A Probabilistic Cohort-Component Model for Population Forecasting – The Case of Germany

Abstract

The future development of population size and structure is of importance since planning in many areas of politics and business is conducted based on expectations about the future makeup of the population. Countries with both decreasing mortality and low fertility rates, which is the case for most countries in Europe, urgently need adequate population forecasts to identify future problems regarding social security systems as one determinant of overall macroeconomic development. This contribution proposes a stochastic cohort-component model that uses simulation techniques based on stochastic models for fertility, migration and mortality to forecast the population by age and sex. We specifically focused on quantifying the uncertainty of future development as previous studies have tended to underestimate future risk.

The model is applied to forecast the population of Germany until 2045. The results provide detailed insight into the future population structure, disaggregated into both sexes and age groups. Moreover, the uncertainty in the forecast is quantified as prediction intervals for each subgroup.

Appendices

Appendices

Appendix A: Forecast Functions of Principal Components

Migration Model

Labor Market Index (Principal Component 1):

$$ l\left({y}_l\right)=-\mathrm{23,943.514}+\mathrm{1,942.419}\ast {y}_l+\mathrm{3,646.207}\ast \sin \left(0.698\ast {y}_l-3.316\right)+{u}_l\left({y}_{l-1}\right)+{e}_l\left({y}_l\right) $$

• y_l being the year under study, with y_l = 0 corresponding to the year 1990

• u_l(y_l) being the autoregressive part in the ARIMA model specifying the difference between the observation in y_l and the deterministic long-term trend specified by the model:

  $$ {u}_l\left({y}_l\right):= l\left({y}_l\right)-\left[-\mathrm{23,943.514}+\mathrm{1,942.419}\ast {y}_l+\mathrm{3,646.207}\ast \sin \left(0.698\ast {y}_l-3.316\right)\right] $$

• e_l(y_l) being the nuisance parameter of the ARIMA model with ${e}_l\sim \mathcal{NID}\left(0;{\mathrm{7,307.992}}^2\right)\forall {y}_l$

Crises Index (Principal Component 2):

$$ c\left({y}_c\right)=-\mathrm{11,777.96}+0.687\ast {u}_c\left({y}_{c-1}\right)+{e}_c\left({y}_c\right) $$

with

$$ {e}_c\sim \mathcal{NID}\left(0;{\mathrm{11,180.7}}^2\right)\forall {y}_c $$

Principal Components 3–1463:

$$ {pc}_i^M(y)={pc}_i^M\left(y-1\right)+{e}_i^M(y) $$

with

$$ {e}_i^M\sim \mathcal{NID}\left(0;{\sigma_i^{M^2}}\right),i=3,4,\dots, 1463 $$

Mortality Model

Lee-Carter Index (Principal Component 1):

$$ m\left({y}_m\right)=-66.518-23.881\ast \frac{\exp \left(\frac{y_m}{14.901}\right)}{1+\exp \left(\frac{y_m}{14.901}\right)}+0.618\ast {u}_m\left({y}_{m-1}\right)+{e}_m\left({y}_m\right) $$

with

• y_m = 0 corresponding to the year 1995

  $$ {e}_m\sim \mathcal{NID}\left(0;{0.299}^2\right)\forall {y}_m $$

Behavioral Index (Principal Component 2):

$$ b\left({y}_b\right)=10.3+8.341\ast \frac{\exp \left(\frac{y_b}{14.025}\right)}{1+\exp \left(\frac{y_b}{14.025}\right)}+1.298\ast {u}_b\left({y}_{b-1}\right)+0.405\ast {u}_b\left({y}_{b-2}\right) $$

$$ \kern4.5em -0.702\ast {u}_b\left({y}_{b-3}\right)+{e}_b\left({y}_b\right) $$

with

• y_b = 0 corresponding to the year 1990

  $$ {e}_b\sim \mathcal{NID}\left(0;{0.158}^2\right)\forall {y}_b $$

Principal Components 3–209:

