Smoothing, Decomposing and Forecasting Mortality Rates

Abstract

The Lee–Carter (LC) model represents a landmark paper in mortality forecasting. While having been widely accepted and adopted, the model has some limitations that hinder its performance. Some variants of the model have been proposed to deal with these drawbacks individually, none coped with them all at the same time. In this paper, we propose a Three-Component smooth Lee–Carter (3C-sLC) model which overcomes many of the issues simultaneously. It decomposes mortality development into childhood, early-adult and senescent mortality, which are described, individually, by a smooth variant of the LC model. Smoothness is enforced to avoid irregular patterns in projected life tables, and complexity in the forecasting methodology is unaltered with respect to the original LC model. Component-specific schedules are considered in projections, providing additional insights into mortality forecasts. We illustrate the proposed approach to mortality data for ten low-mortality populations. The 3C-sLC captures mortality developments better than a smooth improved version of the LC model, and it displays wider prediction intervals. The proposed approach provides actuaries, demographers, epidemiologists and social scientists in general with a unique and valuable tool to simultaneously smooth, decompose and forecast mortality.

Background

An accurate knowledge of future mortality patterns and their uncertainty is crucial in almost all areas of a society, including the health-care system, health and life insurance, as well as pension schemes. Since Bonynge (1852), demographers, actuaries and statisticians have developed several methodologies and techniques to predict mortality. However, up until the 1980s, mortality forecasts were based on deterministic scenarios and, generally, some variants would be assumed at the life expectancy level as input for producing population projections. This approach was used almost without exception by international bodies, such as the United Nations and World Bank, and national statistical offices (Booth 2006).

One of the main limitations of this scenario-based approach is that the probability of different scenarios cannot be quantified nor specified a priori, and the structure of mortality developments over age and time is only partially considered. For this reason, the last thirty years witnessed an increasing use of more sophisticated statistical methodologies for projections, which enable the construction of forecast probability distributions and probabilistic prediction intervals, mainly using time changes in age-specific death rates (Booth and Tickle 2008).

The use of stochastic models has flourished in recent decades (for comprehensive reviews, see Booth 2006; Booth and Tickle 2008; Stoeldraijer et al. 2013) and in 1992, a breakthrough was achieved by Lee and Carter, who proposed an elegant and powerful methodology to forecast mortality that later became widely accepted and employed. The Lee–Carter (LC) model is based on linear extrapolations of the logarithms of age-specific death rates, using principal component techniques. In particular, a matrix of logged age-specific mortality rates over time is summarized by two vectors of age-specific parameters ${\alpha }_x$ and ${\beta }_x$, which describe the general shape of mortality and the fixed rate of mortality improvement at age x, respectively, and by a time-varying index ${\kappa }_t$, which captures the general level of mortality. Mortality forecasts are then derived from the projection of ${\kappa }_t$ using standard time series methods.

While the LC model works reasonably well, several limitations have been revealed in the succeeding literature. A central assumption of the LC model is the fixed rate of age-specific mortality improvement over time (Lee and Miller 2001). Already recognized by Alho (1992), this assumption has been violated in several low-mortality countries in recent decades, as rates of mortality improvements have tended to decline over time at younger ages, and they have risen at older ages (Kannisto et al. 1994; Vaupel et al. 1998; Wilmoth and Horiuchi 1999). As a result, projections that ignore this “rotation” of mortality declines will lead to errors, particularly in the projected age patterns of future death rates. To overcome this limitation, Li et al. (2013) proposed a rotation of the ${\beta }_x$ schedule for long-term projections. This allows one to capture the observed trend of deceleration of infant and child mortality decline and the acceleration of old-age mortality reduction in low-mortality countries during the most recent decades. A similar approach has been used in other projection models (Ševčíková et al. 2016; Pascariu et al. 2020). Nevertheless, the approach proposed by Li et al. (2013) relies strongly on a subjective choice of the model parameters, and it does not account for the stochastic process behind mortality.

The original LC model is estimated by minimizing the residual sum of squares using singular value decomposition (SVD), which is as simple as inappropriate when dealing with mortality data (Wilmoth 1993; de Jong and Tickle 2006; Koissi and Shapiro 2006). Brouhns et al. (2002) embedded the LC model within a Poisson setting using maximum likelihood estimation and a Newton–Raphson algorithm to estimate model parameters. Other papers used this Poisson assumption for estimating the LC model (Renshaw and Haberman 2003a; Haberman and Renshaw 2008; Li and Li 2017; Brouhns et al. 2005; Czado et al. 2005). Despite the appropriate estimation procedure, the LC model leads to outcomes which become less smooth and more unrealistic when projected, and age-patterns tend to deviate from any given baseline described by ${\alpha }_x$ (Girosi and King 2007, 2008). To prevent irregular projected life tables, smoothing techniques have been implemented within an LC framework. Whereas Renshaw and Haberman (2003c) suggested smoothing estimated series of ${\beta }_x$, Hyndman and Ullah (2007) employed a functional data approach and smoothed the mortality data as a preliminary step. Alternatively, a penalized likelihood approach has been proposed by Delwarde et al. (2007) and Currie (2013). Their approach allows them to frame the LC model within a Poisson setting and simultaneously obtain smooth ${\beta }_x$ (in both works, and also smooth ${\alpha }_x$ in Currie (2013)) and, consequently, regular forecast age-patterns.

None of the variants and generalizations of the LC model attempted to account for the complex nature of the human mortality age-pattern. Specifically, we refer to the hypothesis that human mortality can be decomposed into three different groups that operate principally upon childhood, middle and old ages, respectively. This idea goes back at least one and a half centuries (Thiele 1871) and it leads to a so-called “bath-tub shaped” mortality age-pattern. In most developed populations, during the first years of life, mortality decreases steeply from a relatively high value at age 0 (Levitis 2011). A minimum is commonly reached at ages around 10-15 (Ebeling 2018). Afterward, especially for men, hazard rates show a hump at young-adult ages (Remund 2015; Goldstein 2011). Starting from about age 30, mortality rises exponentially and eventually it tends to level off at ages above 80 (Preston 1976; Thatcher et al. 1998; Vaupel 1997). The idea of mortality constructed from three competing components has been extensively used for modeling purposes (Siler 1979; Heligman and Pollard 1980; Siler 1983; Kostaki 1992; Rogers and Little 1994; Dellaportas et al. 2001; de Beer and Janssen 2016; Camarda et al. 2016; Remund et al. 2018; Mazzuco et al. 2018). However, very few attempts have been made to forecast the three components. Only Forfar and Smith (1987) employed the Heligman and Pollard model for forecasting, as the large number of parameters limits the model’s potential for projections (McNown and Rogers 1992). In addition, the model’s parameters are typically highly correlated (Hartmann 1987), and these high correlations further compromise their interpretability (Booth and Tickle 2008).

Here, we propose a novel model which could be seen from two distinct viewpoints. On the one hand, we suggest an extension of the LC model that, as in Delwarde et al. (2007) and Currie (2013), is framed within a Poisson setting and enforces smoothness in the outcomes. On the other, we address the drawback of the fixed rates of mortality improvement from a different perspective, i.e., by modeling childhood, early-adult and high-ages mortality simultaneously, but with distinct structures. This last aspect connects our approach to the Sum of Smooth Exponentials (SSE) model introduced by Camarda et al. (2016) where smooth components are summed on the scale of the expected values for decomposing complex series of (death) counts.

In particular, we simultaneously estimate a smooth variant of the LC model for all three components of human mortality. We therefore call our approach the “Three-Component smooth Lee–Carter” (3C-sLC) model. In other words, given a set of observed death counts and exposure-to-risk data, we derive the standard LC parameters ${\alpha }_x$, ${\beta }_x$ and ${\kappa }_t$ for each component, with both ${\alpha }_x$ and ${\beta }_x$ smoothly varying over x. Forecasting is achieved by extrapolating component-specific time indexes via standard Box-Jenkins methods, maintaining the simplicity of the original LC model in projecting future trends. The main advantage of our model is the greater flexibility of mortality patterns, which do not depend on a single fixed rate of age-specific mortality improvement, but rather on the combination of different component-specific schedules. The methodology that we propose also allows us to forecast overall mortality as a sum of future component-specific patterns. Consequently, future trends in childhood, early-adult and old-age mortality can be analyzed independently, adding great explanatory value to the proposed approach.

The presence of three independent components could prompt a reader to establish an association between the proposed 3C-sLC and LC variants with additional ${\beta }_x^{(i)}{\kappa }_t^{(i)}$ interaction terms (Renshaw and Haberman 2003b; Hyndman and Ullah 2007). However, the two approaches present substantial differences in both methodological and demographic perspectives. Whereas the 3C-sLC attempts to decompose the mortality pattern into additive independent components, the LC model (with more interaction terms) tries to approximate it with additional terms that have decreasing explicative power. In statistical terms, the 3C-sLC models the expected values of the Poisson process, while the LC variants the logarithm of the expected values.

This paper is organized as follows. In Section 2, we start by introducing the data that we use in the paper, which are derived from the Human Mortality Database (2020). We present the original LC model and its smooth version in Section 2.1; then, Section 2.2 introduces the 3C-sLC model, as well as the procedures for estimating its parameters and obtain mortality forecasts. Throughout Section 2, we present outcomes of the smooth LC and 3C-sLC models on a specific population for illustrative purposes. In Section 3, we show and compare the results of fitting and forecasting mortality in four populations using the two models. Section 4 provides a discussion of our methodology and its related results, and Section 5 concludes the paper. A larger comparison with alternative forecasting methods, application on ten populations as well as out-of-sample forecasts will be presented in Appendix A. Finally, Appendix B will present a version of the model in which only two components are estimated.

