- Original Article
- Open Access
Mortality and life expectancy forecast for (comparatively) high mortality countries
- Ahbab Mohammad Fazle Rabbi^{1}Email author and
- Stefano Mazzuco^{1}
https://doi.org/10.1186/s41118-018-0042-x
© The Author(s) 2018
- Received: 27 February 2018
- Accepted: 20 September 2018
- Published: 1 November 2018
Abstract
Background
The Lee–Carter method and its later variants are widely accepted extrapolative methods for forecasting mortality and life expectancy in industrial countries due to their simplicity and availability of high quality long time series data.
Objective
We compared and contrasted mortality forecasting models for higher mortality regimes that lack long time series data of good quality, which is common in several Central and Eastern European (CEE) countries.
Data and methods
We utilized seven different variants of the Lee–Carter method and coherent mortality forecasts of various CEE countries, and the Bayesian Hierarchical Model used by the United Nations to produce probabilistic forecasts. The data of nine CEE countries with comparatively higher mortality have been considered.
Results
The performance of the forecasting models for the nine CEE countries was found to be lower than that observed for low-mortality countries. No model gives uniquely best performance for all the nine CEE countries. Most of the LC variants produced lower forecasts of life expectancies than current life expectancy values for Belarus, Russia, and Ukraine. A coherent mortality forecast could not overcome the limitations of single population forecasting techniques due to increasing mortality differences between these countries over the fitting period (mortality divergence). In the same context, the use of the probabilistic forecasting technique from the Bayesian framework resulted in a better forecast than some of the extrapolative methods but also produced a wider prediction interval for several countries. The more detailed analysis for Hungary indicates that a better fit of certain forecasting methods may occur in the later part of the life span rather than the whole life span.
Conclusion
These findings imply the necessity of inventing a new forecasting technique for high-mortality countries.
Keywords
- Mortality forecast
- Lee–Carter method
- Coherent forecasting
- Mortality in Eastern Europe
- Bayesian hierarchical model
Introduction
Improved mortality has been observed globally during the twentieth century and is always considered a positive change for the socio-economic advancement of a country. This change has brought new requirements to support systems for the elderly, such as improved health care and pension provision. Aging has become the greatest population problem for many industrialized countries since the 1970s (Brouhns et al. 2002). This has resulted in increasing interest among government policy makers and planners in accurately modeling and forecasting age-specific mortality rates. Policymakers rely greatly on the possibility of forecasting future population structures. Population aging is increasing not only in low-mortality industrialized countries but also in several Eastern European countries including Russia. These countries have a slower pace of mortality improvement in several stages of the life span compared to low-mortality countries, which delayed the aging problem (Gavrilova and Gavrilov 2009). This leads to the necessity for research on mortality forecasting from the perspective of these countries.
Several parametric and nonparametric methods have been proposed over the years in order to forecast age-specific mortality rates and life expectancy. Booth and Tickle (2008) reviewed all existing mortality forecasting methods under three broad classes in terms of expectation, extrapolation, and explanation. The simplest way of parametric forecasting is to parameterize the available series of life tables and extrapolate each of the parameters separately to obtain the forecast using the assumed model (Keyfitz 1991). In addition to all subjective approaches, a groundbreaking approach to probabilistic forecasting was proposed by Lee and Carter (1992). The advantages of the Lee–Carter (LC) method are its simplicity, as it has few parameters and a straightforward explanation, and robustness in situations where age-specific log mortality rates have linear trends (Booth et al. 2002). To increase the precision of the LC method in the presence of an irregular mortality schedule, later studies restricted the fitting period to post-war years, along with other modifications (Lee and Miller 2001). More generally, Booth et al. (2002) noticed the length of the fitting period might greatly affect point forecast accuracy. Later nonparametric methods of mortality forecasting were introduced by Hyndman and Ullah (2007) and were based on the LC framework. Nonparametric methods were found to be more robust, as they are more efficient than other LC variants, even in presence of outliers, and provide more accurate forecasts (Shang et al. 2011). Numerous other methods have been proposed for mortality forecasting, although many are extensions of the basic LC method. Renshaw and Haberman (2000) used a generalized linear model to model mortality reduction factors and identified the conditions under which the underlying structure of the proposed model is identical to that of the LC method. A further extension by Renshaw and Haberman (2003) accounts for the cohort effect. Other stochastic models have also been introduced to integrate the cohort dimension in mortality, for example that by Cairns et al. (2011). In addition to several LC variants, the application of the Bayesian framework to the LC methodology has become popular. A few studies have proposed extensions of the LC method using a Bayesian framework (see Wiśniowski et al. 2015, for example). Mortality forecasts might also be affected by a different distribution of causes of death across countries (see Booth and Tickle 2008) and risk factors (see Janssen et al. 2013, for an example on smoking status). De Beer (2012) introduced a new relational model for both smoothing and projecting age-specific mortality rates. Mortality forecasting based on the distribution of death has recently become more popular. These approaches are gaining traction because they are independent of the LC framework and are able to overcome some of the basic limitations of the LC methodology (De Beer et al. 2017). Nevertheless, these approaches are newer, and LC variants are the most applied techniques for mortality forecasting (Bohk-Ewald and Rau 2017).
