The effect of education, family size, unemployment and childcare availability on birth stopping and timing

Abstract

Using data from Portugal’s Fertility and Family Survey, I analyze childbearing decisions up to the third birth using a split-population (SP) model. The advantage of this approach is the separability of the covariates’ impact on birth timing and birth stopping. This paper is the first to apply an SP model to investigate the effect of unemployment and the availability of childcare. I also address how education, family size, age at previous birth of the woman and sex composition of existing children influence childbearing decisions, and provide empirical support for each of these. Comparing these with estimates obtained using survival models that do not include a regression on birth stopping suggest that the results of the latter tend to be unreasonable.

Appendix

This section explains how to obtain the log-likelihood function of the SP model. The uncensored observations’ contribution to the likelihood is the probability distribution function f(t) and the right-censored observations’ contribution is S(t). I split the time span into one-year-long episodes and assume that the value of a time-varying explanatory variable is constant through a calendar year. The probability of entering year twith t ₁>t≥t ₀ is S(t), where t ₀ refers to the beginning of the episode (1^st January). Now the density of one observation can be written as

$$f(t)^{dS(t)^{(1-d)}S(t_{0})^{-1}}, $$

where dis the censoring indicator with right-censoring if d=0. Taking the logarithm of the expression the contribution to the log-likelihood becomes

$$d\ln h(t)+(1-d)\left(\ln S(t)-\ln S(t_{0})\right) $$

and the sample log-likelihood function can be written as

$$\ln L=\Sigma d\ln h(t)+(1-d)\left(\ln S(t)-\ln S(t_{0})\right). $$

Survival and hazard functions from the non-mixture (Eqs. 1–2) or the mixture model (Eqs. 3–4) with the selected link (Eqs. 6–7) and either the lognormal (Eq. 8) or the gamma density function (Eq. 9) shall be substituted for the numerical maximization.