Sexspecific mortality forecasting for UK countries: a coherent approach
Abstract
This paper introduces a gender specific model for the joint mortality projection of three countries (England and Wales combined, Scotland, and Northern Ireland) of the United Kingdom. The model, called 2tier Augmented Common Factor model, extends the classical Lee and Carter [26] and Li and Lee [32] models, with a common time factor for the whole UK population, a sex specific period factor for males and females, and a specific time factor for each country within each gender. As death counts in each subpopulation are modelled directly, a Poisson framework is used. Our results show that the 2tier ACF model improves the insample fitting compared to the use of independent LC models for each subpopulation or of independent Li and Lee models for each couple of genders within each country. Mortality projections also show that the 2tier ACF model produces coherent forecasts for the two genders within each country and different countries within each gender, thus avoiding the divergence issues arising when independent projections are used. The 2tier ACF is further extended to include a cohort term to take into account the faster improvements of the UK ‘golden generation’.
Keywords
Mortality projection LeeCarter Common factor Coherent forecast Cohort term1 Introduction
The last three decades have witnessed tremendous developments in the area of mortality modelling and forecasting, beginning with the LeeCarter (LC) proposed in [26]. This pioneering paper rapidly gained popularity and credit due to its simplicity and ability to capture most of the variation in mortality rates. Over time, various extensions and variants of the basic LC model have been put forward, see for instance [2, 27, 35] and [5, 16] for a review and comparison. All these models focus on a single population. When they are applied independently in modelling multiple related subpopulations with similar demographic trends, they would generally lead to divergent forecasts.
Diverging trends over time for closely related subpopulations is usually not a desirable outcome. For example, due to genetic and biological reasons, male mortality rates have constantly been higher than female rates, see [23]. However, if male mortality improvements are faster than female ones and the two genders are projected independently, the model may forecast male mortality rates eventually lower than females. As noted in [Section 5.3,[8]], independent projection methodologies have to be adjusted in order to avoid divergence issues. It is also intuitively true that the mortality of populations that are geographically close or otherwise related is driven by a common set of factors such as socialeconomic conditions, health and care system, and the general environment. Therefore, nondivergent or ‘coherent’ models are sought to address the issue of divergence. The augmented common factor model (ACF) of [32] is an extension of the LC model and is an important step in producing a model that captures both the shortterm divergence and longterm coherence among related populations (subpopulations). The ACF model, which we may also call 1tier ACF, uses a common factor to depict the longterm overall trend of the total population, with additional specific factors included to capture the shortterm discrepancy from the common trend for each subpopulation. Several mortality models for multiple populations have been proposed in the last decade, see for instance [6, 11, 12, 13, 21, 22, 24, 28, 29, 30, 38, 39, 41]. See also [10, 14] for a review and comparison. However, most of the multipopulation models introduced so far, including the ACF, have focused on achieving consistent forecasts among populations differentiated according to a single dimension  either gender or geographical difference, but not both.
In the UK, apart from age and gender being the traditional differentiating mortality factors, the socialeconomic differences among three countries (England & Wales combined, Scotland, and Northern Ireland) have led to notably different mortality trends, at least in the shortterm. The aim of this paper is to introduce an extension to the ACF model, which we call 2tier ACF, where a common factor models the trend for the aggregated UK population, a sex specific factor captures the discrepancy between each gender and the total population, and a country/sex specific factor captures the discrepancy of a gender in a specific country from the overall trend of that gender. This specification ensures to achieve coherence of forecasts in both dimensions  mortality by gender within each country and mortality by country within each gender. The 2tier ACF model is then further extended to include a genderspecific cohort term (2tier ACFC), allowing for the fact that UK mortality experience in the past century cannot be explained by age and period factors only but requires terms depending on the year of birth, see [40].
