Parity Transition Probabilities

Definition

The parity transition probability $p_{j,t}$ is the probability that a woman of birth parity $j$ (having had exactly $j$ children) gives birth to her $(j+1)$th child during year $t$. It is formally the age-weighted average of the age-parity-specific transition rate $p_{j,t}(a)$ — the probability that a parity-$j$ woman of age $a$ has a next birth in $[t, t+1)$:

$$p_{j,t} = \frac{\sum_{a=15}^{49} p_{j,t}(a)\, m_{j,t}(a)}{M_{j,t}}$$

where $m_{j,t}(a)$ is the number of women at parity $j$ and age $a$ at the start of year $t$, and $M_{j,t} = \sum_a m_{j,t}(a)$ is the total parity-$j$ stock.

Key Ideas

• Core identity: Because $B_{j+1,t} = \sum_a p_{j,t}(a) m_{j,t}(a) = p_{j,t} M_{j,t}$, births of order $j+1$ equal the transition probability times the parity-$j$ stock. Forecasting requires forecasting both $p_{j,t}$ and $M_{j,t}$.
• Observability: $p_{j,t}$ is estimated as the ratio of births of order $j+1$ to the count of women at parity $j$ — both available from vital statistics and census data.
• Stability: Parity transition probabilities are substantially more stable over time than age-specific fertility rates, which are sensitive to tempo shifts (the timing of births within a cohort's life course). This makes them better candidates for autoregressive integrated moving average (ARIMA)-based time series forecasting.
• Collapsibility: Empirically (U.S. 1917–1982), the ARIMA models fitted to $p_{j,t}$ for orders $j = 2, 3, \ldots, 7+$ are nearly identical in structure, coefficients, and error variance. Orders 3 and higher can be collapsed into a single $2+$ series without loss of forecasting accuracy (Al-Osh 1986).

How It Works

Given estimates of $p_{j,t}$ up to year $t$, ARIMA models are fitted to each series and used to forecast $p_{j,t+h}$ for horizon $h = 1, 2, \ldots$. Separately, a demographic recursion tracks the parity distribution of women $M_{j,t}$ forward using survival rates and the predicted $p_{j,t}$ values. Combining both gives birth order-specific forecasts $\hat{B}_{j+1,t+h} = \hat{p}_{j,t+h} \hat{M}_{j,t+h}$, which sum to total birth forecasts with explicit probability limits.

Why It Matters

The parity transition probability framework unifies several prior fertility forecasting approaches under a single model:

• Feeney (1983): constant $p_{j,t}$ over time is exactly Feeney's parity progression model.
• Lee (1974), Saboia (1977), McDonald (1979): the framework reduces to their renewal equation $B_t = \sum_k \Phi_t(k) B_{t-k}$ when $p_{j,t}$ and mortality are held fixed.

In forecasting performance, parity transition probability models achieve root mean squared error (RMSE) roughly $4\times$ lower than age-parity rate methods (Passel 1976) and $10\times$ lower than Census Bureau projections on U.S. data (Al-Osh 1986).

Open Questions

• Whether the ARIMA stability of $p_{j,t}$ holds outside the U.S. or in countries with more volatile parity distributions (e.g., low-fertility countries below total fertility rate (TFR) 1.3 where third-order and higher $p_{j,t}$ collapse toward zero).
• How tempo adjustments (Bongaarts-Feeney) interact with the parity-specific framing; the two approaches address the same instability from different angles.
• Whether parity-specific data availability limits the approach in countries without complete birth registration by order.

Related

• Lee-Carter Model
• Thompson et al. 1989 — Multivariate Time Series Projections of Parameterized Age-Specific Fertility Rates
• Kohler and Ortega 2004 — Old Insights and New Approaches Fertility Analysis and Tempo Adjustment in the Age-Parity Model
• Mohamed Al-Osh

Sources

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