Available online at www.sciencedirect.com International Journal of Forecasting 27 (2011) 1241–1247 www.elsevier.com/locate/ijforecast Robust backward population projections made possible Dalkhat M. Ediev ∗ Vienna Institute of Demography of the Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Vienna, Austria Abstract Based on formal results for population dynamics under varying fertility and mortality levels, this paper presents a new approach to backward population projection. Unlike other methods in the literature, the method presented here is robust and accurate in both the short and long run. The method and the theory behind it contribute to the knowledge about dynamic populations and may find applications in population modeling and reconstruction. c ⃝ 2011 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. Keywords: Backward population projection; Back projection; Inverse projection; Population dynamics; Ergodic property; Reproductive value; Demographic potential; Dynamically stable population; Historical demography 1. The problem of backward population projection Population projection looking backward would have been a useful tool in a variety of research contexts of historical demography. Typically, we have only scarce non-standardized population data in the past and a more recent population census. The ability to project the population backward from the census year based on demographic scenarios would help in assessing the plausibility of the scenarios and selecting those which are the most consistent with the rare historical data. The very possibility of projecting a population backward is of theoretical interest. Unfortunately, backward population projection has remained an unsolved problem so far. Formally ∗ Tel.: +43 1515817728; fax: +43 1515817730. E-mail address: Dalkhat.Ediev@oeaw.ac.at. speaking, the population projection matrix L is singular, and therefore the usual matrix equation of forward population projection P t +1 = LP t , where P t is the vector consisting of 1 January population numbers by single years of age, may not be inversed (as there is no L −1 for obtaining P t = L −1 P t +1 ). An easy way of overcoming this limitation by truncating the projection matrix at the last age at reproduction and then inversing the truncated matrix does not work either, due to the instability of the results (Keyfitz, 1977). The fundamental cause of this instability lies in the spectral properties of the population projection matrix (of the renewal operator in the continuous model): while the real eigenvalue (which determines the population intrinsic rate) dominates the spectrum of the forward projection matrix, it is dominated by all other (complex) eigenvalues in the backward projection. The backward 0169-2070/$ - see front matter c ⃝ 2011 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.ijforecast.2010.09.008