# Stochastic Forecasting of Regional Age-Specific Fertility Rates: An Outlook for German NUTS-3 Regions

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## Abstract

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## 1. Introduction

#### 1.1. Background and Motivation

#### 1.2. Regional Aspects of Fertility in Germany

## 2. Materials and Methods

#### 2.1. Data Source and Preparation

#### 2.2. Model Choice by Backtesting

- Greeks: the parameters estimated based on the 1996–2008 data via OLS;
- ${p}_{k,y}$: the value of $P{C}_{k}$ in year y;
- $y=0:$ the year 1996.

#### 2.3. Stochastic Forecast Approach

- ${\mathbf{S}}_{y,t}$: the simulation matrix $(1000\times 2370)$ of the logit-DASFRs for year y in trajectory t;
- ${\mathsf{\Pi}}_{y,t}$: the simulation matrix $(1000\times 2370)$ of PCs for year y in trajectory t;
- ${\mathsf{\Lambda}}^{-1}$: the inverse $\left({2370}^{2}\right)$ of the loading matrix.

- ${s}_{d,a,y,t}$: the simulated logit-DASFR for females in age group a living in district d at the end of year y in trajectory t;
- ${\varphi}_{d,a,y,t}$: the simulated DASFR for females in age group a living in district d at the end of year y in trajectory t.

## 3. Results

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ASFR | Age-specific fertility rate |

CFR | Cohort fertility rate |

DASFR | District- and age-specific fertility rate |

NUTS | Nomenclature des unités territoriales statistiques |

OLS | Ordinary least squares |

PC(A) | Principal component (analysis) |

PI | Prediction interval |

$SMAP{E}_{m}$ | Symmetric mean absolute percentage error of Model m |

TFR | Total fertility rate |

## Appendix A. German Federal States

## References

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**Figure 3.**Time series of the first two principal components based on pre-2009 data (source: own computation and illustration).

Principal Component | Individual Share | Cumulative Share |
---|---|---|

1 | 71.1 | 71.1 |

2 | 7.7 | 78.8 |

3 | 2.9 | 81.8 |

4 | 2.0 | 83.7 |

5 | 1.1 | 84.8 |

6 | 1.1 | 85.9 |

7 | 1.0 | 86.9 |

8–2370 | <1.0 | 100.0 |

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**MDPI and ACS Style**

Vanella, P.; Hassenstein, M.J.
Stochastic Forecasting of Regional Age-Specific Fertility Rates: An Outlook for German NUTS-3 Regions. *Mathematics* **2024**, *12*, 25.
https://doi.org/10.3390/math12010025

**AMA Style**

Vanella P, Hassenstein MJ.
Stochastic Forecasting of Regional Age-Specific Fertility Rates: An Outlook for German NUTS-3 Regions. *Mathematics*. 2024; 12(1):25.
https://doi.org/10.3390/math12010025

**Chicago/Turabian Style**

Vanella, Patrizio, and Max J. Hassenstein.
2024. "Stochastic Forecasting of Regional Age-Specific Fertility Rates: An Outlook for German NUTS-3 Regions" *Mathematics* 12, no. 1: 25.
https://doi.org/10.3390/math12010025