port of the Social Security Administration (http://www.ssa.gov/oact/tr/2011/index.html) and the small adjustments to data circa 2008 described earlier. The least-squares fit provided by the singular value decomposition of the 1917–2010 series of ASFR yields i trajectories:
where ax = ASFRx,2010i and Σ|bx| = 1 and Kti is a constrained autoregressive moving average (1,1) process with autoregressive coefficient 0.9673 and moving average coefficient 0.5367. The process is constrained so that the long-run average total fertility rate is 2.0 (see Lee and Tuljapurkar, 1994, for further details).
Net migration flows into the population follow the assumptions used in the deterministic population projections by age and sex (see Table A-1). Each stochastic trajectory experiences the same migration pattern.
A population trajectory i begins with the initial launch population by age and sex in 2009, Ns,x,2009i. To bring this population to 2010, losses from death during the year and additions from new births are determined using rates Ms,x,2009i and ASFRs,x,2009i applied to the population. Finally, immigrants are added according to the baseline schedule.
The SR for the i th trajectory,
is calculated along the trajectory based on the stochastic population counts. The consumption schedule c(x,t) is normalized so that the support ratio in 2007 is unity.