Data and Methods

We assume that we have data on deaths and exposures to the risk of death, arranged in two matrices, $\mathit{Y}=(y_{x,t})$ and $\mathit{E}=(e_{x,t})$, each $m\times n$ where rows and columns are classified by single age at death, $\mathit{x},m\times 1$, and single year of death, $\mathit{t},n\times 1$, respectively. We aim to forecast mortality for future years ${\mathit{t}}_F,n_F\times 1$. We assume that the number of deaths $y_{x,t}$ are realizations of the random variable $Y_{x,t}$, which follows a Poisson distribution:

$$\begin{array}{c}\hfill Y_{x,t}\sim \mathcal{P}(e_{x,t}{\mu }_{x,t}),\end{array}$$ 1

i.e., the expected value is the product of exposures and the force of mortality (Brillinger 1986). Description of the mortality development over age and time is commonly provided by portraying the so-called death rates: $m_{x,t}=y_{x,t}/e_{x,t}$.

In the following, we will present outcomes from the proposed model for four different populations. Specifically, we analyze both female and male populations of the USA and Switzerland (CHE). Data are taken from the Human Mortality Database (2020). We model mortality from ages 0 to 100 over the period 1960–2018, forecasting up to 2050.

In this section and only for illustrative purposes, methodology is presented on Swiss males data. Figure 1 shows mortality rates on a log scale over ages for all years as well as time trends for selected ages over time. The typical mortality age-pattern is portrayed in the left panel: a rapid declining mortality from birth, a roughly log-linear trajectory starting from ages 30–40 and an intermediate phase in early-adult ages often described as “accident hump.”

Fig. 1.

Actual death rates over age and time on a log scale for Swiss males. Ages 0–100, years 1960–2018. Left panel: rates over age for all years. Selected years are depicted by thicker lines. Right panel: selected ages over years

Time trends are also identifiable in the right panel of Fig. 1. A general mortality improvement is clear with decreasing rates over time. However, differences in the speed of mortality reductions are evident for different ages, and some ages have also experienced stagnation and increasing mortality, e.g., age 30 from 1960 to early 1990s. Whereas looking at death rates helps to have a good image of mortality developments, it is essential to develop statistical models to obtain an accurate and parsimonious description as well as a reliable prediction of the observed trends. The following pages will be devoted to this purpose.

The Lee–Carter Model

Lee and Carter (LC, 1992) pioneered an elegant and powerful methodology to model and forecast mortality based on a simple formula for the logged death rates:

$$\begin{array}{c}\hfill ln(m_{x,t})={\alpha }_x+{\beta }_x{\kappa }_t+{\epsilon }_{x,t},\end{array}$$ 2

where ${\alpha }_x$ describes the average shape of the age profile, ${\beta }_x$ the rate of mortality improvement at age x, and ${\kappa }_t$ the general level of mortality at time t. The ${\epsilon }_{x,t}$ are error terms that reflect residual age-specific historical influences not captured by the model. To ensure model identification, the parameters are commonly subjected to the following constraints:

$$\begin{array}{c}\hfill \sum _x{\beta }_x=1\mathrm{and}\sum _t{\kappa }_t=0.\end{array}$$ 3

The original LC model is estimated by minimizing the residual sum of squares:

$$\begin{array}{c}\hfill \sum _{x,t}{\left(ln,(m_{x,t}),-,{\alpha }_x,-,{\beta }_x,,{\kappa }_t\right)}^2.\end{array}$$ 4

In particular, the authors employ a singular value decomposition (SVD) method to find the least squares solution of (4); as such, the estimated ${\widehat{\alpha }}_x$ is the average of the observed $ln(m_{x,t})$, while ${\widehat{\beta }}_x$ and ${\widehat{\kappa }}_t$ are the first left- and right-singular vectors of the SVD of the matrix $ln(m_{x,t})-{\widehat{\alpha }}_x$. Furthermore, the parameter ${\widehat{\kappa }}_t$ is adjusted in a second-step estimation so that the resulting fitted deaths match the total number of deaths observed in the data at each year t.

An advantage of the LC model is that its time index ${\kappa }_t$ can be easily forecast since it is able to capture the linear decline of the observed mortality development: see the right panel of Fig. 2. In particular, forecasting in the LC model is performed by modeling and extrapolating fitted values of ${\kappa }_t$ by an autoregressive integrated moving average (ARIMA) process. A Random Walk with Drift (RWD) is used almost exclusively in applications with low-mortality populations, and in the following, we will adopt this approach. Finally, the extrapolated ${\kappa }_t$ and its uncertainty are combined with the previous estimations to forecast the future death rates and associated prediction intervals.

Fig. 2.

Estimated parameters from a smooth variant of the Lee–Carter. Swiss males, ages 0–100, years 1960–2018, time index $\mathit{\kappa }$ forecast up to 2050 by RWD with 80% prediction intervals. Left panel: $\mathit{\alpha }$, average shape of the age profile. Central panel: $\mathit{\beta }$, age-specific average rates of mortality decline. Right panel: $\mathit{\kappa }$, level of mortality at time t

With respect to the estimation procedure, the main drawback of the ordinary least-squares estimation via SVD is that the errors are assumed to be homoskedastic and normally distributed, which is a fairly unrealistic assumption for human mortality (Alho 2000). Indeed, the logarithm of the observed force of mortality is much more variable at older than at younger ages because of the much smaller number of deaths (Brouhns et al. 2002).

To overcome this limitation, Brouhns et al. (2002) embed the LC model in a Poisson setting. The logarithm of the force of mortality in (1) can then be written as follows:

$$\begin{array}{c}\hfill ln({\mu }_{x,t})={\alpha }_x+{\beta }_x{\kappa }_t\end{array}$$ 5

or in matrix notation:

$$\begin{array}{c}\hfill ln\mathit{M}=\mathit{\alpha }{\mathbf{1}}_n^{\prime }+\mathit{\beta }{\mathit{\kappa }}^{\prime }\end{array}$$ 6

where $\mathit{M}=({\mu }_{x,t})$ is the matrix of the underlying force of mortality, $\mathit{\alpha }$, $\mathit{\beta }$ and $\mathit{\kappa }$ denote the LC parameter vectors and ${\mathbf{1}}_n$ is a $n\times 1$ matrix of 1s. Estimates of the parameters can be achieved by maximizing the associated log-likelihood function:

$$\begin{array}{c}\hfill ln\mathcal{L}\left(\mathit{\alpha },,,\mathit{\beta },,,\mathit{\kappa },|,,\mathit{Y},,,\mathit{E}\right)\propto \sum _{x,t}\left[y_{x,t},,ln,\left({\mu }_{x,t}\right),-,e_{x,t},,{\mu }_{x,t}\right].\end{array}$$ 7

The model parameters are subject to the same constraints as in (3) and the log-likelihood in (7) is maximized by a uni-dimensional iterative Newton–Raphson method. The advantage of this assumption is that the error terms in the model now follow a nonadditive heteroskedastic error structure, which is more appropriate for modeling human mortality. Moreover, maximum-likelihood estimation does not require any second-step adjustment of $\widehat{\mathit{\kappa }}$, as the fitted number of deaths already matches the observed deaths (Brouhns et al. 2002, p. 379).

Regardless of the estimation procedure, irregular age-patterns in future mortality have been observed when the LC model has been used, and increasing loss of smoothness over ages is a well known feature of this model (Girosi and King 2007, 2008). This is mainly due to the $\mathit{\beta }$, which often exhibits a fluctuating pattern that results in irregular projected life tables. Instead of smoothing the estimated $\mathit{\beta }$ without modifying $\mathit{\alpha }$ and $\mathit{\kappa }$, Delwarde et al. (2007) proposed a penalized likelihood approach to obtain smooth $\mathit{\beta }$ within the estimation procedure. In addition, we follow Currie (2013) and smooth the parameter vector $\mathit{\alpha }$ since, especially in small populations, the simple average mortality age-pattern might present variable behavior that can contribute to produce unsmooth future mortality patterns. In contrast, the time index $\mathit{\kappa }$ is free to vary over the years to better capture mortality fluctuations and provide an appropriate measurement of the uncertainty when forecasting is involved.

Smooth LC estimates could be achieved by adding a penalty term in the log-likelihood (7) which penalizes differences between adjacent $\mathit{\alpha }$ and $\mathit{\beta }$:

$$\begin{array}{c}\hfill ln{\mathcal{L}}^P\left(\mathit{\alpha },,,\mathit{\beta },,,\mathit{\kappa },|,,\mathit{Y},,,\mathit{E}\right)=ln\mathcal{L}\left(\mathit{\alpha },,,\mathit{\beta },,,\mathit{\kappa },|,,\mathit{Y},,,\mathit{E}\right)-\frac{1}{2}{\lambda }_{\alpha }{\mathit{\alpha }}^{\prime }{\mathit{D}}^{\prime }\mathit{D}\mathit{\alpha }-\frac{1}{2}{\lambda }_{\beta }{\mathit{\beta }}^{\prime }{\mathit{D}}^{\prime }\mathit{D}\mathit{\beta }\end{array}$$ 8

where the smoothing parameters ${\lambda }_{\alpha }$ and ${\lambda }_{\beta }$ control the amount of smoothness in the vector $\mathit{\alpha }$ and $\mathit{\beta }$ and $\mathit{D}$ is the second-order difference $m\times (m-2)$ matrix:

$$\begin{array}{c}\hfill \mathit{D}=\left[\begin{array}{cccccc}1\hfill & -2\hfill & 1\hfill & 0\hfill & 0\hfill & \cdots \hfill \\ 0\hfill & 1\hfill & -2\hfill & 1\hfill & 0\hfill & \cdots \hfill \\ 0\hfill & 0\hfill & 1\hfill & -2\hfill & 1\hfill & \cdots \hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill & \ddots \hfill \end{array},\right].\end{array}$$ 9

The order of difference in $\mathit{D}$ does not play a relevant role in this context. Unless specified otherwise, in the following we use second-order differences. Moreover, we can use the same $\mathit{D}$ matrix for both parameter vectors since they both act on the age dimension.

