Box 1. Box 1 Different approaches to probabilistic population projections

The cohort component method of projection is taken as a standard; thus, the differences between alternative approaches discussed in this box refer only to the modelling of future fertility, mortality and migration rates. Here we can distinguish between the specific process chosen for representing the time series of rates, and the basis for the specific assumptions made about the future range of uncertainty.

Model In the literature there are essentially two methods of specifying the series of vital rates: (1) processes with annual fluctuations^{10, 13, 14, 15}; and (2) piece-wise linear scenarios^{1, 8, 16}. Whereas method (2) has the advantage of conforming to the current practice of scenario definition in statistical offices around the world (including the UN)², method (1) can produce realistic annual fluctuations given that the possible levels are bounded. We have chosen the following moving-average model with annual fluctuations, in order to avoid the argument that our model underestimates variance¹⁰.

Let v be a vital rate to be forecasted for periods 1 through T and v_t the forecasted value at time t. v_t = _t + _t, where the mean of v_t, _t, and its standard deviation at time t, (_t), are determined according to the assumptions in the text. Let {x_2-n,...,x_T} be the values of T + n - 1 independent draws from a standard normal distribution and n be the number of periods in the moving average. Then _t = [(_t)/n]^t_{i = t-n+1}x_i. A more detailed description of the model is given in the Supplementary Information.

Assumptions The literature suggests three approaches for deriving assumptions about the future range of uncertainty of the components: (1) to compute a measure of the future error from the ex post analysis of past projections^{17, 18, 19}; (2) to apply time series models^{10, 13}; and (3) to have well informed experts make assumptions based on explicitly stated substantive arguments⁹. These three approaches are not mutually exclusive, and approaches (1) and (2) also include expert judgement.

Here we use a synthesis of the three approaches. Our process specification uses a time series model. We have explicitly considered existing national-level parameter estimates^{13, 14} given that, at the level of world regions, empirical estimation is impossible owing to lack of data. The ex post analysis of past errors enters our study in two ways: the substantive assumptions made on fertility and mortality changes are informed by the analysis of past errors in those components^{11, 18}, and our results at the regional level have been compared to the results of an ex post error analysis of global UN projections documented in the NRC report. Because we preferred to err on the side of higher variance (that is, lower probability of population growth ending this century), we followed the general rule of producing intervals that are at least as large as those in the NRC report at the level of major world regions¹¹. Combining this with argument-based expert judgement¹², we saw substantive reasons for assuming a larger uncertainty in many regions as a result of new factors such as HIV/AIDS, the new situation in the former USSR and the indeterminacy of long-range post-transitional fertility levels that will affect an increasing number of countries.

The 95 per cent interval resulting from the NRC ex post error analysis is inserted in Fig. 2 as a vertical line in 2050 (the latest year given in the NRC report). It corresponds to roughly 80 per cent of our distribution, which clearly indicates that our method produces a broader uncertainty range than the ex post error approach.