Modelling Mortality with Common Stochastic Long-Run Trends
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Abstract
Modelling mortality and longevity risk is critical to assessing risk for insurers issuing longevity risk products. It has challenged practitioners and academics alike because of first the existence of common stochastic trends and second the unpredictability of an eventual mortality improvement in some age groups. When considering cause-of-death mortality rates, both aforementioned trends are additionally affected by the cause of death. Longevity trends are usually forecasted using a Lee-Carter model with a single stochastic time series for period improvements, or using an age-based parametric model with univariate time series for the parameters. We assess a multivariate time series model for the parameters of the Heligman-Pollard function, through Vector Error Correction Models which include the common stochastic long-run trends. The model is applied to circulatory disease deaths in U.S. over a 50-year period and is shown to be an improvement over both the Lee-Carter model and the stochastic parameter ARIMA Heligman-Pollard model.
Keywords
mortality trends Heligman-Pollard model Lee-Carter model VECM causes of death mortality forecastsIntroduction
Longevity and mortality improvement has been driven by many factors over the last century. These factors have impacted different ages to varying extents resulting in common improvements across age groups. As a result, longevity and mortality risk across ages is expected to include common stochastic trends. The Lee-Carter model,^{1} and variants of it, is often used for mortality and longevity risk applications by industry practitioners and academics. This model is highly parameterized and includes a single common stochastic trend across ages fitted using a time series model. An alternative approach^{2} is to fit a parametric model to the age structure of mortality and apply time series models to the parameters. This provides for a more parsimonious model but does not capture common stochastic trends across the age structure of mortality. Our first attempt to improve these models was to use a Vector AutoRegression (VAR) for the parameters which captures dependence between the parameters. This was further used in an analysis of mortality trends in a number of developed countries.^{3} However, VAR models do not include long-run relationships between the parameters that will be important in maintaining the relative levels of mortality for different ages. Our second attempt was then to use Vector Error Correction Models (VECM) that are applied in econometrics to model multivariate dynamic systems. They include time dependency between economic variables, common stochastic trends and long-run equilibrium relationships.
Modelling cause-of-death mortality rates is increasingly important in forecasting trends in aggregate mortality improvement.^{4} Models that better capture trends and that can be applied to cause-of-death mortality rates will provide practitioners with an improved understanding of mortality risks and factors determining future mortality trends. Indeed, causes of death reflect underlying socio-economic factors and provide important insights into future trends and volatility for longevity and mortality modelling.^{5} Cause elimination models as well as cause-delay models have been developed^{6} assuming independent causes of death. Projections of mortality rates by cause of death have also been considered.^{7} The Lee-Carter model has been used for cause of death data.^{8} In general, models used for mortality at a population level are not appropriate for cause-specific rates as demonstrated by an analysis of six causes of death on Italian data.^{9} Different causes of death have different age patterns, different time trends and these vary by country.^{10} New approaches are then needed to model mortality by cause of death. A flexible model that can capture different age patterns and maintain consistent patterns while allowing trends to differ by age is required for a cause of death approach.
This paper models the age structure of cause-specific mortality rates with a parametric mortality model, the Heligman-Pollard model.^{11} The parameters for this model are readily interpreted according to their impact on different age ranges. These parameters are modelled with a VECM allowing for time dependency and long-run trends between the parameters. To demonstrate the model performance, it is estimated over the period 1950–2000 for United States circulatory system deaths. The VECM approach is shown to be an improvement over the Lee-Carter model and approaches using an AutoRegressive Integrated Moving Average (ARIMA) process for the parameters and allows a more realistic quantification of risk.
This paper provides practitioners with new methods for modelling mortality, in particular, cause-of-death mortality rates. The paper demonstrates the benefits of incorporating common trends in longevity and mortality risk models to be used by insurers issuing longevity and mortality risk products. The model assessed in the paper improves existing models and this is demonstrated in the improved forecasting performance.