The contribution of this paper is twofold. On one hand, we aim at introducing a model that, as described above, guarantees consistency across several dimensions, gender and country. On the other hand, we apply this model to the mortality experience of six subpopulations of the UK, consisting of two genders and three countries within each gender, England and Wales combined, Scotland and Northern Ireland. We use data from the Human Mortality Database for the period between 1975 and 2011 and project mortality rates up to the year 2050. The results from the 2tier ACF and 2tier ACFC are compared with the LeeCarter model independently applied to each of the six subpopulations and the 1tier ACF model applied independently to each couple of gender based populations within each country of UK. The fitting period is chosen so as to make sure that the period index of the common factor is reasonably linear. Due to the high volatility of mortality rates at the very old ages, we have excluded ages above 100 from the analysis.
2 Forecasting models
2.1 LeeCarter and augmented common factor models
2.2 2tier augmented common factor model
In this section, we introduce a new twotier extension to the ACF model including a second additional factor for each specific country within each gender, resulting in a joint doublelayer model for different sex and countries. We call this model the 2tier Augmented Common Factor model (2tier ACF).

fit \(\widehat{a}_{x,i,j}+\widehat{B}_x \widehat{K}_t\);

conditional on that, fit \(\widehat{b}_{x,i} \widehat{k}_{t,i}\);

conditional on the previous two stages, fit \(\widehat{b}_{x,i,j} \widehat{k}_{t,i,j}\).
2.3 2tier augmented common factor model with cohort extension
3 Comparison of the models

the LC model (1) applied to each of the six subpopulations independently;^{1}

the 1tier ACF model (2) applied to each of the three couples of gender specific subpopulations within each country independently;^{2}

the 2tier ACF with a common factor for the total UK population, a gender specific factor, and a gendercountry specific factor;

the 2tier ACFC model with a genderspecific cohort extension on top of the 2tier ACF model.
3.1 Model fitting
BIC, AIC, MAPE and ER of LC, ACF, 2tier ACF and 2tier ACFC. \(f=\) females, \(m=\) males
Fitted model  

Metrics  Country  LeeCarter  1tier ACF  2tier ACF  2tier ACFC 
BIC  Overall  214355  216459  205253  200153 
AIC  Overall  202953  201786  190580  183300 
MAPE  England and Wales (f)  0.05797  0.05787  0.05196  0.05045 
Scotland (f)  0.14843  0.14452  0.14229  0.14060  
Northern Ireland (f)  0.26665  0.25882  0.26243  0.26364  
England and Wales (m)  0.05546  0.05387  0.04437  0.04303  
Scotland (m)  0.12964  0.12711  0.12945  0.12357  
Northern Ireland (m)  0.21604  0.20522  0.20633  0.20672  
Overall  0.14570  0.14124  0.13872  0.13800  
ER  England and Wales (f)  0.96856  0.96358  0.98057  0.99140 
Scotland (f)  0.91073  0.90641  0.91944  0.93077  
Northern Ireland (f)  0.85879  0.86411  0.85709  0.86290  
England and Wales (m)  0.98400  0.98488  0.98914  0.99555  
Scotland (m)  0.95979  0.96199  0.96229  0.96797  
Northern Ireland (m)  0.90242  0.91782  0.91493  0.91587  
Overall  0.97839  0.97731  0.98579  0.99369 
The AIC and BIC consistently rank the 2tier ACFC, despite its relative complexity, above the other models. Independent specification of each gender in each country, or of each couple of genders within each country, does not seem to provide any substantial benefit compared to the aggregate modelling of all countries and genders. The addition a gender cohort term results in a further stark improvement of both indices.
It should be noted, from the values of MAPE and ER, that all models fit better to the mortality experience in England and Wales, less so to Scotland, and fit least well to Northern Ireland. This is due to the fact that populations with larger exposures have more stable historical mortality patterns and hence they are easier to fit using Poissontype models that implicitly weigh populations according to their exposure. England and Wales is the largest population among the three countries; therefore the model best fits its experience, followed by Scotland and then Northern Ireland. It is clear from Table 1 that the 2tier ACF fits better the historical experience than the independent LC or 1tier ACF models according to both MAPE and ER, while the 2tier ACFC further improves the model fitting, its extent varying from moderate to substantial depending on the country and gender. One notable exception is Northern Ireland, where the 1tier ACF slightly outperform the other models, confirming nonetheless the need of country specific period terms common to both genders.
