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This paper gives an overview of the Lee Carter method and reiterates the feasibility of using it to construct mortality forecast for the population data. In a first step, the model is fitted in a traditional way and used to extrapolate forecast of the time-varying mortality index. The observed pattern of the mortality rates shows a different variability at different ages, highlighting that the homoscedasticity hypothesis is quite unrealistic. Thus, in a second step, the paper aims to produce more reliable mortality forecasting, focusing on the errors in the estimation of the model parameters. The robustness of the estimated parameter is analysed throughout an experimental strategy which allows to assess the robustness of the Lee Carter model by inducing the errors to satisfy the homoscedasticity hypothesis. The graphical and numerical results are tested by means of a comparison in terms of prediction accuracy.

The actuarial literature has developed a number of approaches to make objective projections on mortality rates [

Let us recall the traditional LC model analytical expression (4):

ln ( m x , t ) = α x + β x κ t + ε x , t (1)

where m x , t are the log-mortality rates and α x , β x and k t are an age-spe- cific parameter independent of time, a coefficient describing the tendency of mortality to change and a time-varying parameter, respectively. The error term ε x , t is assumed to be homoscedastic (with mean 0 and variance σ ε 2 ).

To find a Least Squares Solution to the LC analytical expression, we use the close approximation to the Singular Value Decomposition (SVD) method proposed by Lee and Carter [

1) We estimate α x as the logarithm of the geometric mean of the crude mortality rates, averaged over all t , for each x :

α x = 1 h ∑ t = t 1 t n ln m x t = ln [ ∏ t = t 1 t n m x t 1 h ]

In other words, the α x coefficients must be simply the average values over time of the ln ( m x , t ) values for each x .

2) We compute k t as the sum over age of ( ln ( m x , t ) − α x ) .

3) We estimate β x from ( ln m x t − α x ) = β x k t ( 1 ) + ε ′ x t (where k t ( 1 ) refers to the k t estimated at step 2) using the least squares estimation, i.e. choosing β x

to minimize ∑ x , t ( ln m x t − α x − β x k t ( 1 ) ) 2 Math_26#.

By way of these steps, we find each β x by regressing ( ln ( m x , t ) − α x ) on k t , without a constant term, separately for each age group x .

The estimation introduced in Section 2.1, is a first stage estimation based on logs of death rates rather than the death rates themselves. To guarantee that the fitted death rates will lead to the actual numbers of deaths, when applied to given population age distribution, it is necessary to estimate k t in a second step, taking the α x and β x estimates from the first step. In particular, we use an iterative method to adjust the estimated k t , so that the actual total observed deaths

∑ x = x 1 x k d x t equal the total expected deaths ∑ x = x 1 x k e x t e ( α x + β x k t ) , for each year t .

The iterative method proceeds as follows:

1) We compare the total expected deaths ∑ x = x 1 x k e x t e ( α x + β x k t ( 1 ) ) to the actual total observed deaths ∑ x = x 1 x k d x t in each period.

2) With this comparison, we can find one of three possible states:

i) If ∑ x = x 1 x k e x t e ( α x + β x k t ( 1 ) ) > ∑ x = x 1 x k d x t , we need to decrease the expected deaths, adjust-

ing the estimated k t so that the new estimate of k t , say k t ( 2 ) , will be: k t ( 2 ) = k t ( 1 ) ( 1 − d ) , if k t ( 1 ) > 0 (where k t ( 1 ) is the first estimate of k t ); k t ( 2 ) = k t ( 1 ) ( 1 + d ) , if k t ( 1 ) < 0 , where d is a small number.

ii) If ∑ x = x 1 x k e x t e ( α x + β x k t ( 1 ) ) = ∑ x = x 1 x k d x t , we stop here the iterations.

iii) If ∑ x = x 1 x k e x t e ( α x + β x k t ( 1 ) ) < ∑ x = x 1 x k d x t , we need to increase the expected deaths adjusting

the estimated k t so that : k t ( 2 ) = k t ( 1 ) ( 1 + d ) , if k t ( 1 ) > 0 ; k t ( 2 ) = k t ( 1 ) ( 1 − d ) , if k t ( 1 ) < 0 .