The paper begins with a summary of the theoretical background for a VECM analysis in the next section, following which the data used is described. Long-run equilibriums between age-based risk factors are estimated and discussed in the later section. An improved mortality forecasting approach using a stochastic parameter Heligman-Pollard VECM is provided in the subsequent section. The improved accuracy of this new forecasting approach over the Lee-Carter model and ARIMA processes is then demonstrated for United States circulatory system deaths in the penultimate section. The final section concludes.
VAR and VECM, theoretical background
VAR model dynamic interactions between a set of variables. They capture dependence through time and between variables. This multivariate time series approach has been popular among economists for several decades and is often used to model time series of economic variables. VECM, an extension of VAR, include long-run equilibrium relationships between variables, also referred to as common stochastic trends, using the concept of cointegration.
where Ω is a symmetric positive definite covariance matrix.
Estimates of the parameters of a VAR(p) and the associated asymptotic distributions assume stationarity of the process. A VAR(p) is (weakly) stationary if its first and second moments are constant over time, that is E(y_{t}) and E(y_{t}y′_{t−j}) are independent of time t, although they will usually differ with the time lag j. In this case the process has a constant mean (no trend) and its variance does not change over time. However, many time series do have a trend so that non-stationarity must be considered as well.
then the variable is referred to as integrated of order one, denoted I(1). If the process is integrated of order one, differencing removes the non-stationarity and a VAR(p) can then be fitted. However, differencing will lose any information about long-run trends in the levels of the data. Even if the variables are non-stationary, they may move together with common stochastic trends. These common trends can be captured by a long-run equilibrium relationship. A linear combination of these variables will then exist such that the relation is stationary even if each variable is not.
where z_{t} is a stochastic variable representing that deviation.^{12} If a long-run equilibrium exists, then z_{t} will be stationary. In this case these integrated variables are referred to as cointegrated and the above relation as the cointegration relation.
with β_{i} the ith cointegration relation, for i=1, 2, …, r. The stationary vector β′y_{t} contains the r linearly independent cointegrated relations of the n variables under study.^{13} If the columns of β represent all the linearly independent relations, that is all other cointegrated relations are a linear combination of the columns of β, then there are exactly r cointegrating relations among the elements of y_{t} and (β_{1} β_{2} … β_{r}) forms a basis of the space of cointegrating vectors.
The cointegrated relations, given by the matrix β, are not uniquely defined. Each cointegrating vector could be multiplied by a constant and the relationship will remain the same. Broadly speaking, for any non-zero vector v with r elements, v′β′y_{t} also represents the r cointegrated relations. As a result constraints are required for some values of the matrix β.
- Π=
−(I_{n}−Φ_{1}−⋯−Φ_{p});
= αβ′;
= matrix of rank r;
- α=
a (n × r) loading matrix;
- β=
a (n × r) matrix containing the r vectors forming a basis of the space of cointegration;
- ξ_{i}=
−(Φ_{i+1}+⋯+Φ_{p}) for i=1, …, p−1.
which is known as the VECM of the cointegrated system. Each element is stationary as the first difference of an I(1) process is stationary as are the cointegration relations. The loading matrix α indicates which cointegrated relation has an impact on which variable and to what extent. For example, the element α_{ij} measures the effect of the cointegrated relation j (j=1, …, r) on the variable i (i=1, …, n).
The rank of the matrix Π indicates the number of cointegrated relations among the variables of the process. Three different cases are possible: There is no cointegrated relation (r=0). A VAR(p−1) may be applied on the first differences of the variables. Or all linear combinations are stationary (r=n). Thus, all the variables in the process are stationary. Or there are r cointegrated relations (0<r<n), such that Π=αβ′. In this case, the cointegrated relations are included in the error correction term.
Johansen's approach is used to estimate the number of cointegrated relations in a process as well as the parameters in the matrices α, β, c and ξ_{i} for i=1, 2,…, (p−1). As a summary, the following steps are used to estimate a VECM:
- 1
Lag order of the VAR, p: Use selection criteria, such as Akaike's Information Criteria (AIC), Hannan-Quinn Criterion (HQ), Schwarz Criterion (SC), Final Prediction Error (FPE), to select the lag order of the VAR.