All these methods predict the future mortality of a single population without considering impact of neighbor countries (geopolitically, socio-economically, or by any other common criteria) which may significantly impact the mortality of that population (Hyndman et al. 2013). The first approach to coherent mortality forecasting was introduced by Li and Lee (2005) as an extended hierarchical interface of the LC method. Later, Hyndman et al. (2013) extended the nonparametric approach of Hyndman and Ullah (2007) for coherent forecasting; Ahmadi and Li (2014) used generalized linear modeling and Bergeron-Boucher et al. (2017) used compositional data analysis on distribution of death for coherent forecasting. In another approach, Janssen et al. (2013) considered the smoking epidemic for coherent forecasting.
Given the existence of so many forecasting methods, it is particularly important to assess which model is the most useful in specific contexts. Such an assessment has been conducted by Shang et al. (2011), even though only a specific context was considered, that is, industrialized countries characterized by low mortality, high life expectancies, lower adult and early senescence mortality, a stable pattern of mortality transition over time, and high data quality. Similarly, only low-mortality, industrialized countries have typically been considered in coherent forecasting (see Seligman et al. 2016, for example). Fewer attempts have been made to consider high-mortality Central and Eastern European (CEE) countries or a large number of different mortality forecasting methods (see Bohk and Rau 2015, for example). These countries were also mentioned before for increasing mortality differences over time (the divergence of mortality regime), whereas the developed countries in Western Europe have almost opposite scenario (Vallin and Meslé 2001; Gavrilova and Gavrilov 2009). Even though the mortality pattern is still different from that of Western European countries (see Vallin and Meslé 2001), there are also some similarities (Bálint and Kovács 2015), including increasing population aging (see Gavrilova and Gavrilov 2009). Due to limitations of long time series data (with good quality), it is possible that high-mortality regime forecasts will not be the same as those obtained by Shang et al. (2011) or Shang (2012). This expectation is somewhat confirmed by Bohk and Rau (2015) who compared and contrasted forecasting techniques to evaluate the impact of the recent financial crisis on some of the CEE countries and stated that irregular mortality developments are particularly difficult to forecast due to major changes in long-term trends. Therefore, the CEE countries would benefit from an accurate mortality forecasting by comparing the outcomes of different forecasting techniques.
The aim of this paper is to assess the performance of selected forecasting methods for countries characterized by a higher mortality regime vis-\(\grave {\mathrm {a}}\)-vis Western countries. We compared and contrasted the performance of the mortality forecasting models for nine CEE countries—Belarus, Bulgaria, Estonia, Hungary, Latvia, Lithuania, Russia, Slovakia, and Ukraine. The selected CEE countries differ from Western European and other non-Eastern European countries in five ways: (a) higher mortality, (b) irregular mortality trends, (c) increasing mortality differences between countries over time (the divergence of mortality regime), (d) shorter time series data, and (e) lower quality mortality data. Following Shang et al. (2011), we limited our main interest on LC variants. As mentioned, LC variants are widely used because of their simplicity; many European countries use LC variants for official forecasts (Stoeldraijer et al. 2013). Basic LC method is the main one used for comparisons due to its widespread acceptability (see Booth et al. 2012, Hyndman and Ullah 2007, Shang 2002, for example). Moreover, we considered two more approaches for comparison—coherent mortality forecasting and life expectancy forecasting using a Bayesian Hierarchical model adapted by the United Nations (UN). We extended this comparison to a coherent setup because on contrary to the concept of coherent mortality forecasting, these countries have diverging mortality patterns compared to those of low-mortality countries (Li and Lee 2005). Thus, this extension can provide more insight regarding the assessment of coherent mortality forecasting. The UN adapted the probabilistic approach for forecasting life expectancy using a Bayesian hierarchical model over the previously used deterministic approach (Raftery et al. 2013). UN projections are widely accepted even for countries with limited data, therefore, we extended our comparison to include this technique. Overall, this sort of comparison of mortality forecasting for high-mortality countries may help researchers to better understand the scope of developing mortality forecasting models and may help policymakers in terms of policy implications relating to age- and cause-specific mortality. Unlike the study by Shang (2012), comparing and contrasting on the abovementioned comparatively high mortality countries may give us better insight regarding the impact of recent mortality improvements on future mortality.
This paper consists of four parts. In the next section, we review the forecasting models used in this study and the source of data. The “Results” section consists of two subsections: in the first one, we compared the findings of the models for selected countries with high-mortality regimes, while in the second one, we discuss the country-specific results for Hungary to gain more insight regarding the fitted models. The last section consists of the concluding remarks.
Data and methodology
Data
Fitting periods for the countries and life expectancy at birth (HMD 2018)
Country | Starting year | End year | e_{0} (female) | e_{0} (male) |
---|---|---|---|---|
Belarus | 1970 | 2014 | 78.43 | 67.81 |
Bulgaria | 1950 | 2010 | 77.25 | 70.31 |
Estonia | 1959 | 2013 | 81.33 | 72.72 |
Hungary | 1960 | 2014 | 79.24 | 72.26 |
Latvia | 1970 | 2013 | 78.73 | 69.26 |
Lithuania | 1959 | 2013 | 79.37 | 68.52 |
Russia | 1970 | 2014 | 76.48 | 65.26 |
Slovakia | 1962 | 2014 | 80.32 | 73.25 |
Ukraine | 1970 | 2013 | 76.21 | 66.31 |
For the Bayesian probabilistic model (UN forecast), the life expectancy at birth data are taken from the HMD on a 5-year basis rather than complete life tables used for the extrapolative models. We also contrast this method using the data of World Population Prospects 2012 (UN 2013).
