COVID-19 epidemic is spreading at a rapid rate, getting people afraid of being infected. According to the Korea Disease Control and Prevention Agency (2020.12.30., KDCA), the confirmed and death cases continue to occur and those total cumulative numbers of the countries whose cumulative confirmed cases are over 590,000 are 67,628,461 and 1,536,244 respectively, leading to the total fatality rate of 2.27% as of 2020.12.30. The fatality rates are different by ages, and they increase rapidly for the elderly (
The most commonly used model for forecasting mortality is Lee-Carter model (LCM) because the model is simple to use and the performance is quite good. There are various research papers regarding LCM and related models such as
LCM is developed by Lee and Carter in yr1992. In this model log mortality rate is equal to average log mortality rate plus the product of an age dependent sensitivity value and a period dependent trend value plus an error term as in
where
where
4-PFM (
It is designed such that 4 factors (
The average shapes of the loading functions by using the Korea life-tables from yr1983 to yr2018 are in
The estimation process of 4-PFM in this paper consists of two levels. Level 1 fits the model to the data by least squares method, and level 2 forecasts mortality rates by analyzing time trends of the fitted factors. The estimation methods in level 1 are a little different between for ‘fitted accuracy’ and ‘forecast accuracy’. For level 1 in ‘fitted accuracy’ test, the parameters are estimated by minimizing the sum of squares for all years at once. It is expressed in (
where
Then, the factors are estimated each year from yr1983 to yr2017 life-tables given the fixed parameters estimated in (
In level 2, the fitted factor values are forecasted by vector auto-regression (VAR) analysis and then put in the model producing the forecasted mortality rates. The parameters estimated for ‘forecast accuracy’ test could be used for ‘fitted accuracy’ test, but the forecast accuracies are not better than by using the parameters estimated by the most recent year (e.g., yr2017).
In Section 3, the results of fitted and forecasted values of the models are shown and the accuracies between 4-PFM and LCM are compared.
The accuracy implies how models reflect the real data. The accuracies are measured by root mean square error (RMSE) as in
where
‘Accuracy’ is divided into two parts, ‘fitted accuracy’ and ‘forecast accuracy’. The former is the accuracy between real data and the fitted data produced by the models, and the latter is the accuracy between real data and the forecasted data produced by analyzing time trends of the fitted data. Thus ‘forecast accuracy’ somewhat depends on ‘fitted accuracy’.
The Korea life-tables from yr1983 to yr2018 are used to test ‘fitted accuracy’(
The fitted variables of LCM are shown in
For ‘fitted accuracy’ test 4-PFM is fitted in level 1 as mentioned in Section 2. The limited memory BFGS procedure (L-BFGS-B) via the R package ‘optim’ is used for the fitting as in
Next, the time trends of the 4 fitted factors are shown in
The result of the average ‘fitted accuracy’ from yr1983 to yr2018 is shown in
To test ‘forecast accuracy’, forecasted results are produced by time trend analysis for the fitted data estimated in level 1 and differences between real log mortality rates and forecasted results are computed. Since future mortality rates are not known, a point of time A in the past is assumed as a current point, and forecast is performed based on the assumed current point A.
‘Forecast accuracy’ can be affected by the data characteristics at some specific point of time. For example, the accuracy would not be good though the forecast by the time trend analysis is accurate if the data at some specific point to measure the accuracy may not follow the past time trend, or the accuracy might be good by luck if the data at some specific point are fortunately close to the forecasted data though the time trend analysis is not accurate. To reduce coincidence, two tests are performed based on two different point of time. Test 1 assumes yr2017 as a current point of time and forecasts mortality rates at yr2018 and computes RMSE between forecasted data and real data at yr2018. Test 2 assumes yr2013 as a current point of time and forecasts mortality for the next 5 years and computes RMSE between the 5 year-forecasted data and the 5 year-real data from yr2014 to yr2018.