- 2
Unit root tests on the variables: For a process to be stationary, the characteristic polynomial of its VAR should have all its roots outside the complex unit circle.^{14} Therefore, if this polynomial has a root equal to unity, some or all the variables are integrated of order one and there might be cointegrated relations among them. Unit root tests, such as the Kwiatkowski-Phillips-Schmidt-Shin test (KPSS), the Augmented Dickey-Fuller test (ADF) or the Phillips-Perron test (PP), are useful tools to check for the stationarity of the variables. KPSS tests the null hypothesis that the variable is level or trend stationary, while ADF and PP test the null hypothesis of a unit root, and thus, the null hypothesis of non-stationarity.
- 3
If the variables are stationary, denoted I(0), a VAR(p) is suitable. If the variables are I(1), the Johansen's procedure is applied in order to find the number of cointegrated relations. Two test statistics are commonly used in order to find the number of cointegrated relations: the trace test and the maximum-eigenvalue test.
- 4
If the variables are I(1) and if there is no cointegration, a VAR(p−1) on the first difference is fitted. Otherwise, the appropriate VECM should be found.
- 5
Model validation: test for residual autocorrelations and non-normality.
Data
The data used are for five-year age group cause-specific mortality rates collected from the Mortality Database administered by the WHO.^{15} Mortality rates were determined as the number of persons for each age group and sex who die in a particular year of a specified cause, divided by the number of persons of that age group and sex alive at the beginning of the year. Mid-year population provided by the WHO database is used as an approximation for the beginning-of-the-year population. Indeed, as the population increases over time, mid-year population is an appropriate approximation, partly including migrations. Besides, diseases of the circulatory system for females in the United States over the period 1950–2005 are used and are known to be reliable. Circulatory diseases are recognised as the most important cause of death in developed economies, especially for ages above 30–50.^{10} They are important in modelling future aggregate mortality rates as well as capturing the most recent trends in mortality improvement. This cause represented around 40 per cent of total deaths in United States in 1955, 56 per cent in 1970, 50 per cent in 1985 and 41 per cent in 2000.
Causes of death are defined by the International Classification of Diseases (ICD), which ensures consistencies between countries. The ICD changed three times between 1950 and 2006, from ICD-7 to ICD-10, in order to take into account progresses in science and technology and to refine the classification. In the Unites States, the ICD changed in 1968, 1979 and 1999. The raw data is then not directly comparable for different periods. To make it comparable, comparability ratios are computed. The aim is to smooth mortality rates across the classifications. The average of the mortality rates over the last two years of a classification is required to coincide with the average of mortality rates over the first two years of the next classification. A comparability ratio is defined as the sum of the probabilities of dying in the first two years of a new classification divided by the sum of the probabilities of dying in the last two years of the previous classification.
In order to obtain data comparable over the complete period under observation, the number of deaths in a new classification is divided by the comparability ratio linking this classification with the previous one and previous comparability ratios where appropriate. Discontinuities in the mortality rates at the junction points between two classifications have been adjusted by these comparability ratios. The data used are the adjusted mortality rates.
Long-run equilibrium among age-based risk factors
The Heligman-Pollard (H-P) model has been used to model mortality rates in the United States over the period 1900–1985^{16} and the multi-exponential model for cause-specific mortality rates in the United States over the period 1960–1985.^{17} In previous research, the resulting time series of the parameters have been modelled with ARIMA processes and these models are used to forecast the value of the parameters and hence the complete age profile of mortality. Although univariate models are assumed for the parameters, the authors recognise the potential improvement from incorporating the covariation between the parameters.