Mortality projection models
The models used in this study are briefly reviewed in the following subsections. We used seven different variants of the LC method along with a coherent mortality forecasting method and the Bayesian hierarchical model used by the UN to produce probabilistic forecasts. We considered four parametric LC variants those by Lee and Carter (1992), Lee and Miller (2001), Booth et al. (2002), and Brouhns et al. (2002), and three nonparametric variants of the Hyndman–Ullah method, robust Hyndman–Ullah method, and weighted Hyndman–Ullah method (Hyndman and Ullah 2007).
Lee–Carter method (1992) and variants
The LC method makes a second stage estimate of k_{t} by finding the value of k_{t}, which for a given population age distribution and previously estimated a_{x} and b_{x} produces exactly the observed number of total deaths for the fitting period of the model (Lee and Carter 1992). An ARIMA(0,1,0) with drift is then fitted for adjusted \(\hat {k}_{t}\) and used to forecast future mortality. Later, Lee and Miller (2001) proposed three modifications of the basic LC model: (i) the fitting period is restricted from 1950 and onward to reduce structural shifts, (ii) the adjustment of k_{t} is done by matching life expectancy, and (iii) the “jump-off error” is eliminated by forecasting forward from observed (rather than fitted) rates.
The LC model can be extended by including higher-order terms instead of the rank-1 approximation considered in the earlier two approaches. Higher-order terms were modeled by Booth et al. (2002), and forecasts were later developed using univariate ARIMA processes (Renshaw and Haberman 2003). The key modifications of this method are (i) the fitting period is determined by a statistical “goodness of fit” criterion, under the assumption that k_{t} is linear; and (ii) the adjustment of k_{t} involves fitting to the age distribution of deaths rather than to the total number of deaths in the basic LC model. This model is a significant development in the research of forecasting mortality, as it slightly eliminates a shortcoming of the LC model. The LC model assumes invariant b_{x}, whereas evidence of substantial age–time interaction is common (Shang 2012).
Nonparametric approaches: the Hyndman and Ullah (2007) methods
This w_{t} is the new weight defined by Hyndman and Ullah (2007) for 0<κ<1, a geometrically decaying weight parameter. The optimal value of κ is chosen by minimizing an overall forecast error measure within the validation data set among a set of possible candidates. Details of the methods can be found elsewhere (Hyndman and Shang 2009).
Coherent mortality forecasting
where i stands for a specific country in the group, and a_{x,i} is the country-specific average log mortality rate. The terms B_{x} and K_{t} are relative speed of change in mortality at each age x and a mortality index capturing the main time trend for the reference population, respectively. Li and Lee (2005) mentioned the term B_{x}K_{t} as a common factor because this quantity is common for all the countries of the group. The error term of Eq. (5) is the country-specific estimate of the error. To obtain the country-specific estimates of the Li–Lee model, first the Lee and Miller (2001) model is fitted to a reference population. The reference group is constructed by adding all the populations of the group from which the common factor is extracted to use in the country-level mortality forecasting mentioned in Eq. (5). Both for K_{t} and k_{t,i}, random walk with drift is used for forecasting in the current study. Following Lee and Miller (2001), actual data is used for mortality forecasting (rather than fitted data) to avoid jump-off error. Choosing a reference population remains one of the biggest problem in coherent forecasting. Several approaches have tried different strategies for using particular countries as reference populations considering geographic and economic similarities, and other criteria (Kjærgaard et al. 2016). In the current study, we combined all populations together as a reference group for each of these countries. Li and Lee (2005) also considered a group of low-mortality countries as a reference for making coherent forecasts for high-mortality countries with the optimistic assumption that these countries will catch-up with the low-mortality countries in future.
Besides these abovementioned variants of LC and HU method, there are other approaches for mortality forecasting. Most of these techniques are largely based on the original LC method. For example, Tuljaparker et al. (2000) used the LC model without any adjustment of the time component and started model fitting from 1950; Girosi and King (2006) considered more than one component in the LC model, and later in another study, they extended the model to incorporate age–period–cohort effects. Using the parameter estimation technique of Brouhns et al. (2002), several approaches have been proposed on Bayesian framework. For the sake of simplicity and wide applicability, we restrict our comparison to major variants of the LC and HU methods only.
Bayesian probabilistic projections (UN life expectancy forecast)
Raftery et al. (2013) defined proper prior for all 13 parameters of the model in such a way that the prior distributions are more diffuse than the posterior distributions. Thus, the abovementioned hierarchical model turned into a Bayesian hierarchical model. This method has one advantage over any other parametric or nonparametric methods; it is flexible for choosing the prior to obtain fast, slow, or medium pace of change in level of life expectancy, per the usual UN projection. The UN method has been proposed to forecast life expectancy using data of World Population Prospects or similar sources. As a consequence, one of the major disadvantages of this method is that it does not forecast considering whole life tables like the previous methods. This makes it complicated to compare the outcomes with those of LC variants. This method takes a single value for life expectancy for each of the 5 years and also returns the forecast as a single value for a period of 5 years (as a median life expectancy for 5 years) utilizing a Bayesian hierarchical model. We considered this method for comparison to evaluate whether LC variants are more optimistic than those used in forecasting only life expectancy at birth. None of the previous approaches compared their findings with UN projections.