In other words, the larger the smoothing parameters ${\lambda }_{\alpha }$ and ${\lambda }_{\beta }$, the smoother the estimates of $\mathit{\alpha }$ and $\mathit{\beta }$, respectively. Consequently fitted rates will show smoother patterns. Alternatively small $\lambda$s lead to estimates close to the original LC model. Here, we optimized both smoothing parameters by minimizing the Bayesian information criterion (BIC, Schwarz 1978).

Gray lines in Fig. 2 present estimated parameters from this smooth variant of the Lee–Carter model for Swiss males. The vector $\mathit{\alpha }$ shows the average mortality age-pattern, and the accident-hump is evident for early adult ages. This average trajectory is modulated over age and time by the parameters $\mathit{\beta }$ and $\mathit{\kappa }$, respectively. A general linear time-trend is visible in the right panel and the average mortality improvement described by ${\beta }_x$ presents large changes in younger ages and about age 70.

The time index is then extrapolated using a RWD and associated prediction intervals can be computed (gray areas in the right panel of Fig. 2). From estimated and future values of ${\kappa }_t$ as well as estimated $\mathit{\alpha }$ and $\mathit{\beta }$, the whole mortality development over both age and time is computed.

Figure 3 shows estimated and forecast death rates on a log scale for Swiss males. Whereas the smooth Lee–Carter is able to capture the general mortality improvement, future age-patterns still show fluctuating behavior. Moreover, the fixed age-specific average rates of mortality improvement, described by the vector $\mathit{\beta }$, lead to unreasonable time-trends, i.e., all age groups show a constant decline in mortality that has not been observed in the past decades. These features are ultimately translated into future age-patterns that produce a poor description of past trends; see especially the fitted mortality for ages 0 and 30.

Fig. 3.

Actual, estimated and forecast death rates by a smooth variant of the Lee–Carter over age and time on a log scale for Swiss males. Ages 0–100, fitted years 1960–2018, forecast years 2019–2050. Left panel: rates over age for all years. Selected years are depicted by thicker lines. Right panel: selected ages over years

The Three-Component smooth Lee–Carter model

Outcomes from the classic LC model have shown that the assumption of a single shape of mortality improvement over age described by $\mathit{\beta }$ can often be wrong for mortality data. In other words, within a plain Lee–Carter framework, we assume that age-specific improvements are fixed throughout the observed period as well as in the forecast horizon. Available approaches attempted to modify the shape in $\mathit{\beta }$ accounting for the projected life expectancy at birth and assuming relationships between estimated values at different ages (Li et al. 2013; Ševčíková et al. 2016; Pascariu et al. 2020). Here, we opt for a different approach which does not require prior knowledge of future life expectancy and is not based on a subjective choice of the model parameters. Moreover, it allows us to simultaneously decompose and forecast the whole of childhood, early-adulthood and senescent mortality.

In formulas, let ${\mathit{\gamma }}_C$, ${\mathit{\gamma }}_A$ and ${\mathit{\gamma }}_S$ denote the vectors representing childhood, early-adulthood and senescent mortality, respectively. The length of these vectors is equal to the product of the number of ages and years in the original data, $mn$. In a 3C-sLC model, each component is defined as a Lee–Carter model:

$$\begin{array}{c}\hfill ln{\mathit{\gamma }}_k=\mathtt{vec}({\mathit{\alpha }}_k{\mathbf{1}}_n^{\prime }+{\mathit{\beta }}_k{\mathit{\kappa }}_k^{\prime })\text{for}k=C,A,S.\end{array}$$ 10

Here, the $\mathtt{vec}$ operator stacks the columns of a matrix in column order on top of each other. We note that with this definition the age suffix varies faster than the year suffix in all ${\mathit{\gamma }}_k$.

Let $\mathit{A}=({\alpha }_{x,k})=[{\mathit{\alpha }}_C:{\mathit{\alpha }}_A:{\mathit{\alpha }}_S]$ denote a $m\times 3$ matrix combining the average mortality age-patterns for each component. Similarly, we build matrices with ${\mathit{\beta }}_k$ and ${\mathit{\kappa }}_k$. The overall underlying force of mortality can thus be written as follows:

$$\begin{array}{c}\hfill \mathit{M}=({\mu }_{x,t})=\sum _{k}exp\left({\alpha }_{x,k},+,{\beta }_{x,k},,{\kappa }_{t,k}\right).\end{array}$$ 11

This formulation could recall LC variants with additional set of principal components as in Renshaw and Haberman (2003b) and Hyndman and Ullah (2007). However, these approaches are substantially different since they aim to approximate the matrix of centered log-mortality with additive bi-linear terms:

$$\begin{array}{c}\hfill \mathit{M}=({\mu }_{x,t})=exp\left({\alpha }_x,+,\sum _k,{\beta }_x^{(k)},,{\kappa }_t^{(k)}\right).\end{array}$$

With our approach, three sets of $\mathit{\alpha }$, $\mathit{\beta }$, and $\mathit{\kappa }$ will be estimated. The advantages of this are twofold: on the one hand, we will have component-specific mortality improvement patterns which will mitigate the LC model drawback of a fixed $\mathit{\beta }$ and, on the other, we will be able to extract forecast mortality trajectories independently for childhood, middle-age and senescent components. These important enhancements come at a price: the 3C-sLC model is extremely flexible and highly nonlinear. Consequently, prior knowledge of population-specific mortality development would be required to define the necessary amount of flexibility.

As in the classic LC model, future mortality can be obtained by extrapolating estimated ${\mathit{\kappa }}_k$ by an ARIMA process and computing component-specific death rates by (10) using associated ${\mathit{\alpha }}_k$ and ${\mathit{\beta }}_k$. The sum of the three components provides the overall mortality. In the following, we opt to keep the original LC approach and apply a RWD to each time index. Alternative procedures such as the selection of ${\mathit{\kappa }}_k$-specific ARIMA models would simply add an additional layer of complexity without improving the final outcomes, and lead to a disproportionate increase of prediction intervals.

Instead of decomposing the whole mortality data and successively estimating independent LC models, we opt for a simultaneous estimation of the three components. For the purpose of regression, we arrange the matrix of deaths by column order into a vector, that is, $\mathit{y}=\mathtt{vec}(\mathit{Y})$. Likewise we arrange the matrix of exposures $\mathit{e}=\mathtt{vec}(\mathit{E})$. We then express the expected values of the Poisson distribution in (1) as a sum of the three components:

$$\begin{array}{c}\hfill \mathit{y}\sim \mathcal{P}(\mathit{e}\ast \mathit{\mu }=\mathit{C}\mathit{\gamma }),\end{array}$$ 12

where $\mathit{\gamma }=\mathtt{vec}({\mathit{\gamma }}_C:{\mathit{\gamma }}_A:{\mathit{\gamma }}_S)$ and $\ast$ denotes the element-wise product. The matrix $\mathit{C}$ additively combines the ${\mathit{\gamma }}_k$ and also incorporates the exposures:

$$\begin{array}{c}\hfill \mathit{C}={\mathbf{1}}_{1,3}\otimes \mathtt{diag}(\mathit{e}).\end{array}$$ 13

where ${\mathbf{1}}_{1,3}$ is a $1\times 3$ matrix of ones, $\mathtt{diag}(\mathit{e})$ is the diagonal matrix of the exposure population and $\otimes$ denotes the Kronecker product (Harville 1997, p. 333).

Within this system, the force of mortality in each component is simultaneously multiplied by the associated exposures and summed up to obtain the final expected value. We thus hold the Poisson assumption, each component ${\mathit{\gamma }}_k$ incorporates the associated force of mortality as described in (10) and the sum of the three components is done automatically by matrix multiplication.

Estimation by a Composite Link Model Approach

The 3C-sLC model in (12) can be viewed as a Composite Link Model (CLM). Introduced by Thompson and Baker (1981) as an extension of the generalized linear model (GLM; McCullagh and Nelder , 1989), CLM is an elegant framework for modeling expected values which are described as a sum of components. A demographic application of this class of models is given in Remund et al. (2018). In their paper, the authors employed the Sum of Smooth Exponentials model (SSE, Camarda et al. 2016) for decomposing young adult excess mortality by cause of death. Specifically, the SSE model can be seen as a generalization of the 3C-sLC where decomposition is achieved in a completely nonparametric setting. However, none of these approaches deal with mortality forecasting which could be obtained by describing each component with an LC structure as in (10).

Unlike the SSE, each component of the 3C-sLC model is not linear with respect to all unknown parameters. Consequently, estimation cannot be carried out directly by a penalized Iterative re-Weighted Least Squares (IWLS) algorithm as proposed by Eilers (2007). Nevertheless, we linearize the system of equations with respect to each series of parameters and iteratively solve the associated IWLS. Similarly, Currie (2013) proposed estimating a classic LC model as a sequence of two constrained GLMs.

