashed", color = slate_gray) +
  scale_color_manual(values = c(slate_gray, secondary_teal), name = "") +
  annotate("text", x = -0.5, y = 16, label = "Treatment", color = slate_gray, size = 4) +
  labs(
    title = "Parallel Pre-Trends",
    x = "Period", y = "Outcome"
  ) +
  theme_health_econ(base_size = 14)

Rule 2: No One Peeked into the Future

Violations:

• Buying a house today because you expect a housing subsidy next month
• Hospitals changing staffing before a merger is announced
• Patients stocking up on medications before a policy change

Why It Matters:

If the pre-period is already affected, we don’t have a clean baseline for comparison

Show R Code

antic_data <- tibble(
  time = rep(-4:4, 2),
  group = rep(c("Control", "Treated (with anticipation)"), each = 9),
  outcome = c(
    10 + 0.5 * (-4:4),  # Control
    c(12, 12.5, 13, 14, 15, 17, 18, 19, 20)  # Treated - anticipates at t=-2
  )
)

ggplot(antic_data, aes(x = time, y = outcome, color = group)) +
  geom_line(linewidth = 1.5) +
  geom_point(size = 3) +
  geom_vline(xintercept = -0.5, linetype = "dashed", color = slate_gray) +
  annotate("rect", xmin = -2.5, xmax = -0.5, ymin = 9, ymax = 21,
           fill = accent_coral, alpha = 0.15) +
  annotate("text", x = -1.5, y = 20.5, label = "Anticipation\nContamination",
           color = accent_coral, size = 4, fontface = "bold") +
  scale_color_manual(values = c(slate_gray, accent_coral), name = "") +
  labs(
    title = "Anticipation Violates No-Anticipation",
    x = "Period", y = "Outcome"
  ) +
  theme_health_econ(base_size = 14)

Rule 3: The Universes Don’t Leak

From Rules to Violations

If all three rules hold, DiD identifies the ATT. But in practice, each rule can fail — and the violations have distinctive signatures. Let’s see what happens when the universes aren’t truly parallel.

Semmelweis’s Critics Broke Rule 1

His critics claimed improved ventilation — not hand-washing — caused mortality to drop.

The Shapeshifter: When Your Window Changes Shape

Example: Hong et al. (2013) — Napster and Music Sales

• Study uses Consumer Expenditure Survey data on music spending
• Compares internet users vs. non-users before/after Napster (June 1999)
• Problem: Who uses the internet changed dramatically over this period!

Hong et al.: The Window Shifted

The Demographics Tell the Story

Demographic characteristics by period

The Crystal Ball: When Policy Sees the Future

Examples:

• States with rising obesity rates pass soda taxes
• Hospitals with declining quality join health systems
• Cities with rising crime adopt new policing strategies

The Issue:

The very trends that prompted the policy violate parallel trends

Show R Code

endo_data <- tibble(
  time = rep(-5:5, 2),
  group = rep(c("Control", "Treated"), each = 11),
  outcome = c(
    10 + 0.3 * (-5:5),  # Control - steady growth
    c(12, 13, 14.5, 16.5, 19, 21, 22, 22.5, 23, 23.5, 24)  # Treated - was growing faster, policy slows it
  )
)

ggplot(endo_data, aes(x = time, y = outcome, color = group)) +
  geom_line(linewidth = 1.5) +
  geom_point(size = 3) +
  geom_vline(xintercept = -0.5, linetype = "dashed", color = slate_gray) +
  annotate("text", x = -3, y = 18, label = "Pre-existing\ntrend difference",
           color = accent_coral, size = 4, fontface = "bold") +
  scale_color_manual(values = c(slate_gray, accent_coral), name = "") +
  labs(
    title = "Policy Endogeneity",
    subtitle = "Treated group was already on a different trajectory",
    x = "Period", y = "Outcome"
  ) +
  theme_health_econ(base_size = 14)

The Ghost: Invisible Forces Push the Universes Apart

Common scenarios:

• Economic shocks hit regions differently
• Demographic changes vary across areas
• Other policies implemented simultaneously

From Violations to Diagnostics

We can’t directly observe the parallel universe. But we can check whether the universes were aligned before the split — and event studies are our diagnostic tool.

Can You Trust Your Window?

