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A combined Brass-random walk approach to probabilistic household forecasting: Denmark, Finland, and the Netherlands, 2011–2041

Abstract

Probabilistic household forecasts to 2041 are presented for Denmark, Finland, and the Netherlands. Future trends in fertility, mortality and international migration are taken from official population forecasts. Time series of shares of the population in six different household positions are modelled as random walks with drift. Brass’ relational model preserves the age patterns of the household shares. Probabilistic forecasts for households are computed by combining predictive distributions for the household shares with predictive distributions of the populations, specific for age and sex. If current trends in the three countries continue, we will witness a development towards more and smaller households, often driven by increasing numbers of persons who live alone. We can be quite certain that by 2041, there will be between two and four times as many persons aged 80 and over who live alone when compared with the situation in 2011.

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Data sources: 1981–2001 register data; 2011 census data, Eurostat 2014, 2021–2041 model extrapolations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4

Data sources: 1991–2001 register data; 2011 census data, Eurostat 2014; 2021–2041 model extrapolations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5

Data sources: 2001 register data; 2011 census data, Eurostat 2014, 2021–2041 model extrapolations. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

Source: Eurostat (2014). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7

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Acknowledgments

This work was supported by the European Commission’s Seventh Framework Programme under Grant FP7-SSH-2012-1/No. 320333. We acknowledge useful comments by Coen van Duin, Juha Alho, and members of the WP2 Research Team of the MOPACT (“Mobilising the Potential of Active Ageing in Europe”) project.

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Correspondence to Nico Keilman.

Appendix: Back transformation from ξ to α

Appendix: Back transformation from ξ to α

In “Modelling household shares” section the shares α j are transformed into fractions ξ k . In this “Appendix” we outline the back transformation from ξ k to α j . We suppress indices for age, sex, time, and country. The starting point is the set of expressions that transform the shares α j into fractions ξ k .

$$\begin{aligned} \xi_{2} & = {\text{logit}}\left( {\left( {\alpha_{3} + \alpha_{4} } \right)/\left( {1 \, - \alpha_{1} } \right)} \right) \\ \xi_{3} & = {\text{logit}}\left( {\alpha_{4} /\left( {\alpha_{3} + \alpha_{4} } \right)} \right) \\ \xi_{4} & = {\text{logit}}\left( {\left( {\alpha_{2} + \alpha_{7} } \right)/\left( {\alpha_{2} + \alpha_{5} + \alpha_{6} + \alpha_{7} } \right)} \right) \\ \xi_{5} & = {\text{logit}}\left( {\left( {\alpha_{2} } \right)/\left( {\alpha_{2} + \alpha_{7} } \right)} \right) \\ \xi_{6} & = {\text{logit}}\left( {\alpha_{5} /\left( {\alpha_{5} + \alpha_{6} } \right)} \right) \\ \end{aligned}$$

There are many equivalent expressions for the α j written as functions of the ξ k . One of these is the following set

$$\begin{aligned} \alpha_{2} & = {{\left( {1 \, {-}\alpha_{1} } \right)\exp (\xi_{4} )\exp (\xi_{5} )} \mathord{\left/ {\vphantom {{\left( {1 \, {-}\alpha_{1} } \right)\exp (\xi_{4} )\exp (\xi_{5} )} {\left\{ {\left( {1 \, + \, \exp \left( {\xi_{2} } \right)} \right)\left( {1 \, + \, \exp \left( {\xi_{4} } \right)} \right) \, \left( {1 \, + \, \exp \left( {\xi_{5} } \right)} \right)} \right\}}}} \right. \kern-0pt} {\left\{ {\left( {1 \, + \, \exp \left( {\xi_{2} } \right)} \right)\left( {1 \, + \, \exp \left( {\xi_{4} } \right)} \right) \, \left( {1 \, + \, \exp \left( {\xi_{5} } \right)} \right)} \right\}}} \\ \alpha_{3} & = {{\left( {1 \, - \alpha_{1} } \right)\exp \left( {\xi_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {1 \, - \alpha_{1} } \right)\exp \left( {\xi_{2} } \right)} {\left\{ {\left( {1 \, + \, \exp \left( {\xi_{2} } \right)} \right)(1 \, + \, \exp \left( {\xi_{3} } \right)} \right\}}}} \right. \kern-0pt} {\left\{ {\left( {1 \, + \, \exp \left( {\xi_{2} } \right)} \right)(1 \, + \, \exp \left( {\xi_{3} } \right)} \right\}}} \\ \alpha_{4} & = \alpha_{3} \exp \left( {\xi_{3} } \right) \\ \alpha_{6} & = {{\left( {1 \, {-}\alpha_{1} {-}\alpha_{3} {-}\alpha_{4} } \right)} \mathord{\left/ {\vphantom {{\left( {1 \, {-}\alpha_{1} {-}\alpha_{3} {-}\alpha_{4} } \right)} {\left\{ {\left( {1 \, + \, \exp \left( {\xi_{4} } \right)} \right)(1 \, + \, \exp \left( {\xi_{6} } \right)} \right\}}}} \right. \kern-0pt} {\left\{ {\left( {1 \, + \, \exp \left( {\xi_{4} } \right)} \right)(1 \, + \, \exp \left( {\xi_{6} } \right)} \right\}}} \\ \alpha_{5} & = \alpha_{6} \exp \left( {\xi_{6} } \right) \\ \alpha_{7} & = {{\alpha_{6} \exp \left( {\xi_{4} } \right)\left( {1 \, + \, \exp \left( {\xi_{6} } \right)} \right)} \mathord{\left/ {\vphantom {{\alpha_{6} \exp \left( {\xi_{4} } \right)\left( {1 \, + \, \exp \left( {\xi_{6} } \right)} \right)} {\left( {1 \, + \, \exp \left( {\xi_{5} } \right)} \right)}}} \right. \kern-0pt} {\left( {1 \, + \, \exp \left( {\xi_{5} } \right)} \right)}} \\ \end{aligned}$$

By assumption, α 1 is independent of ξ k (k = 2, 3,…6).

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Keilman, N. A combined Brass-random walk approach to probabilistic household forecasting: Denmark, Finland, and the Netherlands, 2011–2041. J Pop Research 34, 17–43 (2017). https://doi.org/10.1007/s12546-016-9175-y

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Keywords

  • Probabilistic household forecast
  • Brass relational method
  • Random walk with drift
  • Random share method
  • One-person households
  • Cohabiting couples
  • Married couples
  • Lone parents
  • Institutions