3.2 Residual plots
In this section, we inspect the residual plots of the four models against cohorts to assess the models’ capacity in capturing systematic variations by cohort. The residual plots against age and calendar year are fairly similar among different models and are included in Appendix 1.
3.3 Longterm projection
Figure 5 gives the central estimates of logscale mortality rates by age in year 2050 in the four models. Firstly, the 2tier ACF forecasts much smoother age to age mortality rates as compared to the independent LC and 1tier ACF. The independent LC projection for Scotland male even shows decreasing mortality by age at around age 40. Lack of crossage smoothness of the LC model has long been highlighted in research, see for instance [5], as it uses only one age modulator \(b_x\) to measure the age sensitivity to mortality improvement for the specific subpopulation and assumes that it remains constant. Over time, small differences between nearby \(b_x\) terms lead to large discrepancies in mortality forecasts between neighbouring ages, causing in turn lack of smoothness. The independent 1tier ACF model, despite the presence of a common bilinear term, seems to be affected by the same issue. However, in the 2tier ACF model, for each subpopulation the mortality improvement trend is decomposed into tiers  the common trend of total population, the trend of a specific gender, and the trend of the specific subpopulation. Overall, the more finely grained model produce an age pattern displaying smoother crossage mortality improvement. The cohort factor in the 2tier ACFC however, adds slightly more crossage volatility to ages between 30 and 40 than the 2tier ACF model, as now mortality at a specific age in a calendar year is also dependent on the variations from the year of birth. The difference, however, is negligible.
Figure 6 isolates the projection of agespecific mortality rates according to the 2tier ACFC model, together with confidence bounds. For England and Wales and Scotland, relatively narrow confidence intervals reflect the sizes of the corresponding populations. For young adult (age 20 to 40) the confidence regions of males and females are separated, implying that, even over a long horizon, mortality convergence between sexes will be observed only at young and old ages. For Northern Ireland, slightly wider confidence bounds are obtained as a consequence of its smaller population. In this country, the apparent lack of smoothness across age of the projection is put into the right perspective when comparing it with the corresponding projection in Figure 5 under the independent LC or 1tier ACF model. The presence of period terms spanning the three countries helps in dramatically reducing the agetoage variation of mortality rates forecast.
The differences among the four models become more obvious in Figure 8, when life expectancy at retirement age 65 is projected. The independent LC model even forecasts an increasing gap between Northern Ireland and England and Wales for males, and the independent 1tier ACF extends this undesirable pattern to females as well. Although Northern Ireland’s higher rate of suicide, maternal and infant conditions and cancers have historically contributed to the male life expectancy gap, since 1980–82 Northern Ireland’s life expectancy has been improving at a faster pace than England and Wales, see [25]. A slowly narrowing gap allowing for shortterm disparities, as forecasted by the 2tier ACFC, provides a much more reasonable outlook.
In Figure 9, the malefemale mortality ratio (on a square root scale) for England & Wales is plotted against age for a selection of years, and the results are in line with the understanding that sex differences in mortality are mainly contributed by the high mortality of very young and middle aged males, see [23]. It can be seen that for the LC projection, as also found by [21], at the very young ages, when the number of deaths is very small, undesirable projection outcomes of sex ratios less than 1 may occur. The coherent projections under the 1tier and 2tier ACF do not have such issues. The independent LC produce increasing sex ratios up to as high as 2 in 2050 for age groups between 30 and 50, again showing the undesirable features of divergence in longterm projections, while sex ratios from the 2tier ACF model remain stable and constrained. However, sex ratios in England and Wales follow a stable pattern for the successive 40 years under the 1tier and 2tier ACF model, which is unlikely to be true. Compared to the 2tier ACF, projecting the cohort factor of each gender independently introduces some additional variation over the forecast years for the 2tier ACFC, while keeping the sex ratios constrained in a stable and reasonable range. Figure 10 isolates the malefemale mortality ratio (on a square root scale) for the 2tier ACFC model for the three countries, together with confidence intervals. Again, the uncertainty around mortality ratios reflect the corresponding population sizes, with Northern Ireland dominating Scotland which in turn dominates England and Wales. It is remarkable that, for Northern Ireland, mortality for some young adult males is forecast to be as high as four times as the corresponding female mortality.