In general, we need to solve the following system of equations (Eilers 2007, p. 244):

$$\begin{array}{c}\hfill ({\breve{\mathit{X}}}^{\prime }\tilde{\mathit{W}}\breve{\mathit{X}}+\mathit{P})\mathit{\theta }={\breve{\mathit{X}}}^{\prime }(\mathit{y}-\mathit{\mu })+{\breve{\mathit{X}}}^{\prime }\tilde{\mathit{W}}\breve{\mathit{X}}\tilde{\mathit{\theta }},\end{array}$$ 14

where $\mathit{\theta }$ denotes the combined vector with one of the triplets of the LC parameters: ${\mathit{\theta }}^{\prime }={({\mathit{\alpha }}_C,{\mathit{\alpha }}_A,{\mathit{\alpha }}_S)}^{\prime }$, ${\mathit{\theta }}^{\prime }={({\mathit{\beta }}_C,{\mathit{\beta }}_A,{\mathit{\beta }}_S)}^{\prime }$ and ${\mathit{\theta }}^{\prime }={({\mathit{\kappa }}_C,{\mathit{\kappa }}_A,{\mathit{\kappa }}_S)}^{\prime }$. Moreover, we have that:

$$\begin{array}{c}\hfill \breve{\mathit{X}}={\mathit{W}}^{-1}{\mathit{C}}_{\mathit{\theta }}\mathbf{\Gamma }{\mathit{X}}_{\mathit{\theta }};\mathit{W}=\mathtt{diag}(\mathit{e}\ast \mathit{\mu });\mathbf{\Gamma }=\mathtt{diag}(\mathit{\gamma }).\end{array}$$ 15

A tilde, as in $\tilde{\mathit{\theta }}$, indicates the current approximation to the solution. Note that only composite and design matrices (${\mathit{C}}_{\mathit{\theta }}$ and ${\mathit{X}}_{\mathit{\theta }}$) change with respect to the series of parameters we currently estimate.

We start by fixing all triplets of parameters ${\mathit{\kappa }}_k$ and ${\mathit{\beta }}_k$ and estimate ${\mathit{\theta }}^{\prime }={({\mathit{\alpha }}_C,{\mathit{\alpha }}_A,{\mathit{\alpha }}_S)}^{\prime }$. Composite and design matrices for solving (14) are thus given by:

$$\begin{array}{cc}\hfill {\mathit{C}}_{\mathit{\alpha }}=& [{\mathit{u}}_C({\mathbf{1}}_n\otimes {\mathit{I}}_m):{\mathit{u}}_A({\mathbf{1}}_n\otimes {\mathit{I}}_m):{\mathit{u}}_S({\mathbf{1}}_n\otimes {\mathit{I}}_m)]\hfill \\ \hfill {\mathit{X}}_{\mathit{\alpha }}=& {\mathit{I}}_{3m}\hfill \end{array}$$ 16

where ${\mathit{u}}_k=\mathtt{diag}[\mathit{e}\mathtt{vec}(exp({\mathit{\beta }}_k{\mathit{\kappa }}_k^{\prime }))]$ for $k=C,A,S$ and ${\mathit{I}}_m$ is the identity matrix with m rows and columns.

Analogously, we then fix the other triplets of parameters and we solve (14) by changing the composite and design matrices:

$$\begin{array}{c}\hfill {\mathit{C}}_{\mathit{\beta }}={\mathit{C}}_{\mathit{\kappa }}=[{\mathit{u}}_C:{\mathit{u}}_A:{\mathit{u}}_S]\\ \hfill {\mathit{X}}_{\mathit{\beta }}=\mathtt{diag}[{\mathit{\kappa }}_C\otimes {\mathit{I}}_m,{\mathit{\kappa }}_A\otimes {\mathit{I}}_m,{\mathit{\kappa }}_S\otimes {\mathit{I}}_m]\\ \hfill {\mathit{X}}_{\mathit{\kappa }}=\mathtt{diag}[{\mathit{I}}_n\otimes {\mathit{\beta }}_C,{\mathit{I}}_n\otimes {\mathit{\beta }}_A,{\mathit{I}}_n\otimes {\mathit{\beta }}_S]\end{array}$$ 17

where ${\mathit{u}}_k=\mathtt{diag}[\mathit{e}\mathtt{vec}(exp({\mathit{\alpha }}_k{\mathbf{1}}_n^{\prime }))]$ for $k=C,A,S$.

Smoothness of the parameters is achieved via the penalty matrix $\mathit{P}$ in (14). As in the smooth LC variant in (8), smoothness is enforced only for triplets ${\mathit{\alpha }}_k$ and ${\mathit{\beta }}_k$. Time indexes ${\mathit{\kappa }}_k$ are free to change over time. In general, the penalty term takes a block diagonal structure as follows:

$$\begin{array}{c}\hfill \mathit{P}=\mathtt{diag}({\mathit{P}}_{{\mathit{\theta }}_C},{\mathit{P}}_{{\mathit{\theta }}_A},{\mathit{P}}_{{\mathit{\theta }}_S})\end{array}$$ 18

where ${\mathit{P}}_{{\mathit{\theta }}_k}={\lambda }_{{\mathit{\theta }}_k}{\mathit{D}}^{\prime }\mathit{D}$. Difference matrices $\mathit{D}$ are constructed as in (9) and are used to measure roughness in the parameter vectors. We choose second-order differences, except for ${\mathit{\alpha }}_A$, whose log-concave shape of the early-adult component suggests using $d=3$.

As in the smooth LC variant, the smoothing parameter ${\lambda }_{{\mathit{\theta }}_k}$ controls the roughness of the vector ${\mathit{\theta }}_k$. Given the complexity and the flexibility of the 3C-sLC model, information criteria (e.g., BIC) do not lead to reasonable outcomes in terms of component-specific patterns. We thus establish and suggest a rule of thumb which has worked for most of populations in the Human Mortality Database (2020). First, in order to reduce the computational burden, we assume that the smoothing parameter is the same within each triplet of parameters, i.e., ${\lambda }_{{\mathit{\alpha }}_C}={\lambda }_{{\mathit{\alpha }}_A}={\lambda }_{{\mathit{\alpha }}_S}$ and ${\lambda }_{{\mathit{\beta }}_C}={\lambda }_{{\mathit{\beta }}_A}={\lambda }_{{\mathit{\beta }}_S}$. However, we do not impose restrictions on ${\lambda }_{{\mathit{\theta }}_k}$ across different triplets, i.e., in general ${\lambda }_{{\mathit{\alpha }}_k}\ne {\lambda }_{{\mathit{\beta }}_k}$. Then, we start from really large smoothing parameters (e.g., ${10}^8$) and gradually reduce their values until the procedure provides reasonable outcomes without encountering convergence problems.

We believe that allowing users to tune the amount of smoothness, and consequently the effective dimension of the estimated 3C-sLC, is not a major drawback of approach. Users have full control on the flexibility of the model, and they can adapt the model to diverse situations modifying the mentioned procedure for selecting the smoothing parameters. In other words, we acknowledge that, at the current stage, we have not identified a general and objective rule for tuning smoothness. Selection of ${\lambda }_{{\mathit{\theta }}_k}$ largely depends upon several factors such as variability in the data, strong/weak evidence of the mortality components and user’s prior knowledge about mortality developments in a specific population.

In demography, age 0 is commonly treated differently, e.g., in classic life table construction (Chiang 1984). This age constitutes a discontinuity in the mortality trajectory and we incorporate this feature, allowing discontinuity in the childhood component for the parameters operating over ages: ${\mathit{\alpha }}_C$ and ${\mathit{\beta }}_C$. This feature is obtained by not penalizing both ${\alpha }_{1,C}$ and ${\beta }_{1,C}$.

Additional demographic knowledge is included in the model without enforcing a specific parametric structure of the parameter vectors. In particular, we impose the average shape of the age profile for early-adult (${\mathit{\alpha }}_A$) and senescent components (${\mathit{\alpha }}_S$) to be strictly log-concave and increasing with age, respectively. Asymmetric penalties as in Bollaerts et al. (2006) are necessary to enforce these shapes. In formulas penalty terms in (18) are modified when employing for ${\mathit{\alpha }}_A$ and ${\mathit{\alpha }}_S$:

$$\begin{array}{c}\hfill {\breve{\mathit{P}}}_{{\mathit{\alpha }}_A}={\mathit{P}}_{{\mathit{\alpha }}_A}+{\lambda }_s{\mathit{D}}_2^{\prime }{\tilde{\mathit{V}}}_{{\mathit{\alpha }}_A}{\mathit{D}}_2\text{and}{\breve{\mathit{P}}}_{{\mathit{\alpha }}_S}={\mathit{P}}_{{\mathit{\alpha }}_S}+{\lambda }_s{\mathit{D}}_1^{\prime }{\tilde{\mathit{V}}}_{{\mathit{\alpha }}_S}{\mathit{D}}_1\end{array}$$

where ${\mathit{D}}_1$ and ${\mathit{D}}_2$ forms first- and second-order differences of the coefficients in ${\mathit{\alpha }}_A$ and ${\mathit{\alpha }}_S$. Both matrix ${\tilde{\mathit{V}}}_{{\mathit{\alpha }}_A}$ and ${\tilde{\mathit{V}}}_{{\mathit{\alpha }}_A}$ are diagonal with diagonal elements equal to 1 when, respectively, ${\alpha }_{i,A}\le 2{\alpha }_{i-1,A}-{\alpha }_{i-2,A}$ and ${\alpha }_{i,S}\le {\alpha }_{i-1,S}$, and 0 otherwise. These penalties exert influence only when the shape constraints are violated, and the diagonal elements in ${\tilde{\mathit{V}}}_{{\mathit{\alpha }}_A}$ and ${\tilde{\mathit{V}}}_{{\mathit{\alpha }}_A}$ are computed iteratively in (14). The value of ${\lambda }_s$ regulates how strictly the shape constraints are enforced and we chose a really large value, ${\lambda }_s={10}^{12}$.