By using ‘auto.arima’ function in R package for test 1, the appropriate ARIMA model for
The result of ‘forecast accuracy’ for yr2018 by the time trend analysis is shown in
Now we move to test 2 which is to test the forecast for yr2014~yr2018. The time trend analysis for LCM shows that ARIMA(2, 1, 0) is appropriate for male, and ARIMA(1, 2, 1) is appropriate for female. For 4-PFM, there exist 1st order autocorrelation for both sexes, and the result of cointegration analysis shows that there is no cointegration effect for male but there is one cointegration relation for female as in
The results of ‘forecast accuracy’ for yr2014~2018 by the time trend analysis are shown in
As of December 30, 2020, the cumulative number of confirmed case of COVID-19 in Korea was about 115.3 per 100,000 people, which was significantly lower than 5,765.1 in US, 992.0 in Singapore, 3,477.2 in UK, and 2,020.3 in Germany. However, it is hard to forecast how long COVID-19 exists, and how much more it spreads out in the future (
COVID-19 does not affect the fatality rates equally for all ages. The fatality rates increase rapidly with age. According to KDCA, the cumulative number of confirmed cases and death from COVID-19 as of December 30, 2020 were 59,773 and 879, respectively, for all ages, 2,958 and 486 for ages over 80, 4,702 and 250 for ages of 70~79, and 9,458 and 103 for ages of 60~69.
Although the difference between the fatality rates by COVID-19 and the mortality rates of the life-tables has been calculated in a short term, it can last for a long time or even increases because COVID-19 might not be controlled in a short period of time and furthermore the virus can damage body function such as lungs, which may not cause deaths in a short period but many years later.
In this section, the accuracies between the two models are compared by scenarios assuming that the mortality rates are impacted by COVID-19. The accuracies for the scenarios are also compared with those for the life-tables to test which model is more reliable to the impact.
To set up the scenarios, the ratios of fatality rates of the confirmed people over the mortality rates of the yr2018 life-table are produced and applied to the life-tables. The reason why the ratios are applied to the life-tables and the fatality rates are not used directly is because the fatality rates are only the current rates but we need time trend rates. By applying the ratios to the life-tables in different years, we can obtain the scenarios of time trend data. The ratios computed by using the two death rates by age are in
Three scenarios are set up by applying
Scenario 1: The ratios of 1.53 for age 65, 2.49 for age 75, and 1.95 for age 90 are multiplied to the life-tables from yr2001 to yr2018 identically, then the impacted mortality rates are smoothed.
Scenario 2: The ratios are set up with decrease as time passes, that is, the same ratios as in scenario 1 are multiplied to the yr2001 life-table, then the ratios reduce gradually untill 1 is multiplied to the yr2018 life-table. Thus the scenario mortality in yr2018 is the same as that of the yr2018 life-table.
Scenario 3: The ratios are multiplied to the life-tables such that they increase from yr2001 to yr2009 in which the ratios are peaked to the same as in scenario 1, then from yr2009 turn to decrease until yr2018 where the scenario mortality rates are the same as in the yr2018 life-table.
In scenario 1, the log mortality rates decrease as the year passes as in the life-tables because the same ratios are multiplied from yr2001 to yr2018. Thus there is no time trend distortion. However, there are age trend distortions as you can see the shapes of the impacted curves. When compared with the life-tables, the impacted curves are located above the life-tables since the ages of 60.
In scenario 2, the scenario curves since the ages of 60 are located above the life-tables in yr2001 and yr2009, but return to the life-table in yr2018. Since the multiplied ratios reduce, there are some time trend distortions such as a large gap between yr2009 and yr2018 scenario curves. There are also the age trend distortions as in scenario 1.