Assessing the performance of the mortality forecasting techniques
We assess the performance of the mortality forecasting techniques in general for all the countries using three classes of forecasting techniques: LC and HU variants, coherent forecasts, and UN projections. As coherent forecasting and UN projections are different than LC and HU variants in terms of interpretation and implications, we assess them in different subsections of the “Results” section. The problem of the shortened fitting periods is illustrated in the subsection on coherent forecasting. We fitted the models for 18 series of life tables for males and females separately. The analysis performed in this study are implemented by “Demography” package of R for all of the LC and HU variants. For Bayesian forecast, we used the “bayesLife” package in R. For illustration, we presented the results for females only. To evaluate a forecast technique, we considered two criteria—how optimistic the forecast is in the long run and the accuracy level during the out-of-sample evaluation period (except for the UN method). For high-mortality countries with a jagged pattern of mortality improvement over the years, it is possible that the forecast could be lower than the last observed life expectancy (Booth et al. 2002). We considered it a failure of the model to capture the mortality trend for that particular country because a forecast showing future decline in the mortality pattern is contrary to the basic assumption of the models regarding future mortality improvement (Lee and Miller 2001). Moreover, lower estimates of life expectancy may occur due to a seasonal jump of life expectancy during the out-of-sample evaluation period or in the short run, which is not the case for the long forecast horizon (Shang 2012). Although we have included the UN forecast technique to compare with our findings, we still limit the main topic of discussion to the LC and HU variants.
Here, y_{x,j} represents the observed mortality rate for age x in year j, and \(\widehat {y}_{x,j}\) represents the forecast. Unlike Shang et al. (2011) or Shang (2012), we choose the mean squared forecast error over the mean forecast error as a measure of forecast accuracy. Most of the time, the mean forecast error can be misleading because it may conceal forecast inaccuracy due to the offsetting effect of large positive and negative forecast errors or very low error in forecasting (actual). We used available mortality data from the last 10 years for the out-of-sample evaluation of the forecasting technique. Using the data in the fitting period, we made the one-step-ahead and ten-step-ahead point forecasts and determined the forecast accuracy by comparing the forecasts with the holdout data in the out-of-sample evaluation period.
Due to a shorter fitting period for most of these countries, we kept the forecast horizon for 40 years only; this is the approximate available length of the fitting period for all of these countries. For high-mortality countries, this ground also seemed safe to us like the previous studies conducted on low-mortality countries (e.g., Janssen et al. 2013). Here, LC stands for the basic (Lee and Carter 1992); LC_{P} stands for the model with a Poisson regression (Brouhns et al. 2002); LM stands for the modified LC model proposed by Lee and Miller (2001); BMS stands for the modified LC model proposed by Booth et al. (2002); HU stands for the nonparametric approach proposed by Hyndman and Ullah (2007); HU_{R} stands for the robust (Hyndman and Ullah 2007); HU_{W} stands for the weighted (Hyndman and Ullah 2007); LL stands for the coherent mortality forecast proposed by Li and Lee (2005); and UN stands for the UN life expectancy forecast using a Bayesian hierarchical model (Raftery et al. 2013). These notations are used throughout the paper. Except for the coherent forecasting, the fitting periods are the same for the rest of the models (see Table 1). To keep the comparison of models in same the time frame as the Lee and Miller (2001), we started the fitting from 1950. For coherent forecasting, we used the common fitting period of 1970 to 2010 for all the countries. We used the life tables constructed up to age 100 for model fitting and forecasting due to the lack of data for most of the countries at later age groups.
Results
Comparison of mortality projection models for the nine selected CEE countries
Comparison of 40 years ahead life expectancy forecast for selected countries with high mortality
Country | e _{0} | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL | UN |
---|---|---|---|---|---|---|---|---|---|---|
Belarus | 78.43 | – | – | 78.43 | – | – | – | 80.79 | – | 79.95 |
Bulgaria | 77.25 | 82.16 | 79.77 | 80.24 | 79.76 | 78.49 | 80.83 | 82.61 | 77.55 | 80.63 |
Estonia | 81.33 | 85.98 | 85.99 | 86.19 | 91.00 | 84.19 | 82.74 | 84.95 | 84.71 | 85.70 |
Hungary | 79.24 | 83.16 | 83.05 | 83.68 | 85.98 | 84.64 | 82.99 | 87.81 | 82.60 | 83.54 |
Latvia | 78.73 | 81.87 | 81.76 | 81.92 | 86.37 | 81.30 | 81.76 | 82.31 | 80.10 | 82.43 |
Lithuania | 79.37 | 82.31 | 81.05 | 81.93 | 85.20 | 80.37 | 80.09 | 82.70 | 81.63 | 82.76 |
Russia | 76.48 | – | – | – | – | – | 77.82 | – | – | 79.08 |
Slovakia | 80.32 | 84.91 | 84.65 | 84.66 | 86.24 | 83.93 | 83.64 | 84.17 | 82.37 | 84.44 |
Ukraine | 76.21 | – | – | – | – | – | 80.19 | 78.64 | – | 79.24 |
LC and HU variants
Although HU_{W} produced an optimistic forecast for Belarus and Ukraine, still the results are the subject to analyze. HU_{W} is employed for countries with long time series data, which was not the case for these countries. Similarly, despite providing more optimistic forecasts and greater forecast accuracy than the other methods, forecasts using HU_{R} are affected by unusual improvements in age-specific mortality patterns more than forecasts using the other two HU variants (see the country-specific example of Hungary for more details). It should be noted that although we tried to utilize all available data from the HMD, it was not possible for all these countries. Due to caution notes from the HMD, we started the fitting period only from the best available years mentioned in the HMD (Table 1), making data quality a restriction for the application of the models. BMS also has some flaws regardless of providing the optimistic forecast for the Baltic counties. BMS considers only the best fitting period instead of taking into account all the observed data. For all nine countries, the data of the last 20 years was found to be more significant for forecasting during model fitting.
