ASSESSING THE SUITABILITY OF THE LEE-CARTER MODEL IN MODELLING MORTALITY DATA WITH VARYING 𝑏 𝑥 PARAMETER

The Lee-Carter model, developed by Lee and Carter in 1992 is one of the most influential model among others that is used for mortality projection. Although the model's performance has so far been examined in a variety of situations, its effectiveness in modeling mortality data with varying speeds of change in mortality b x across ages and the ability to detect trends in mortality index k t more precisely over time has not been studied. The method traditionally used in the Lee-Carter model shows obvious drawbacks in describing the shape of future mortality. The effectiveness of the Lee-Carter model to model mortality data with varying speeds of change in mortality b x was investigated. The model was applied to mortality data with both constant and varying speeds of change for female sub population. Results show that the Lee-Carter model is not good for mortality data with varying speeds of change.


INTRODUCTION
Understanding the assumptions of future mortality and longevity risk is crucial for people in actuarial science and the government in general. The welfare and productive life expectancies of millions of people are increased by longevity breakthroughs, but these same advancements also increase the cost of pension systems, endangering the long-term creditworthiness of financial institutions owing to upsurges in unexpected future obligations. Additionally, public health expenses are impacted if unhealthy life expectancy is prolonged as a result of increases in mortality rates at later ages. Both academics and professionals have realized the significance and necessity of accurately studying future mortality in order to deal with these rapid changes and avoid undesirable consequences. There has been a lot of work put into researching and creating more accurate mathematical models and methods for predicting mortality. The need to forecast into the future and have a good understanding of what future mortality rates and life expectancy will look like in the future should be a major importance to researchers. Numerous methods have been developed recently to project mortality using stochastic models (McNown and Rogers, 1989;Lee and Carter, 1992;Alho, 1990;Currie et al., 2004;Cairns et al., 2006;Renshaw and Haberman, 2006). The Lee-Carter (LC) model, created by Lee and Carter in 1992, is one of the most often used models for projecting mortality rates among the several models that have been used to forecast mortality. The LC model has garnered a lot of interest and has come to be regarded as the benchmark for mortality modeling and projection. Even though the model has received a lot of criticism, it continues to dominate mortality forecasts. The LC model uses linear extrapolations of the age-specific death rates to project mortality. In particular, a logged matrix of age-specific mortality rates over time, and describes the mortality profile and the constant improvement of mortality at age respectively, and also a time varying index that captures the level of mortality generally. The projected of is used to generate forecasts using conventional time series approaches. One major weakness of the LC model is the assumption of a fixed age-component over time, i.e., the change of mortality over all ages is constant. The mortality of the industrialized countries of the world has considerably shown age-time interaction. Therefore, it is appropriate to note that these two dominant trends have impacted different age groups. The LC model continues to play a leading role in mortality forecasts, despite the fact that there have been many models employed for mortality forecasting throughout the years. The assumption of a fixed age component over time, however, is a significant issue with LC model (Lee and Miller 2001). This assumption of constant mortality variability over time seems to be unlikely and unworkable. Of course, we believe that the mortality trends seen in various countries over time show a significant age-time interaction, with mortality affecting different age groups at different times. Horiuchi and Wilmoth (1995) discovered that mortality reduces more quickly as people get older than at younger ages in various countries around the world. In their research, Lee and Carter (1992) did not consider this possibility. This research suggests that a fixed change in mortality may not adequately describe the mortalities of developing nations like Nigeria.

MATERIALS AND METHODS
The World Population Prospect (WPP) database of the United Nations was used to collect the data for this study. From 2009 to 2020, age-specific mortality rates data for Nigerian females were utilized for both overall mortality and piecewise mortality (Adults). Lee and Carter (1992) proposed a method for projecting future mortality rates. The main components of the Lee-Carter approach are the base model and the Autoregressive Integrated Moving Average (ARIMA) model. In the first stage,

Lee-Carter model
, and are estimated using the actual mortality surface. In the second stage, the Box and Jenkins (1970) method is used to model and extrapolate the fitted values of using an autoregressive integrated moving average process.
is given as the matrix of the observed age-specific death rate at age x during year t. is the average shape of mortality by age. is the relative speed of change or variability in mortality by age as changes.
is the time trend for the general mortality. It is also be referred to as the mortality index. is the error term or residual at age and time .