Parameters of the Heligman-Pollard model
Age range | Parameter | Demographic meaning |
---|---|---|
Childhood | A∈(0, 1) | Level of infant mortality |
B∈(0, ∞) | Mortality change between age zero and age one (the larger the value, the smaller the change) | |
C∈(0, 1) | Speed at which the mortality rates decline at young ages | |
Middle ages: | D∈(0, 1) | Intensity of the accident hump |
Accident hump | E∈(0, ∞) | Age range at which the accident hump occurs (varies inversely with the spread of the hump) |
F∈(0, 150) | Age at which the hump is the highest | |
Older ages | G∈(0, 1) | General level of mortality rates |
H∈(0, ∞) | Steepness of the curve | |
K∈(−∞, ∞) | Curvature at the last ages (a positive value indicates that the age pattern of mortality on a semi-logarithmic scale is concave downward; a negative value indicates a concave upward curve) |
The model estimation process in Figure 1 is applied to the mortality rates due to diseases of the circulatory system for females in the United States. The parameters of the H-P function are first estimated by weighted least squares, the weights used implying a minimisation of the relative error as suggested by Heligman and Pollard.^{11} Because of issues regarding multicollinearity, irregular changes of the parameter estimates from one year to another may occur even if the mortality rates of adjoining years are similar.^{18} This multicollinearity makes it more difficult to fit the H-P function to any given data set. Without setting constraints on the parameters, convergence problems occur during the non-linear minimisation, as parameter B tends to increase to values greater than ten, even if it was expected to be smaller than one based on previous studies,^{19} and parameter F tends to reach the upper bound of 150.
Fixing the value of some of the parameters, either a priori or estimated from the data, has been proposed as a solution,^{20} thus reducing the number of stochastic factors in the model. Since parameter B converges to an optimum value of one for a few years, without setting any constraints during the minimisation process, we fixed it to be equal to one. This assumption does not have an important effect on the fit of the H-P function on mortality as parameter B has a negligible impact at ages other than zero. Parameter F is also fixed at its median value of 33.85. By fixing these two parameters, the non-linear minimisation converges without any difficulties to an optimum value. Only seven parameters are then required in the VECM and forecasted to give the complete age profile of mortality rates.
Lag order selection
A lag order of one was found to provide a best fit to the parameters for the mortality rates, which is consistent with the autoregressive nature of mortality. This is determined based on AIC, HQ, SC and FPE tests performed on the data. These four criteria have the smallest value at a lag order of one, even with a constant or a trend included in the VAR for the parameters.
Unit root tests
KPSS, ADF and PP tests reveal similar results. The only parameter clearly trend stationary according to the three tests is parameter C, even at a 1 and 10 per cent significance levels. KPSS test does not reject the null hypothesis of stationarity at a 10 per cent significance level. ADF and PP tests reject the null hypothesis of non-stationarity at a 1 per cent significance level. Results are less conclusive for parameter D. ADF and PP tests reject the null hypothesis of non-stationarity at a 5 per cent significance level, but do not reject it at a 1 per cent significance level. KPSS test does not reject the null hypothesis of stationarity at a 1 per cent significance level but rejects it at the 5 per cent level. All three tests confirm the five remaining parameters as non-stationary.
In order to allow for this, two models are fitted, one assuming only parameter C is trend stationary and the other assuming parameters C and D are trend stationary. Normality tests on the residuals of both models as well as autocorrelations among the residuals show that the best results are obtained with parameter D modelled as non-stationary. This model is the one used. Cointegrated relations model then common stochastic trends in the six non-stationary parameters A, D, E, G, H and K.
Cointegrated relations
Tests for the number of cointegrated relations, 1950–2005, Circulatory system, U.S.A., females (a) Trace test; (b) Maximum-eigenvalue test
r | Trace stat | Critical values | |||
---|---|---|---|---|---|
10% | 5% | 2.50% | 1% | ||
(a) | |||||
5 | 0.66 | 2.69 | 3.76 | 4.95 | 6.65 |
4 | 7.80 | 13.33 | 15.41 | 17.52 | 20.04 |
3 | 25.14 | 26.79 | 29.68 | 32.56 | 35.65 |
2 | 48.59 | 43.95 | 47.21 | 50.35 | 54.46 |
1 | 88.91 | 64.84 | 68.52 | 71.8 | 76.07 |
0 | 142.96 | 89.48 | 94.15 | 98.33 | 103.18 |
r | Eigen stat | Critical values | |||
10% | 5% | 2.50% | 1% | ||
(b) | |||||
5 | 0.66 | 2.69 | 3.76 | 4.95 | 6.65 |
4 | 7.14 | 12.07 | 14.07 | 16.05 | 18.63 |
3 | 17.34 | 18.6 | 20.97 | 23.09 | 25.52 |
2 | 23.45 | 24.73 | 27.07 | 28.98 | 32.24 |
1 | 40.32 | 30.9 | 33.46 | 35.71 | 38.77 |
0 | 54.06 | 36.76 | 39.37 | 41.86 | 45.1 |
The maximum-eigenvalue statistic tests the null hypothesis of r cointegrated relations against the hypothesis of r+1 cointegrated relations. Table 2(b) shows that the null hypothesis of two cointegrated relations is not rejected, while the null hypothesis of one cointegrated relation is rejected indicating that two cointegrated relations exist. These two long-run equilibrium relationships for the parameters of the Heligman-Pollard model determine how changes in the age-based risk factors move relative to each other. They allow for stochastic trends in the mortality curve, while maintaining long-run relationships between ages.
