2. Economic model

This section presents an economic model of father absence and adolescent criminal behavior. Borrowing from Becker (Reference Becker1968), the model views adolescent crime as having costs and benefits, and thus, the decision to engage in crime is the result of a utility maximization process. The main purpose of the model is to establish that family-specific attributes, especially those difficult to observe in household surveys, likely simultaneously affect both father absence and adolescent criminal proclivity. The presence of those family-specific attributes, in turn, produces ambiguous conclusions as to whether father absence directly affects criminal behaviors.

Suppose that an adolescent receives utility from legally-obtained goods and services, L, as well as illegally-obtained goods and services, I. (The term I could include euphoria from breaking the law or approval from peers.) Assuming Cobb–Douglas utility with constant returns to scale, the adolescent's utility is

(1)$$L^{\alpha }I^{1-\alpha }$$

where 0 < α < 1 to satisfy diminishing marginal utility.

Borrowing inspiration from Cameron et al. (Reference Cameron, Trivedi, Milne and Piggott1988), who use a similar setup to model the simultaneity between medical insurance and health care usage, the adolescent must “produce” the illegally-obtained goods and services by engaging in criminal activities, denoted c. Let that production function be Cobb–Douglas

(2)$$I = c^{\sigma }$$

where 0 < σ < 1 to satisfy diminishing returns in production. The exponent in the production function depends on father absence, f, and other family-specific attributes, u, according to the function σ = g(f,  u). That dependence on f and u can be justified by the argument that the ease with which adolescents translate crimes into tangible benefits likely depends on, among other things, parental supervision, which, in turn, likely depends on socioeconomic traits, including father absence.

The adolescent's budget constraint is

(3)$$L + pc = Y$$

where Y is disposable income. With the price of L normalized to unity, the term p represents the “price” of criminal activities, which incorporates the probability of being caught. The price might be monetary, as with a fine, or it might be translated into monetary terms, as with jail time or parental punishment. Either way, the budget constraint highlights that the adolescent must expend resources to engage in crime. The idea that crime has a “price” borrows from Becker's (1968) seminal work on the economics of crime.

The adolescent seeks to maximize (1), where I is produced according to (2), subject to the constraint in (3). The solution to that problem (shown in Appendix A), yields a demand equation for criminal activities

(4)$$c = {Y\over p}\left({( \alpha -1) \sigma \over \alpha \sigma -\alpha -\sigma } \right).$$

Note that ∂c/∂σ > 0, implying that larger values of σ lead to more crime. Recalling that σ = g(f,  u), let f = 1,  0 denote, respectively, father absence and presence. Then if g(1,  u) − g(0,  u) > 0, absent fathers lead to increases in adolescent crime.

However, father absence, itself, is the outcome of an optimization process, and one that likely depends on family attributes, u. Borrowing from random utility theory, let f* be a father's propensity to be absent. That propensity can be written as

$$f^{\ast } = h( u) + V$$

where h is some function of family attributes, and V denotes other things that affect that propensity. Then, if f* exceeds some threshold, the father becomes absent.

Suppose that certain family attributes included in u, such as financial stress or medical problems, increase the father's propensity to be absent, ∂f*/∂u > 0. Suppose, also, that those same attributes, which appear in σ = g(f,  u), alter the adolescent's production of crime, such that ∂σ/∂u > 0. If, as assumed here, u affects both f* and σ, then family traits affect father absence and adolescent criminal behaviors, making it difficult to determine whether father absence causes adolescent criminal activities. Put differently, because f and u are likely jointly distributed across households, it becomes difficult to isolate the pure affect of f. The main aim of this paper is to present empirical methods that seek to isolate the effect of f.

3. Data

Data used in this study come from the National Longitudinal Survey of Youth 1997 (NLSY97). The NLSY97 provides a nationally-representative sample of approximately 9,000 individuals between ages 12 and 16 on December 31, 1996. The first wave of the survey took place in 1997, with subsequent waves occurring annually. This paper considers individuals present in the six annual waves covering the years 1998–2003. The reason for limiting the analysis to those years is that, during those years, all NLSY97 respondents were between ages 14 and 23, prime ages for adolescent criminal behaviors. Furthermore, those 6 years seem to have the most complete information on household structures and criminal activities. (The analyses presented in this paper do not consider the oversample of economically disadvantaged non-black, non-Hispanic respondents, although similar conclusions emerged with that oversample included.) The final estimation sample includes 2,289 unique individuals, each observed for 6 years, for a total of 13,734 persons/year observations.

The main variables involved in this study fall into two categories: adolescent criminal activity and father absence. The criminal behavior measures are binary:

• • Has the respondent used any illegal drug since the last interview?

