Firearms, Local Policing, and Lethal Force

Evidence from Brazil’s Disarmament Statute

Professor Julian E. Gerez

May 14, 2026

What is causal inference?

https://xkcd.com/552/

Causal inference = “what if” questions

1. What would be my headache pain level if I took ibuprofen?
2. How rich would a country be if it were democratic?
3. How would police violence change if police are unarmed?

• Core goal of causal inference: accurate description of causal relationships
  • A change in \(X\) produces a change in \(Y\)
    • E.g., wearing a seatbelt reduces the probability of dying in a car crash
  • But \(X\) and \(Y\) can merely move together without a causal link
    • E.g., ice cream sales and drowning deaths both rise in summer
  • We need to compare observed and counterfactual outcomes

How is correlation different from causation?

How is correlation different from causation?

Example: hospital spending and mortality, where \(X\) is spending per patient, \(Y\) is mortality rate, underlying factor is type of hospital

Counterfactuals represent causality

• Causation: difference between observed and counterfactual outcomes
• Causal inference: learn about counterfactuals from observed outcomes

Potential outcomes and causal effects

• Each unit \(i\) is either treated or not: \(T_i = 1\) if treated, \(T_i = 0\) otherwise
• Each unit has two potential outcomes:
  • \(Y_i(1)\): outcome if treated
  • \(Y_i(0)\): outcome if not treated
• The causal effect for unit \(i\) is simply: \(\tau_i = Y_i(1) - Y_i(0)\)
• But we only ever observe one: \[Y_i = \begin{cases} Y_i(1) & \text{if } T_i = 1 \\ Y_i(0) & \text{if } T_i = 0 \end{cases}\]
• Fundamental problem of causal inference: \(\tau_i\) is never directly observed

How do we recover causal effects?

• We can’t observe counterfactual outcomes: we need to estimate them
  • \(Y_i(0)\) for treated units and \(Y_i(1)\) for control units
• The gold standard: a randomized experiment
  • Random assignment ensures treated and control units are comparable
  • Control group average is a valid counterfactual for the treatment group
    • … in the absence of the treatment
• But we often can’t randomize: ethically, practically, or politically
• Instead we look for natural experiments
  • Today: a threshold that determines whether police are armed

Why randomization is the gold standard

• Randomization produces balanced groups on average
  • Across all observed and unobserved dimensions
• Any outcome differences between groups must be due to the treatment

Brazil’s Estatuto do Desarmamento (2003)

• Brazil has some of the highest rates of police killings in the world
• Guardas municipais: local police forces under municipal governments
  • Distinct from state military police more visible in smaller cities
• 2003 Statute created population-based rules for arming them:
  • Municipalities ≥50,000 inhabitants: guards permitted to carry firearms
  • Municipalities <50,000 inhabitants: guards restricted from carrying
  • (Metro areas and capitals were exempt; rules applied on duty only)
• In 2018, Brazil’s Supreme Court suspended the population threshold
  • All guards could now be armed regardless of city size

The headline numbers are dominated by big cities and states — over 6,000 people killed by police in 2023, with Black and mixed-race Brazilians making up over 80% of victims — but that’s mostly the metros.

Violence has been moving inward. Homicides declined in larger municipalities but increased in smaller ones with populations under 25,000, a pattern researchers call “interiorization.” So police violence in your non-metro municipalities isn’t necessarily low — it may actually be a growing problem

Economic development without policing. The leading explanation is that smaller cities experienced greater economic dynamism, but this was not accompanied by increased policing — essentially, money arrived before security institutions could keep up. ScienceDirect Organized crime diffusion. Organized crime factions migrated from capitals to neighboring smaller municipalities where security infrastructure is weaker, with drug trafficking disputes over routes and sales points driving most deaths. MOVISAFE Both explanations are actually quite relevant to your paper — they suggest the mechanism behind rising violence in smaller municipalities is organized crime and weak institutional capacity, not weapon availability per se. Which is consistent with your null result: arming municipal guards wouldn’t do much against that structural dynamic.

Regression discontinuity design

• A natural experiment where treatment is assigned by crossing a threshold
  • Units just below: control; units just above: treated
• Key assumption: potential outcomes vary smoothly through the cutoff
  • Any jump in outcomes at the threshold = causal effect of treatment
• Our setting:
  • Running variable \(X_i\): municipality population
  • Cutoff \(c\): 50,000 inhabitants
  • Treatment: municipal guards permitted to carry firearms
  • Outcome \(Y_i\): police killings per 100,000 inhabitants

Regression discontinuity design illustration

Regression discontinuity design illustration

Results

• Main result: no evidence arming authorization increased killings
• Validity checks: no discontinuities in other characteristics at the threshold
  • Municipalities just above and below are comparable
• Temporal dimension: not just above vs. below
  • But whether the gap changes when the law is binding vs. not
• Why? Survey data shows the law didn’t consistently translate into arming
  • Changing the rules doesn’t change behavior?
• Help me think through ideas!

Questions?