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Crime, Employment and Social Welfare: An Individual-Level Study on Disadvantaged Males

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Abstract

Objectives

We seek evidence for economic and social mechanisms that aim to explain the relationship between employment and crime. We use the distinctive features of social welfare for identification.

Methods

We consider a sample of disadvantaged males from The Netherlands who are observed between ages 18 and 32 on a monthly time scale. We simultaneously model the offending, employment and social welfare variables using a dynamic discrete choice model, where we allow for state dependence, reciprocal effects and time-varying unobserved heterogeneity.

Results

We find significant negative bi-directional structural effects between employment and property crime. Robustness checks show that only regular employment is able to significantly reduce the offending probability. Further, a significant uni-directional effect is found for the public assistance category of social welfare on property offending.

Conclusion

The results highlight the importance of economic incentives for explaining the relationship between employment and crime for disadvantaged individuals. For these individuals the crime reducing effects from the public assistance category of social welfare are statistically equivalent to those from employment, which suggests the importance of financial gains. Further, the results suggest that stigmatizing effects from offending severely reduce future employment probabilities.

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Notes

  1. The omitted variables problem, or selection problem, arises when variables are omitted from the offending-employment/welfare regression that are not randomly correlated with the outcome variable. The corresponding OLS regression parameter estimates become biased whenever this occurs, see Davidson and MacKinnon (2004, Chapter 8). The simultaneity problem, or reverse causality problem as discussed by Ehrlich (1973), arises when offending, employment and welfare outcomes have mutual causal effects on each other. The corresponding OLS regression parameter estimates, resulting from a one-way regression of employment and welfare outcomes on offending outcomes, become biased whenever this occurs.

  2. The other category that is included in the 1.4 million is maternity leave (42,800), which is not included in the current study.

  3. Modeling the actual probabilities \(\pi _{i,t}\) is difficult as these are restricted to lie between zero and one. The logistic transformation ensures that the transformed probabilities \(\theta _{i,t}\) are unrestricted. Note that the \(3\times 1\) vector \(\theta _{i,t}\) has elements \(\log \left[ \pi _{C,i,t} / (1+ \pi _{C,i,t})\right] , \log \left[ \pi _{E,i,t} / (1+ \pi _{E,i,t})\right]\) and \(\log \left[ \pi _{W,i,t} / (1+ \pi _{W,i,t})\right]\).

  4. The functional form of this specification can be relaxed to allow for different age-dependent correlations

    $$\delta = \delta _{0} + \sum _{t=1}^T X_{i,t} \delta _{t} + \lambda Y_{i,0},$$

    which is the specification adopted in Chamberlain (1980). Here the coefficients \(\delta _{t}\) capture the correlation between the random part of the factor structure and the explanatory variables at each monthly period. We experimented with such extensions and the results did not qualitatively change.

  5. In our empirical study we experimented with different ways of including the detention variable. No qualitative changes in the structural effects were found for different constructions of the detention variable.

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Appendices

Appendix 1: Estimation Method

In this appendix we discuss the Monte Carlo maximum likelihood method that is used to estimate the parameters of the logistic trivariate panel data model that is discussed in “Statistical Model” section. The methodology is discussed for a more general panel data model in Mesters and Koopman (2014). All parameters are placed in the vector \(\psi\), which includes the unrestricted elements of \(\Gamma , \beta , \delta _0, \delta _1, \lambda , \Sigma _v\) and the knots of the splines.

We summarize the vector of dependent variables for individual i in time period t by \(Y_{i,t} = (C{i,t},E_{i,t},W_{i,t})^{\prime}\), which is thus a \(3\times 1\) vector of binary variables. The loglikelihood for the observations \(Y=\{Y_{i,t}\}_{i=1,\ldots ,N \ t=1,\ldots ,T}\) is defined as \(\ell (\psi ;Y) = \log p(Y;\psi )\), where \(p(Y;\psi )\) is the joint density of all observations. In the presence of the random effects \(\mu _i\), defined in (3), we can express the joint density as a high dimensional integral as follows

$$p(Y;\psi ) =\int _{\mu } p(Y,\mu ;\psi ) \ {\mathrm{d}}\mu = \int _{\mu } p(Y|\mu ;\psi ) p(\mu ;\psi ) \ {\mathrm{d}}\mu ,$$
(9)

where \(\mu = \{\mu _i\}_{i=1,\ldots ,N}\) and \(p(\mu ;\psi )\) is defined in Eq. (3). The conditional density \(p(Y|\mu ;\psi )\) for the trivariate model can be written as

$$p(Y|\mu ;\psi ) = \prod _{i=1}^N \prod _{t=1}^T p\left( Y_{i,t} | \mu _i ;\psi \right) ,$$

where

$$\log p\left( Y_{i,t} | \mu _i ;\psi \right) = \sum _{j=C,E,W} Y_{j,i,t} \theta _{j,i,t} -\log \left( 1 + \exp \theta _{j,i,t}\right) ,$$

where \(\theta _{i,t} = (\theta _{C,i,t},\theta _{E,i,t},\theta _{W,i,t})\) is given in (2) and \(Y_{j,i,t}\) corresponds to the outcome variables \(Y_{C,i,t} = C{i,t}, Y_{E,i,t} = E_{i,t}\) and \(Y_{W,i,t} = W_{i,t}\) (Durbin and Koopman 2012, e.g., Section 10.3).