These specialized penalties have proven useful in other demographic applications (Camarda et al. 2016; Remund et al. 2018; Camarda 2019). Moreover, incorporating additional knowledge about early-adulthood and senescent components is necessary to ensure the identification of a complex model such as the 3C-sLC. However, these constraints may not be sufficient and concerns about identifiability remain, particularly with small smoothing parameters.

Finally, the choice of the starting values is important, but not as crucial as the nonlinearity of the 3C-sLC model would suggest. Initial values of ${\mathit{\beta }}_k$ and ${\mathit{\kappa }}_k$ are set to ${\mathbf{1}}_m/m$ and to a simple linear trends over time, respectively. As in Currie (2013), ${\mathit{\kappa }}_k$ must be centered (since ${\sum }_t{\kappa }_{t,k}=0,\forall k$) and then scaled (since ${\sum }_x{\beta }_{x,k}=1,\forall k$). Extra care is necessary for the component-specific average patterns. Inspired by Heligman and Pollard (1980) and Siler (1983), we suggest starting parameters that roughly describe human mortality components. We recommend a linear decreasing age-pattern for the childhood component (${\mathit{\alpha }}_C^{(0)}=-6-.55\mathit{x}$), a quadratic curve for the adulthood part (${\mathit{\alpha }}_A^{(0)}=-15+.7\mathit{x}-.013{\mathit{x}}^2$) and a linear increasing age-pattern for the senescence component (${\mathit{\alpha }}_S^{(0)}=-10+.1\mathit{x}$). Other options may be equally good, though the suggested initial values have worked well for all populations and smoothing parameters used in this study.

Inference and Computation

Point estimates are only part of the results produced by a forecasting method. Uncertainty about the future needs to be measured properly. Conventionally, LC prediction intervals are obtained by applying uncertainty in future values of the extrapolated vector $\mathit{\kappa }$: upper and lower bounds for $\mathit{\kappa }$ in ${\mathit{t}}_F$ (see gray area in right panel of Fig. 2) are combined with estimated $\mathit{\alpha }$ and $\mathit{\beta }$ to compute prediction intervals for future death rates. Alternatively, instead of focusing on the variability in the time-varying parameter, Brouhns et al. (2005) and Koissi et al. (2006) proposed incorporating variability from all LC parameters using a bootstrap approach.

Here, we combine both options to account for both sources of uncertainty, as previously suggested by Keilman and Pham (2006) and incorporated in recent forecasting approaches (see, e.g., Bergeron-Boucher et al. 2017; Basellini et al. 2020). Uncertainty in future mortality comes from variability in the estimated ${\mathit{\alpha }}_k$, ${\mathit{\beta }}_k$ and ${\mathit{\kappa }}_k$ as well as from variability in the ARIMA model when extrapolation is performed on time indexes ${\mathit{\kappa }}_k$.

In order to achieve this exhaustive measure of uncertainty, we first carry out a residual bootstrap procedure as in Koissi et al. (2006) for the LC framework, and Ouellette et al. (2012) and Camarda (2019) within a smoothing context. This allows us to simulate bootstrapped matrices of death counts ${\mathit{Y}}_{(b)}$. We replicate this step 50 times and for each new ${\mathit{Y}}_{(b)}$, together with the original matrix of exposure $\mathit{E}$, we estimate the 3C-sLC model. This procedure yields 50 new series of ${\mathit{\alpha }}_k$, ${\mathit{\beta }}_k$ and ${\mathit{\kappa }}_k$. Confidence intervals for the estimated parameters as well as for the fitted values during the past observed years can be obtained within this step.

We then apply a RWD model to each new vector ${\mathit{\kappa }}_k$. Instead of taking upper and lower bounds from the 50 estimated RWD models, we simulate 100 future ${\mathit{\kappa }}_k$ based on the fitted drift parameter. By combining the 50 estimated time indexes from the bootstrap procedure and the 100 extrapolated ${\mathit{\kappa }}_k$ from the RWD simulation step, we obtain 5000 new ${\mathit{\kappa }}_k$. From the distributions of future ${\mathit{\kappa }}_k$, we compute 5,000 matrices of future death rates for both overall and component-specific mortality. Empirical percentiles and confidence intervals can be extracted for death rates, as well as for any desirable summary measure of mortality, e.g., life expectancy at birth (cf. Sect. 3).

Finally, routines developed to model, decompose and forecast mortality data by a 3C-sLC model were implemented in R (R Development Core Team 2019) and are publicly available in an open source framework repository on this link https://osf.io/rgf8y/. Using only basic R packages and for a single population, fitting a 3C-sLC model takes about 75 seconds, and it takes about 35 minutes to run the whole procedure to obtain confidence and prediction intervals for future mortality trends (portable personal computer, Intel i5-6300U processor, 2.4 GHz $\times$ 4 and 6 Gbytes random-access memory). With regard to computer data storage, a file of about 18 MB can store the R workspace with all outcomes for a single population.

Results

In this Section, we present the outcomes obtained from the proposed 3C-sLC model. First, we start by showing the estimated parameters and the age-specific results for the Swiss male population analyzed in Sect. 2. Next, we present fitted and forecast summary measures of mortality for the four populations investigated in this article. Finally, we perform an exercise that demonstrates one demographic advantage of the 3C-sLC model: we estimate the potential gains derived from the hypothetical scenario of eliminating one or more mortality components from the age-pattern of mortality. In Appendix A, we analyze more populations and compare the 3C-sLC with alternative forecasting models in terms of both goodness-of-fit and out-of-sample forecasting performance.

Figure 4 shows the estimated parameters of the 3C-sLC for Swiss males. Here, we compare these results with the smooth variant of the Lee–Carter (LC) model (Delwarde et al. 2007), as previously shown in Fig. 2. In the left panel, the average shape of mortality over age, captured by the parameter $\mathit{\alpha }$ in the LC model, is decomposed into component-specific average patterns. The three ${\mathit{\alpha }}_k$ display the expected shapes over age: decreasing, log-concave and increasing mortality with age for the childhood, early-adulthood and senescent components, respectively. In the central and right panels, component-specific rates of mortality improvements ${\mathit{\beta }}_k$ and time indexes ${\mathit{\kappa }}_k$ are presented, together with estimates from the LC model. Here, it is interesting to observe the similarity between the LC estimates and the senescent ones. Furthermore, the childhood and adulthood parameters capture additional dimensions of mortality developments that are overlooked and conflated in the LC framework.

Fig. 4.

Estimated parameters from 3C-sLC model and a smooth variant of the Lee–Carter. Swiss males, ages 0–100, years 1960–2018, time index(es) $\mathit{\kappa }$ forecast up to 2050 by RWD with 80% prediction intervals. Left panel: $\mathit{\alpha }$, average shape of the age profile. Central panel: $\mathit{\beta }$, age-specific average rates of mortality decline. Right panel: $\mathit{\kappa }$, level of mortality at time t. For 3C-sLC parameters, 80% confidence intervals are also computed for the observed years

Combining estimates and forecasts of the 3C-sLC parameters allows us to derive death rates for all ages and years of interest. Figure 5 shows estimated and forecast 3C-sLC death rates on a log scale for Swiss males. This figure can be directly compared with Fig. 3, where the respective results are shown for the smooth LC model. Several observations can be drawn from comparing the two figures. In the left panel, fitted and forecast mortality age-profiles of the 3C-sLC seem more plausible than LC ones, lacking the variable behavior and the constant age-specific rate of mortality decline of the smooth LC model. In the right panel, the superior fit of the 3C-sLC is evident, particularly at younger ages; this results from relaxing the assumption of a constant $\mathit{\beta }$ in the LC framework. Age-specific mortality improvements are more flexible and produced by the combination of the component-specific parameters. The increased flexibility further translates into wider prediction intervals, which capture the uncertainty behind mortality developments at different ages.

Fig. 5.

Actual, estimated and forecast death rates by 3C-sLC over age and time on a log scale for Swiss males. Ages 0–100, fitted years 1960–2018, forecast years 2019–2050. Left panel: rates over age for all years. Selected years are depicted by thicker lines. Right panel: selected ages over years

Figure 6 displays the fitted and forecast childhood, early-adulthood and senescent components on a log scale for Swiss males. The shapes observed in the ${\mathit{\alpha }}_k$ parameters are directly carried into the mortality components. Furthermore, the central panel clearly shows the increase in early-adulthood mortality during the years 1980–1990s, related to the strong HIV epidemic that hit this population at that time (Csete and Grob 2012).

Fig. 6.

