Thompson et al. 1989 — Multivariate Time Series Projections of Parameterized Age-Specific Fertility Rates

Summary

Thompson et al. (1989) solve the dimensionality problem in age-specific fertility forecasting by fitting a shifted gamma probability density to each year's fertility schedule, reducing 32 concurrent age-specific time series to three demographically interpretable parameters — the total fertility rate (TFR), mean age at childbearing (MACB), and standard deviation of age at childbearing (SDACB) — and then modeling those three parameters jointly via a restricted multivariate autoregressive integrated moving average ARIMA(0,1,0) with an autoregressive AR(3) correction for TFR. The method was used directly in the U.S. Census Bureau's 1988 national population projections, with short-term time-series forecasts blended with judgmentally determined ultimate levels. The paper establishes a methodological template — compress the age schedule to a low-dimensional parameterization, model the parameters as a time series — that Lee and Carter (1992) applied to mortality, substituting the single index $\kappa_t$ for Thompson et al.'s three gamma parameters.

Key Claims

The age-specific fertility schedule for U.S. white women (1921–1984) is well-approximated in each year by a scaled, shifted gamma density with just three parameters: TFR (level), MACB (mean/timing), and SDACB (spread/shape). Dimensionality is reduced from 32 age-specific rates to 3 parameters.
The three parameters are modeled jointly by a restricted multivariate ARIMA: an unconstrained AR(1) matrix $\Phi_1$ plus a restricted AR(2) and AR(3) that only include the (1,1) element (the TFR autocorrelation). The model is estimated after adjusting for World War II (1942–1947) outlier effects.
The only strong relationship among the three parameters is the contemporaneous correlation between MACB and SDACB. The dynamic off-diagonal elements of $\Phi_1$ are at most marginally significant. TFR is largely independent of the age distribution — the fertility level shifts separately from the timing and spread.
A bias adjustment is required for short-term accuracy: the deviations of actual rates from the fitted gamma curve are modeled as random walks and projected forward as constant offsets. This substantially improves 1–5 year ahead forecasts but is negligible at horizons beyond ~10 years.
Long-run point forecasts for all three parameters level off (due to first differencing and no constant term), facilitating blending with judgmentally set ultimate demographic levels — exactly how Census Bureau 1988 projections were constructed.
67% forecast intervals for TFR reach replacement level (2.1) by 1988–1992, indicating substantial short-run fertility uncertainty. The 95% lower limit falls to 0.78 and upper limit rises to 3.78 by 2020 — a three-child range far wider than the traditional one-child range between Census Bureau high/low scenarios, reflecting model-based uncertainty that demographers judge as too wide given the baby-boom-and-bust was unlikely to repeat.

Concepts Introduced or Extended

Entities Mentioned

Ronald Lee (Lee 1974, 1975; Carter and Lee 1986 cited as prior TFR time-series work)

My Take

The paper's intellectual contribution is the dimensionality-reduction template: fit a parametric curve to summarize the age schedule, then model the parameters in a time series. The direct demographic application (fertility) is less important for this wiki than its status as the acknowledged precursor to Lee-Carter. The finding that TFR and MACB/SDACB are nearly uncorrelated undermines the Rogers (1986) approach of projecting the age distribution as a function of TFR — a useful cautionary note. The paper also documents the inherent tension between pure time-series forecast widths (±1.5 children by 2020) and demographically credible ranges, a tension that Tuljapurkar and Boe (1999) later analyze formally through the F* structural uncertainty framework.