Fitted VECM
where z_{1t} and z_{2t} are two stochastic variables representing the deviation from the equilibrium. These two variables are stationary. The sign associated with each parameter is of significance in these relations.^{21} The first cointegrated relation shows that as mortality around age zero decreases, that is parameter A decreases, then either the accident hump will decrease or impact a smaller age range, so that parameter D will decline or E will increase, or mortality for the elderly will increase, with parameter K declining, parameter G or parameter H increasing, or a combination of these impacts. This relationship reflects the historical data and the relative changes in the mortality curve across all ages. The VECM assumes these estimated long-run relationships will continue when used for forecasting. The curve changes stochastically but is constrained by the long-run equilibrium relationships between the parameters given by the cointegrating vectors.
Increases in mortality at older ages will occur in the Heligman-Pollard model if there is an increase in either or both parameters G and H or a decrease in parameter K. This is captured in both cointegrating vectors as the coefficients for parameters G and H have the same sign, while the coefficient of K is of an opposite sign.
In a comparison of mortality for males and females in Switzerland and in the United States using the Heligman-Pollard model,^{22} mortality rates at older ages are shown to be lower, but the age pattern steeper, in Switzerland than in the Unites States, since parameter G is smaller and parameter H higher in Switzerland. Similarly, female mortality at older ages is lower and increases at a faster pace than male mortality in both countries. This link between these two parameters is reflected in the cointegrations, as a decrease in parameter G may produce an increase in parameter H for the relation to stay at the equilibrium.
Model validation
Tests on residuals of the fitted VECM, 1950–2005, Circulatory system, U.S. females
Type of test | Name of the test | Statistic value | p-value |
---|---|---|---|
Autocorrelation | Portmanteau (15 lags) | 776.68 | 0.07 |
Portmanteau (25 lags) | 1224.33 | 0.39 | |
Normality | Skewness | 8.39 | 0.30 |
Kurtosis | 8.05 | 0.33 | |
Both | 16.44 | 0.29 |
Tests for normality are based on the third and fourth central moments (skewness and kurtosis) of a normal distribution.^{24} The test statistic labelled both in Table 3 is a joint test of skewness and kurtosis. The three tests clearly show that the null hypothesis of normality is not rejected. The fitted model residuals for the parameters of the Heligman-Pollard model satisfy the assumption of normally distributed and no-autocorrelation. The model is a good representation of the data generating process and thus captures the trends and age dependence of mortality rates for deaths due to the circulatory system in the United States.
Projections
The fitted VECM provides a good fit to the data with 33 parameters for 19 age groups over 56 years based on 1,064 observations. Parameters are used to model stochastic trends and to ensure that long-run relationships between different ages are maintained through cointegration. These are very important features of the model approach.
To illustrate the model forecasting performance, the model is fitted to a shorter data period and projections are performed in order to compare the forecasts with the actual data. At least 50 years of observations are usually needed in order to reliably estimate a VECM. The model is fitted to data over the period 1950–2000 and then forecasted for the following 20 years. Parameter B is assumed fixed at a value of one and parameter F is assumed fixed at the median value over the period 1950–2000 of 25.42. Since actual data is only available until 2005, the first five years of forecasted mortality rates are compared with actual mortality.
