Assessing the use of sample selection models in the estimation of fertility postponement effects

Several studies have shown that at the individual level there exists a negative relationship between age at first birth and completed fertility. Using twin data in order to control for unobserved heterogeneity as possible source of bias, Kohler et al. (2001) showed the significant presence of such "postponement effect" at the micro level. In this paper, we apply sample selection models, where selection is based on having or not having had a first birth at all, to estimate the impact of postponing first births on subsequent fertility for four European nations, three of which have now lowest-low fertility levels. We use data from a set of comparative surveys (Fertility and Family Surveys), and we apply sample selection models on the logarithm of total fertility and on the progression to the second birth. Our results show that postponement effects are only very slightly affected by sample selection biases, so that sample selection models do not improve significantly the results of standard regression techniques on selected samples. Our results confirm that the postponement effect is higher in countries with lowest-low fertility levels.     

The analysis of age-specific fertility patterns via logistic models

In this paper, we introduce logistic models to analyse fertility curves. The models are formulated as linear models of the log odds of fertility and are defined in terms of parameters that are interpreted as measures of level, location and shape of the fertility schedule. This parameterization is useful for the evaluation, and interpretation of fertility trends and projections of future period fertility. For a series of years, the proposed models admit a state-space formulation that allows a coherent joint estimation of parameters and forecasting. The main features of the models compared with other alternatives are the functional simplicity, the flexibility, and the interpretability of the parameters. These and other features are analysed in this paper using examples and theoretical results. Data from different countries are analysed, and to validate the logistic approach, we compare the goodness of fit of the new model against well-known alternatives; the analysis gives superior results in most developed countries.     

Birth forecasting based on birth order probabilities,with application to U.S. data

"A model for birth forecasting based on prediction of the so-called 'birth order probabilities' is constructed. The relation between this model and recent models of fertility prediction is derived. Birth forecasts with approximate probability limits for the U.S. for the period 1983-1997 are generated. The performance of the proposed model in predicting future fertility is tested by fitting time series models to part of the available series (1917-1982) and ultimately generating birth forecasts for the remainder of the period, then comparing these forecasts with the actual data." The accuracy of the fertility forecasts made are compared with those made by other methods.     

Model averaging procedure for partially linear single-index models

This paper is concerned with model selection and model averaging procedures for partially linear single-index models. The profile least squares procedure is employed to estimate regression coefficients for the full model and submodels. We show that the estimators for submodels are asymptotically normal. Based on the asymptotic distribution of the estimators, we derive the focused information criterion (FIC), formulate the frequentist model average (FMA) estimators and construct proper confidence intervals for FMA estimators and FIC estimator, a special case of FMA estimators. Monte Carlo studies are performed to demonstrate the superiority of the proposed method over the full model, and over models chosen by AIC or BIC in terms of coverage probability and mean squared error. Our approach is further applied to real data from a male fertility study to explore potential factors related to sperm concentration and estimate the relationship between sperm concentration and monobutyl phthalate.     

Forecasting aggregate period-specific birth rates: the time series properties of a microdynamic neoclassical model of fertility

"This article demonstrates the value of microdata for understanding the effect of wages on life cycle fertility dynamics. Conventional estimates of neoclassical economic fertility models obtained from linear aggregate time series regressions are widely criticized for being nonrobust when adjusted for serial correlation. Moreover, the forecasting power of these aggregative neoclassical models has been shown to be inferior when compared with conventional time series models that assign no role to wages. This article demonstrates that, when neoclassical models of fertility are estimated on microdata using methods that incorporate key demographic restrictions and when they are properly aggregated, they have considerable forecasting power." Data are from the 1981 Swedish Fertility Survey.     

A new behavioural model for fertility schedules

Modelling age-specific fertility rates is of great importance in demography because of their influence on population growth. Although we have a variety of fertility models in the demographic literature, most of them do not have any demographic interpretation for their parameters. It is generally expected that models with behavioural interpretation are more universal than those without any interpretation. Even though the famous Gompertz model has some behavioural interpretation it suffers from other drawbacks. In the present work, we propose a new fertility model, which has its genesis in the generalization of logistic law. The proposed model has good behavioural interpretation, alongside having nice parameter interpretations.     