Estimated and forecast childhood, adulthood and senescent components by 3C-sLC over age on a log scale for Swiss males. Ages 0–100, fitted years 1960–2018, forecast years 2019–2050. Axes of the left and central panels have been truncated for better displaying the mortality components

Next, we show the results of applying the 3C-sLC and smooth LC models to the four populations introduced in Sect. 2, namely females and males in Switzerland (CHE) and the USA. We present outcomes is terms of two standard and complementary summary measures of mortality. We use life expectancy at birth $e_0$ as a traditional measure of longevity and population health (Preston et al. 2001), and the average number of life years lost at birth $e_0^{†}$ (Vaupel and Canudas-Romo 2003) as a measure of lifespan disparity. In formulas:

$$\begin{array}{c}\hfill e_0=\frac{{\sum }_0^{\omega }{\ell }_x}{{\ell }_0}\text{and}{e}_0^{†}=\frac{{\sum }_0^{\omega }{e}_x{d}_x}{{\ell }_0},\end{array}$$ 19

where ${\ell }_x$ denotes the life-table probability of surviving from birth to age x, ${\ell }_0$ is the life-table radix, and $\omega$ is the highest age attained in the life table. The vectors $e_x$ and $d_x$ denote the remaining life expectancy and the age-at-death distribution at age x, respectively. Whereas life expectancy at birth has a long tradition in measuring longevity, $e_0^{†}$ has been recently employed to assess the degree of lifespan inequality within and between populations (see, e.g., Shkolnikov et al. 2011; Vaupel et al. 2011; Aburto and van Raalte 2018), and it has also been used to evaluate mortality forecasts (Bohk-Ewald et al. 2017; Camarda 2019).

Figure 7 shows the actual $e_0$ and $e_0^{†}$ for the four populations, together with the estimated and forecast values with 80% prediction intervals for the smooth LC and 3C-sLC models. The recent flattening in $e_0$ as well as the increase in $e_0^{†}$ in the USA populations, mostly caused by the stagnating decline in cardiovascular disease mortality (Mehta et al. 2020) and the increase in drug overdoses and suicides (Barbieri 2019), is evident from the figure. The wider prediction intervals of the 3C-sLC death rates (cf. Fig. 5) translate into slightly larger intervals also for the summary mortality measures. Moreover, it is interesting to observe that forecast $e_0$ from both models are very similar, whereas forecast $e_0^{†}$ differ: specifically, the 3C-sLC predicts slower mortality compression (i.e., higher lifespan disparity) than the smooth LC. Finally, the figure shows that the models successfully capture the observed trends in $e_0$, while they are less successful for $e_0^{†}$: nevertheless, the 3C-sLC performs better than the smooth LC, particularly for the Swiss populations.

Fig. 7.

Actual, estimated and forecast life expectancy at birth ($e_0$, left panels) and lifespan disparity ($e_0^{†}$, right panels) by smooth Lee–Carter (red lines and shades) and 3C-sLC (dark green lines and shades) with 80% prediction intervals for females and males in Switzerland (CHE) and the USA. Ages 0–100, fitted years 1960–2018, forecast years 2019–2050

In order to emphasize the uniqueness of the 3C-sLC model in terms of possible outcomes, we evaluate the potential gains resulting from the hypothetical scenarios of removing one or two mortality components from the age-pattern of mortality. Specifically, we estimate the potential gains in $e_0$ and reductions in $e_0^{†}$ derived from the elimination of: (i) the accident hump alone, and (ii) the accident hump together with the childhood component. Within the 3C-sLC, these scenarios can be readily implemented by removing corresponding components (cf. Fig. 6) from the overall mortality pattern. Figure 8 shows the estimated and forecast 3C-sLC summary measures of mortality for males (which have a more pronounced accident hump) in Switzerland and the USA, as well as the measures computed in the two hypothetical scenarios. Removing only the accident hump would result in an average increase over the fitted periods of 0.76 and 0.65 years in $e_0$ for males in Switzerland and the USA, respectively, and in an average reduction of 0.59 and 0.48 years in $e_0^{†}$. If the childhood component were eliminated too, gains in $e_0$ would rise to 1.71 and 1.78 years in the two populations, respectively, and reductions in $e_0^{†}$ would amount to 1.41 and 1.42 years.

Fig. 8.

Estimated and forecast life expectancy at birth ($e_0$, left panels) and lifespan disparity ($e_0^{†}$, right panels): (i) 3C-sLC (green lines), (ii) 3C-sLC removing the early-adulthood component (blue lines), and (iii) 3C-sLC removing the childhood and early-adulthood components (purple lines). The gray inset plots zoom in the forecast area and further show 80% prediction intervals. Males in Switzerland (CHE) and the USA, ages 0–100, fitted years 1960–2018, forecast years 2019-2050

Moreover, the 3C-sLC allows us to decompose summary measures in future years. For instance, we forecast USA males overall life expectancy in 2050 to reach 80.65 years, with a 80% prediction interval of [79.48, 81.76]. The mean value of $e_0$ would increase to 81.29 if we removed only the early-adulthood component, and to 81.55 if childhood was excluded too. Smaller reductions are forecast for Swiss males: total life expectancy in 2050 is projected to 86.15 years [84.89, 87.31], which would increase to 86.37 years if the accident hump was removed, and to 86.53 if both components were theoretically eradicated.

Discussion

Models to predict the future course of mortality of rather large populations have flourished during the most recent decades, stimulated by the influential contribution of Lee and Carter (1992). Although the number of available approaches to forecast mortality keeps increasing by the day, the Lee–Carter (LC) model arguably remains the most widely known and adopted methodology for this purpose. At the root of this success lie two important factors, namely power and simplicity, which have made LC the benchmark approach in mortality forecasting.

Despite its almost universal recognition and adoption, the original LC model suffers from some shortcomings (cf. Sect. 1). Several extensions of the model have been proposed to mitigate its principal limitations (e.g., Brouhns et al. 2002; Lee and Miller 2001; Booth et al. 2002; Li et al. 2013; Renshaw and Haberman 2006). However, no attempts have been made to overcome all the recognized issues simultaneously.

In this article, we have introduced a novel generalization of the LC model that tackles several of its main limitations at once. Starting from the hypothesis that human mortality can be conceived as the sum of three independent components (Thiele 1871), we propose a methodology in which mortality is simultaneously smoothed, decomposed and forecast within an LC framework. As such, we call our approach “Three-Component smooth Lee–Carter” (3C-sLC) model.

The 3C-sLC is embedded in a Poisson setting, which provides a reasonable assumption for the underlying mortality process. Smoothness in the outcomes is enforced within the estimation of the 3C-sLC parameters by employing a penalized likelihood approach. Overall mortality developments are described by the combination of three mortality components: childhood, early-adulthood and senescent mortality. Each component is modeled and forecast with a smooth version of the LC model. As such, mortality developments are more flexible than in the original LC model, because they do not depend on the rather unreasonable assumption of a fixed rate of age-specific mortality improvement.

This large flexibility prompts identification issues especially when smoothing parameters are relatively small and/or one of the component is not particularly pronounced in the overall mortality age-pattern, e.g., early-adult component among female populations. It is noteworthy that this limit is intrinsic in the nonlinearity of the 3C-sLC model and visual inspection of the outcomes is a crucial tool for identifying meaningless estimated mortality components. In other words, an extremely flexible model, such the 3C-sLC, cannot be used as a mere black-box and a critical view of the estimated components is a paramount step before any further forecast.

We have shown the results of fitting and forecasting mortality with the 3C-sLC model in four populations, namely females and males in Switzerland and the USA. We have compared the 3C-sLC outcomes with those of the improved smooth version of the LC model (Delwarde et al. 2007). In addition, a more extensive comparative study that includes an out-of-sample forecast exercise is provided in Appendix A. It should be noted that all these comparisons could have been made only at the overall mortality level: all LC variants as well as other existing forecasting models do not allow one to simultaneously forecast and decompose mortality into meaningful components. For the same reasons, the additional comparisons and out-of-sample exercises presented in Appendix A assess performances of the 3C-sLC only for one of its features, underrating its great explanatory value. A relevant exception is given by the 3C-STAD model (Basellini and Camarda 2020) that, in a rather different setting, also allows to decompose and forecast mortality.

The 3C-sLC forecast rates are characterized by wider prediction intervals, which result from the enhanced flexibility of the model. This is a potential advantage of the 3C-sLC model, as LC prediction intervals have been criticized for being too narrow (Alho 1992). Although the 3C-sLC generally fits well the observed mortality data, alternative approaches can be preferred by the BIC criterion, especially for the female populations (see Appendix A). A potential explanation for this is that the accident hump is not a relevant feature of the mortality pattern for females analyzed in this article. In Appendix B, we proposed a version of the 3C-sLC in which we model and forecast female mortality using only two components (i.e., childhood and senescence).

We have further computed and compared two complementary summary measures of mortality: life expectancy at birth ($e_0$) and lifespan disparity at birth ($e_0^{†}$). The two measures provide information on the longevity and lifespan inequality of the population. Both smooth LC and 3C-sLC models are characterized by rather similar fitted and forecast values of $e_0$. However, difference between the two models clearly emerges from $e_0^{†}$. The 3C-sLC better captures observed trends in lifespan inequality, and it forecasts slower mortality compression than the smooth LC model. This is another advantage of the 3C-sLC, since LC forecasts have been shown to produce a (possibly) too fast mortality compression (see, e.g., Bardoutsos et al. 2018; Basellini and Camarda 2019). In terms of prediction intervals, the widening observed in 3C-sLC rates only partially carries forward to forecast $e_0$ and $e_0^{†}$; the reason is that larger 3C-sLC intervals occur at younger ages especially, whose influence on summary mortality measures is rather limited.

Comparing the two models’ estimated parameters (cf. Fig. 4) provides two interesting points for discussion. First, the 3C-sLC estimates of the senescent component closely mirror those of the smooth LC model (with the exception of the $\widehat{\mathit{\alpha }}$ values at young and early-adult ages). This hints at the fact that the LC model is driven by the senescent component of mortality, which carries most weight in terms of death counts. Within the standard (or improved) LC framework, mortality developments of childhood and early-adulthood components are thus conflated with senescent ones, resulting in a loss of fitting accuracy. The second point worthy of discussion concerns the variability of the 3C-sLC model. Figure 4 shows that confidence intervals for the ${\mathit{\alpha }}_k$ estimates are very narrow. Hence, uncertainty in the 3C-sLC model depends on the variability of the ${\mathit{\beta }}_k$ and ${\mathit{\kappa }}_k$ estimates.