3) Go back to Step 1.

Once we obtain the new time-varying parameter k t , we can model it as a stochastic process. To this aim, we use the standard Box and Jenkins methodology (identification-estimation-diagnosis) and choose an appropriate ARIMA (p, d, q) model for the mortality index k t [

Let us state with M ˜ x , t the matrix holding the mean centred log-mortality rates, given by ln ( m x , t ) − α x . We can express the LC model as follows:

M ˜ x , t = ln ( m x , t ) − α x = β x κ t + ε x , t (2)

We have seen that, in the original LC model, the errors are supposed to be homoscedastic with respect to the different age-groups. This hypothesis can be seriously different from reality and can affect the robustness of the mortality index k t . As we know (4, Appendix B), the LC model incorporates different sources of uncertainty: uncertainty in the demographic model and uncertainty in forecasting. The uncertainty in the demographic model can be incorporated by considering errors in fitting the original matrix of mortality rates, while forecast uncertainty arises from the errors in the forecast of the mortality index. Aim of this contribution is to take into consideration the demographic component in order to focus on the sensitivity of the estimated mortality index. To achieve this aim, we propose an experimental strategy to force the fulfilment of the homoscedasticity hypothesis and to assess its effect on the LC estimates.

To induce the errors to satisfy the homoscedasticity hypothesis, we propose the following scheme:

1) We express the residual term ε ^ x , t as the difference between the matrix M ˜ x , t , referring to the mean centred log-mortality rates and the product between β_{x} and k_{t}, deriving from the LC model estimation:

ε ^ x , t = M ˜ x , t − β ^ x κ ^ t (3)

2) We explore the residuals by means of statistical indicators such as: Range, Interquartile Range, Mean Absolute Deviation (MAD) of a sample of data, Standard Deviation, Box-plot, etc., in order to find some non-conforming age- groups.

3) We find those age-groups which show higher variability in the errors and rank the non-conforming age-groups according to decreasing non-conformity, i.e. from the more widespread to the more homogeneous one.

4) For each selected age-group:

4.1) we reduce the variability dividing the entire range in several quantiles, leaving aside each time the fixed α% of the extreme values;

4.2) we substitute the extreme values with uniform random values ranging from the α^{th} to the 100-α^{th} percentiles, with α given by: 0.05; 0.10; 0.15; 0.20; 0.25; 0.30.

5) For each age-group and for each percentile, we define a new error matrix ε ⌢ x , t which is used for computing a new data matrix M ^ x , t , from which it is possible to derive the correspondent k t .

6) We replicate each running under the same conditions a large number of times (i.e.: 1000).

By way of this experiment, we investigate the residuals heteroscedasticity deriving from two factors: the age-group effect and the amount of altered values in each age-group. Throughout the successive running, we obtain more and more homogeneous error terms, which allow to determine the hypothetical pattern of k t . Thus, under these assumptions, we investigate the changes in k t which can be derived from every simulated error matrix.

From the relation:

[ M ˜ x , t − ε ⌢ x , t ] = M ⌢ x , t → β x κ t (4)

we obtain a new matrix M ⌢ x , t , where M ˜ x , t is the matrix holding the actual data and ε ⌢ x , t the matrix holding the mean of altered errors. Assuming fixed the β_{x}, the k t are obtained as the Ordinary Least Square (OLS) coefficients of a multivariate regression model:

Κ ′ ( 1 × years ) = ( β ′ β ) ′ ( 1 × 1 ) β ′ ( 1 × a g ) M ⌢ ( age × years )

We fit the LC model to a data matrix of male Italian death rates, supplied by the Human Mortality Database [

We model the new time-varying parameter k t , as a stochastic process, following the standard Box and Jenkins methodology (identification-estimation- diagnosis). In a first step, we analyse the general pattern of the time series, noticing a decreasing linear trend (see

By using the Akaike and Schwarz Information Criterion (AIC and SIC) per model, together with examination of autocorrelations and partial autocorrelations, we select an ARIMA (0, 1, 0) process to model the male k t index, i.e.:

Male ARIMA (0, 1, 0) | ||||
---|---|---|---|---|

Variable | Coefficient | Std. Error | t-Statistic | Prob. |

λ | −0.424882 | 0.137488 | −3.090321 | 0.0033 |

K t = K t − 1 + λ + ε t

In

Basing on the period 1950-2000, we make use of the ARIMA (0, 1, 0) model to forecast the index of mortality k t for the next 25 years. The results will be compared in the next to the k t forecast derived by the experimental strategy.