One-step-ahead forecast bias of the fitted models in terms of MAE
Country | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL |
---|---|---|---|---|---|---|---|---|
Belarus | – | – | 0.144 | – | – | – | 0.102 | – |
Bulgaria | 0.160 | 0.144 | 0.128 | 0.122 | 0.137 | 0.131 | 0.110 | 0.131 |
Estonia | 0.248 | 0.246 | 0.322 | 0.259 | 0.324 | 0.292 | 0.273 | 0.267 |
Hungary | 0.181 | 0.164 | 0.146 | 0.132 | 0.119 | 0.118 | 0.115 | 0.157 |
Latvia | 0.186 | 0.185 | 0.204 | 0.186 | 0.174 | 0.196 | 0.175 | 0.146 |
Lithuania | 0.330 | 0.167 | 0.209 | 0.187 | 0.167 | 0.177 | 0.168 | 0.210 |
Russia | – | – | – | – | – | 0.079 | – | – |
Slovakia | 0.181 | 0.177 | 0.257 | 0.186 | 0.200 | 0.212 | 0.195 | 0.214 |
Ukraine | – | – | – | – | – | 0.087 | 0.066 | – |
One-step-ahead forecast bias of the fitted models in terms of MSE
Country | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL |
---|---|---|---|---|---|---|---|---|
Belarus | – | – | 0.079 | – | – | – | 0.033 | – |
Bulgaria | 0.044 | 0.037 | 0.033 | 0.038 | 0.047 | 0.045 | 0.034 | 0.034 |
Estonia | 0.141 | 0.137 | 0.283 | 0.151 | 0.214 | 0.204 | 0.176 | 0.167 |
Hungary | 0.065 | 0.057 | 0.049 | 0.052 | 0.048 | 0.047 | 0.042 | 0.057 |
Latvia | 0.110 | 0.110 | 0.140 | 0.109 | 0.115 | 0.134 | 0.107 | 0.051 |
Lithuania | 0.262 | 0.079 | 0.128 | 0.096 | 0.074 | 0.086 | 0.079 | 0.127 |
Russia | – | – | – | – | – | 0.010 | – | – |
Slovakia | 0.112 | 0.111 | 0.1967 | 0.130 | 0.169 | 0.163 | 0.162 | 0.109 |
Ukraine | – | – | – | – | – | 0.015 | 0.008 | – |
Coherent forecasting
Comparison of 40 years ahead life expectancy forecast for selected countries with high-mortality for the harmonized fitting period (1970–2010)
Country | e _{0} | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL |
---|---|---|---|---|---|---|---|---|---|
Belarus | 76.49 | – | – | – | – | – | – | 78.58 | – |
Bulgaria | 77.25 | 79.92 | 80.09 | 79.95 | 80.09 | – | 79.17 | 77.43 | 77.55 |
Estonia | 80.55 | 84.73 | 84.66 | 84.82 | 88.23 | – | – | 80.57 | 84.71 |
Hungary | 78.34 | 82.30 | 82.42 | 82.78 | 84.79 | 85.71 | 86.42 | 86.44 | 82.60 |
Latvia | 77.39 | 80.28 | 79.60 | 79.70 | 79.75 | 78.62 | - | 79.83 | 80.10 |
Lithuania | 78.73 | 80.96 | 79.76 | 80.52 | 79.66 | 79.41 | 78.96 | 81.45 | 81.63 |
Russia | 74.86 | – | – | – | – | – | – | 74.89 | – |
Slovakia | 79.15 | 83.50 | 83.45 | 83.49 | 84.67 | 82.69 | 82.67 | 83.45 | 82.37 |
Ukraine | 75.19 | – | – | – | – | – | – | 78.90 | – |
The performance of the coherent mortality forecast was greatly affected for three reasons: (i) combination of mortality rates of a population with large exposure to mortality rates of a population with comparatively smaller exposure size, (ii) high adult male mortality for several of these countries, and (iii) irregular trend of life expectancy of joint mortality data because of (i) and (ii). Although LL performed reasonably for most of these high-mortality countries, it failed to do so for Belarus, Russia, and Ukraine. During estimation of the common factor, countries with high mortality dominated the comparatively low-mortality countries. This might be due to mixing large exposure with smaller exposure or combining mortality rates of populations with different age- and cause-specific mortality patterns. Because country-level mortality has two parts—common factor from the reference population and country-specific estimate of b_{x,i}k_{t,i}—the country-specific forecast is affected because the common factor of the reference population greatly affects completely different mortality patterns. This is another consequence of using coherent forecasting for population groups with increasing mortality differences over time (mortality divergence). LL adjusts the time component of the common factor according to the estimated life expectancy of combined mortality data. The estimated life expectancy for our high mortality countries has irregular trend that is different than the country-specific individual trends of life expectancy in most of the cases. Characteristics of the combined mortality rates of these countries and the key outcomes of the fitted model using joint mortality rates are given in the Appendix.