• • Has the respondent stolen anything since the last interview?

• • Has the respondent attacked anyone since the last interview?

• • Has the respondent been arrested since the last interview?

As for the other important category, a father is considered “absent” if the adolescent lives with his or her biological mother, but not with his or her biological father, or if the adolescent lives with non-parental relatives, including grandparents. (The estimation sample does not include the relatively small number of adolescents living with adoptive parents, foster parents, or undetermined parental situations.)

The top portion of Table 1 shows sample means for the criminal activity measures, partitioned according to father absence. The numbers show that, for each of the four measures, father absence correlates with larger, and statistically significant, propensities to engage in criminal activities. However, those differences cannot be interpreted as direct causal consequences of father absence, because adolescents with absent fathers differ from their counterparts along other dimensions that might, themselves, correlate with criminal behaviors. For example, the bottom portion of Table I, which presents a limited number of socioeconomic measures, shows that adolescents with absent fathers are more likely to be black and less likely to be currently enrolled as students. And the fact that the two sample partitions show differences across some observable dimensions raises the possibility of further differences across unmeasurable attributes.

Table 1. Sample means

**“Father absent” mean differs from “father present” mean at p < 0.05.

*“Father absent” mean differs from “father present” mean at p < 0.10.

The methods described in the following section, especially those that employ fixed effects, require sufficient intra-person variation across years in the criminal activity and father absence measures. Although few guidelines exist regarding what constitutes sufficient variation, those measures do appear to exhibit non-trivial intra-person movements over time. To that point, Table 2 shows within-person coefficients of variation (within-person standard deviations divided by overall means). The criminal measures show large within-person variation; all four measures have within-person standard deviations that exceed their respective means by 127–387%. The father absent measure, meanwhile, shows less, but still non-trivial, within-person variation, with the within-person standard deviation about 42% the magnitude of the mean.

Table 2. Within-person coefficients of variation (within-person standard deviation divided by overall mean)

4. Standard panel methods

All empirical models presented in this paper use linear probability setups for which the dependent variable, denoted y _it, equals 1 if adolescent i engaged in a criminal activity during sample year t, and 0 otherwise. The main explanatory variable of interest, labeled A _it, equals 1 if the adolescent's father was absent during sample year t, and 0 otherwise.

4.1. Simple ordinary least squares

With those variables defined, the main linear probability setup assumes the form

(5)$$y_{it} = {\bf X}_{it}^{\prime }{\boldsymbol \beta } + \gamma A_{it} + \varepsilon _{it}$$

where X_it is a vector of observed socioeconomic controls, some of which vary across time, with estimable coefficient β, and the term ɛ _it represents white noise error. The main parameter of interest, γ, captures the extent, if any, to which father absence affects the propensity to engage in criminal behaviors.

Simple ordinary least squares (OLS) estimates of equation (5), reported below, not surprisingly uncover large, and statistically significant, estimates of γ. But those estimates cannot be interpreted as pure causal effects, as many unmeasured attributes likely correlate with both y _it and A _it. For example, financial stability, and the attendant reduced stress associated with financial stability, might keep fathers around while also reducing adolescent behavior problems. Such unobserved heterogeneity becomes absorbed into the error term and exerts upward bias on γ, leading one to erroneously believe that absent fathers directly cause adolescent criminal problems.

4.2. Fixed effects

To reduce the possibility of such bias, the panel structure of the data is exploiting by the setup

(6)$$y_{it} = c_{i} + {\bf X}_{it}^{\prime }{\boldsymbol \beta } + \gamma A_{it} + \varepsilon _{it}$$

where the individual-specific random intercept c _i captures unobserved heterogeneity common to adolescent i. Those unobserved traits likely correlate with other right-hand side variables (i.e., X_it and A _it); allowing such correlation often leads to the random intercepts being labeled “fixed effects.”

However, the fixed effects only account for time-invariant unobserved heterogeneity, as indicated by the lack of time subscript attached to c _i. That restriction presents a problem for this study, as many forms of heterogeneity that destabilize families and entice adolescent misbehavior, such as job loss or health problems, are inherently time-varying. Such time-varying heterogeneity becomes absorbed into the error term ɛ_it and imparts bias on the main parameter of interest γ, similar to equation (5).