Forecasting births in post-transition population: stochastic renewal with serially correlated fertility

Time series analysis of fertility can improve demographic forecasts. The optimal forecast and its variance for births to an age-structured population subject to serially correlated random fertility are developed. The general case in which the fertility process had arbitrary autoregressive structure is dealt with and then the 4 special cases of white noise, 1st-order autoregressive, 2nd-order autoregressive, and random walk are considered. Consequently, it is determined that the predictions and their variances are highly sensitive to the autoregressive structure of fertility and, therefore, if stochastic models are to be used for prediction, they must emphasize this aspect of the problem. Preliminary empirical efforts to model the time series of U.S. fertility from 1917 to 1972 proved unsuccessful, but it is obvious that at least a 2nd-order autoregressive scheme is require d. The analysis proveded should be helpful in: 1) any application of the procedures requires a successful parameterization of the fertility process; 2) fertility variations could be decomposed into the effects of nuptiality and marital fertility and then births and marriages could be jointly predicted; and 3) the simplifying approximations should be dropped and each age-specific rate could be analyzed and predicted.     

Modelling individual fertility levels in Malawian women: a spatial semiparametric regression model

We applied semiparametric spatial Poisson models to analyse fertility decisions at individual level in Malawi. We used the 2000 Malawi Demographic and Health Survey (MDHS) to investigate determinants of fertility, in the model that allowed for nonlinear, fixed and spatial risk factors. Inference was based on the Bayesian approach. The unstructured spatial effects were modelled using the exchangeable prior, while for the structured spatial effects we used the intrinsic conditional autoregressive models. Nonlinear effects were modelled using P-splines. Results showed non-linear declining trends of fertility with year of marriage and increasing trends with age at marriage. We also observed significant unstructured effects, however, no significant spatial autocorrelated effects were displayed. Overall, total spatial effects were significantly different at district level.     

Survival models for fertility evaluation

"Two proportional hazards models for cohort fertility evaluation are constructed. Time-dependent covariates describe sources of heterogeneity between and within women regarding fertility characteristics. In the first model, U.S. birth rates specific to maternal age, race, parity, and birth cohort are used as underlying hazard rates. Covariate effects are estimated by maximizing the full likelihood. In the second model, covariate effects are estimated via Cox regression with stratified underlying hazard rates regarded as unknown nuisance parameters." The authors illustrate the models "with an evaluation of the fertility histories of the wives of workers at a manufacturing plant with potential for hazardous exposure. Adjustments to the U.S. birth rates for maternal age and parity zero experience are required with the first approach. Then, despite differences in the model-specific estimation procedures, the point estimates of the exposure effect and the estimated standard errors from the two models are practically equivalent."     

Bayesian inference and data cloning in the calibration of population projection matrices

Discrete time models are used in Ecology for describing the dynamics of an age-structured population. They can be introduced from a deterministic or from a stochastic viewpoint. We analyze a stochastic model for the case in which the dynamics of the population is described by means of a projection matrix. In this statistical model, fertility rates and survival rates are unknown parameters which are estimated by using a Bayesian approach and also data cloning, which is a simulation-based method especially useful with complex hierarchical models.

Both methodologies are applied to real data from the population of Steller sea lions located in the Alaska coast since 1978–2004. The estimates obtained from these methods show a good behavior when they are compared to the nonmissing actual values.     

Modeling household fertility decisions: a nonlinear simultaneous probit model

"This article proposes new methods for modeling household fertility decisions....Specifically, we model the trivariate distribution of wife's stated desire for additional children, husband's stated desire for additional children, and subsequent fertility. In the model, the stated desire of the husband (wife) is viewed as an indicator of the husband's (wife's) latent disposition toward subsequent fertility. The husband's (wife's) disposition is allowed to depend on the wife's (husband's) disposition. The two dispositions are then combined to generate the couple's propensity for subsequent fertility. We show how such models can be estimated and tested and how the parameters can be used to assess the relative influence of each partner on the propensity." The model is tested using U.S. data from the Princeton Fertility Study for the 1950s and 1960s. The results indicate that "both husband's disposition score and wife's disposition score affect the propensity score, and, under some additional assumptions, that husbands and wives have equal relative influence on the propensity."     

The Effects of Variations in the Components of Change on the Size and Structure of The Australian Population: 1911-66

The main aim of the study was to estimate separately the effects of the variations in the components of population change and the effects of their mutual interactions on the size and age structure and other characteristics of the Australian population during 1911-66. The method proposed, called here the factorial projection method, is to project the population over the period under consideration, under different assumptions according to a 2×2×2 factorial design, utilizing the observed variations in fertility, mortality and migration. It was found that, apart from the effects of variations in the components occurring individually, the two factor interaction effects– the interaction effects of the changes occurring simultaneously in two of the components (viz. fertility and mortality or mortality and migration or fertility and migration)–on the population size, were considerable. The contribution of migration to the working age group was greater than its effect on the total size, and mortality improvement played a more prominent role in increasing the size of the old age population, while the reduction in fertility caused a higher proportion in the old age group and thus increased the mean age of the population both in the case of males and females. The two factor interaction effects were not negligible in changing the size of the school age population, population in the working age group and also in the old age group. But the interaction of migration and the decline in fertility was the only interaction effect which was important in changing the percentage age distribution and the mean age of the male and female population.     