Our coherent forecasting results also indicates necessity of implying a rigid assumption for choosing a similar group of countries (Kjærgaard et al. 2016). The six countries for which the coherent mortality forecast of life expectancies is higher than the last observed life expectancies are all currently member states of the European Union. In the current methodology of coherent forecasting, the fitting period of combined mortality data might be shorter for individual countries with longer time series data; this may affect the forecast as well.
One-step-ahead forecast bias of the fitted models in terms of MAE considering harmonized fitting period
Country | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL |
---|---|---|---|---|---|---|---|---|
Belarus | – | – | – | – | – | – | 0.113 | – |
Bulgaria | 0.120 | 0.119 | 0.120 | 0.118 | - | 0.119 | 0.151 | 0.131 |
Estonia | 0.348 | 0.289 | 0.327 | 0.293 | - | - | 0.235 | 0.267 |
Hungary | 0.130 | 0.182 | 0.184 | 0.148 | 0.139 | 0.172 | 0.115 | 0.157 |
Latvia | 0.337 | 0.185 | 0.139 | 0.138 | 0.114 | 0.121 | 0.112 | 0.146 |
Lithuania | 0.287 | 0.209 | 0.395 | 0.207 | 0.208 | 0.209 | 0.190 | 0.210 |
Russia | – | – | – | – | – | – | 0.046 | – |
Slovakia | 0.142 | 0.142 | 0.142 | 0.153 | 0.163 | 0.156 | 0.149 | 0.214 |
Ukraine | – | – | – | – | – | – | 0.061 | – |
One-step-ahead forecast bias of the fitted models in terms of MSE considering harmonized fitting period
Country | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL |
---|---|---|---|---|---|---|---|---|
Belarus | – | – | – | – | – | – | 0.037 | – |
Bulgaria | 0.039 | 0.039 | 0.039 | 0.038 | – | 0.039 | 0.062 | 0.034 |
Estonia | 0.320 | 0.197 | 0.283 | 0.194 | – | – | 0.122 | 0.167 |
Hungary | 0.067 | 0.063 | 0.072 | 0.056 | 0.046 | 0.059 | 0.040 | 0.057 |
Latvia | 0.075 | 0.040 | 0.044 | 0.042 | 0.031 | 0.033 | 0.028 | 0.051 |
Lithuania | 0.227 | 0.119 | 0.307 | 0.114 | 0.141 | 0.136 | 0.124 | 0.127 |
Russia | – | – | – | – | – | – | 0.003 | – |
Slovakia | 0.041 | 0.040 | 0.042 | 0.049 | 0.062 | 0.060 | 0.059 | 0.109 |
Ukraine | – | – | – | – | – | – | 0.007 | – |
UN forecasting
Unlike most of the previous comparisons, we extend the comparison of the mortality forecasting models with probabilistic forecasting obtained through a Bayesian framework (UN forecast). It should be noted that the UN life expectancy forecasts shown in Table 2 refer to years 2050–2055, as this technique uses 5 years age groups. For the nine countries considered in this study, we utilized the life expectancy at birth in 5-year intervals from the HMD instead of using the UN data. We project the life expectancy at birth up to year 2100, to compare with the LC and HU variants, the results are shown up to 40 years. In addition to the data from HMD, we also used the data from the World Population Prospects (UN 2013); the comparison between these two different data sources is given in the Appendix. The simulation was done with 160,000 iterations (10,000 burn-in); the thinning interval was 10, and the number of chains was three. It was already mentioned in the previous section that an irregular trend in mortality is visible in the case of several of these countries. The trend of life expectancies computed on a 5-year basis was also similar to that; fluctuations in the trend for life expectancies remained for Belarus, Russia, and Ukraine during the fitting period. Like HMD, a similar pattern was also observed in case of data from World Population Prospects. Nevertheless, UN forecasts showed an increasing trend for life expectancy at birth for all these countries (both for HMD and World Population Prospects). Unlike several LC variants and coherent forecasting, the forecast of life expectancies was higher than the last observed one for all three high-mortality countries—Belarus, Russia, and Ukraine. For all other countries, the forecast produced using the UN forecast technique fell between the other forecast methods. The UN forecast was lowest for Hungary and Latvia among all the methods.
One shortcoming of the UN forecasting technique is that it is not based on life table like the previous LC or HU variants. This prevents age-specific mortality forecasting and comparing forecast accuracy in out-of-sample evaluation period, as we have done for the LC and HU variants or for the coherent forecasting. Instead, we plotted the 95% prediction interval to forecast life expectancy at birth for all nine countries (see Appendix). These prediction intervals are showed here over the whole projection period (up to year 2100). Belarus, Russia, and Ukraine have wider prediction intervals compared to the other countries. The diverging mortality pattern affects the forecast of life expectancy as well.