Singular value Decomposition
Singular Value Decomposition (SVD) was introduced in 1991 by Bell and Monsell. It is a process for decomposing a matrix into each of its several constituent matrices, revealing many of the interesting and valuable properties of the original matrix. Its foundation is a linear algebraic theorem that states that a matrix can be reduced to the product of three matrices: an orthogonal matrix , a diagonal matrix Σ, and the transpose of an orthogonal matrix . The matrix should ideally be divided into a set of factors that are optimal according to some criterion. Matrix can be uniquely decomposed as: = Σ (2) The matrices and are both defined to be orthogonal matrices. Matrix Σ is a diagonal matrix. The elements along the diagonal of Σ are known as the singular values of the matrix . The columns of are known as the left-singular vectors. The right-singular vectors are the columns of .
Since there are no provided regressors on the right side of the estimate, the LC model cannot be fitted using conventional regression techniques. The estimators of , and will be denoted by ̂,̂ and ̂ respectively. Firstly, lets define the ̂ estimate as the averaging ln(̂, ) over a period of time ̂1 = ∑ (3) The parameters and are calculated by applying Singular Value Decomposition to the matrix Z. where;

RESULTS AND DISCUSSION
The Lee-Carter model was applied to female mortality data to examine the impact on various age groups. As seen in figures 1, 2 and 3, there was no regular pattern emerging from the plots, but rather we had non-linear trends. It is observed that the variability of mortality is not the same across all ages. Also, the contribution of some particular age groups, i.e. infant, child, and mortality at old age, to the overall mortality of the population is different. The high incidence and prevalence of childhood diseases and deaths at old age could have been majorly responsible for the changes in the shape of the parameter. Thus, as against what is obtainable in the original Lee-Carter model where the overall mortality is studied from age 0, the study will also consider piecewise or segmented mortality (adult mortality) to check the assumption of constant variability in mortality change. Therefore, our proposed model is given as:  . 1, February, 2023, pp 293 -296   295 i.
Obtain the logarithm of mortality rates that is ( ) ii.
Obtain the parameter ̂which is a column vector, it is calculated as average over time of the logarithm of mortality rates ̂= 1 ∑ ( ) =1 iii. Create a matrix for estimating and where = ( ) −̂ iv.
Decompose the matrix into a product of matrices using the singular value decomposition method ( ) = Σ After applying the techniques stated in above, we obtained the following result: From the decomposition of matrix we got the parameter ̂ which is the mortality index, it is a vector that captures the overall mortality change overtime. A standard time series model selection procedure was carried out to identify the best ARIMA model for Nigerian mortality data using AIC, BIC and some error measures. It can be seen that from Table 1, ARIMA (1,1,0) seems to be the best ARIMA model for female Nigeria mortality data as against a random walk (0,1,0) that was used by Lee and Carter in 1992 and also widely used by other researchers. Thus, ARIMA (1,1,0) was used to model Nigeria's female mortality data for both the overall mortality profile and piecewise with specific age intervals.  Table 2 above shows the forecasted values of ̂ for overall female mortality profile. The LC model is then updated with these predicted values to project future age-specific mortality. Next is to compare the performance of LC model on mortality data with varying parameter for both overall mortality data and adulthood mortality.

LC Model for adult mortality data with constant parameter
Here the LC model is given as 15 − 19,20 − 24,25 − 29, … 55 − 59 and = 2009,2010, … 2020. The model parameters ̂,̂ and ̂ were estimated using the SVD.    Table 3 shows the forecasted values of ̂ for adulthood mortality. The LC model is then updated with these predicted values to project future age-specific mortality.  Table 4. Shows that piecewise mortality (adulthood mortality) has minimum errors as compared to overall mortality.

CONCLUSION
Though the performance of the LC model in modelling the entire mortality profile has been studied in some literature (Li and Chan, 2007;Nan et al. 2015;Taruvinga et al. 2017), there is no concrete research in terms of piecewise mortality modelling for different ages. For this reason, in this study, the LC model was utilized within the perspective of overall mortality modelling and piecewise modelling. We take into consideration the demographic data for Nigeria. Results show noticiable defference between the two approaches to modelling moratlity data with varying and moratlity data with constant . As seen in Table 4, the adulthood mortality modelling approach gives minimum errors with regards to the forecast of the mortality rate of age 15 to 59 for female population. Mortality data with a constant gives a more accurate forecast in terms of mortality rates. This supports our suggestion that for some mortality data, the whole age span has a varying speed of change in mortality and thus may not be modelled well by the original LC model.