As explained in the previous section, the typical result of applying the LC model is a time series indicating the mortality trend.

Following the scheme of our procedure, after expressing the error term ε ^ x , t as the difference: M ˜ x , t − β ^ x κ ^ t , we carry on an analysis of the residuals’ variability in order to find some “non-conforming” age-groups. We explore the residuals by means of some dispersion indices, in the matter in question Interquartile Range, MAD, Range and Standard Deviation, to determine the age-groups in which the model hypothesis does not hold (

In

Age | IQ Range | MAD | Range | STD |
---|---|---|---|---|

0 | 0.107 | 0.059 | 0.3 | 0.075 |

1 - 4 | 2.046 | 0.99 | 4.039 | 1.139 |

5 - 9 | 1.2 | 0.565 | 2.318 | 0.653 |

10 - 14 | 0.165 | 0.083 | 0.377 | 0.099 |

15 - 19 | 1.913 | 0.872 | 3.615 | 1.007 |

20 - 24 | 0.252 | 0.131 | 0.51 | 0.153 |

25 - 29 | 0.856 | 0.433 | 1.587 | 0.498 |

30 - 34 | 0.536 | 0.25 | 1.151 | 0.299 |

35 - 39 | 0.24 | 0.186 | 0.868 | 0.239 |

40 - 44 | 0.787 | 0.373 | 1.522 | 0.424 |

45 - 49 | 0.254 | 0.126 | 0.436 | 0.145 |

50 - 54 | 0.597 | 0.311 | 1.29 | 0.367 |

55 - 59 | 0.196 | 0.151 | 0.652 | 0.187 |

60 - 64 | 0.247 | 0.17 | 0.803 | 0.212 |

65 - 69 | 0.207 | 0.119 | 0.604 | 0.147 |

70 - 74 | 0.294 | 0.171 | 0.739 | 0.202 |

75 - 79 | 0.23 | 0.117 | 0.485 | 0.133 |

80 - 84 | 0.346 | 0.187 | 0.835 | 0.227 |

85 - 89 | 0.178 | 0.099 | 0.482 | 0.124 |

90 - 94 | 0.307 | 0.153 | 0.701 | 0.186 |

95 - 99 | 0.071 | 0.042 | 0.22 | 0.051 |

residuals in the age-groups 1 - 4; 5 - 9; 15 - 19 and 25 - 29 show the highest variability. The age - groups 30 - 34; 40 - 44; 50 - 54 could also be suspected and should be examined. To choose the age-groups to enter in our experiment, we provide also a graphical analysis. In Fig. 2 we display the boxplot of the residuals’ variability for each age-group in order to see if these residuals are in compliance with the expected ones.

If we have a look at the age-group 1 - 4; 15 - 19, which show the largest widespread, we can notice that the range goes from −2 to 2. We rank these non- conforming age-groups according to decreasing non-conformity, i.e. from the more widespread to the more homogeneous one. The ultimate aim is to analyze at what extend the estimated k t ’s are affected by such a variability.

On the basis of the previous analysis, we conclude that the age-groups 1 - 4; 5 - 9; 15 - 19 and 25 - 29 are far away from being homogeneous and will be sequentially entered in the experiment. For each of the four age-groups, we reduce the variability dividing the entire range into 6 quantiles: 5%, 10%, 15%, 20%, 25%, 30%, leaving aside each time a fixed 5% of the extreme values. We generate

1000 random replications and define, for each age-group and for each quantile, a new error matrix used for computing a new data matrix. From the replicated errors, we compute the estimated k t and then we extrapolate the 24 average of the 1000 simulated k t (

By comparing the 24 averaged k t (in red) to the original one (in black), we can see the effect of the changes in homogeneity on the k t for each of the four age-groups. In particular, we can notice that as the homoscedasticity in the errors increases, k t ’s tend to be flatter than the original one. In other words, more regular residuals lead to a flatted pattern of the k t ’s. For sake of comparison, in

In

Our aim is to compare the results obtained from the LC model fitted in a traditional way to the results obtained processing the residuals with the proposed experimental strategy. In the last case, our findings showed that a more regular residual matrix leads to a flatter k t . Taking into consideration the new k t (let’s call it “experimental k t ”), our aim is to find an appropriate ARIMA time series model for the mortality index and then use that mortality model to generate forecasts of the mortality rates. The ultimate purpose is to compute life expectancy at birth from forecasted mortality rates in both cases and compare them to the actual one.