Country-specific illustration: Hungary
Past mortality trends
Forecast of life expectancies
Forecast of life expectancies at birth for Hungarian males and females
Forecast period | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL |
---|---|---|---|---|---|---|---|---|
Male, 2034 | – | – | 73.93 | – | 77.35 | 77.53 | 77.96 | 73.19 |
Female, 2034 | 81.17 | 81.06 | 81.58 | 82.90 | 82.22 | 80.85 | 83.86 | 80.96 |
Male, 2044 | – | – | 74.42 | – | 79.33 | 79.70 | 80.42 | 74.23 |
Female, 2044 | 82.18 | 82.07 | 82.65 | 84.50 | 83.47 | 81.94 | 85.91 | 82.00 |
Male, 2054 | – | – | 74.82 | – | 81.03 | 81.63 | 82.97 | 75.24 |
Female, 2054 | 83.16 | 83.05 | 83.68 | 85.98 | 84.64 | 82.99 | 87.81 | 83.01 |
Forecast of remaining life expectancies at age 65 for Hungarian males and females
Forecast period | LC | LC_{P} | LM | BMS | HU | HU_{R} | HU_{W} | LL |
---|---|---|---|---|---|---|---|---|
Male, 2034 | 21.04 | – | 17.14 | 14.77 | 16.65 | 17.29 | 17.42 | 15.21 |
Female, 2034 | 20.29 | 20.19 | 20.26 | 20.93 | 20.59 | 19.35 | 21.72 | 19.49 |
Male, 2044 | 22.84 | – | 18.61 | 15.26 | 17.71 | 18.49 | 19.10 | 15.77 |
Female, 2044 | 21.15 | 21.06 | 21.18 | 22.12 | 21.52 | 20.07 | 23.35 | 20.17 |
Male, 2054 | 25.02 | – | 20.27 | 15.76 | 18.73 | 19.67 | 21.10 | 16.35 |
Female, 2054 | 22.00 | 21.91 | 22.08 | 23.27 | 22.41 | 20.82 | 24.90 | 20.87 |
Although LC and BMS failed to produce optimistic forecasts of life expectancies at birth, the forecast of e_{65} was higher than the last observed e_{65} for both methods; which was not the case for LC_{P}. In the other cases, all the models performed well, although higher forecasts are observed from HU_{W} for females compared to the other methods. Moderate improvements in the mortality of people aged 65 years or over were observed in previous studies as well, unlike the pattern observed for the middle-aged population (Bálint and Kovács 2015).
Forecast of mortality rates
Here, years are plotted again using a rainbow palette, so the beginning years of the fitting period are in red, followed by orange, yellow, green, blue, and indigo, with the most recent years plotted in violet. All the models forecast a rapid and sharp decline in infant and adult mortality, except for coherent forecasting (Fig. 6). Except for HU and HU_{W}, all other methods are affected by the recent short-run improvement of early-aged mortality around age 5 to 15. For Hungarian female mortality, this improvement occurred only for a few of the very recent years (Fig. 3) which highly affects the forecasts of BMS, HU_{R}, and possibly LL. Combined mortality rates of the reference population were higher across the life span compares to observed female mortality rates for Hungary. As a consequence, the coherent forecasts for the Hungarian females are more pessimistic than those of the other methods (Tables 8 and 9). We already mentioned it as a drawback of current settings for choosing reference population in coherent forecasting. Clearly, the increasing mortality differences over time between the countries of the reference group have high impact on individual country-specific forecasts. Forecasts of senescence mortality are virtually similar for all methods except for coherent forecasting due to this reason.
Conclusion and discussion
Summary of results
We compared and contrasted mortality forecasting models for higher mortality regimes that lack long time series data of good quality, which is common in several Central and Eastern European (CEE) countries. The performance of the forecasting models for the nine CEE countries was found to be lower than that observed for low-mortality countries. No model gives uniquely best performance for all the nine CEE countries included. All studied LC variants, with the exception of the weighted version of the (Hyndman and Ullah 2007) method, resulted in lower future life expectancy than current observed life expectancy values for Belarus, Russia, and Ukraine. Coherent mortality forecasting could not overcome the limitations of the LC variants completely by producing more optimistic forecasts or more accurate forecasts during out-of-sample evaluation. The UN forecasting technique (through a Bayesian hierarchical model) can provide a more optimistic forecast than some LC variants for Belarus, Russia, and Ukraine, but it also produced a wider prediction interval for several of these countries. The country-specific forecast for Hungary revealed that some of the models can overcome the shortcomings of pessimistic forecasts of the earlier part of the life span.
Explanations of the main findings
The performance of the forecasting models for the nine CEE countries was found to be lower than that of the observed for low-mortality countries. In addition to a high-mortality regime, the lower performance of the forecasting models for the nine CEE countries is highly attributable to the irregular mortality trends and a shorter fitting period. Except for weighted and robust variants of the HU method, most of the LC variants were not appropriate for some of the high-mortality countries. The mentioned two methods succeed to produce optimistic forecast for these countries due to application of different weighting techniques, which was not the case for the other methods. Except for Hungary and Slovakia, all of these countries suffered from mortality crisis during 1990s, and it was particularly severe for Belarus, Russia, and Ukraine (Shkolnikov et al. 1998). The accuracy obtained for weighted HU method during out-of-sample evaluation is thus subject to analyze as this method gives more weight on recent period than distant past (Hyndman and Ullah 2007).