To assess the performance of the VECM approach, two other models are fitted over the period 1950–2000 and used to forecast mortality rates until 2005. The first is a variant of the well-known Lee-Carter model, which has been successfully applied at the population level for United States data in the past. The second model fits the Heligman-Pollard model for mortality rates and uses univariate time series, traditional ARIMA processes, to model the parameters.
VECM
The VECM fitting process is applied for the time period 1950–2000. AIC, HQ, SC and FPE tests confirm a lag order of one. ADF and PP tests reject the null hypothesis of non-stationarity at a 1 per cent significance level for parameters C, D and E, while KPSS test does not reject the null hypothesis of trend stationarity at a 10 per cent significance level only for parameter C and at a 1 per cent significance level for parameter E. Parameters C, D and E are assumed trend stationary based on support for this from at least two out of the three tests.
Tests for the number of cointegrated relations, 1950–2000, Circulatory system, U.S. females (a) Trace test; (b) Maximum-eigenvalue test
r | Trace stat | Critical values | |||
---|---|---|---|---|---|
10% | 5% | 2.50% | 1% | ||
(a) | |||||
3 | 4.27 | 2.69 | 3.76 | 4.95 | 6.65 |
2 | 12.71 | 13.33 | 15.41 | 17.52 | 20.04 |
1 | 40.04 | 26.79 | 29.68 | 32.56 | 35.65 |
0 | 88.70 | 43.95 | 47.21 | 50.35 | 54.46 |
r | Eigen stat | Critical values | |||
10% | 5% | 2.50% | 1% | ||
(b) | |||||
3 | 4.27 | 2.69 | 3.76 | 4.95 | 6.65 |
2 | 8.44 | 12.07 | 14.07 | 16.05 | 18.63 |
1 | 27.34 | 18.6 | 20.97 | 23.09 | 25.52 |
0 | 48.66 | 24.73 | 27.07 | 28.98 | 32.24 |
These are similar to the cointegrated relations for the VECM fitted over the full period 1950–2005 with parameters G and H having coefficients with similar signs, while the coefficient of K is of opposite sign. The middle age parameters D and E are no longer included.
Tests on residuals of the fitted VECM, 1950–2000, Circulatory system, U.S. females
Type of test | Name of the test | Statistic value | p-value |
---|---|---|---|
Autocorrelation | Portmanteau (15 lags) | 872.36 | 8.7 × 10^{−5} |
Portmanteau (25 lags) | 1451.54 | 2.01 × 10^{−6} | |
Normality | Skewness | 12.19 | 0.09 |
Kurtosis | 11.02 | 0.14 | |
Both | 23.21 | 0.06 |
Lee-Carter model
- μ_{x,t}
force of mortality at age x, and year t;
- α_{x}
mean value over time, at age x, of the logarithm of the force of mortality;
- β_{x}
deviation from the mean value α_{x} at age x.
Reflects the impact of the time trend represented by κ_{t} on age.
The higher its absolute value, the larger the changes in mortality at that age;
- κ_{t}
mortality rates trend over time;
- ɛ_{x,t}
random changes including those not captured by the model;
= errors assumed to have mean zero and variance σ^{2} (homoscedasticity).