The Italian Labour Force Survey to estimate fertility

The own-children method (OCM) applied to the Italian Labour Force Survey (ILFS) is an alternative way to give information on fertility for the years before the survey. By deriving children information and the population at risk on the basis of parents’ characteristics, a large-scale dataset for fertility analysis in Italy becomes available, also to reconstruct event histories. The quality assessment provided by comparing the total fertility rate (TFR) calculated on ILFS with the official regional and national TFRs by ISTAT gives us usable outcomes.     

Births in remarriages (using data from sample surveys)

Data from sample surveys conducted between 1978 and 1981 are used to examine the fertility of women in second and subsequent marriages in the USSR. The results indicate that women up to age 25 who have been married more than once have higher fertility than women in a first marriage. However, total fertility is higher for women in uninterrupted marriages. The analysis is presented separately for various cohorts and for socioeconomic characteristics such as educational status and rural or urban residence.     

Model-based estimation of individual fitness

Fitness is the currency of natural selection, a measure of the propagation rate of genotypes into future generations. Its various definitions have the common feature that they are functions of survival and fertility rates. At the individual level, the operative level for natural selection, these rates must be understood as latent features, genetically determined propensities existing at birth. This conception of rates requires that individual fitness be defined and estimated by consideration of the individual in a modelled relation to a group of similar individuals; the only alternative is to consider a sample of size one, unless a clone of identical individuals is available. We present hierarchical models describing individual heterogeneity in survival and fertility rates and allowing for associations between these rates at the individual level. We apply these models to an analysis of life histories of Kittiwakes ( Rissa tridactyla ) observed at several colonies on the Brittany coast of France. We compare Bayesian estimation of the population distribution of individual fitness with estimation based on treating individual life histories in isolation, as samples of size one (e.g. McGraw & Caswell, 1996).     

The dynamics and differentials of fertility in the USSR (survey results)

The results of a fertility survey carried out in the USSR in 1978 are presented. The survey included 33,076 women aged 18 to 59. Data are included on fertility rates by region and Union Republic and by urban or rural area, and on expected fertility of women aged 18 to 44. Changes in actual and desired fertility over time are compared for five-year periods from 1945 to 1978. Differences in fertility are analyzed by type of settlement, educational status, and nationality.     

Why is the number of births increasing?

The author examines whether the unexpectedly high number of births recorded in Poland in 1982 and 1983 is evidence of a change in fertility patterns. It is suggested that the increase in the gross reproduction rate that occurred was due to lower standards of living and fewer opportunities to acquire material possessions or travel abroad as an alternative to having children. Some of the increase may also be due to new pro-natalist measures such as prolongation of paid leave of absence for mothers. The author suggests that the increase in fertility is temporary and that fertility will soon decline to its former level.     

Some statistical analysis issues at the World Fertility Survey

"The World Fertility Survey carried out cross-sectional probability surveys of fertility in more than 40 developing countries between 1972 and 1984. Statistical issues in regression analysis of the data are reviewed, including treatment of interactions, the selection of regressor variables, and appropriate linear models for rate variables. Similar issues arise in many other applications of regression to observational data."     

A random-effects model for cycle viability in fertility studies

The authors describe a random-effects fertility model based upon the assumption that the menstrual cycle viability probability varies from couple to couple according to a beta distribution. An EM algorithm is used to fit the model. The proposed estimating procedure is fully expandable to allow covariate effects on the beta variate. The method can be applied generally whenever dependency among Bernoulli trials is induced by a susceptibility state and the outcomes can be observed only in the aggregate. Based upon data from a cohort of 221 couples with no known fertility problems who were attempting pregnancy, cycle viability was found to be heterogeneous among couples. Stratification on the presence or absence of prenatal exposure of the woman to her mother's cigarette smoking revealed a statistically significant difference in the two-cycle viability distributions. Differences are discussed in the interpretation of the beta model compared to the marginal approach based upon generalized estimating equations.     

Differentiation of female fertility by educational attainment

The author examines fertility differentials by educational attainment for Polish women using official vital statistics on births as modified in 1974. Comparisons are made with fertility recorded in previous studies based on census or sample survey data