4 Further discussions and concluding remarks
4.1 Critical appraisal of the 2tier ACFC model.
This approach is analogous to that of [41] to fit cohort extensions of the Poisson Common Factor Model (PCFM) of [28], but is fundamentally different from the method proposed by [36] where, when extending the LC model to include a cohort term, the latter is fitted together with the period factor. However, the approach in [36] cannot be readily applied into the ACF framework, as the multiple bilinear components of the ACF are arranged in hierarchy, so that common trends are fitted prior to fitting individual subpopulation trends. Therefore, the term \(g_{tx,i}\) would have to be placed within this hierarchy and should be fitted after the term \(a_{x,i,j}+B_x K_t\) but before the bilinear term \(b_{x,i,j} \kappa _{t,i,j}\), for the model to make sense. This is another key feature of the 2tier ACFC: including a cohort term still gives rise to a coherent forecast in terms of differences in mortality among subpopulations, because the common trend of the entire population is prioritised while the term \(g_{tx,i}\) is modelled as a stationary process. It may be argued that a common gender cohort factor \(g_{tx}\) could be fitted, together with the bilinear term \(B_x K_t\), so as to maintain the coherence property. However, the residual plots from the 1tier ACF suggests that cohort patterns do differ between different genders, which is consistent with the findings in [40].
The approach used in this paper also fits \(b_{x,i} \kappa _{t,i}\) prior to fitting \(g_{tx,i}\), setting in this way the priority of period factors over cohort factors. This is consistent with the assumption that mortality depends more on the calendar year than on the year of birth when fitting the idiosyncratic trend for each gender. Some research findings, however, disagree with this assumption. In [37] it is suggested that, when fitting the mortality rates of the elderly population in the UK, the cohort effect is more prominent than the period effect. This may suggest alternative orderings when fitting the different components of the ACFC model  one might choose to fit the cohort factor \(g_{tx,i}\) prior to fitting any bilinear term \(b_{x,i} \kappa _{t,i}\), or at least to jointly fit them in a single step when minimising the deviance function. [15] also suggest that the order of model fitting in ageperiodcohort models makes a huge difference to parameter shapes. Further research may therefore be able to identify more elegant ways of including the cohort extensions within the 2tier ACF hierarchy.
It should also be noted that it only makes sense to extrapolate \(g_{tx,i}\) as stationary process when \(a_{x,i,j}+B_x K_t+b_{x,i} \kappa _{t,i}\) is prioritised in the fitting process, as it is the residuals after fitting these components that drive the shape of \(g_{tx,i}\). The plots of cohort factors produced by [41] are much more erratic compared to those in [36]. This is primarily because the PCFM, as used by [41], uses up to five sexspecific bilinear terms to capture the trends of a gender departing from the overall combined population, and if the whole PCFM model is fitted prior to fitting any cohort extension, the residuals used to fit such cohort term are already very erratic. However, since we impose that the term \(g_{tx,i}\) is fitted after the component \(a_{x,i,j}+B_x K_t+b_{x,i} \kappa _{t,i}\) but before the term \(b_{x,i,j} \kappa _{t,i,j}\), the cohort factor turns out to be less erratic (Figure 12) and easier to interpret. If the cohort factor shows a negative slope, it means that mortality in that cohort is improving at a faster pace than implied by the 1tier ACF model. One can easily spot in Figure 12 the golden generation of those born between 1925 and 1945, especially for females, which is consistent with [40]. Another merit of the current approach is that it generally avoids the issues in the two steps method adopted by [36] that the fitting algorithm may not converge for certain combinations of data, parameters and identifiability constraints, which makes the cohort factor harder to interpret, as is pointed out by [20].
