Effects of child support and welfare policies on nonmarital teenage childbearing and motherhood

Abstract

This paper is an assessment of the impact of child support enforcement and welfare policies on nonmarital teenage childbearing and motherhood. We derive four hypotheses about the effects of policies on nonmarital teenage childbearing and motherhood. We propose that teenage motherhood and school enrollment are joint decisions for teenage girls. Based on individual trajectories during ages 12–19, our analysis uses an event history model for nonmarital teenage childbearing and a dynamic model of motherhood that is jointly determined with school enrollment. We find some evidence that child support policies indirectly reduce teen motherhood by increasing the probability of school enrollment, which, in turn, reduces the probability of teen motherhood. This finding suggests that welfare offices may wish to place greater weight on outreach programs that inform more teenagers of the existence of strong child support enforcement measures. Such programs might reduce nonmarital teen motherhood further and thus reduce the need for welfare support and child support enforcement in the long run.

Technical Appendix

This appendix shows the relationships between multinomial logit parameters based on a cross-classification of two endogenous variables and parameters that fit the transition of each of the two endogenous variables under Yamaguchi’s model (1990). The method converts multinomial logit parameters into parameters of interest and their standard errors.

To begin, suppose we have two dichotomous random variables A and B that take the value of either −1 or 1. Cross-classifying the two variables, we have four categories:

		B
		−1	1
A	−1	P 00	P 01
	1	P 10	P 11

For simplicity, we have one covariate, X, and the two lagged endogenous variables with δ‘s being their parameters. A multinomial logit specification is then:

$$ \begin{gathered} log\left( {\frac{{\mathop p\nolimits_{{\text{01}}} }} {{\mathop p\nolimits_{{\text{00}}} }}} \right) = \delta _{20} + \delta _{21} A_{t - 1} + \delta _{22} B_{t - 1} + \delta _{23} X \hfill \\ log\left( {\frac{{\mathop p\nolimits_{{\text{10}}} }} {{\mathop p\nolimits_{{\text{00}}} }}} \right) = \delta _{30} + \delta _{31} A_{t - 1} + \delta _{32} B_{t - 1} + \delta _{33} X \hfill \\ log\left( {\frac{{\mathop p\nolimits_{{\text{11}}} }} {{\mathop p\nolimits_{{\text{00}}} }}} \right) = \delta _{40} + \delta _{41} A_{t - 1} + \delta _{42} B_{t - 1} + \delta _{43} X. \hfill \\ \end{gathered} $$

(1)

A = −1 & B = −1 is the reference category and its parameters, δ₁’s, are constrained to 0.

Following Yamaguchi, let μ, φ ^A, φ ^B and φ ^AB represent a set of parameters that are defined as the following:

$$ \begin{gathered} log\mathop p\nolimits_{{\text{00}}} = \mu - \mathop \phi \nolimits^A - \mathop \phi \nolimits^B + \mathop \phi \nolimits^{AB} \hfill \\ log\mathop p\nolimits_{{\text{01}}} = \mu - \mathop \phi \nolimits^A + \mathop \phi \nolimits^B - \mathop \phi \nolimits^{AB} \hfill \\ log\mathop p\nolimits_{{\text{10}}} = \mu + \mathop \phi \nolimits^A - \mathop \phi \nolimits^B - \mathop \phi \nolimits^{AB} \hfill \\ log\mathop p\nolimits_{{\text{11}}} = \mu + \mathop \phi \nolimits^A + \mathop \phi \nolimits^B + \mathop \phi \nolimits^{AB} \hfill \\ \end{gathered} $$

(2)

where φ ^A fits the log-odds of A, φ ^B fits the log-odds of B, and φ ^AB fits the A and B association. Through algebraic manipulation of equations in (2), we get:

$$ \begin{gathered} 4\phi ^A = - \log \left( {\frac{{\mathop P\nolimits_{01} }} {{\mathop P\nolimits_{00} }}} \right) + \log \left( {\frac{{\mathop P\nolimits_{10} }} {{\mathop P\nolimits_{00} }}} \right) + \log \left( {\frac{{\mathop P\nolimits_{11} }} {{\mathop P\nolimits_{00} }}} \right) \hfill \\ 4\phi ^B = \log \left( {\frac{{\mathop P\nolimits_{01} }} {{\mathop P\nolimits_{00} }}} \right) - \log \left( {\frac{{\mathop P\nolimits_{10} }} {{\mathop P\nolimits_{00} }}} \right) + \log \left( {\frac{{\mathop P\nolimits_{11} }} {{\mathop P\nolimits_{00} }}} \right) \hfill \\ 4\phi ^{AB} = - \log \left( {\frac{{\mathop P\nolimits_{01} }} {{\mathop P\nolimits_{00} }}} \right) - \log \left( {\frac{{\mathop P\nolimits_{10} }} {{\mathop P\nolimits_{00} }}} \right) + \log \left( {\frac{{\mathop P\nolimits_{11} }} {{\mathop P\nolimits_{00} }}} \right) \hfill \\ \end{gathered} $$

(3)

φ ^At, φ ^Bt and φ ^ABt each is a function of the covariate X and the cross-lagged variables, with λ _s being the corresponding parameters. We multiply each side of the equations by 4:

$$ \begin{gathered} {4\phi ^{A_t } = 4\lambda _0^A + 4\lambda _1^A \mathop A\nolimits_{t - 1} + 4\lambda _2^A \mathop B\nolimits_{t - 1} + 4\lambda _3^A X} \hfill \\ 4\phi ^{B_t } = 4\lambda _0^B + 4\lambda _1^B \mathop A\nolimits_{t - 1} + 4\lambda _2^B \mathop B\nolimits_{t - 1} + 4\lambda _3^B X \hfill \\ {4\phi ^{AB_t } = 4\lambda _0^{AB} + 4\lambda _1^{AB} \mathop A\nolimits_{t - 1} + 4\lambda _2^{AB} \mathop B\nolimits_{t - 1} + 4\lambda _3^{AB} X} \hfill \\ \end{gathered} $$

(4)

where $\lambda _1^A$ is the first-order state-dependence for A _t , $\lambda _2^A$ is the lagged causal effect of B _t-1 on A _t , $\lambda _3^A$ is the parameter related to the covariate. Similarly, $\lambda _1^B$ is the lagged causal effect of A _t-1 on B _t , $\lambda _2^B$ is the first-order state-dependence for B _t , $\lambda _3^B$ is the parameter related to the covariates. λ ^AB’s are parameters for the association between A _t and B _t .

Substituting Eqs. (1) and (4) into Eq. (3), we can solve for the parameters corresponding to the covariate. For example, for the parameters corresponding to A_t−1, we have Eq. (5) where the parameters that fit A, B, and their association AB can be expressed as linear combinations of the multinomial logit parameters:

$$ \begin{gathered} 4\lambda _1^A = - \delta _{21} + \delta _{31} + \delta _{41} \hfill \\ 4\lambda _1^B = \delta _{21} - \delta _{31} + \delta _{41} \hfill \\ 4\lambda _1^{AB} = - \delta _{21} - \delta _{31} + \delta _{41} \hfill \\ \end{gathered} $$

(5)

Since the variance of a linear combination of random variables is the sum of the variance of each variable and twice the covariance among them (adjusting for the scaling and the signs), the corresponding variance for a particular parameter can be expressed as linear combinations of elements in the variance-covariance matrix of the multinomial logit estimation. For the variances of parameters corresponding to A_t−1, we get:

$$ \begin{gathered} Var(4\lambda _1^A ) = Var(\delta _{21} ) + Var(\delta _{31} ) + Var(\delta _{41} ) - 2Cov(\delta _{21} ,\delta _{31} ) - 2Cov(\delta _{21} ,\delta _{41} ) + 2Cov(\delta _{31} ,\delta _{41} ) \hfill \\ Var(4\lambda _1^B ) = Var(\delta _{21} ) + Var(\delta _{31} ) + Var(\delta _{41} ) - 2Cov(\delta _{21} ,\delta _{31} ) + 2Cov(\delta _{21} ,\delta _{41} ) - 2Cov(\delta _{31} ,\delta _{41} ) \hfill \\ Var(4\lambda _1^{AB} ) = Var(\delta _{21} ) + Var(\delta _{31} ) + Var(\delta _{41} ) + 2Cov(\delta _{21} ,\delta _{31} ) - 2Cov(\delta _{21} ,\delta _{41} ) - 2Cov(\delta _{31} ,\delta _{41} ) \hfill \\ \end{gathered} $$

(6)

We can obtain the variance of other parameters in the same manner.