4.3. Dynamic model

An alternative to the fixed effects setup is a dynamic specification of the form

(7)$$y_{it} = \rho y_{i, t-1} + {\bf X}_{it}^{\prime }{\boldsymbol \beta } + \gamma A_{it} + \varepsilon _{it}$$

where the lagged-dependent variable on the right-hand side controls for unobserved heterogeneity, to the extent that unmeasured traits that affected criminal activity last year persist into the current year. But unlike fixed effects, those unmeasured traits are permitted to vary across periods. The lagged-dependent variable setup also make sense if, following a change in a person's propensity to engage in criminal activity, the dependent variable returns partly, but not entirely, to its original state. Such a pattern would be expected if, for example, an introduction to criminal activities tends to beget further illicit behavior.

The dynamic setup seems appropriate in light of the strong correlations in the data between current and past criminal acts. The following table shows sample correlations for y _it and y _i,t−1, along with p-values for the null hypothesis that the correlation equals zero.

However, neither the dynamic model nor the fixed effect setup nests the other, meaning that each model potentially misses forms of heterogeneity that the other captures. As a formal explanation of the non-nested nature of the two approaches, Angrist and Pischke (Reference Angrist and Pischke2009, pp. 246–247) demonstrate that equations (6) and (7) “bracket” the true effect by showing that, in terms of probability limits, when the dynamic setup in equation (7) represents the true data generating process, but a fixed effects setup like equation (6) is mistakenly used, the estimate of γ tends to be too large. On the other hand, if model (6) is true, estimates of γ based on equation (7) tend to be too small. Indeed, results presented below show that estimates based on models (6) and (7) point to widely conflicting findings. An ideal model would include both dynamics and fixed effects, which the following section attempts to address.

5. Dynamic fixed effects model

Blending the models from the two previous subsections, consider

(8)$$y_{it} = \rho y_{i, t-1} + c_{i} + {\bf X}_{it}^{\prime }{\boldsymbol \beta } + \gamma A_{it} + \varepsilon _{it}$$

where the right-hand side includes the lagged-dependent variable and a fixed effect. This setup allows two channels for the aforementioned serial correlation in criminal activity, each with different explanations and policy conclusions. First, as highlighted in the previous subsection, perhaps past criminal behavior begets future criminal behavior, in which case ρ > 0. In that case, policymakers can reduce future criminal acts by preventing current ones. On the other hand, even if ρ = 0, criminal behavior still might exhibit serial correlation if time-invariant traits, captured by the fixed effect c _i, exert nontrivial influence on criminal activities. In that case, policymakers can reduce criminal acts by identifying those traits and either reducing them or shrinking their ties to criminal acts.

However, the inclusion of both lagged-dependent variables and fixed effects in the same model introduces (potentially severe) statistical bias [Nickell (Reference Nickell1981)]. Cameron and Trivedi (Reference Cameron and Trivedi2005, p. 764) provide an algebraic explanation of that bias. The standard correction for that bias is the Arellano–Bond estimator, which uses further lags on y _it as instruments for y _i,t−1 [Anderson and Hsiao (Reference Anderson and Hsiao1981); Holtz-Eakin et al. (Reference Holtz-Eakin, Newey and Rosen1988); Arellano and Bond (Reference Arellano and Bond1991)].

Unfortunately, the Arellano–Bond estimator can produce varied conclusions, both in terms of estimates and efficiency, depending on the number and forms of those instruments. Furthermore, construction of those instruments requires many time periods, which might pose problems for the relatively short panels often available in micro surveys. Lancaster (Reference Lancaster2002) further argues that, whether statistically valid or not, those lagged instruments do not contain suitable information required for model identification. For attempts to improve upon the Arellano–Bond approach, see Ahn and Schmidt (Reference Ahn and Schmidt1995); Blundell and Bond (Reference Blundell and Bond1998), and Arellano and Honore (Reference Arellano, Honore, Heckman and Leamer2001).

To sidestep the bias inherent in dynamic fixed effects models, this paper employs an estimator developed by Lancaster (Reference Lancaster2002). The method is related to the transformed likelihood approach proposed by Hsiao et al. (Reference Hsiao, Pesaran and Tahmiscioglu2002), but in contrast to that model, the Lancaster approach does not require information about criminal activities at time t = 0. Since household surveys almost always being surveying respondents only after some dynamic process has already begun, models that require “initial conditions” also require assumptions about what those initial conditions might have been, with wrong assumptions leading to biased estimates.