4.2 Limitations of the 2tier ACF/ACFC models.
Firstly, the method fundamentally belongs to the class of models described as ‘extrapolative’, so it can only capture trends well embedded in the historical data and lack the ability to project more uptodate information such as medical progresses, environmental and socialeconomic changes such as, for example, the increasing female participation in the workforce, see [18].
Secondly, the 2tier ACF/ACFC models are extensions of the LC model. A major issue of such class of models is that they neglect the existence of an agetime interaction. More specifically, rates of mortality change \(b_x\), \(b_{x,i}\), and \(b_{x,i,j}\) are assumed to remain constant over time, whereas substantial agetime interactions have been identified in actual experience, see [27]. This results in the fact that the models tend to underestimate the life expectancy. In [7], a possible extension of the LC method accounting for the changing age sensitivity to mortality improvement by applying the LC method is proposed. This extension could be potentially applied to the 2tier ACF/ACFC models to consider the evolving pattern of age modulating terms.
Another issue of the 2tier ACF framework is that it assumes homogeneity at different levels. When the \(B_x K_t\) term is fitted, homogeneity is assumed for all lives aged x in year t, but when \(b_{x,i} \kappa _{t,i}\) is estimated, homogeneity is assumed for all lives aged x in year t with the same gender, and the assumption is further relaxed when the model is extended to the country dimension. It should be noted that homogeneity assumptions were embedded in the basic LC model, and methods to build in heterogeneity into the framework has been suggested by [31].
Throughout this research, we have proposed to fit, for simplicity purposes, an AR(1) or random walk to all the mortality period indices, instead of other higher order ARIMA models which may fit better past experience. Moreover, the mortality indices in the model have been extrapolated independently. Despite the fact that \(\kappa _{t,i}\) and \(\kappa _{t,i,j}\) may be correlated and a vector approach may further improve the model forecasting, see for instance [21], each period index in the ACF/ACFC framework represents a trend of a subpopulation that departs from the general trend of the aggregated population, justifying therefore the independent extrapolation used in the paper. Moreover, if a vector approach were considered, correlations among time indices would have to be estimated, compromising the simplicity of the model. Similarly, an AR(1) was chosen to the fit cohort terms in the 2tier ACFC model, which are then extrapolated independently. Although historically females and males have displayed different cohort patterns in their mortality improvements, there could be interactions between the cohort effects of the two genders, since inevitably females and males born in the same year are exposed to similar socialeconomic context and healthcare facilities. Therefore, a more sensible approach may consist in fitting and extrapolating the cohort factors using a vector time series.
Most of the results considered in this paper are point estimates for future mortality rates. Further research should look into the statistical errors of estimates, which are primarily driven by standard errors of parameters in fitting the mortality time indices. The 2tier ACF/ACFC model could be potentially extended to include more tiers to form coherent estimates in several dimensions, for instance taking into account regional inequalities within each country. However, further division within each sex and country means that the sample size of each subpopulation would be smaller and may produce less statistically significant results. Using a different perspective, one may wonder whether the role played by the two factors used to disaggregate mortality improvements, namely gender and country, could be interchanged, i.e. interpret i as the country index and j as gender index. This reversed 2tier ACF model would then fit a common bilinear term, followed by a country specific term and finally a gender specific term within each country. In the current example based on three countries of UK, the two alternatives are bound to produce similar results, as both are rich enough to represent (implicitly or explicitly) sex differences between countries and country differences within each sex. The reverse ACF would only require one additional bilinear term. In a more general example where I countries were to be modelled, the direct 2tier ACF based on gender first/country second will require \(1+2(1+I)\) bilinear terms. The reversed 2tier ACF based on country first/gender second will need \(1+3I\) bilinear terms. As the number of countries I grows, the eventual benefit of adopting the reversed approach would be overshadowed by the increased number of parameters to be estimated.