In scenario 3, the levels of the scenario curves are more complicated than scenario 1 and 2. As in scenario 2, the scenario curves of yr2001 and yr2009 are located above the life-tables in yr2001 and yr2009 but the scenario curve of yr2018 is folded on the yr2018 life-table. The levels between the two scenario curves of yr2001 and yr2009 are different depending on ages. The yr2001 scenario curve is located above the yr2009 scenario curve as in scenario 1 and 2 when the age is below 67, but the level is reversed when the age is above 67. It is because the multiplied ratios are different between the two years. They are larger for yr2009 than for yr2001. Thus there are distinctive time trend distortions such as level reversion between yr2001 and yr2009 in scenario 3. There are also the age trend distortions as in both scenario 1 and 2.
The estimation process of the models for the scenarios is the same as in section 3 and the variable estimation results are in
In Section 4.2, the results of the accuracies between two models are compared by scenario. First, ‘fitted accuracy’ and ‘forecast accuracy’ are shown, and second, the reason why the accuracies of the scenarios are changed is explained.
First, ‘fitted accuracy’is shown in
However, in terms of the level of aggravation in ‘fitted accuracy’ as the mortality structure changes, 4-PFM is less aggravated than LCM as you can see in
Next, ‘forecast accuracy’ is compared in two separate tests. Test 1 is performed by forecasting to yr2018, and test 2 is by forecasting to yr2014~yr2018 as in Section 3. The results are shown in
In terms of the level of aggravation in ‘forecast accuracy’ as the mortality structure changes, 4-PFM is less aggravated than LCM as you can see in
Thus we can conclude that 4-PFM is more reliable in performance to the structural changes than LCM based on the both ‘fitted accuracy’ and ‘forecast accuracy’ tests.
Second, the reason why the performance of LCM is aggravated when mortality structure is distorted is explained as follows. LCM applies SVD to the model estimation. The SVD method decomposes the log mortality matrix, after subtracted by average log mortality vector (
where
Moreover, since the inaccurately fitted values of
On the other hands, 4-PFM uses 4 factors with each factor fitted for the mortality rates by some range of ages. Since the factors would be adjusted for the distorted parts, the distortion of mortality structure does not affect significantly the model’s performance.
In this paper the accuracy between LCM and 4-PFM is compared, assuming that COVID-19 impacts the mortality structure. The accuracy tests were performed in two levels, one with ‘fitted accuracy’ and the other with ‘forecast accuracy’.
Both accuracy tests showed that the performance of LCM was better than 4-PFM when the life-table data were used, but was not always better when the mortality structure was changed. The performance of LCM was better for ‘fitted accuracy’ test, but 4-PFM worked better for forecast accuracy’ test. The level of the aggravation also showed that 4-PFM was relatively more reliable than LCM.
The performance of LCM was explained by defining
Currently the most popular mortality model is LCM. However, the performance of LCM is not always reliable. We cannot say that COVID-19 has influenced the mortality rates such that their structure is changed so far, because the number of confirmed people and the number death from COVID-19 hold small proportions over the population. It is possible, however, that if COVID-19 spreads out or various mutants occur, the mortality structure could be changed. Therefore, more in-depth analyses regarding the performance of mortality models are required.
Finally, 4-PFM needs to be improved. The model is more reliable than LCM, but the accuracy in a normal situation is not good enough compared with LCM. Designing new PFM adding more factor loadings to current 4-PFM to increase the accuracies could be the next research project.