$$ {pc}_j^D(y)={pc}_j^D\left(y-1\right)+{e}_j^D(y) $$

with

$$ {e}_j^D\sim \mathcal{NID}\left(0;{\sigma_j^{D^2}}\right),j=3,4,\dots, 209 $$

Fertility Model

Tempo Index (Principal Component 1):

$$ t\left({y}_t\right)=-7.592+0.155\ast {y}_t+0.742\ast \ln \left({y}_t\right)+{u}_t\left({y}_{t-1}\right)+{e}_t\left({y}_t\right) $$

with

• ln() denoting the natural logarithm

• y_t = 0 corresponding to the year 1967

  $$ {e}_t\sim \mathcal{NID}\left(0;{0.289}^2\right)\forall {y}_t $$

Quantum Index (Principal Component 2):

$$ q\left({y}_q\right)=-17.186+5.758\ast \frac{\exp \left(\frac{y_q}{5.167}\right)}{1+\exp \left(\frac{y_q}{5.167}\right)}+{u}_q\left({y}_{q-1}\right)+{e}_q\left({y}_q\right) $$

with

• y_q = 0 corresponding to the year 2010

  $$ {e}_q\sim \mathcal{NID}\left(0;{0.216}^2\right)\forall {y}_q $$

Principal Components 3–37:

$$ {pc}_k^F(y)={pc}_k^F\left(y-1\right)+{e}_k^F(y) $$

with

$$ {e}_k^F\sim \mathcal{NID}\left(0;{\sigma_k^{F^2}}\right),k=3,4,\dots, 37 $$

Appendix B: Selected Forecast Results for Three Age Groups

Table 1 Forecast Population (in millions) for Selected Years and Three Age Groups with 75% PIs

Explanations:

• “Young” population means the population younger than age 20

• “Working Age” addresses the population aged 20–66

• “Old” population means persons aged 67 and older

Appendix C: Backtest for Forecast Accuracy

We have used the population data described in the data section until the year 2008 only, the year before the big migration influx described in the introduction occurred. Avoiding the structural break, which occurred in the data due to the 2011 Census (see also Section 3), we base the test on the last census before that, which occurred in 1987. The base population is the age- and sex-specific population on December 31, 2008. Since the dataset provided does not include detailed information for the population aged 95 and older, we estimated the distribution of the population in that age group by using the population estimates for December 31, 1999 and updating the respective cohorts using the age- and sex-specific numbers of deaths by cohort for the years 2000–2008. Migration in this case is ignored for the sake of simplicity. All of these data have been provided by Destatis on demand (Destatis 2016b, 2017c). Our model is now used for forecasting the age- and sex-specific population until December 31, 2018, following the same approach as that described in Section 3. To put the results in perspective, some selected results of Destatis’ (2009) 12th coordinated population projection, namely, the stated most probable middle assumptions alongside the two extreme scenarios, are compared with the hypothetical population numbers. We stress that this is not the actual population number, but rather the estimate of what the population number would have been, based on updating the 1987 Census estimate. To update this estimate, we annually increase the population by birth numbers and net migration, while subtracting the death numbers, starting at our estimate for the 2008 population numbers.

Figure 17 provides all these estimates for the period 2009 to 2018 alongside our model estimates of the median population and the 90% PIs.

Our median forecast would have performed similarly as poor as Destatis’ projection (not even their extreme scenarios have been able to capture any of the real population developments). However, Fig. 17 underlines the advantage of our probabilistic approach, as eight of the ten population estimates from the update are captured by our 90% PI, and even the two values outside the interval in 2015 and 2016 are just slightly outside of our interval bounds. Our model even managed to identify the extreme migration of the year 2015 as a possible, yet unlikely, scenario. This stresses the message advocated by us: It is extremely difficult to predict the future demographic development, especially when with regard to migration, but an appropriate stochastic approach covers all possible outcomes and quantifies them.

Fig. 17

Backtest Results for 2009–2018 with Destatis Projection and Population Update. (Sources: Destatis 2009, 2016b, 2017c; Own calculation and design)