Inverse Projection and Back Projection: A Critical Appraisal,and Comparative Results for England, 1539 to 1871

Inverse projection and back projection are two methods for exploiting long historical series of births and deaths to produce estimates of population size and age structure, net migration, and vital rates. While inverse projection requires extraneous information on population size at scattered dates, back projection does not. In this paper I argue that back projection attempts an impossible task, and can only arbitrarily select one demographic past from among an infinite set of equally plausible and acceptable ones, which are also consistent with the input data. Inverse projection, on the other hand, is more modest in its goal, but is robust and straightforward. In an important and outstanding book, Wrigley and Schofield use back projection to reconstruct English demographic history from 1539 to 1871. In this paper, inverse projection is used to replicate their reconstruction under assumptions that are in important respects weaker, although these estimates are contingent on independent population size estimates for 1541 and 1696. The results buttress Wrigley and Schofield's reconstruction. However, it is argued that their data and reconstruction cannot offer independent evidence for the general levels of population before the mid-eighteenth century; rather, they help us to interpolate among benchmarks for which we have extraneous evidence, and contingent on these benchmarks, fill in the rich details of the demographic past.