The decomposition of mortality into its three independent components has received considerable interest during recent decades (cf. Sect. 1). However, only a few methodologies have been proposed to take into account the three components when projecting mortality. The well-established model of Heligman and Pollard (1980) has only been employed by Forfar and Smith (1987) in mortality projections due to the high number of parameters to forecast. Moreover, Bardoutsos et al. (2018) employed the ten-parameter CODE model (de Beer and Janssen 2016) to forecast the full age range of mortality. The authors find that accounting for the compression and delay (i.e., shift) of mortality improves point forecast accuracy with respect to the LC model when the modal age at death increases linearly. Finally, Basellini and Camarda (2020) proposed the 3C-STAD model to analyze and forecast component-specific age-at-death distributions derived from a nonparametric decomposition of the mortality pattern. The authors show that 3C-STAD projections for high-longevity populations are more accurate in terms of point forecasts and prediction intervals than the LC model and its variants.

The mortality decomposition provides interesting demographic insights into fitted and forecast mortality schedules. The increased quantity of information on component-specific patterns allows one to obtain a more thorough overview on mortality developments. Furthermore, hypothetical exercises can readily be performed within this framework. For example, it is possible to estimate the potential gains in $e_0$ and reductions in $e_0^{†}$ resulting from the elimination of one or more mortality components. We have performed this exercise for the male populations analyzed in this article: for the observed period, we estimate an average 1.71 and 1.78 additional years of longevity in Switzerland and the USA, respectively, and an average 1.41 and 1.42 reduced years of lifespan disparity by theoretically eradicating the childhood and early-adulthood components. We present similar analysis for the forecast years and in Appendix A these outcomes are given for 10 populations too.

Two additional findings related to these exercises deserve mention. Firstly, Fig. 8 shows that the childhood and early-adulthood components have a significant effect on $e_0$ and $e_0^{†}$ for both male populations during the fitted years only. In forecast years, the effect of the two components varies by population: in Switzerland, the two forecast scenarios converge to the baseline mortality pattern, while in the USA differences between the scenarios do not disappear. Forecast childhood and early-adulthood components thus seem to maintain a significant role only in the USA. Secondly, the bottom panels of Fig. 8 highlight that lifespan disparity fluctuations are driven by the childhood and early-adulthood components: removing them from the age-pattern of mortality smooths the trends of $e_0^{†}$ in both populations.

Conclusions

The proposed 3C-sLC aims to provide demographers and actuaries with a new extension of the LC model that overcomes several of its main drawbacks while maintaining the factors that underpin its success. Conceptually, the 3C-sLC can be thought of as the combination of three LC models, each one targeted to model and forecast the childhood, early-adulthood and senescent components of mortality. We thus keep the simplicity of the LC framework, namely the interpretability of the three set of parameters ${\alpha }_x$, ${\beta }_x$ and ${\kappa }_t$ (one triplet for each mortality component). Furthermore, forecasting is performed as in the original LC model, without introducing additional layers of complexity. Although the formulas developed to fit the 3C-sLC are somewhat more elaborate than the original SVD decomposition suggested by Lee and Carter (1992), we provide the routines needed to fit and forecast mortality with our proposed model in the Supplementary Material to this article. Our hope is that the 3C-sLC could be employed by researchers and statistical agencies that currently use the LC model in an effort to obtain more desirable outcomes within the same LC framework.

Appendix A Comparison in Terms of Methods and Populations

This appendix will be devoted to a broader comparison among different mortality forecasting models and populations. We will asses performance of the proposed Three-Component smooth Lee–Carter (3C-sLC) model against several competitors by means of goodness-of-fit assessments and out-of-sample forecast exercises.

Specifically, we compare the proposed 3C-sLC model with other four alternative forecasting methods:

• the smooth Lee–Carter variant by Delwarde et al. (2007): sLC

• the functional data approach of the LC by Hyndman and Ullah (2007): HU

• the CP-splines introduced by Camarda (2019): CP-S

• the three-component segmented transformation age-at-death distributions proposed by Basellini and Camarda (2020): 3C-STAD

We selected these methods since they represent a wide range of approach in modeling and forecasting mortality, they all model mortality in a single population by single year of age, and routines for estimating these models are freely available (Hyndman et al. 2019; Camarda 2019; Basellini and Camarda 2020).

Nevertheless, it is noteworthy that these approaches have rather different modeling features, rationales and assumptions. Whereas all of them yield future overall mortality patterns, the proposed 3C-sLC is the only (with the exception of the 3C-STAD model) that can provide summary measures as well as age-specific deaths rates over time decomposed in a demographic meaningful manner.

Concerning the data, we used deaths and exposures for five countries: USA, Switzerland (CHE), France (FRA), England and Wales (ENW), and Australia (AUS). We assessed all methods on both males (m) and females (f).

For illustrative purposes, Fig. 9 shows the Poisson deviance residuals of all 5 models for Swiss males. It displays how each model is able to describe observed mortality data over age and time. A good model should show in this plot values about zero (i.e., perfect fit) and without any specific pattern. It is clear that Hyndman–Ullah model and the CP-splines clearly outperform their competitors. However, the 3C-sLC displays smaller residuals than the smooth LC and 3C-STAD model, especially at childhood and early-adult ages. The red and blue clouds (corresponding to model misfit) at ages 20–40 are especially evident for the smooth LC, and they are considerably reduced in the 3C-sLC model.

Fig. 9.

Poisson deviance residuals of all models for Swiss males. Ages 0–100, fitted years 1960–2018

As for the case of Swiss males, we model all other populations over the period 1960–2018 and we forecast up to 2050. The first three columns in Table 1 present outcomes in terms of model accuracy to observed data. Goodness of fit is measured by deviance (Dev, McCullagh and Nelder 1989), and it is a summary measure for all populations of the residuals shown in Fig. 9 for Swiss males only. The effective dimension (ED) gives a measure of the complexity of the model and the Bayesian information criterion (BIC) balances these two statistics allowing us to evaluate precision in describing the data. With this table, we also aim to emphasize differences in terms of model rigidity. When comparing the ED of the models, it is evident that only 3C-sLC, sLC and CP-S approaches let the data determine the complexity of the estimated model, while the HU and 3C-STAD employ a fixed number of parameters.

Table 1.

Numerical outcomes from the estimated models as well as for the out-of-sample forecast exercise

		Observed			Out-of-sample: 1960–2008 $\to$ 2009–2018
		1960–2018			$e_0$		$e_0^{†}$		$ln\mathit{M}$
		Dev	ED	BIC	RMSE	ME	RMSE	ME	RMSE	ME
USAm	3C-sLC (8,5)	198960	325	201783	0.370	0.040	0.450	0.315	0.124	− 0.018
	sLC	197575	225	199531	0.445	− 0.164	0.753	0.669	0.132	0.007
	HU	31343	1049	40462	0.416	− 0.143	0.657	0.520	0.108	0.012
	CP-S	34481	298	37069	0.411	− 0.154	0.520	0.394	0.106	0.022
	3C-STAD	184702	94	185515	0.595	− 0.344	0.379	0.223	0.148	0.004
CHEm	3C-sLC (6,4)	8054	341	11016	0.316	0.238	0.126	− 0.092	0.292	− 0.113
	sLC	10376	164	11802	0.375	0.314	0.169	− 0.145	0.345	− 0.135
	HU	6644	1049	15763	0.473	0.425	0.101	− 0.059	0.274	− 0.110
	CP-S	7073	102	7960	0.226	0.153	0.152	− 0.128	0.248	− 0.063
	3C-STAD	10122	94	10936	0.191	0.094	0.121	0.047	0.255	− 0.041
ENWm	3C-sLC (6,3)	31584	344	34576	0.404	0.347	0.219	0.195	0.168	− 0.036
	sLC	37213	182	38792	0.324	0.250	0.342	0.328	0.178	− 0.005
	HU	15241	1049	24360	0.477	0.399	0.112	0.018	0.156	− 0.080
	CP-S	16094	213	17945	0.507	0.478	0.128	− 0.091	0.126	− 0.058
	3C-STAD	45266	94	46079	0.676	0.650	0.131	− 0.100	0.256	− 0.111
FRAm	3C-sLC (7,3)	32423	349	35454	0.394	0.357	0.086	− 0.072	0.151	− 0.070
	sLC	45260	183	46854	0.428	0.396	0.094	− 0.081	0.204	− 0.079
	HU	11433	1049	20552	0.202	− 0.017	0.232	0.217	0.129	0.017
	CP-S	13017	221	14943	0.317	0.248	0.046	0.023	0.105	0.001
	3C-STAD	53538	94	54352	0.315	0.272	0.134	0.105	0.162	− 0.044
AUSm	3C-sLC (6,6)	17568	294	20124	0.333	0.315	0.283	0.239	0.194	− 0.020
	sLC	17013	187	18636	0.344	0.327	0.273	0.227	0.203	− 0.030
	HU	8499	1049	17618	0.135	0.039	0.305	0.233	0.151	− 0.021
	CP-S	9982	125	11070	0.261	0.238	0.142	0.057	0.133	− 0.019
	3C-STAD	21397	94	22210	0.186	0.040	0.207	0.131	0.180	0.030
USAf	3C-sLC (7,4)	100138	330	103009	0.246	− 0.024	0.324	0.240	0.137	0.024
	sLC	101829	246	103964	0.243	− 0.020	0.343	0.265	0.140	0.016
	HU	26369	1049	35488	0.293	0.245	0.458	0.382	0.116	0.013
	CP-S	28321	253	30520	0.225	0.129	0.376	0.310	0.113	0.020
	3C-STAD	81640	94	82453	0.271	− 0.109	0.179	0.099	0.135	0.015
CHEf	3C-sLC (7,5)	8728	309	11414	0.260	− 0.224	0.077	0.012	0.335	− 0.074
	sLC	8002	159	9387	0.182	− 0.128	0.093	− 0.050	0.417	− 0.163
	HU	6572	1049	15691	0.198	− 0.143	0.080	0.022	0.340	− 0.098
	CP-S	6975	97	7822	0.212	− 0.167	0.147	− 0.129	0.329	− 0.067
	3C-STAD	8706	94	9519	0.363	− 0.289	0.167	0.136	0.317	− 0.006
ENWf	3C-sLC (6,5)	24853	310	27548	0.401	0.346	0.104	− 0.071	0.142	− 0.014
	sLC	26836	180	28399	0.358	0.296	0.077	0.009	0.132	0.002
	HU	13642	1049	22760	0.580	0.550	0.085	0.024	0.131	− 0.062
	CP-S	15241	209	17061	0.401	0.370	0.125	− 0.089	0.111	− 0.035
	3C-STAD	36324	94	37138	0.314	0.048	0.142	0.099	0.225	0.114
FRAf	3C-sLC (7,4)	22646	314	25379	0.374	− 0.287	0.242	0.224	0.155	− 0.014
	sLC	25137	170	26619	0.378	− 0.293	0.277	0.261	0.186	− 0.033
	HU	9260	1049	18378	0.191	0.082	0.080	0.054	0.124	− 0.010
	CP-S	13847	211	15681	0.301	− 0.213	0.085	0.045	0.109	0.026
	3C-STAD	30758	94	31572	0.379	− 0.237	0.069	0.014	0.201	− 0.045
AUSf	3C-sLC (6,3)	10532	325	13358	0.127	0.023	0.149	0.098	0.194	0.019
	sLC	10057	182	11639	0.130	0.031	0.146	0.091	0.198	0.001
	HU	8819	1049	17937	0.157	− 0.080	0.196	0.158	0.194	0.002
	CP-S	9151	118	10175	0.120	0.042	0.097	0.045	0.175	0.001
	3C-STAD	16919	94	17733	0.252	− 0.124	0.111	0.048	0.264	0.119

