Composition and Decomposition in Nonmarital Fertility

Abstract

Ermisch (2009) criticized Gray, Stockard, and Stone (2006), arguing that they incorrectly tested a model positing a nonlinear relationship between the nonmarital fertility ratio and the proportion of unmarried women. I identify a different problem, which is that even if this model were to hold for a particular population, it would not in general hold for subgroups of this population; likewise, were it to hold for subgroups, it would not hold for aggregations of these subgroups.

In this comment, I identify a key difficulty with a model proposed by Gray, Stockard, and Stone (2006; hereafter GSS) and criticized by Ermisch (2009). GSS proposed a stylized period model under which both marital and nonmarital fertility are hypothesized to be causally related to a single aspect of population composition—the proportion of women who are unmarried in a given calendar year. Ermisch argued that GSS incorrectly specified tests of their model by ignoring nonstationarity in key model parameters. I identify a different problem, which is that even if the model were to be true for a population, it would not in general be true for subgroups of this population due to nonlinearities in the model; likewise, were it to hold for population subgroups, it would not in general hold for aggregations of these subgroups.

Ermisch and I agree that under assumptions outlined by GSS, the period non marital fertility ratio (NFR), the proportion of births occurring to unmarried women in a given calendar year, takes the remarkably simple and parsimonious form Su², where Su denotes the proportion of unmarried women in a given calendar year. Both Ermisch and GSS then tested this model using regressions of the form NFR = b₀ + b₁Su² + e, specified separately for white and black women aged 20–24, 25–29, 30–34, and 35–39. GSS interpreted the close fit of their estimates with period trends in the NFR as providing strong support for their compositional model. Ermisch identified persistent nonstationary deviations of model predictions from the data, which he interpreted as “point[ing] to rejection” of the GSS model (p. 196).

For demographers, both composition and decomposition matter. Implicit in any decomposition exercise is the idea that if a model is true for an entire population, it should also be true for population subgroups; likewise, if the model holds for subgroups, it should also hold for aggregations of these subgroups. To see that this is violated by the GSS model, consider two mutually exclusive and exhaustive population subgroups; then let Su = (U₁ + U₂) / N, with N denoting the number of women in the population and U_i denoting the numbers of unmarried women in group i, i = 1,2 respectively. Letting Su_i =U_i / N, we have

But because 2Su₁Su₂ ≠‚ 0 when Su₁ ≠‚ 0 and Su₂ ≠‚ 0, (1) cannot be reconciled with the GSS model for each subgroup:

where NFR_i denotes the ratio of unmarried births in group i to all births. By contrast, suppose that some transformation g(NFR) were to be linear in Su so that g(NFR) = Su = Su₁ + Su²; then subgroup means for g(NFR) will, by virtue of linearity, be automatically related to the subgroup proportions Su_i. Clearly, the above generalizes trivially to the case of more than two subgroups. Note also that ln(NFR) = 2ln(Su₁ + Su₂) is likewise nonlinear in Su. This shows that even if the GSS model holds in a population, it would not in general hold for population subgroups; likewise, if it holds for mutually exclusive subgroups, it would not in general hold for aggregates of these subgroups.

GSS (but not Ermisch) noted that d(NFR) / dSu is linear in Su, but when analyzing this relationship, GSS presented no regression results and only one figure. (Their analysis was somewhat ad hoc, estimating d(NFR) / dSu by the difference in NFR and Su between two time points, 1974 and 2000, and estimating Su by its mean value during the period 1974–2000.)

It is noteworthy that both Ermisch and GSS restricted attention to nonteen women, with GSS noting that their model fits poorly for teens. This poses a substantive puzzle, given the overlap between teen and nonmarital births noted by Ermisch (see also Wu, Bumpass, and Musick 2001). Although GSS defended the exclusion of teens by arguing that in many states, teens could not legally marry, this ignores a key behavioral endogeneity—whether and for whom a teen pregnancy is taken to term—an issue of particular relevance in the period following the 1973 Roe v. Wade decision. Similarly, one of the stronger and arguably least plausible of GSS’s “deliberately strong” model assumptions is that the “propensity” of a woman to bear a child can be modeled via a single parameter γ that does not vary with other characteristics, such as a woman’s parity, age, or marital status. Note in particular that GSS’s assumptions about γ produce the model nonlinearities that in turn give rise to the difficulties I identify.

In summary, both Ermisch and GSS examined a model in which one aspect of population composition—the proportion of women who are unmarried—is hypothesized to be the key causal factor explaining the proportion of births occurring to unmarried women. Ermisch faulted the adequacy of the GSS model on statistical grounds. I identify a different problem, which is that even if this model were to be true for a population, it would not in general be true for subgroups of this population; likewise, were it to hold for population subgroups, it would not in general hold for aggregations of these subgroups. Additional grounds for doubting the utility of the GSS model can be found in the poor fit of the model to teen births and in its strong homogeneity assumptions concerning women’s propensity to bear children.