1. Did European fertility forecasts become more accurate in the past 50 years? Nico Keilman

2. Background Data assembled in the framework of the UPE project “Uncertain population of Europe” Stochastic population forecasts for each of the 17 EEA countries + Switzerland

3. Analysed empirical forecast performance of subsequent population forecasts in 14 European countries Predictive distribution of (errors in) fertility, mortality, migration http://www.stat.fi/tup/euupe/

4. Scope Official forecasts in 14 European countries: Austria, Belgium, Denmark, Finland, France, Germany/FRG, Italy, Luxembourg, Netherlands, Norway, Portugal, Sweden, Switzerland, United Kingdom Focus on Total Fertility Rate (TFR) (#ch/w)

5. Scope (cntnd) Forecasts produced by statistical agencies between 1950 and 2002 Compared with actual values 1950-2002

6. Measuring forecast accuracy absolute forecast error (AE) of TFR |obs. TFR – forec. TFR| accuracy/precision, not bias

10. Regression model to explain AE Independent variables: launch year forecast duration forecast year (year to which forecast applies) country forecast variant stability in observed parameter (slope & trend)

11. Model  F f forecast (launch year) effect P p period effect D(d) duration, parameterized (linear & square root) C c country effect V v variant effect

12. Perfect multicollinearity forecast year = launch year + forecast duration solution: -duration effect parameterized -effects of forecast year and launch year were grouped into five-year intervals

13. “Panel”, but strongly unbalanced Repeated measurements for each - country - launch year - calendar year but many missing values http://folk.uio.no/keilman/upe/upe.html e.g. Italy (165), Denmark (1014)

14. Estimation results for errors in Total Fertility Rate (TFR) forecasts The dependent variable is ln[0.3+abserror(TFR)]. The figure shows estimated forecast effects in a model that also controls for period, duration, country, and forecast variant. Launch years 2000-2001 were selected as reference category for the forecast effects. R 2 = 0.578, N = 4847.

15. Interpretation of estimated forecast effects The forecast effect F f for launch years f equals ln[0.3 + AE(f)] – ln[0.3 + AE(ref)] with AE(ref) the error for the reference launch years 2000- 2001. AE(ref) arbitrary -- Choose 0.7 Then AE(f) = exp(F f ) – 0.3 and estimated forecast effects vary between 0.4 (1975-79) and 1.13 ch/w (1965-69) -- relative to 0.7 in 2000-2001

16. TFR No improvement in accuracy since 1975-79 TFR forecasts became worse!

17. Problems 1. Only fixed effects 2. Autocorrelated residuals 1. Include random effects Mixed model 2. Include AR(1) process

18. Random effects for countries

19. For country c, there are n c observations, N = Σ c n c. y c is the (n c x 1) data vector for country c, c = 1, 2, …, m. y c = X c β + Z c b c + e c. β is an unknown (p x 1) vector of fixed effects X c is a (n c x p) matrix with ind. variables for country c b c is an unknown r.v. for the random effect, b c ~ N(0,δ 2 ) the variance δ 2 is the same for all countries Z c is a (n c x 1) vector [1 1 … 1]’ e c is a (n c x 1) vector of intra-country errors, e c ~ N(0, σ 2 I), assuming iid residuals b c and e c are independent

20. Cov(y c ) = σ 2 I + Z c δ 2 Z c ’

21. Estimated forecast effects MixedFixed F65-69 0.327 (.0787) 0.328 (0.0788) F70-74-0.094 (.0701)-0.094 (0.0701) F75-79-0.298 (.0626)-0.299 (0.0626) F80-84-0.281 (.0553)-0.281 (0.0553) F85-89-0.232 (.0486)-0.233 (0.0486) F90-94-0.199 (.0444)-0.199 (0.0444) F95-99-0.131 (.0420)-0.132 (0.0420) F00-02 0 0

22. Country st. dev. 0.112 Residual st. dev.0.258 (Fixed effects residual st. dev. 0.258)

23. Including random country effects does not change the conclusion based on simple fixed effects model Random period effects?

24. Estimated forecast effects MixedFixed F65-69 0.405 (.0818) 0.581 (0.0325) F70-74-0.037 (.0936) 0.128 (0.0310) F75-79-0.250 (.0721)-0.106 (0.0305) F80-84-0.246 (.0656)-0.119 (0.0306) F85-89-0.206 (.0599)-0.103 (0.0318) F90-94-0.183 (.0550)-0.099 (0.0341) F95-99-0.122 (.0499)-0.060 (0.0368) F00-02 0 0

25. Calendar year st. dev. 0.167 Residual st. dev.0.256 (Fixed effects residual st. dev. 0.258)

26. Conclusion Random effects for country or calendar year do not change conclusion that forecast accuracy became worse since 1970s

27. Next Include AR(1) in (fixed effects) model Estimate AR(1) parameter ρ from residuals Transform data (e.g. Cochrane/Orcutt or Prais/Winsten) and re-estimate model