In the Lee-Carter model, a single common factor across ages determines the general level of mortality improvement over time. This improvement is modelled with a single time series (κ_{t}) that is projected in order to forecast the complete age profile of mortality. Lee and Carter suggest the use of a simple random walk with drift, that is an ARIMA(0, 1, 0) model. Our analysis confirms an ARIMA(0, 1, 0) process with a drift of −0.36 as the best model.^{26}
ARIMA processes
As another alternative for comparison, the parameters of the Heligman-Pollard model fitted over the period 1950–2000 are modelled with ARIMA processes, instead of a VECM. We follow the approach applied in previous works,^{27} that is trends are removed from the variables by first or second differencing. According to ADF, PP and KPSS tests, first differencing on each parameter is necessary and sufficient to assure stationarity. ARIMA(k, 1, (k−1)) models are successively fitted for each parameter of the Heligman-Pollard model, increasing k by one, as suggested by Pandit and Wu.^{26} The F-criterion is used to decide which model is the most adequate between an ARIMA(k, 1, (k−1)) and an ARIMA((k+1), 1, k). The F-criterion is a test to check for improvement in the residual sum of squares of the error ɛ_{t}. It tests the assumption that some of the coefficients in a model are restricted to zero.^{26}
Fitted ARIMA(p, 1, q) processes, Circulatory system, U.S. females
A: ARIMA (2, 1, 2) | C: ARIMA (2, 1, 1) | D: ARIMA (2, 1, 0) | E: ARIMA (3, 1, 2) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Values | CI 95% | Values | CI 95% | Values | CI 95% | Values | CI 95% | |||||
φ_{1} | −0.66 | −1.01 | −0.31 | −1.36 | −1.58 | −1.13 | −0.72 | −1.00 | −0.44 | 0.37 | 0.16 | 0.59 |
φ_{2} | −0.55 | −0.82 | −0.27 | −0.58 | −0.81 | −0.35 | −0.36 | −0.63 | −0.08 | −0.24 | −0.46 | −0.02 |
φ_{3} | −0.65 | −0.86 | −0.43 | |||||||||
θ_{1} | 0.76 | 0.57 | 0.96 | 1.00 | 0.87 | 1.13 | −1.27 | −1.42 | −1.12 | |||
θ_{2} | 1.00 | 0.80 | 1.20 | 1.00 | 0.81 | 1.19 | ||||||
∑ɛ_{t}^{2} | 6.58·10^{−9} | 4.15·10^{−3} | 7.56·10^{−10} | 713.81 |
G: ARIMA (2, 1, 1) | H: ARIMA (2, 1, 1) | K: ARIMA (2, 1, 0) | |||||||
---|---|---|---|---|---|---|---|---|---|
Values | CI 95% | Values | CI 95% | Values | CI 95% | ||||
φ_{1} | 0.49 | 0.21 | 0.77 | −1.48 | −1.69 | −1.28 | −0.71 | −0.98 | −0.45 |
φ_{2} | 0.48 | 0.22 | 0.75 | −0.64 | −0.85 | −0.43 | −0.29 | −0.55 | −0.03 |
θ_{1} | −0.84 | −1.04 | −0.65 | 1.00 | 0.88 | 1.12 | |||
∑ɛ_{t}^{2} | 5.00·10^{−12} | 1.65·10^{−4} | 56.71 |
Model comparison
These figures show that the Lee-Carter model tends to produce relatively poor forecasts by over-estimating the decline of mortality at young ages. The Heligman-Pollard model along with ARIMA processes underestimates mortality improvement, particularly at older ages. The VECM provides better forecasts and does not exhibit either of these features.
Mean absolute percentage error and ratio of mean square error compared to no-change forecast, Circulatory system, U.S. females
Year | VECM | Lee-Carter | ARIMA | |||
---|---|---|---|---|---|---|
MAPE (%) | vs no-change forecast | MAPE (%) | vs no-change forecast | MAPE (%) | vs no-change forecast | |
2001 | 9.56 | 3.21 | 13.04 | 0.44 | 9.90 | 3.51 |
2002 | 9.33 | 1.27 | 14.15 | 0.30 | 11.08 | 1.25 |
2003 | 11.71 | 0.87 | 17.85 | 0.40 | 10.71 | 0.78 |
2004 | 11.84 | 0.43 | 19.33 | 0.55 | 15.37 | 0.97 |
2005 | 13.46 | 0.33 | 24.03 | 0.57 | 17.06 | 0.59 |
The forecasts from the VECM are closer to the actual mortality rates than the forecasts of the Lee-Carter model or the forecasts performed with the Heligman-Pollard model and ARIMA processes. The forecast error (MAPE) with the VECM is lower and increases more slowly with time than the other two models. This is expected to apply for projections over longer time horizons. The VECM performance comes from its ability to capture relationships between the ages better than these alternative models.
