Carter and Lee 1986 — Joint Forecasts of US Marital Fertility, Nuptiality, Births, and Marriages

Summary

Carter and Lee (1986) present an index-based approach to jointly forecasting U.S. marital fertility, nuptiality, births, and first marriages using Box-Jenkins time series methods. They define a scalar marital fertility index $g(t)$ and nuptiality index $h(t)$, each convolved with fixed demographic age/duration schedules to generate birth and marriage counts, then compare five modeling strategies (univariate autoregressive integrated moving average (ARIMA), two transfer-function variants, vector autoregressive moving average (ARMA) conditional, vector ARMA exact). Nuptiality is best modeled by the univariate ARIMA; fertility by a transfer function. The combined system implies an ultimate net reproductive rate (NRR) of $0.94$ and, by forecast year 14, a $95\%$ confidence interval (CI) spanning $\pm 100\%$ of the forecast level—forecast uncertainty grows without bound.

Key Claims

• The marital fertility index $g(t)$ and nuptiality index $h(t)$ each can be indexed by a single forecastable scalar, reducing joint fertility/nuptiality forecasting to a bivariate time series problem on rates rather than event counts.
• For the nuptiality index $h(t)$, the best forecast model is the univariate ARIMA (Model A); the vector ARMA (Model D) is rejected because it implies implausible negative nuptiality-fertility feedback inconsistent with demographic theory.
• For the fertility index $g(t)$, the transfer function (Model B) outperforms the univariate ARIMA with slightly smaller root mean squared error (RMSE); using nuptiality as an input to fertility forecasting yields a marginal improvement.
• Forecast uncertainty grows rapidly and without bound: by 1996 (forecast year 14), the $95\%$ CI for births spans $\pm 100\%$ of the forecast level; by 2000 (year 18) the error correlation between birth and marriage forecasts grows to $0.86$, reflecting demographic propagation of nuptiality shocks into births.
• The ultimate NRR implied by the long-run index forecasts is $0.94$ ($= 0.9066 \times 2.75 \times 0.8968 \times 0.86/2.05$), indicating eventual population decline under the forecast scenario.
• The "index + fixed age/duration schedule" architecture — scalar time-varying intensity $g(t)$ multiplied by fixed schedule $f(x)$ — directly foreshadows the Lee-Carter (1992) mortality model, where the scalar $k(t)$ is multiplied by fixed $b(x)$ loadings.

Concepts Introduced or Extended

• Stochastic Fertility Forecasting
• Lee-Carter Model

Entities Mentioned

• Lawrence Carter
• Ronald Lee

Quotes

"Within 5 years, the 95% interval may be ±30%, and within 15 years, ±50%." "...implies an ultimate net reproductive rate of .94 (= .9066 × 2.75 × .8968 × .86/2.05) and, therefore, imply eventual population decline."

My Take

This paper is the clearest methodological predecessor to Lee and Carter (1992). The "index + fixed schedule" structure — a scalar time series $g(t)$ multiplied by a fixed age/duration distribution to generate rates — is precisely the architecture used in Lee-Carter, with singular value decomposition (SVD) replacing the index construction heuristic. The paper's finding that vector ARMA models are rejected for nuptiality because they imply implausible feedbacks anticipates a recurring theme in the stochastic demography literature: adding cross-series information rarely beats the univariate random walk for long-run demographic forecasting. The unbounded forecast variance (uncertainty $\pm 100\%$ at year 14) is the same structural feature that makes Lee-Carter intervals very wide at 75-year horizons, though that limitation receives less attention in the mortality context.