Parameter estimation of 4-PFM
Sex | ||||
---|---|---|---|---|
Male | 1.1 | 13.9 | 1.3 | 10.8 |
Female | 1.0 | 15.3 | 1.8 | 10.6 |
Parameter estimation of 4-PFM
Sex | ||||
---|---|---|---|---|
Male | 1.1 | 13.3 | 1.3 | 10.7 |
Female | 1.7 | 14.7 | 1.8 | 10.5 |
Parameter estimation of 4-PFM
Sex | ||||
---|---|---|---|---|
Male | 1.1 | 13.1 | 1.3 | 10.7 |
Female | 1.6 | 14.1 | 1.8 | 10.5 |
Fitted parameter values of 4-PFM
Parameter | ||||
---|---|---|---|---|
Male | 0.5 | 13 | 1.1 | 20 |
Female | 0.7 | 3 | 1.7 | 27 |
‘fitted accuracy’ (RMSE): 4-PFM vs. LCM, averaged from yr1983 to yr2018
Male | Female | ||||
---|---|---|---|---|---|
4-PFM(1) | LCM(2) | (1) / (2) | 4-PFM(1) | LCM(2) | (1) / (2) |
0.104 | 0.055 | 1.891 | 0.130 | 0.079 | 1.646 |
‘Fitted accuracy’ is measured by RMSE.
Result of Johansen test of 4-PFM for test 1 (yr2018)
Number of cointegration | Test | Significance level | |||
---|---|---|---|---|---|
Male | Female | 10% | 5% | 1% | |
1.83 | 1.83 | 6.50 | 8.18 | 11.65 | |
7.59 | 6.07 | 15.66 | 17.95 | 23.52 | |
16.84 | 21.61 | 28.71 | 31.52 | 37.22 | |
40.20 | 62.91 | 45.23 | 48.28 | 55.43 |
R is number of cointegration. Life-tables of yr1983~yr2017 are used.
Accuracy (RMSE) for test 1 (yr2018): 4-PFM vs. LCM
Accuracy | Male | Female | ||||
---|---|---|---|---|---|---|
4-PFM(1) | LCM(2) | (1) / (2) | 4-PFM(1) | LCM(2) | (1) / (2) | |
Fitted accuracy( | 0.095 | 0.084 | 1.130 | 0.130 | 0.093 | 1.394 |
Forecast accuracy( | 0.124 | 0.082 | 1.518 | 0.181 | 0.158 | 1.143 |
The values of ‘fitted accuracy‘ are those in yr2017 in Table 2.
Result of Johansen test of 4-PFM for test 2 (yr2014~yr2018)
Number of cointegration | Test | Significance level | |||
---|---|---|---|---|---|
Male | Female | 10% | 5% | 1% | |
0.03 | 0.07 | 6.50 | 8.18 | 11.65 | |
6.01 | 4.15 | 15.66 | 17.95 | 23.52 | |
18.63 | 16.07 | 28.71 | 31.52 | 37.22 | |
35.77 | 75.31 | 45.23 | 48.28 | 55.43 |
R is number of cointegration. Life-tables of yr1983~yr2013 are used.
Accuracy (RMSE) for test 2 (yr2014~yr2018): 4-PFM vs. LCM
Accuracy | Male | Female | ||||
---|---|---|---|---|---|---|
4-PFM(1) | LCM(2) | (1) / (2) | 4-PFM(1) | LCM(2) | (1) / (2) | |
Fitted accuracy( | 0.083 | 0.049 | 1.695 | 0.133 | 0.073 | 1.824 |
Forecast accuracy( | 0.120 | 0.102 | 1.168 | 0.200 | 0.155 | 1.289 |
^{*}The values of ‘fitted accuracy’ are those in yr2013 in Table 2.
^{**}‘Forecast accuracy’ is the average RMSE from yr2014 to yr2018.
Ratios of the two death rates by age: confirmed people of COVID-19 over life-table
Age | Mortality rate of life-table (1) | Fatality rate of confirmed people (2) | Ratio (2/1) |
---|---|---|---|
60~69 | 0.7131% | 1.09% | 1.53 |
70~79 | 2.1403% | 5.32% | 2.49 |
80+ | 8.4077% | 16.43% | 1.95 |
Source: Life-table (yr2018)-Statistics, Korea (2020) KDCA (Korea Disease Control and Prevention Agency 2020.12.30).
