Probabilistic Forecasting Using Stochastic Diffusion Models, With Applications to Cohort Processes of Marriage and Fertility

The conceptual similarity between stochastic differential equations (SDE) and our results is masked by the fact that we work in an exclusively discrete setup.

The tendency to treat complex processes as linear is occasionally criticized, as in the “general linear reality” paradigm in which the timing and order of events are irrelevant for the outcome and there is no feedback from the outcome to the effect (see Abbot 1988). The diffusion models we use take this critique into account by allowing for dynamic feedback over time.

Deterministic time trend with transient disturbances appears to be contrary to the idea of diffusion processes in which the past influences the future. The difference-stationary specification, which has “long memory” (Raffalovich 1994), seems to fit better for diffusion processes.

Deviations from linearity in g signal deviations from model assumptions. In principle, one can use standard methods to test the linearity (Harvery and Leybourne 2007; Hinich 1982). In demography, the number of observations is typically small, resulting in low test power. Therefore, visual inspection of the process may be preferred over formal testing, as in Wu (1990). These remarks apply also to the Gompertz and logistic models.

The prediction Eq. (6) is preferred because of its simplicity and linearity in exp(g _t ). Alternatives include , which follows directly from Eq. (2), and a prediction that is based on solving from . This quadratic equation arises from the approximation . Empirically, these result in similar predictions. The predictions have a small bias resulting from a discrete growth factor being applied to , whereas optimally, one would apply a continuous growth factor from to . The bias could be reduced by a midpoint correction, analogous to the Euler method for solving differential equations numerically (see Griffiths and Smith 1991), in which the growth factor is averaged over two observations. The bias, however, is small if the step length is small, and it was negligible in simulations in which the step length corresponded to one year. These remarks apply also to the Gompertz and logistic models: a small bias resulting from the discretization is present and could be reduced using the midpoint correction, but in practice, such a correction is not needed with one-year age groups.

The prediction intervals are estimated in a standard way using as the variance estimate for k-step-ahead prediction . Because of the discretization (Eq. (4)), g ₂₀ is not observed.

Billari and Toulemon (2006) used the Hernes model to forecast cohort childlessness and based the forecast uncertainty on uncertainty in the model parameters, in the same spirit as Goldstein and Kenney (2001).

In empirical applications, the starting age may influence the results. For example, if very young ages were included, the process of adopting the modeled behavior could in these young ages differ from that of the older ages, resulting in nonlinearities in the underlying process g. As a rule of thumb, a reasonable starting age is when g starts to behave linearly.

We first estimate the drift parameter for each cohort using data up to age 30. We denote this cohort-specific drift parameter by . For each cohort, we construct predictions for ages 31–45. We estimate the parameter IFC by minimizing , where the weights are w _t  = (45 – t – 1) / (15 ⋅ 8). With this specification, the weights decline linearly so that w ₃₁ = 1 / 8 and w ₃₁ = 1 / (15 ⋅ 8). Such a weighting gives more weight to young ages, whose contribution to fertility matters more than that of older ages. We then find the best-fitting IFC using a grid search with IFC ranging from 0 to 1.500 with step length 0.001.

The prediction interval for 1955 cohort is 33 % wider than for cohort 1950 even though the predictions both start at age 30, and the ultimate completed fertility rates are similar. The reason for this is that the 1955 cohort had children later than the 1950 cohort, which means that from the model’s perspective, predictions for the 1955 cohort start earlier.