Is Less More? Data-Driven Dimensionality Reduction in Parametric ASFR Models

The timing of fertility behaviour can be summarized by the schedule of age-specific fertility rates (ASFR). Numerous parametric models for indexing empirical ASFR schedules by a moderate number of parameters have been proposed, allowing for more convenient interpolation, inference, and specification of projection assumptions. Reducing the number of parameters involves a number of trade-offs, not least between expressiveness and parsimony. Recent proposals have essentially converged conceptually to using 4 parameters to specify the scale, location of peak, and the shape of the ascending and descending slopes. The difference rests largely in whether the slopes are specified as polynomial splines [7], exponential [4], or logistic [6]. A competitive 3-parameter ASFR model, however, has remained elusive, because the most obvious path to eliminate one of the generic parameters above, namely assuming a symmetric or at least deterministic relationship between the two slopes, clearly fails to capture the diversity in empirical schedules. Motivated by the insight that a more concise model would need to exploit complex dependencies between the above parameters, and that unaided human cognition is notoriously poor at high-dimensional reasoning, I propose to push for more parsimonious ASFR models through algorithmic methods. Concretely, I explore a novel application of a data-driven dimensionality reduction technique, namely Independent Component Analysis (ICA), to embed a ‘reduced’ ASFR model of low parametric dimension within an arbitrary parametric ASFR ‘full’ model of higher dimension. This yields two functions: one to project the parameters of the full model onto a subspace, and another to conversely express a model described by the ICs in terms of the original, higher-dimensional parametrization. Doing so combines the statistical benefits of parsimony of the reduced model with the expressiveness and transparent interpretation of the full one. This approach frees human experts from the intrinsically algorithmic task of identifying and avoiding statistical redundancy. Instead, they may focus on devising ASFR models that are meaningful with respect to the substantive demographic concepts and able to reproduce a wide range of human fertility timing behavior, without excessive concern for concision. The subsequent data-driven techniques can exploit statistical dependencies between parameters in the full model much more efficiently than manual ‘fine tuning’. In contrast to other efforts to identify the ‘optimal’ number of parameters for ASFR modeling according to general model selection criteria such as the AIC [1], the aim is not to balance fit (to empirical ASFR schedules) and model dimension, but to maximize fit given the constraint that the number of parameters should be 2 or 3 at most. This reflects a particular kind of application, where a robustly estimable model with reasonable fit is preferable to a better fitting model that cannot be reliably estimated in practice. In particular, this includes situations where fertility rates are provided only for five-year age groups, implying the need to estimate the model given only 7 data points, or possibly even fewer, and if the data are messy and incomplete, say historical data from developing countries.