The 2tier ACFC model can be improved in several directions. In particular, the common age effect model introduced recently by [24] is worth mentioning. Unlike the common factor paradigm, under this approach different populations feature different period trends but share some of the corresponding age modulating parameters. The idea is that, while mortality improvements are free to vary between subgroups, the corresponding age specific changes will be in common between the same subgroups. This could be very convenient when some of the subgroups have small exposures, negatively affecting the properties of the corresponding age response term estimators. Inheriting the age terms from subgroups with larger size will provide a relief against this issue, adding up to the overall benefit coming from the reduction in the number of parameters. This approach has been taken up in [17] in the context of basis risk assessment in longevity transfers, where the mortality of (small) pension schemes relative to the national population needs to be assessed. In the application considered in this paper, the 2tier ACFC could be complemented by letting some of the subgroups at gender/country level share the age response term with other subgroups. This could help, for instance, in reducing the lack of smoothness in some projections such as those of Northern Ireland as evidenced in Figures 6,10. A similar idea has been pursued in [14], where the Li and Lee model is simplified by restricting some of the age response terms, relative to the country specific or to the overall period effects, to be equal.
5 Conclusions
We have extended the ACF model proposed by [32] to a 2tier structure in order to model subpopulations of different genders and countries jointly and coherently. A Poisson structure similar to that in [28] is applied to introduce a robust statistical framework for testing the accuracy of model fitting. The 2tier ACF model fits better the historical mortality experience of the six subpopulations than the independent LC model and the independent Li and Lee models applied to each couple of genders separately. For longterm projections, the 2tier ACF model produces coherent results for both gender difference within each country and country differences within each gender. The 2tier ACF model is also extended to the 2tier ACFC by including a cohort factor, which further improves model fitting, removes significant patterns displayed in the plots of cohort residuals, and maintains the coherence property in longterm projection.
Footnotes
 1.
 2.
When fitting the independent 1tier ACF models, death counts are specified through (3) coupled with (2) deprived of the error term, and completed with the first two identifiability conditions in (5). Finally, each common period term is modelled as random walk with drift, while gender specific period terms are modelled as AR(1) time series.
 3.These metrics are defined as follows:Here, \(n_p\) is the number of parameters net of the number of constraints, \(n_d\) is the number of actual observations, \(\widehat{\ell }\) is the maximized loglikelihood, \(\widehat{m}_{x,t,i,j}\), \(d_{x,t,i,j}\) and \(\widehat{d}_{x,t,i,j}\) are respectively the fitted mortality rate, observed death count and fitted death count at age x, year t, gender i and country j. The fitted death counts are defined by \(\widehat{d}_{x,t,i,j}=E_{x,t,i,j}\widehat{m}_{x,t,i,j}\), where the fitted mortality rate \(\widehat{m}_{x,t,i,j}\) is given by (1) deprived of the error term for the independent LC models; by (2) deprived of the error term for the independent 1tier ACF model; by (4) or (9) for the 2tier ACF, respectively 2tier ACFC models. In each case parameters are replaced with their estimates. See [4] for the definition and properties of AIC and BIC, and [28] for the use of MAPE and ER in the context of mortality forecasting.$$\begin{aligned}&{\text {AIC}}=2\widehat{\ell }+2 n_p,\qquad {\text {BIC}}= 2\widehat{\ell }+n_p \log n_d,\\&{\text {MAPE}}=\frac{1}{n_d} \sum _{x,t,i,j}\left \frac{\widehat{d}_{x,t,i,j}d_{x,t,i,j}}{d_{x,t,i,j}} \right ,\\&{\text {ER}}=1\frac{\sum_{x,t,i,j}\left[ d_{x,t,i,j}\widehat{d}_{x,t,i,j}\right] ^2}{\sum _{x,t,i,j}[d_{x,t,i,j}E_{x,t,i,j} \exp {(a_{x,i,j})}]^2}. \end{aligned}$$
 4.
When fitting the cohort terms, weights \(\omega _{x,t}\) taking values 0 or 1 are added to zeroise cohorts for which only few observations are available. In the present application, zero weight was assigned to the five youngest and oldest cohorts.
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