Bayesian panel quantile regression for binary outcomes with correlated random effects: an application on crime recidivism in Canada

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 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 novel data from 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.

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Notes

  1. 1.

    Some Classical techniques include simplex method (Dantzig 1963; Dantzig and Thapa 1997, 2003; Barrodale and Roberts 1973; Koenker and d’Orey 1987), interior point algorithm (Karmarkar 1984; Mehrotra 1992) and smoothing algorithm (Madsen and Nielsen 1993; Chen 2007). Bayesian methods using Markov chain Monte Carlo (MCMC) algorithms for estimating quantile regression was introduced in Yu and Moyeed (2001) and refined, among others, in Kozumi and Kobayashi (2011). A non-Markovian simulation-based algorithm was proposed in Rahman (2013). See also Soares and Fagundes (2018) for interval quantile regression using swarm intelligence.

  2. 2.

    For other developments on panel data quantile regression see Lamarche (2010), Canay (2011), Chernozhukov et al. (2013), Galvao et al. (2013), Galvao and Kato (2017), Harding and Lamarche (2017), Graham et al. (2018), Galvao and Poirier (2019), and Gu and Volgushev (2019) to mention a few. We thank a referee for suggesting the last reference.

  3. 3.

    Baltagi et al. (2003) suggested an alternative pretest estimator based on the Hausman–Taylor (HT) model. This pretest alternative considers an HT model in which some of the variables, but not all, may be correlated with the individual effects. The pretest estimator becomes the random effects estimator if the standard Hausman test is not rejected. The pretest estimator becomes the HT estimator if a second Hausman test (based on the difference between the FE and HT estimators) does not reject the choice of strictly exogenous regressors. Otherwise, the pretest estimator is the FE estimator.

  4. 4.

    A body of work related to quantile regression for discrete outcomes include, but is not limited to, Kordas (2006), Benoit and Poel (2010), Alhamzawi (2016), Omata et al. (2017), Alhamzawi and Ali (2020) and Rahman and Karnawat (2019).

  5. 5.

    In the Bayesian literature, the elements of \(\beta \) do not differ across individuals and are referred to as fixed effects, whereas the \(\alpha _i\)’s are referred to as random effects. This terminology differs from the one used in econometrics. In the latter, the \(\alpha _i\)’s are treated either as random variables, and hence referred to as random effects, or as constant but unknown parameters and thus referred to as fixed effects [see Greenberg (2012), Baltagi et al. (2018)].

  6. 6.

    The quantile regression objective function appears in the exponent of the AL distribution. Therefore, the minimization of the quantile loss function is equivalent to the maximization of the log-likelihood from an AL distribution. There does not exist any other known distribution which has a one-to-one correspondence between the coefficients of classical quantile regression and Bayesian quantile regression. We thank a referee for suggesting that we be more explicit about the choice of the AL distribution.

  7. 7.

    We thank a referee for this suggestion.

  8. 8.

    For the 3 quantiles p = 0.25, 0.5, 0.75, we have \(F^{-1}_{v}\left( 0.25\right) = -0.6745\), \(F^{-1}_{v}\left( 0.5\right) = 0\) and \(F^{-1}_{v}\left( 0.75\right) = 0.6745\).

  9. 9.

    Starting in 2012, the government enacted a series of legislations that made prison conditions more austere; imposed lengthier incarceration periods; significantly expanded the scope of mandatory minimum penalties; and reduced opportunities for conditional release, parole, and alternatives to incarceration.

  10. 10.

    Recidivism is a yearly dummy variable equal to one the year at which the new incarceration begins and zero otherwise. Recidivism may be equal to one in consecutive years so long as the repeat offenses occurred after the end of the previous sentence. Reincarcerations while on parole or on conditional release are not considered repeat offenses.

  11. 11.

    Obviously, detainees who entered the sample on or after 2012 have had less time to reoffend. Yet, in our sample as many as 34% of detainees are reincarcerated within 12 months upon release, and as many as 43% within 2 years. Hence, the sharp decline in repeat offenses in the post-2012 period is unlikely due to the sampling frame. See Lalande et al. (2015).

  12. 12.

    To the extent the new legislation has indeed lowered the recidivism rates, it not clear whether it did so through deterrent or incapacitative effects. Yet, see Bhuller et al. (2020) for US evidence according to which deterrence dominates incapacitation.

  13. 13.

    Thinning has been criticized by some (MacEachern and Berliner 1994; Link and Eaton 2012), while others acknowledge that it can increase statistical efficiency (Geyer 1991). See Owen (2017) who claims that the arguments against thinning may be misleading.

  14. 14.

    Note that the time-varying covariates (Age, Schooling and Unemployment rate) have been “demeaned” and that Age has been divided by 10. The parameter estimates must thus be interpreted accordingly.

  15. 15.

    Recall from Table 4 that very few men are married. In addition, next to none report a change in their marital status in between incarcerations. Further, since the marital status of nonrepeaters is not observed in the data, we are constrained to use the information at entry in the panel.

  16. 16.

    The marginal effects for Age correspond to 1/10 of an additional year relative to the mean. Those for Unemployment and Schooling correspond to one additional year and one additional percentage point relative to their individual means, respectively. The remaining marginal effects correspond to a change in the indicator variables.

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Acknowledgements

This paper is written in honor of Professor Badi H. Baltagi for his valuable contributions to econometrics. We are grateful to Bernard Chéné, Senior Advisor, Programs Directorate, Public Safety (Québec), for his advice and for granting us access to the data used in the paper. We are also grateful to William Arbour, Steeve Marchand, and Ivan Jeliazkov for their advice and numerous discussions. Finally, we are indebted to the editors Qi Li, Vasilis Sarafidis, and Joakim Westerlund and to two anonymous referees for their useful comments and suggestions which helped us in substantially improving the manuscript. The usual disclaimers apply.

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Bresson, G., Lacroix, G. & Rahman, M.A. Bayesian panel quantile regression for binary outcomes with correlated random effects: an application on crime recidivism in Canada. Empir Econ (2020). https://doi.org/10.1007/s00181-020-01893-5

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Keywords

  • Bayesian inference
  • Correlated random effects
  • Crime
  • Panel data
  • Quantile regression
  • Recidivism

JEL Classification

  • C11
  • C31
  • C33
  • C35
  • K14
  • K42