Models estimated over years 1960–2018 and ages 0–100. Deviance (Dev), effective dimension (ED) and Bayesian information criterion (BIC) are presented to compare performances of the models in observed years. Lower values of BIC (in bold) indicate a better fit. Out-of-sample forecast is carried out with fitting period 1960–2008, forecast up to 2018 and compared to observed values in 2009–2018. Models are compared using root mean square error (RMSE) and mean error (ME). Accuracy measures are computed on $e_0$, $e_0^{†}$ and $ln\mathit{M}$. Lower values of the RMSE, and ME closer to zero (in bold) correspond to greater accuracy. Populations: USA, Switzerland (CHE), England and Wales (ENW), France (FRA) and Australia (AUS), both males and females

In terms of BIC, the CP-S model outperforms all other competitors: the model has indeed high flexibility (a by-product of the P-splines), and the smoothing procedure selects the optimal number of parameters given mortality developments and population size. The second most successful approach is the HU, which employs a larger number of parameters thereby obtaining an excellent fit to the data (the deviance is minimized for all populations). Nonetheless, the use of such a high number of parameters is penalized by the BIC criterion. This is particularly evident for the smallest populations considered (CHE), where the more parsimonious 3C-sLC and 3C-STAD are preferred to the HU model.

In Table 1, we also show the smoothing parameters ${\lambda }_{\mathit{\alpha }}$ and ${\lambda }_{\mathit{\beta }}$ employed in fitting the 3C-sLC model for all ten populations. Specifically, in rows associated with 3C-sLC model the two numbers in brackets identify the exponents of the power of ten used as smoothing parameters, e.g., (7, 4) denotes $({\lambda }_{\mathit{\alpha }},{\lambda }_{\mathit{\beta }})=({10}^7,{10}^4)$.

Instead of the conventional long table, we graphically present outcomes based on previous estimations. In particular, Fig. 10 shows life expectancy at birth ($e_0$) and average number of life years lost at birth ($e_0^{†}$) in 2050 for all 50 combinations of populations and models. As mentioned for both 3C-sLC and 3C-STAD, we are able to decompose these measures in the hypothetical scenarios of removing one or two mortality components from the age-pattern of mortality.

Fig. 10.

Forecast life expectancy at birth ($e_0$, top panels) and lifespan disparity ($e_0^{†}$, bottom panels) in 2050 by populations, sex and estimated models. For both 3C-sLC and 3C-STAD models, outcomes are presented also for hypothetical scenarios without the adulthood component, and for the senescence component only. Populations: USA, Switzerland, England and Wales, France and Australia. Models estimated over years 1960–2018, ages 0–100

The 3C-sLC forecasts of $e_0$ in 2050 are similar (albeit slightly smaller) to those of the sLC; similar results from the two models emerge in terms of $e_0^{†}$, with the 3C-sLC forecasting less compression (i.e., greater disparity) than the sLC. Furthermore, the HU and CP-S generally predict similar or lower levels of $e_0$ than the 3C-sLC, while 3C-STAD life expectancy forecasts are generally the most optimistic. In terms of $e_0^{†}$, the 3C-STAD and CP-S approaches generally project less compression of the age-at-death distribution in 2050 than the 3C-sLC.

Finally, we assess the forecasting accuracy of the all five models by out-of-sample forecast exercises against the observed trends. We fit all models from 1960 to 2008, and we forecast mortality 10 years ahead (2009–2018), comparing forecast values to those observed over that decade. Table 1 presents these results in terms of life expectancy at birth ($e_0$), average number of life years lost at birth ($e_0^{†}$) and log-mortality over all ages and years ($ln\mathit{M}$). We measure the accuracy of the models by computing root mean square error (RMSE) and mean error (ME). Whereas ME describes the forecast bias due to the model, the RMSE assesses the performance of the model on the same units as the measured variable (Chatfield 2000).

From these exercises, it appears that the CP-S produces the most accurate point forecasts, minimizing the two indicators in 20 instances (out of 60). The 3C-STAD is the second most accurate approach with 15 indicators, followed by the HU (11), sLC (10) and 3C-sLC (5).

Appendix B No Evidence of Adulthood Mortality: The 2C-sLC Model

Whereas the steep mortality decrease during childhood and the exponential increase in the risk of dying from adult ages are generally accepted and evident in all human populations, the presence of an adulthood component can be questionable. As consequence, any mortality model, including the 3C-sLC, attempting to describe an uncertain component may run into identifiability issues. Most importantly, these types of models in these circumstances will be unreasonable from a demographic perspective.

In this appendix, we propose a Two-Component smooth Lee–Carter (2C-sLC) model: a simplification of the proposed approach in which only two components act simultaneously to estimate overall mortality. Again, both components will be described by distinct Lee–Carter models with smooth parameter vectors. Only few adjustments are necessary with respect to what presented in the paper, and two time indexes will be extrapolated for forecasting future mortality.

Computationally more stable with respect to the initial model, this simplified version loses most of the original explanatory value, i.e., whereas a first component will still describe mortality during childhood, the second component will capture all “Residual” mortality.

We estimate this version of the model on Swiss females in which early-adulthood component is not as marked as for males. Figure 11 presents the estimated parameters of the 2C-sLC for Swiss females. As in the original version of the model, average shape of mortality over age, captured by the parameter $\mathit{\alpha }$, is decomposed into two component-specific average patterns. Note that ${\mathit{\alpha }}_R$ for the residual component is sufficiently flexible to capture mortality during early-adulthood which is not exponentially increasing. Component-specific rates of mortality improvements ${\mathit{\beta }}_k$ and time indexes ${\mathit{\kappa }}_k$ are presented in central and right panels of Fig. 11, respectively.

Fig. 11.

Estimated parameters from 2C-sLC model. Swiss females, ages 0–100, years 1960–2018, time index(es) $\mathit{\kappa }$ forecast up to 2050 by RWD with 80% prediction intervals. Left panel: $\mathit{\alpha }$, average shape of the age profile. Central panel: $\mathit{\beta }$, age-specific average rates of mortality decline. Right panel: $\mathit{\kappa }$, level of mortality at time t. Confidence intervals also computed for the observed years

Figure 12 shows mortality age-patterns for two selected years (1960 and 2050) with the associated childhood and residual components. The model is able to describe well mortality. The childhood component can be interpreted as in the manuscript, but it is clear that the “Residual” component merges both senescence and early-adulthood mortality features.

Fig. 12.

Left panel: Observed, fitted and forecast mortality rates (in log scale) in 1960 and 2050. The force of mortality is decomposed into two components (childhood and residual) via the 2C-sLC model. Left panel: Observed, fitted and forecast life expectancy at birth ($e_0$) with 80% prediction intervals obtained by both version of the proposed model. Life expectancy at birth without childhood component is presented for the 2C-sLC model only. Swiss females, ages 0–100, fitted years 1960–2018, forecast years 2019-2050

Finally, life expectancy over time for both Two- and Three-Component smooth Lee–Carter are portrayed on the right panel of Fig. 12. In terms of overall mortality, both versions of the model provide very similar estimates and differences in future trends are negligible: flexibility of the “Residual” component is so ample to include (and predict) most of the early-adulthood mortality.