Bioestadistica1

State space models for estimating and forecasting fertility
Departamento de Estad´stica e Investigaci´ n Operativa, Universidad de Valladolid, Paseo Prado de la Magdalena, s/n, ı o 47005 Valladolid, Spain

Abstract We introduce multivariate state space models for estimating and forecasting fertility rates that are dynamic alternatives to logistic representations for fixed time points.Strategies are provided for the Kalman filter and for quasi-Newton algorithm initialization, that assure the convergence of the iterative fitting process. The broad impact of the new methodology in practice is shown using data series from Spain, Sweden and Australia, and by comparing the results with a recent approach based on functional data analysis and also with official forecasts. Very satisfactory short-and medium-term forecasts are obtained. Besides this, the new modeling proposal provides practitioners with several suitable interpretative tools, and the application here is an interesting example of the usefulness of the state space representation in modelling real multivariate processes. c 2009 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.
Keywords:State space model; Kalman filter; Fertility rates; Demographic forecast; Logistic model; Total fertility rate

1. Introduction The term ‘fertility’ refers to the occurrence of births to an individual, a group or an entire population, and is determined by several biological, economic and social factors. The problem of estimating and forecasting fertility parameters is one that has a long traditionin demography. For example, the population projections derived from the fertility, mortality and migration components have always been of critical importance for policy-making because they set the basis for
∗ Corresponding author.

E-mail addresses: crueda@eio.uva.es (C. Rueda), pilarr@eio.uva.es (P. Rodr´guez). ı

medium- and long-term planning in many fields. In addition, age-fertility ratesare used as inputs in many of the most popular population projection models. Several different approaches have commonly been used for projecting rates in demography. The simplest is to use the average rates from recent years. Another approach is to suppose that the rates in the population to be projected will converge over time with those found in another population, or those chosen by expertjudgement. On the other hand, approaches based on stochastic modeling have also been developed. These approaches have two advantages when compared with the simple extrapolation method: they use more historical information and they provide prediction intervals.

0169-2070/$ - see front matter c 2009 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.ijforecast.2009.09.008

Author's personal copy

C. Rueda, P. Rodr´guez / International Journal of Forecasting 26 (2010) 712–724 ı

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However, projections from time series models, are often strongly affected by the structure of the models themselves, and by the changes in rates that occur during the base period. Therefore, many demographers support the use of mixed procedures, whereexternal judgments and information on historical errors are included in the models. Interesting recent proposals along this line include those of Alho et al. (2006) and Alders, Keilman, and Cruijsen (2007). Nowadays, there is no unanimity about what the best procedure is, because there are important issues with all of the proposals and more statistical research is necessary. This paper is acontribution to this field. The simplest way to make stochastic forecasts is to use univariate time series models to analyse separate age-specific rates. However, when taken together, the separate analyses may not yield a plausible age-pattern (the inconsistency problem). Therefore, it seems desirable to use modeling and forecasting methods that capture the smooth shape over age to produce consistent and...