Al-Osh 1986 — Birth Forecasting Based on Birth Order Probabilities

Filename note: The raw file is labeled 1996, but the paper is Journal of the American Statistical Association (JASA) Vol. 81, No. 395, September 1986, pp. 645–656.

Summary

Al-Osh constructs a fertility forecasting model based on parity transition probabilities — the probability $p_{j,t}$ that a woman of birth parity $j$ has a next child during year $t$ — rather than the age-specific fertility rates (ASFRs) traditionally used. The key demographic identity $B_{j+1,t} = p_{j,t} M_{j,t}$ (births of order $j+1$ equal the transition probability times the number of women at parity $j$) links the probability forecasts to birth counts. Autoregressive integrated moving average (ARIMA) models are fitted to each birth order probability series using U.S. data 1917–1982 and used to generate birth order-specific forecasts for 1983–1997. The model achieves root mean squared error (RMSE) of 5.31% against actual births, compared to 22.36% for the best competing method (Passel 1976) and 55.72% for the U.S. Census Bureau's own projections.

Key Claims

• Parity transition probability: Let $p_{j,t}(a)$ be the probability that a parity-$j$ woman of age $a$ has a birth of order $j+1$ in $[t, t+1)$, and $m_{j,t}(a)$ the number of such women. Then total births of order $j+1$ are $B_{j+1,t} = \sum_{a=15}^{49} p_{j,t}(a) m_{j,t}(a)$.
• Age-averaged simplification: Defining the age-averaged transition probability $p_{j,t} = \sum_{a=15}^{49} p_{j,t}(a) m_{j,t}(a) / M_{j,t}$ (where $M_{j,t} = \sum_a m_{j,t}(a)$) gives the compact identity $B_{j+1,t} = p_{j,t} M_{j,t}$.
• Connection to Feeney (1983): When $p_{j,t}$ are held constant over time, the model reduces exactly to Feeney's parity progression model. When $p_{j,t}$ follow the autoregressive structure Al-Osh estimates, the model is also equivalent to the renewal equation $B_t = \sum_k \Phi_t(k) B_{t-k}$ used by Lee (1974), Saboia (1977), and McDonald (1979) — generalizing all of them.
• ARIMA modeling: One ARIMA model is selected per birth order probability series $p_{j,t}$ for $j = 0, 1, 2, \ldots, 7+$ (Table 1). The model is estimated on U.S. data 1917–1982 and forecasts the $p_{j,t}$ series forward; the demographic recursion then converts these to birth count forecasts.
• Collapsing birth orders: The fitted ARIMA models for orders 2–7+ are nearly identical in structure, coefficients, and error variance — justifying combining orders 3 and higher into a single $2+$ series without loss of accuracy (Table 3 forecasts are comparable to Table 2 with 8 series).
• RMSE comparison: Root mean squared errors for total birth forecasts from a 1967 origin are 55.72% (Census Bureau Series B), 22.36% (Passel 1976), 9.07% (Al-Osh 1982 birth-order model), and 5.31% (this birth order probability model). The precision (width of probability intervals) is slightly lower than Al-Osh (1982) but substantially better than Passel.
• Multivariate extension: An ARIMA(3,1,1) model jointly fitted to the birth order probability series 0, 1, and $2+$ reduces prediction interval width by 13.5% at lead time one but the gain erodes and reverses at longer lead times (by 1997 the multivariate standard deviation exceeds the univariate by 7.36%).
• Migration: Immigration is handled by adding a Census Bureau projection vector of immigrant women (by parity) to the model's stock equation; the parity distribution of immigrants is assumed equal to that of 1976 immigrants.

Concepts Introduced or Extended

• Parity Transition Probabilities — the central construct; how $p_{j,t}$ relates to Feeney's parity progression ratios and to the age-parity-specific transition rates $p_{j,t}(a)$

Entities Mentioned

• Mohamed Al-Osh

Quotes

"The high fluctuations in the age-specific fertility rates, which have been experienced in developed societies since the 1930s, have caused most available birth projections based on these rates to be inaccurate."

"This article has demonstrated the utility of the parity transition probabilities for making fertility predictions, instead of using the age-specific fertility rates traditionally used."

My Take

A compact, methodologically clean paper that does two things at once: introduces a new forecasting construct ($p_{j,t}$) and shows it dominates competitors on RMSE by a wide margin. The theoretical unification — proving that Feeney (1983), Lee (1974), Saboia (1977), and McDonald (1979) all follow from this framework under special assumptions — is the paper's main conceptual contribution. The ARIMA-collapsing result (orders 2–7+ all yield effectively the same model) is a useful empirical finding. The main limitation is that the birth order probability series are not directly observable but are ratios of observed births to an estimated denominator (women at parity $j$ of age $a$), making estimation of $p_{j,t}(a)$ data-intensive; the paper sidesteps the age-specific version and works with age-averaged $p_{j,t}$. The multivariate extension's reversal at long lead times is a cautionary note for multivariate demographic forecasting generally (compare Thompson et al. 1989, who find similar instability). Historically, this approach was superseded by Lee-Carter (1992) and its variants for mortality, and by the Bongaarts-Feeney tempo-adjustment framework for fertility, neither of which uses birth-order-specific series.