Population Projections

The method of population projections is simple and plain. Mathematically, the formula below is used to forecast population in five years:
\[\mathbf{P}^{(n+5)} = \mathbf{M}^{(n+5)} \mathbf{P}^{(n)}\]P's are 36×1 column vectors, consisting of population by sex (male and female) and age group (0–4, 5–9, ..., 80–84, 85+, 18 groups in total). M is the Leslie matrix, with information of birth rates and survival rates (details below). Superscripts indicate year. Specifically, if we have P for population in 2000, P for population in 2050 can be calculated as: \[\mathbf{P}^{(50)} = \mathbf{M}^{(50)}\mathbf{M}^{(45)}\ldots\mathbf{M}^{(5)} \mathbf{P}^{(0)}\]The major task, then, is to estimate M's. M's are 36×36 matrices: \[\bf M=\begin{pmatrix} \\ b_0 & ... & ... & ... & ... & b_{85+} & & 0 & 0 & ... & ... & ... & 0 \\_{5}s_{0} & & &&&&& 0 & 0 & ... & ... & ... & 0 \\ & _{5}s_{5}&&&&&& ...&... & ... & ... & ... & 0 \\ &&...& &&&&...&... & ... & ... & ... & 0 \\ &&&...&&&& ... &...&... & ... & ... & 0 \\ &&&& _{5}s_{80}&_{5}s_{85+} &&...& ...&... & ... & ... & 0 \\& \\ b_0^*& ... & ... & ... & ... & b_{85+}^*&& 0 &0 & ... & ... & ... & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & & _{5}s_{0}^* \\ ... & ... & ... & ... & ... & ... & && _{5}s_{5}^* \\ ... & ... & ... & ... & ... & ... & & && ... \\ ... & ... & ... & ... & ... & ... & & & && ... \\ ... & ... & ... & ... & ... & ... & & & && & _{5}s_{80}^* & _{5}s_{85+}^* \\ & \end{pmatrix}\] b's (w/o asterisk) are female birth rates by age group, s's (w/o asterisk) are female survival rates by age group. b's and s's with asterisks are male counterparts. Historical data of birth and survival rates by sex and age group are used to estimate future birth and survival rates by sex and age group by building ARMA models. BIC method is applied to determine orders for ARMA models. ARMA models contain stochastic terms (white noise). The following figure shows the result of 1,000 stimulation of population forecast for 2005, 2010, ..., 2050, based on 2000 population data and historical data on birth and survival rates. The population in 2050 would be 1.3924 billion, with a 90% confidence interval of 1.3443 to 1.4476 billion. Data Source: Census Bureau, China
Note this forecast model did not take into account the change of onechild policy, which was abolished in 2015. 