Coherent mortality forecasting could not overcome the limitations of the LC variants completely by producing more optimistic forecasts or more accurate forecasts during out-of-sample evaluation. This might be explained by the high heterogeneity across CEE countries in terms of mortality levels and patterns. Such heterogeneity increased during the severe mortality crisis which Belarus, Russia, and Ukraine underwent during the 1990s (Shkolnikov et al. 1998). This lead to pessimistic forecasts even by coherent model.
In this case, UN forecast (using a Bayesian hierarchical model) can provide more optimistic forecasts: life expectancy at bith rather age-specific rates are considered, and in this way, the cross-country differences are mitigated. However, due to irregular trends of life expectancies over time, UN forecasts also produced wider prediction intervals for several of these countries.
The country-specific forecast for Hungary revealed that some of the models can overcome the shortcomings of pessimistic forecasts of life expectancy in earlier part of life. For three LC variants, we observed that the models produce optimistic forecast of life expectancy at later ages despite of failing to do so for life expectancy at birth for Hungarian male.
Limitations
The current study have several limitations. Firstly, we could not consider all possible mortality data. With only nine high-mortality CEE countries we just have a snapshot of the limitations of the mortality forecasting models from a different point of view. This selection was necessary in order to keep data quality at the highest level. Moreover, we did not consider data of Poland or the Czech Republic as their mortality patterns are more similar to low-mortality countries. Secondly, we did not utilize all types of existing mortality forecasting models. For example, we did not consider behavioral impacts on mortality forecast. Impact of smoking and alcohol consumption on life expectancy are mentioned in previous studies (see Luy and Wegner-Siegmundt 2014; Trias-Llimós et al. 2017, for example on smoking and alcohol consumption, respectively). In addition, forecasting considering different causes of deaths could also provide more insight. However, data regarding causes of death or risk factors are not available for some of these countries. Thirdly, we did not compare the interval forecast for LC and HU variants, which could be another topic for further research. Fourthly, due to the unavailability of data for later age groups, we restricted the model fitting to age 100. Such truncation is not suitable for a wide forecast horizon, even for a population with less of an aging problem. Finally, we considered all populations together for coherent forecasting, without following any particular strategies to choose the best reference population (Kjærgaard et al. 2016).
Recommendations for future work
Based on the study findings, future research directions for mortality forecasting can be suggested. A significant improvement would be to develop a new forecasting technique flexible enough to handle this sort of irregularities in mortality trends as most of the LC variants could not perform well in the presence of irregular mortality trends. Most of the existing methods for mortality forecasting are based on the LC method to some extent. Better performances of the HU variants can be attributable to smoothing techniques adopted in those methods. Instead of considering only one-dimensional smoothing (smoothing the age-specific mortality rates of a given year), incorporating the period and/or cohort effect may provide different results than those obtained here. Second-level estimation for fitted parameters in the LC method has a great impact on forecast accuracy for some countries; an alternative estimation approach at this level could also be useful. Coherent mortality forecasting clearly indicates a lack of rigid assumptions for choosing an appropriate set of countries as a reference group. In the current methodology, a shortened, harmonized fitting period for a group of countries also reduces the scope of forecasting. New strategies for choosing a reference population need to address the issues of increasing mortality differences between countries over time (mortality divergence). To overcome the lack of mortality data for later age groups (say after age 100), some extrapolation before forecasting might be useful for longer forecasts. Flexible settings in terms of the fitting period for modeling the joint mortality data may improve the forecast accuracy of country-specific forecasts. Nevertheless, the invention of a new forecasting method for high mortality countries is necessary based on the findings of the current study, as none of the models was completely suitable for the examined nine CEE countries.
Appendix
Key features of coherent forecasting for high mortality countries
Fitted model on joint mortality rates
UN forecast for (comparatively) high mortality countries
Forecast of life expectancies at birth using UN forecast method for HMD-2018 and WPP- 2012^{†}
Data and forecast year | Belarus | Bulgaria | Estonia | Hungary | Latvia | Lithuania | Russia | Slovakia | Ukraine |
---|---|---|---|---|---|---|---|---|---|
HMD, 2034 | 78.672 | 78.901 | 83.381 | 81.165 | 80.361 | 80.938 | 77.463 | 82.109 | 77.507 |
WPP, 2034 | 77.225 | 78.914 | 81.395 | 80.600 | 78.960 | 79.807 | 76.223 | 81.071 | 75.621 |
HMD, 2044 | 79.370 | 79.740 | 84.528 | 82.311 | 81.384 | 81.851 | 78.265 | 83.273 | 78.371 |
WPP, 2044 | 78.022 | 79.858 | 82.371 | 81.592 | 79.724 | 80.645 | 77.204 | 82.057 | 76.219 |
HMD, 2054 | 79.952 | 80.633 | 85.703 | 83.539 | 82.432 | 82.763 | 79.079 | 84.436 | 79.243 |
WPP, 2054 | 78.812 | 80.729 | 83.315 | 82.707 | 80.564 | 81.428 | 77.991 | 83.053 | 76.906 |
Declarations
Acknowledgements
The authors are grateful to Professor Fanny Janssen for her comments regarding the paper and Professor Vladimir Canudas-Romo and Professor Nicola Torelli for their comments regarding the models used in this study.
Availability of data and materials
Data used in study are freely accessible from Human Mortality Database.
Authors’ contributions
Both authors contributed to the conception, analysis, drafting, and revision of the manuscript. Both authors read and approved the final manuscript.
Competing interests
Both authors declare that they have no competing interests.
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