As \(p(Y|\mu ;\psi )\) corresponds to a logistic binary density no closed form solution exists for the high dimensional integral in (9). Instead we follow the conventional literature and solve the integral using Monte Carlo methods. We refer to Cappé et al. (2005) and Durbin and Koopman (2012, Part 2) for general introductions into these methods. A simple Monte Carlo estimate is obtained by drawing S samples of \(\mu\) from \(p(\mu ;\psi )\) and computing the average

$$\hat{p}(Y;\psi ) = S^{-1} \sum _{s=1}^S \sum _{i=1}^N \sum _{t=1}^T p\left( Y_{i,t} | \mu _i^{(s)} ;\psi \right) ,$$

where \(\mu ^{(s)}\) denotes the sth sample from \(p(\mu ;\psi )\). From the law of large numbers it follows that \(\hat{p}(Y;\psi ) \rightarrow p(Y;\psi )\) as \(S \rightarrow \infty\). However, the simple estimate requires many draws S before convergence is achieved. This follows as the density \(p(\mu ;\psi )\) does not account for the observations Y.

More efficiency can be obtained by sampling sequences for \(\mu\) from an appropriate importance density (Ripley 1987). For the construction of an adequate importance density we follow Jungbacker and Koopman (2007) and Mesters and Koopman (2014). The general importance sampling representation for the trivariate model is given by

$$p(Y;\psi ) = \int _{\mu } \frac{p(Y|\mu ;\psi )p(\mu ;\psi )}{g(\mu |Y)} g(\mu |Y) \, {{\mathrm{d}}}\mu ,$$

where \(g(\mu |Y)\) is the importance density. When applying Bayes rule to the right hand side we obtain

$$p(Y;\psi ) = g(Y) \int _{\mu } \frac{p(Y|\mu ;\psi )}{g(Y|\mu )} g(\mu |Y) \, {{\mathrm{d}}}\mu ,$$

where we have imposed \(g(\mu ) = p(\mu )\). A Monte Carlo estimate for the importance sampling representation is given by

$$\hat{p}(Y;\psi ) = g(Y) \sum _{s=1}^S \frac{p(Y|\mu ^{(s)} ;\psi )}{g(Y|\mu ^{(s)})},$$

where samples \(\mu ^{(s)}\) are drawn independently from importance density \(g(\mu |Y)\).

We choose \(g(\mu |Y)\) to follow a Gaussian density with mean equal to the mode of \(p(\mu |Y)\) and variance equal to the curvature around the mode. An instrumental basis for \(g(\mu |Y)\) that allows us to obtain the mode is given by

$$z_i = \mu _i + u_i, \qquad u_i \sim NID(0,D_i),$$

where \(z_i\) and \(D_i\) are obtained by the following Gauss–Newton algorithm.

Algorithm

  1. 1.

    Initialize \(\mu = \mu ^*\);

  2. 2.

    Given \(\mu ^*\); compute

    $$D_i = - \left[ \sum _{t=1}^T \frac{\partial ^2 \log p(Y_{i,t}|\mu _i^*;\psi )}{\partial \mu _i^* \partial \mu ^{*\prime}_i} \right] ^{-1},$$

    and

    $$z_i = \mu _i^* + D_i \sum _{t=1}^T \frac{\partial \log p(Y_{i,t}|\mu _i^*;\psi )}{\partial \mu _i^*},$$

    for \(i=1,\ldots ,N\);

  3. 3.

    Update \(\mu ^*\) by computing \({\mathrm{E}}_{g}(\mu | z)\) based on \(z_i = \mu _i + u_i\) and \(u_i \sim NID(0,D_i)\);

  4. 4.

    Iterate between (2) and (3) until convergence.

Convergence of the algorithm is typically quick (4–5 iterations). The derivatives in step (2) are given in Durbin and Koopman (2012, Part 2). After convergence we have obtained the mode of \(p(\mu |Y;\psi )\) and we can sample S times from the importance density \(g(\mu |Y) \equiv g(\mu |z)\), where \(g(\mu |z)\) is a Gaussian density where the mean and variance are implied by \(z_i = \mu _i + u_i\) and the distribution of \(\mu _i\) given in (3). Using these samples we construct the Monte Carlo likelihood. The resulting likelihood estimate \(\hat{p}(y;\psi )\) is optimized with respect to parameters \(\psi\) by numerical methods (Nocedal and Wright 1999). This is done while using the same random numbers and the same number of draws S in each iteration.

Appendix 2: Factor Splines

In this appendix we provide the details for the construction of the cubic splines that we use to model the factors. More details for methods using splines can be found in Poirier (1976). In principal, it is possible to treat all the factors \(f_{j,t}\) as deterministic parameters and estimate them along with the other parameters. However, since the time series dimension is \(T=168\) this would lead to difficulties in optimizing the likelihood using numerical methods.

To avoid this problem, we make the assumption that the individual preferences and abilities vary smoothly with age. This allows us to fit cubic splines for the factors, which rely on a smaller number of parameters. In particular, we seek a subset of K knots denoted by \(\bar{f}_{R(l),t}\), for \(l=1\ldots ,K\), where \(R(l) \in \{1,\ldots ,T\}\). The locations R(l) of the knots are increasing with age; i.e., \(R(1) < R(2) < \cdots < R(K)\). Between these knots we fit cubic polynomial functions to approximate the factors that lie between the knots. The knots \(\bar{f}_{R(l),t}\) are estimated along with the other parameters. The location of the knots can be determined in a variety of ways (Jungbacker et al. 2014). In this paper we set the locations equal to the first month of every age year. Thus, we take in total 15 knots with are placed at age 18 month 1, age 19 month 1 etc. The final knot is for age 31 month 12.

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Mesters, G., van der Geest, V. & Bijleveld, C. Crime, Employment and Social Welfare: An Individual-Level Study on Disadvantaged Males. J Quant Criminol 32, 159–190 (2016). https://doi.org/10.1007/s10940-015-9258-5

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