The Causal Effect of Educational Attainment on Completed Fertility for a Cohort of Danish Women—Does Feedback Play a Role?

Abstract

This study focuses on the causal effect of educational attainment on completed fertility for a cohort of Danish women born in 1963. The relationship between education and completed fertility (here defined as the number of children at age 41) is the result of a dynamic interplay between the two processes; in particular, educational attainment at a given time might be influenced by previous fertility. This notion is referred to as feedback from the fertility process to the education process. In this study population, feedback turns out to be present, and marginal structural models are used to construct a hypothetical population in which the feedback mechanisms are lifted out of the data. This gives rise to the conclusion that educational differences in completed fertility are to some degree attributable to feedback mechanisms and that they become smaller or are even reversed when this is taken into account. The study is based on data from individual-level, administrative registers.

Appendix

1.1 A.1 Multinomial Logit Models for the Weights

We use a multinomial logistic regression model in which, year by year, we model the probability of either being enrolled in a certain education or having finished a given education, i.e. the probability of belonging to one of the educational categories which are used for the education variable in Model (3). Let π _ikt be the probability that the ith woman belongs to education category k, k=1,…,8 in year t, i.e.

$$\begin{aligned} \pi_{ikt}=\Pr\bigl(E_{i}(t)=k\bigr), \end{aligned}$$

where k denotes one of the 8 categories described previously. Hence, the two models are

$$\begin{aligned} \log \biggl[\frac{\pi_{ikt}}{\pi_{i1t}} \biggr] = \alpha_k+ \beta_k\cdot\mathbf{A}_{it}+\gamma_k\cdot \mathbf{Z}_i, \quad k=2,\ldots, 8 \end{aligned}$$

(6)

and

$$\begin{aligned} \log \biggl[\frac{\pi_{ikt}}{\pi_{i1t}} \biggr] = \alpha_k+ \beta_k\cdot\mathbf{A}_{it}+\gamma_k\cdot \mathbf{Z}_i+\delta _{k1}\cdot B_{i,t-1}+ \delta_{k2}\cdot B_{i,t-2}, \quad k=2,\ldots, 8, \end{aligned}$$

(7)

corresponding to the numerator and the denominator in the factor for year t entering into the weight for subject i, cf. Eq. (4). As a time-dependent covariate, we include age as a restricted cubic spline, and we furthermore include the indicator of having given birth in the year before, B _i (t−1), and the indicator of having given birth the two years before, B _i (t−2). We use the SAS procedures PROC CATMOD and PROC GENMOD [32, 50] to estimate the parameters of this model.

Once the parameter estimates, $\hat{\alpha}_{k}$, $\hat{\beta}_{k}$, $\hat{\gamma}_{k}$ and $\hat{\delta}_{k1}$ and $\hat{\delta}_{k2}$, k=2,…,8, are found, we can calculate $\hat{\pi}_{ikt}$ and use these as building blocks for the estimated weights, $\hat{\operatorname {SW}}_{i}(T+1)$, where T is the year 2004 in which the women reach the age 41.

Note that due to the already vast number of parameters in the model for the weights, we included no explicit conditioning on prior education history up to year t.