The Alignment Check: Event Study Regression

\[ \begin{aligned} Y_{it} = \gamma_i + \lambda_t &+ \sum_{\tau=-q}^{-2} \mu_{\tau} (D_i \times \mathbf{1}\{\tau_t = \tau\}) + \sum_{\tau=0}^{m} \delta_{\tau} (D_i \times \mathbf{1}\{\tau_t = \tau\}) + \varepsilon_{it} \end{aligned} \]

Reading the Alignment Scan

set.seed(123)
event_time <- -4:4
coeff <- c(-0.02, 0.05, -0.03, 0, -0.15, -0.28, -0.35, -0.38, -0.42)
se <- c(0.06, 0.05, 0.04, NA, 0.06, 0.07, 0.08, 0.09, 0.10)

event_data <- tibble(
  period = event_time,
  estimate = coeff,
  se = se,
  lower = coeff - 1.96 * se,
  upper = coeff + 1.96 * se,
  phase = ifelse(period < 0, "Pre-Treatment", "Post-Treatment")
)

ggplot(event_data, aes(x = period, y = estimate)) +
  geom_hline(yintercept = 0, linetype = "dashed", color = slate_gray) +
  geom_vline(xintercept = -0.5, linetype = "dashed", color = slate_gray) +
  geom_errorbar(data = filter(event_data, period != -1),
                aes(ymin = lower, ymax = upper, color = phase), width = 0.2, linewidth = 1) +
  geom_point(data = filter(event_data, period != -1),
             aes(color = phase), size = 4) +
  geom_point(data = filter(event_data, period == -1),
             shape = 1, size = 4, color = slate_gray, stroke = 1.5) +
  scale_color_manual(values = c(secondary_teal, accent_coral), name = "") +
  annotate("rect", xmin = -4.5, xmax = -0.5, ymin = -0.6, ymax = 0.3,
           fill = secondary_teal, alpha = 0.08) +
  annotate("rect", xmin = -0.5, xmax = 4.5, ymin = -0.6, ymax = 0.3,
           fill = accent_coral, alpha = 0.08) +
  annotate("text", x = -2.5, y = 0.25, label = "Pre-trends\n(should be ~0)",
           color = secondary_teal, size = 4, fontface = "bold") +
  annotate("text", x = 2.5, y = 0.25, label = "Treatment effects\n(our estimates)",
           color = accent_coral, size = 4, fontface = "bold") +
  annotate("text", x = -1, y = -0.15, label = "Reference\nperiod",
           color = slate_gray, size = 3.5) +
  labs(
    title = "Anatomy of an Event Study",
    subtitle = "Period -1 is the reference (normalized to zero)",
    x = "Event Time (Periods Relative to Treatment)",
    y = "Coefficient Estimate"
  ) +
  theme_health_econ(base_size = 18) +
  theme(legend.position = "none")

Figure 10: Interpreting event study coefficients

When Everyone Enters Their Universe at Different Times

TWFE Uses Already-Treated Units as Controls

Bacon (2021) showed that TWFE decomposes into three types of comparisons:

Modern Estimators Fix the Staggered Problem

Advanced Upgrades: Inference and Sensitivity

Advanced Upgrades: Triple Differences

Triple Differences (DDD)

Problem: Parallel trends is violated but affects both treatment and placebo groups similarly

Solution: Add a third difference to net out the bias:

\[\delta^{DDD} = \delta^{DiD}_{main} - \delta^{DiD}_{placebo}\]

Example: Compare age-eligible vs. age-ineligible within treatment vs. control states

From Theory to Evidence

We have the toolkit — the bridge, the rules, the alignment check. Now let’s see them work on real health policy questions.

Miller et al. (2019): Discussion Questions

Skim the paper and think about the following:

1. What is the main research question?
2. What is the treatment and what is the comparison group?
3. How do they establish that the “bite” of the policy was real?
4. What falsification test do they run and why?
5. What are the main results and mechanisms?

The Setting: Did Medicaid Expansion Save Lives?

First Check: Did the Policy Actually Change Anything?

The Falsification Test: Looking Through the Wrong Window

The Verdict: Medicaid Expansion Saved Lives

Assembling the Evidence Package

Gaynor et al. (2021): Discussion Questions

Skim the paper and think about the following:

1. What is the main research question about hospital mergers?
2. What makes this study’s data unique (hint: surveys)?
3. How do they establish parallel trends in their event study?
4. What is the “new toy” effect and why might it matter?
5. Do the promised efficiencies of the merger materialize?

The Question: Do Hospital Mergers Deliver on Their Promises?

Paper: The Anatomy of a Hospital System Merger: The Patient Did Not Respond Well to Treatment

The Evidence: Mergers Hurt More Than They Help

Two Studies, One Toolkit

The Regression Table Is the Claim — Evidence Is the Smoking Gun

The Complete Evidence Package: What to Build

The Complete Evidence Package: What to Watch For

The Parallel Universe Principle

DiD works because:

• Policy creates a fork between treated and untreated universes
• The control group’s journey approximates the road not taken
• The gap at the fork is causal — as long as the universes were truly aligned

Looking Ahead

RDD exploited a spatial edge — comparing units on either side of a cutoff. DiD exploits a temporal fork — comparing trajectories before and after treatment.

But what if no single control group follows the same path?