[Paper Review] Bayesian Panel Quantile Regression for Binary Outcomes with Correlated Random Effects: An Application on Crime Recidivism in Canada

Georges Bresson, Guy Lacroix+1 more|arXiv (Cornell University)|Jan 25, 2020

Statistical Methods and Bayesian InferenceMathematics97 references19 citations

TL;DR

This paper proposes a Bayesian panel quantile regression model for binary outcomes with correlated random effects, using an asymmetric Laplace likelihood and MCMC inference to estimate heterogeneous treatment effects across quantiles. Applied to Canadian crime recidivism data, it finds the 'tough-on-crime' policy significantly reduced reoffending, especially among high-risk individuals in lower quantiles.

ABSTRACT

This article develops a Bayesian approach for estimating panel quantile regression with binary outcomes in the presence of correlated random effects. We construct a working likelihood using an asymmetric Laplace (AL) error distribution and combine it with suitable prior distributions to obtain the complete joint posterior distribution. For posterior inference, we propose two Markov chain Monte Carlo (MCMC) algorithms but prefer the algorithm that exploits the blocking procedure to produce lower autocorrelation in the MCMC draws. We also explain how to use the MCMC draws to calculate the marginal effects, relative risk and odds ratio. The performance of our preferred algorithm is demonstrated in multiple simulation studies and shown to perform extremely well. Furthermore, we implement the proposed framework to study crime recidivism in Quebec, a Canadian Province, using a novel data from the administrative correctional files. Our results suggest that the recently implemented "tough-on-crime" policy of the Canadian government has been largely successful in reducing the probability of repeat offenses in the post-policy period. Besides, our results support existing findings on crime recidivism and offer new insights at various quantiles.

Motivation & Objective

• To develop a Bayesian framework for panel quantile regression with binary outcomes and correlated random effects, addressing limitations of standard random-effects models.
• To overcome the challenge of estimating quantile regression in panel data with discrete outcomes and unobserved heterogeneity correlated with regressors.
• To provide a computationally efficient MCMC algorithm that reduces autocorrelation through blocked parameter sampling.
• To enable estimation of marginal effects, relative risks, and odds ratios using posterior MCMC draws for interpretable inference.
• To apply the model to real-world data on crime recidivism in Quebec, Canada, to assess the impact of the 2012 'tough-on-crime' policy.

Proposed method

• Uses an asymmetric Laplace (AL) error distribution to construct a working likelihood for binary outcomes in quantile regression.
• Employs a Bayesian hierarchical model with correlated random effects, allowing individual-specific heterogeneity to be correlated with covariates.
• Proposes two MCMC algorithms, favoring the blocked sampling approach to reduce autocorrelation in posterior draws.
• Leverages the normal-exponential mixture representation of the AL distribution to facilitate Gibbs sampling in the MCMC scheme.
• Derives posterior distributions for coefficients, random effects, and scale parameters, enabling full Bayesian inference.
• Computes marginal effects, relative risks, and odds ratios from MCMC draws to interpret treatment effects at different quantiles.

Experimental results

Research questions

• RQ1 How does the 'tough-on-crime' policy implemented in Canada after 2012 affect the probability of recidivism across different quantiles of the recidivism risk distribution?
• RQ2 To what extent are individual-specific effects correlated with observed covariates in predicting recidivism, and how does this affect model estimation?
• RQ3 How do covariates such as age, education, marital status, and criminal history influence recidivism risk differently across the conditional quantile distribution?
• RQ4 What is the performance of the proposed MCMC algorithm in terms of convergence and autocorrelation compared to alternative schemes?
• RQ5 How do relative risks and odds ratios vary across quantiles, and which covariates have the most substantial impact on recidivism at different risk levels?

Key findings

• The 'tough-on-crime' policy significantly reduced recidivism, with the largest reductions observed in the lower quantiles: recidivism risk decreased by 90–93% for the p10 group.
• For the highest quantile (p90), the policy still reduced recidivism by 61–67% for those post-2012, compared to 76–81% for the pre-2012 group.
• Relative risks showed sharp changes between the 75th and 90th percentiles, indicating that policy effects were most pronounced among high-risk individuals.
• Marital status, Indigenous identity, and crime type (violent, property, other) had significantly different relative risks across quantiles, with strong effects in the upper tail.
• The blocked MCMC algorithm produced lower autocorrelation and better mixing than the alternative, confirming its superiority in posterior simulation.
• The correlated random effects structure was found to be relevant, as it improved model fit and captured heterogeneity in recidivism risk across individuals.