In contrast to the models presented in equations (5)–(7), all of which rely on variants of OLS estimation approaches, the Lancaster method relies on a likelihood-based setup, with parameter estimates obtained by Bayesian methods. This section will not repeat Lancaster's entire exposition, but to grasp his method, consider a model for which the log likelihood expression for individual i, denoted L _i, can be reparameterized such that the fixed effects are “information orthogonal,”

(9)$$E\left({\partial ^{2}L_{i}\over \partial c_{i}\partial \psi }\right) = 0,\; $$

where ψ represents the model's estimable parameters minus the fixed effects. If such a reparameterization can be found, then the fixed effects can be integrated out of L _i by specifying priors on the estimable parameters. [Estimates below use uniform (flat) priors.] This Bayesian approach then yields a marginal posterior for the remaining parameters ψ. Markov chain Monte Carlo (MCMC) methods then are used to draw realizations from the marginal posterior to view distributions of the remaining parameters. (To facilitate comparison with estimates from equations (5)–(7), calculations from this Bayesian method report means and standard deviations of the marginal posterior, which can be interpreted analogously to coefficients and standard errors obtained from estimating equations (5)–(7) by frequentist approaches.)

The main difficulty of this method involves reparameterizing the likelihood according to equation (9), which must be worked out for individual families of models. Lancaster provides the reparameterization for, among others, dynamic linear fixed effects models, which are the main consideration in this paper. The interested reader is referred to Lancaster (Reference Lancaster2002) for the forms of the marginal posterior. (Not surprisingly, for flat priors in a linear setting, the marginal posterior resembles the kernel of a multivariate normal.) The MCMC simulation approach uses the R package OrthoPanels [Pickup et al. (Reference Pickup, Gustafson, Cubranic and Evans2017)].

7. Robustness checks

This section considers several robustness checks to the baseline specifications outlined above. First, the baseline specification, by construction, counts households with nonbiological male parents as having absent fathers. The reason for constructing the absent father dummy in that way is that the presence of another male adult in the household points to at least some household disruption occurring since the birth of the child. But perhaps the presence of any male parental figure provides a similar paternal influence, regardless of biological relationship.

To that end, Table 7 re-estimates all models from Tables 3–6, but redefines the “absent father” dummy such that it equals 0 if a nonbiological male adult resides in the house. (For brevity, Table 7 does not present results for other control variables.) Most of the estimated impacts of absent fathers appear to lose magnitude and statistical significance, but as indicated in the preferred dynamic fixed effects setup, the conclusion is the same: absent fathers do not appear to affect adolescent criminal behavior.

Table 7. Nonbiological male adult in household is not counted as “absent father”

These models include all controls listed in Tables 3–6.

*p < 0.10.

**p < 0.05.

A second robustness check, presented in Table 8, adds a dummy variable for whether family income exceeds 200% of the federal poverty line. The baseline specifications do not include family income for several reasons. First, family income likely strongly correlates with the absent father measure of interest, as having two parents likely boosts household earnings. Another concern is that, in the NLSY97 database, the family income measures contain many missing values. Thus, the measure of exceeding 200% of the poverty line used here is, at best, a blunt and noisy measure of family affluence. Those concerns notwithstanding, estimates in Table 8 are very close to the corresponding numbers presented in Tables 3–6.

Table 8. Add dummy for whether family income >200% poverty line

These models include all controls listed in Tables 3–6.

*p < 0.10.

**p < 0.05.

A third robustness check considers the possibility that absent fathers, while evidently not affecting the probability of criminal behavior, might influence the severity of criminal behavior. Although the NLSY97 database does not contain direct information on the severity of illicit activities, it does record whether adolescents were sentenced to spend time in correctional institutions during each survey period. (Such institutions are defined to include jails, prisons, juvenile halls, reform schools, or training schools.) Assuming that more severe violations are more likely to result in being sentenced to correctional institutions, Table 9 re-estimates the baseline models where the dependent variable has been changed to a dummy for whether the adolescent has been sentenced to a correctional institution during the survey period. Those results appear to be similar to the main conclusions of this paper.

Table 9. Dependent variable: sentenced to correctional institution since past interview

These models include all controls listed in Tables 3–6.

*p < 0.10.

**p < 0.05.

Finally, it is worth noting that the estimation sample in this paper uses a fully-balanced panel. However, if adolescents with missing fathers or criminal records are more likely to attrite from the survey, then such sample attrition could bias the main results. Although difficult to test formally, Table 10 presents sample correlations between (1) whether an adolescent has a “non-interview” period and (2) the paper's main variables—absent father and criminal activity—in the period before the non-interview. Adolescents with absent fathers appear slightly more likely to leave the survey, but the correlation is small in magnitude (0.025). Otherwise, criminal activity does not appear to correlate with subsequently leaving the survey. Thus, although not a formal test for bias, attrition does not appear to contaminate the paper's main findings.

Table 10. Sample correlations with having a “non-interview ”(p-values for H ₀ : ρ = 0 in parentheses)

**p < 0.05.

*p < 0.10.