The main benefit of using the Heligman-Pollard model over the Lee-Carter model is that it smooths mortality rates across ages and reduces the number of parameters for the age structure mortality curve. The Heligman-Pollard model has sufficient flexibility to allow for the full age range of mortality as well as having parameters that are interpreted for their effect on mortality rates. Treating these parameters as stochastic factors allows for more general dynamics than in the Lee-Carter model, even though there is an increase in the number of parameters for the random factors in the model.
The Lee-Carter model appears to be more accurate at older ages during the early forecasted years than a stochastic parameter model based on the Heligman-Pollard model. As mortality rates at older ages are higher than rates at young and middle ages, the absolute error is higher at older ages. The statistic that compares each model with the no-change forecast uses the squared error. In Table 7 the statistic comparing the no-change forecast for the Lee-Carter model is then lower for the first three years than the statistic of the other two models. For the later years this is reversed with the VECM Heligman-Pollard model having lower comparative forecast error. For longer forecast horizons, the age pattern given by the Heligman-Pollard model produces better fits and relationships between ages are maintained by the VECM taking into account long-run stochastic trends for the parameters of the model. The VECM Heligman-Pollard model is expected to be more accurate for longer forecast periods than five years because of this.
Conclusion
This paper presents a modelling and forecasting approach for mortality using VECM concepts for a parametric model for the age structure of mortality rates. Because of its ability to capture changing age structures and to model long-run trends, it has the flexibility to model different causes of death as well as aggregate mortality rates. To demonstrate the application of the model, it is fitted to cause-of-death mortality rates for the circulatory system in the United States over the period 1950–2000 and compared with alternative models previously proposed.
The model is a stochastic parameter Heligman-Pollard model, where the time series of the parameters are modelled with a VECM. The VECM incorporates common stochastic trends through long-run equilibrium relationships. The model captures the age dependence of cause-specific mortality rates through dependence between the parameters. Indeed, the parameters of the Heligman-Pollard model have interpretation as factors impacting specific age ranges—young, middle and older. The demographic meaning of the parameters allows for an analysis of the long-run equilibriums or cointegration relations in the model. The expected relationship between the parameters at older ages is shown to be reflected in these cointegration relations.
The analysis demonstrates for circulatory diseases mortality that a forecasting model based on the Heligman-Pollard model with a VECM for the parameters is an improvement over the well-known Lee-Carter model as well as over a model based on the Heligman-Pollard model with ARIMA processes for the parameters. As the forecast horizon is lengthened, the model based on the Heligman-Pollard model, with a VECM for the parameters, increases in forecasting accuracy. The Heligman-Pollard model uses a smaller number of parameters to model the age structure of mortality rates compared with the Lee-Carter model. The stochastic model for the parameters provides an adequate representation of the randomness of the process and the use of a VECM ensures consistent future mortality curves are forecasted, which is not the case if simpler ARIMA time series models are used. A stochastic parameter VECM form of the Heligman-Pollard model has also been shown to perform well at a national population level based on Australian mortality data.^{28} Finally, new theories on time series models and applications to mortality are being developed.^{29} Standard unit root tests are potentially unreliable in the presence of bounded time series, as they tend to over-reject the null hypothesis of a unit root, even asymptotically. Conventional cointegration tests may also be misleading in the presence of bounded time series. These new theories have the potential to improve the forecasting performance of the Heligman-Pollard model with a VECM for the parameters.
Footnotes
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In this paper, we consider variables that are integrated of order one, in which case, cointegrated relations are necessarily stationary. The general framework is given in Hamilton (1994) and Lütkepohl (2005).
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The size of the coefficients is influenced by the value taken by the parameters. For example, as parameters A, D and G are small, their associated coefficient is high.
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As in Lütkepohl (2005).
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For a detailed description of these tests, see Lütkepohl (2005).
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- 29.
Notes
Acknowledgements
The authors acknowledge the support of ARC Linkage Grant Project LP0883398, Managing Risk with Insurance and Superannuation as Individuals Age with industry partners PwC and APRA. Gaille acknowledges scholarship support from the Swiss National Science Foundation for the project Managing Risk as Individuals Age with Insurance and Superannuation, number PBLAP1-124258.
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