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

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

Analysed empirical forecast performance of subsequent population forecasts in 14 European countries Predictive distribution of (errors in) fertility, mortality, migration

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)

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

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

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)

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

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

“Panel”, but strongly unbalanced Repeated measurements for each - country - launch year - calendar year but many missing values e.g. Italy (165), Denmark (1014)

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 were selected as reference category for the forecast effects. R 2 = 0.578, N = 4847.

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 AE(ref) arbitrary -- Choose 0.7 Then AE(f) = exp(F f ) – 0.3 and estimated forecast effects vary between 0.4 ( ) and 1.13 ch/w ( ) -- relative to 0.7 in

TFR No improvement in accuracy since TFR forecasts became worse!

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

Random effects for countries

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

Cov(y c ) = σ 2 I + Z c δ 2 Z c ’

Estimated forecast effects MixedFixed F (.0787) (0.0788) F (.0701) (0.0701) F (.0626) (0.0626) F (.0553) (0.0553) F (.0486) (0.0486) F (.0444) (0.0444) F (.0420) (0.0420) F

Country st. dev Residual st. dev (Fixed effects residual st. dev )

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

Estimated forecast effects MixedFixed F (.0818) (0.0325) F (.0936) (0.0310) F (.0721) (0.0305) F (.0656) (0.0306) F (.0599) (0.0318) F (.0550) (0.0341) F (.0499) (0.0368) F

Calendar year st. dev Residual st. dev (Fixed effects residual st. dev )

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

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