6.3 Effects of adjustments made The adjustments discussed impact several properties of the model. The times series for the Dutch deviation are more stable The parameter estimates that determine the speed of the convergence (in expected value) for the Dutch deviation time series are now considerably further away from the critical threshold. The introduction of two new constants and the new data points since 2017 also change the estimates of the European trend somewhat, but not by much. The model’s consistency improves An important property of any projection model is time consistency. This means that in a scenario where observed mortality exactly matches a projection, a re-estimate of the model should yield unchanged parameters. In practice minor aberrations will always occur, because after adding new data points there are more observations, altering the estimators’ uncertainty. Adding the two additional parameters (constants) leads to a stronger form of time consistency, that no longer depends on the method applied to rescaling. Moreover, a projection for only the European countries (without adding the Dutch data separately9) then equals the projection for those countries in case the Dutch data are added. This implies that all other European countries in our peer group would find the same European projection as the Netherlands, if they were to apply the AG2020 methodology. 6.4 Parameter estimates Chapter 7 discusses the model results. In this paragraph we discuss the estimation results of AG2020, that drive those model results. The details of the applied estimation procedure are found in Appendix A. The underlying one year mortality probabilities are determined by modelling hazard rates, namely the European hazard rates, x and gender g, and the hazard rate of the Dutch deviation from Europe, x g,EU (t), for year t, age x g,NL (t), for year t, age x and gender g. We model the logarithm of the hazard rates as follows: ln 冠 x ln 冠 x ln g,EU (t) g,NL (t) 冡 = Ax g + Bx g Kt g 冡 = ␣x g + x g t 冠 x g (t) 冡 = ln 冠 x g The hazard rates for the Netherlands, denoted by x g (t), then follow from the equation g,EU (t) 冡 + ln 冠 x g,NL (t) 冡. Graph 6.1 shows the parameter estimates of the hazard rates on a logarithmic scale. The top three graphs (graph 6.1a) show the parameters for Europe and the bottom three graphs (graph 6.1b) show the parameters for the Dutch deviation. The first graph shows the constant age specific effect (Ax second and third graphs yields the age specific time effect (Bx g Kt graph shows the average annual improvement over all ages (Kt g and x g t g and t g): the third g), while the second graph is age specific (Bx g and x g) and represents the degree of change for that age. The interpretation of each of the graphs is linked to the chosen normalisation, but the resulting hazard rate estimates do not depend on this normalisation. 9 – We speak of adding the Dutch data “separately”, because when using European data only, the Dutch data points are in fact included in the aggregated data set. Projections Life Table AG2020 The projection model 21 g and ␣x g respectively). The product of the values in the
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