As first step, by following the methodology illustrated in Section 4.1, we find that, among the others, an ARIMA (0, 1, 0) model is more feasible for the experimental k t series as happened for the k t series derived from the traditional LC model (let’s call it “traditional k t ”). The ARIMA (0, 1, 0) models are then used to generate forecasts of the mortality index for the next 25 years based on the period 1950-2000.

As second step, we build up projected life tables in both cases, by using the traditional and experimental k t . The procedure tracked is the following: after

Years | Experimental Kt_Males | Traditional Kt_Males |
---|---|---|

2001 | −13.0598 | −14.5414 |

2002 | −15.4492 | −14.9663 |

2003 | −17.8387 | −15.3911 |

2004 | −20.2281 | −15.8160 |

2005 | −22.6176 | −16.2409 |

2006 | −25.0070 | −16.6658 |

2007 | −27.3965 | −17.0907 |

2008 | −29.7860 | −17.5156 |

2009 | −32.1754 | −17.9404 |

2010 | −34.5649 | −18.3653 |

2011 | −36.9543 | −18.7902 |

2012 | −39.3438 | −19.2151 |

2013 | −41.7332 | −19.6400 |

2014 | −44.1227 | −20.0648 |

2015 | −46.5121 | −20.4897 |

2016 | −48.9016 | −20.9146 |

2017 | −51.2911 | −21.3395 |

2018 | −53.6805 | −21.7644 |

2019 | −56.0700 | −22.1893 |

2020 | −58.4594 | −22.6141 |

2021 | −60.8489 | −23.0390 |

2022 | −63.2383 | −23.4639 |

2023 | −65.6278 | −23.8888 |

2024 | −68.0172 | −24.3137 |

2025 | −70.4067 | −24.7386 |

obtaining the k t projected series, we construct projected mortality rates. Then we project life expectancy at birth from 2001 up to 2009, by using the traditional k t in the first case and, for the same period, by using the experimental k t . In order to test the validity of our experiment, we compare the resulting life expectancy to the actual life expectancy at birth from 2001 up to 2009. The choice of the period (2001-2009) as the forecast date was due to the consideration of updated projections available for Italy in the HMD.

From

errors. In other words, if we take into account the heteroscedasticity in the errors, we obtain more realistic and reliable survival projections.

The LC mortality forecasting approach has several appreciated properties, but also quite stringent assumptions. A major one considers the errors ε x , t homoscedastic but, in our experience, this is seriously different from reality. Our analysis illustrates the potential utility of considering the homoscedasticity issue of the LC model in survival analysis. When homoscedasticity is found in the residuals, we warn that successive forecast could be biased in some way. For this reason, what we propose is an experimental strategy to force the fulfilment of the homoscedasticity hypothesis by inducing the errors to satisfy it. In the numerical application we find that a more regular residual matrix leads to a more flat k t . We test this result by means of a comparison in terms of prediction accuracy. We project life expectancy at birth from 2001 up to 2009, by using the traditional k t , the experimental k t and comparing them to the actual life expectancy at birth projected for the same period. In terms of predictive performance, for this particular data set, we found that the experimental k t led to more realistic survival projections. In future research we would like to provide a statistical meaning in the k t sloping changes and to provide a general rule in assessing the LC model sensitivity.

The author thanks an anonymous referee whose suggestions improved the original manuscript.

Russolillo, M. (2017) Assessing Actuarial Projections Accuracy: Traditional vs. Experimental Strategy. Open Journal of Statistics, 7, 608-620. https://doi.org/10.4236/ojs.2017.74042