‘fitted accuracy’ by scenario (RMSE): 4-PFM vs. LCM
Scenario | Male | Female | ||||
---|---|---|---|---|---|---|
4-PFM(1) | LCM(2) | (1) / (2) | 4-PFM(1) | LCM(2) | (1) / (2) | |
Life-table | 0.105 | 0.058 | 1.808 | 0.131 | 0.083 | 1.570 |
Scenario 1 | 0.163 | 0.121 | 1.347 | 0.195 | 0.133 | 1.466 |
Scenario 2 | 0.138 | 0.153 | 0.902 | 0.171 | 0.169 | 1.012 |
Scenario 3 | 0.135 | 0.130 | 1.038 | 0.167 | 0.137 | 1.219 |
The results are average RMSE from yr1983 to yr2018.
RMSE ratio over life-table for ‘fitted accuracy’
Scenario | Male | Female | ||
---|---|---|---|---|
4-PFM(1) | LCM(2) | 4-PFM(1) | LCM(2) | |
Scenario 1 | 1.552 | 2.086 | 1.489 | 1.602 |
Scenario 2 | 1.314 | 2.638 | 1.305 | 2.036 |
Scenario 3 | 1.286 | 2.241 | 1.275 | 1.651 |
The numbers are calculated by RMSE of the scenarios divided by RMSE of the life-table.
‘forecast accuracy’ by scenario (RMSE): 4-PFM vs. LCM
Scenario | Male | Female | |||||
---|---|---|---|---|---|---|---|
4-PFM(1) | LCM(2) | (1) / (2) | 4-PFM(1) | LCM(2) | (1) / (2) | ||
Life-tablea | yr2018b | 0.124 | 0.082 | 1.518 | 0.181 | 0.158 | 1.143 |
5 yr avgc | 0.120 | 0.102 | 1.168 | 0.200 | 0.155 | 1.289 | |
Scenario 1 | yr2018 | 0.186 | 0.152 | 1.224 | 0.246 | 0.193 | 1.275 |
5 yr avg | 0.201 | 0.189 | 1.063 | 0.239 | 0.220 | 1.086 | |
Scenario 2 | yr2018 | 0.124 | 0.294 | 0.422 | 0.186 | 0.347 | 0.536 |
5 yr avg | 0.171 | 0.349 | 0.490 | 0.198 | 0.386 | 0.513 | |
Scenario 3 | yr2018 | 0.135 | 0.356 | 0.379 | 0.187 | 0.398 | 0.470 |
5 yr avg | 0.230 | 0.344 | 0.669 | 0.256 | 0.376 | 0.681 |
^{a}The results from Life-tables are from Section 3.
^{b}Computed from forecasted value of yr2018.
^{c}Computed from forecasted value of yr2014~yr2018.
RMSE ratio over Life-table for ‘forecast accuracy’
Scenario | Male | Female | |||
---|---|---|---|---|---|
4-PFM(1) | LCM(2) | 4-PFM(1) | LCM(2) | ||
Scenario 1 | yr2018a | 1.500 | 1.854 | 1.359 | 1.222 |
5 yr avgb | 1.675 | 1.853 | 1.195 | 1.419 | |
Scenario 2 | yr2018 | 1.000 | 3.585 | 1.028 | 2.196 |
5 yr avg | 1.425 | 3.422 | 0.990 | 2.490 | |
Scenario 3 | yr2018 | 1.089 | 4.341 | 1.033 | 2.519 |
5 yr avg | 1.917 | 3.373 | 1.280 | 2.426 |
^{a}RMSE ratio of scenario1 over the life-table for forecasting yr2018.
^{b}Average RMSE ratios of scenario1 over the life-table for forecasting yr2014~yr2018.
Accuracy of decomposition of LCM:
Scenario | Male | Female |
---|---|---|
Life-table | 0.985 | 0.970 |
Scenario 1 | 0.935 | 0.919 |
Scenario 2 | 0.911 | 0.877 |
Scenario 3 | 0.925 | 0.915 |