The authors aim to estimate the causal effect of changes in prison population size on crime rates. Specifically, they investigate whether increasing the number of incarcerated individuals reduces crime, and if so, by how much.
A novice econometrician might run the following OLS regression:
\[ \text{CrimeRate}_{it} = \beta_0 + \beta_1 \cdot \text{PrisonPop}_{it} + \beta_2 \cdot \mathbf{X}_{it} + \epsilon_{it} \]
CrimeRate: Crime rate in region i at time t.
PrisonPop: Prison population size in region i at time t.
X: Vector of control variables (e.g., economic conditions, demographics).
ϵ: Error term.
Running this OLS regression would likely lead to simultaneity bias. This is because crime rates and prison populations are simultaneously determined: higher crime can lead to more incarcerations (increased prison populations), and changes in prison populations can influence crime rates. This reverse causality makes it impossible to disentangle the true causal effect of incarceration on crime using OLS.
The instrument used in the study is prison overcrowding litigation. This exogenous legal factor influences prison populations but does not directly affect crime rates through any channel other than incarceration.
\[ \text{PrisonPopulation}_{it} = \alpha_i + \delta_t + \gamma \cdot \text{Litigation}_t + \epsilon_{it} \]
\(\text{Litigation}_{it}\): A binary variable indicating whether prison overcrowding litigation occurred in region i at time t.
𝛼: State fixed effect, capturing unobserved, time-invariant characteristics specific to each state (e.g., policies or regional factors).
𝛿: Year fixed effect, accounting for time-specific factors affecting all states in a given year (e.g., national policies, economic conditions).
Second-Stage Regression (Estimating Causal Effect):\[\text{CrimeRate}_{it} = \alpha_i + \delta_t + \beta \cdot \widehat{\text{PrisonPop}}_{it} + \epsilon_{it}\]
The instrument is relevant because prison overcrowding litigation directly influences prison population sizes (e.g., by mandating early releases or capping prison capacities). Thus, litigation is strongly correlated with the endogenous variable, \({\text{PrisonPop}}_{it}\).
The instrument is exogenous if litigation affects crime rates only through its impact on prison populations and not through any other channels. For example, litigation might compel states to release prisoners but should not independently alter crime rates.
Exogeneity could be violated if litigation leads to systemic changes unrelated to incarceration, such as increased law enforcement or community monitoring programs, which independently affect crime rates. For example, public pressure from litigation might result in better policing practices, biasing the results.