# [Un1] Specification of Uncertainty

To represent the uncertainty of forecasting, cohort-component book-keeping was applied N = 3,000 times, with stochastically varying values for age-specific mortality, age-specific fertility, and net migration.

The method is based on the so-called scaled model for error (Journal of Official Statistics 13(1997), 203-225.) Within each country, the main characteristics of the model (as used in the forecast) are qualitatively as follows:

- Uncertainty in age-specific mortality and age-specific fertility is treated in the relative (logarithmic) scale, for net-migration uncertainty is treated in the additive scale.
- Uncertainty is assumed to increase with forecast year. Any increasing pattern of error variances can be represented by a suitable choice of the scales of the model.
- Error increments of each age and sex group have a constant non-negative autocorrelation that can be chosen freely.
- Cross-correlation of errors across age are represented by an AR(1) process, whose correlation at lag = 1 is non-negative but otherwise it can be chosen freely.
- Correlation between error increments of males and females, in each age, can be chosen freely.
- Correlation between errors in male and female net migration can be chosen freely.
- Uncertainty in fertility, mortality, and migration were assumed to be independent of each other.
- A normal distribution was used to represent error increments for each age- and sex-group.

For the purpose of combining country-specific forecasts to form a forecast for EU/EEA as a whole,

(a) the total number of net migrants were generated go that they are correlated across countries,

(b) cross-correlation in mortality and fertility across countries was induced via the use of seeds in simulation.

## Mortality

- Scales for error increments were specified so they depended both on age and forecast year.
- Same scales were used for all countries. The scales were estimated from long data series from Austria, Denmark, Finland, France, West Germany, Italy, the Netherlands, Norway, Sweden, Switzerland, and the U.K. The estimates were based on the median level of uncertainty in the past, averaged across countries.
- Autocorrelation of error increments was 0.05.
- Cross-correlation across age was 0.95. Cross-correlation across sexes was 0.85.

## Fertility

- Scales for error increments were specified so they depended on forecast year but not on age. Total fertility rate was used to obtain the estimates.
- Initial values for the scales were estimated, for each country, from the data in 1990-2000, by calculating the standard deviation of first differences (log-scale).
- Eventual value for the scales was obtained from long data series for Denmark, Finland, Iceland, the Netherlands, and Sweden. This value is 0.06 for the total fertility rate. Initial values were connected to the eventual values linearly.
- Autocorrelation of error increments was 0.0.
- The cross-correlation across age was 0.95. The scales estimated for total fertility rates were multiplied by 1.25 to account for less than perfect cross-correlation.

## Net Migration

- Uncertainty in net-migration was specified in terms of total net migration.
- Scales were determined by connecting an estimate of past variability to a judgmentally chosen ultimate value. For countries relying on population registers as the source of population data (Denmark, Finland, Iceland, Netherlands, Norway and Sweden) the uncertainty of net migration was set to zero for year t = 2003.
- Autocorrelation of error increments varied by country, as follows:

Austria | 0.21 |

Belgium | 0.19 |

Denmark | 0.18 |

Finland | 0.34 |

France | 0.13 |

Germany | 0.20 |

Greece | 0.17 |

Iceland | 0.14 |

Ireland | 0.22 |

Italy | 0.26 |

Luxembourg | 0.22 |

Netherlands | 0.29 |

Norway | 0.56 |

Portugal | 0.17 |

Spain | 0.28 |

Sweden | 0.43 |

Switzerland | 0.43 |

United Kingdom | 0.17 |

- Cross-correlation in net migration error between males and females, each year, was assumed to be 0.9.
- A schedule of empirically estimated gross migration levels by age and sex was estimated based on data from Denmark (1998-2002), Norway (2000, 2002) and Sweden (1998-2002). It was used as a multiplier to derive the proportional level of uncertainty by age and sex. Thus, the cross-correlation across age was 1.0.

## Cross-Country Correlations

- Contemporaneous correlations in mortality and fertility were implemented by the "method of seeds" that was developed for this purpose. This method produces a general level of correlation simultaneously for both vital processes.
- The countries were divided into three groups: PS = Portugal and Spain, GI = Greece and Italy, and OTH = other 14 countries. Correlations between countries within each of the three groups, and across the groups, were specified as follows:

PS | 0.70 | ||

GI | 0.30 | 0.30 | |

OTH | 0.15 | 0.30 | 0.30 |

PS | GI | OTH |

These correlations are relevant for the results published for the EU/EEA as a whole, not for the forecasts of the individual countries.

- Contemporaneous correlations in net migration were implemented by generating net-migration uncertainty outside the single state program PEP.
- For this purpose the countries were also divided into three groups: GER = Austria, Germany and Switzerland, MED = Greece, Italy, Portugal, and Spain, and OTH = other 11 countries. Correlation within each of the three groups, and across groups, were specified as follows:

GER | 0.50 | ||

MED | 0.25 | 0.50 | |

OTH | 0.25 | 0.25 | 0.25 |

GER | MED | OTH |

These correlations are also relevant for the results published for the EU/EEA as a whole, not for the forecasts of the individual countries.

Last updated 3.2.2005