Assessing the Impact of the COVID-19 Shock on a Stochastic Multi-Population Mortality Model
2. Data and Notation
3. A Stochastic Multi-Population Mortality Standard of Type Li and Lee
3.1. The Li and Lee Mortality Model
- In a first step, we calibrate the parameters , and in the common mortality trend by assuming that the total number of deaths random variable follows a Poisson distribution with mean . Hereto, we maximize the following Poisson log-likelihood:
- In a second step, we calibrate the country-specific parameters and by assuming that the number of deaths random variable , in the country of interest c, follows a Poisson distribution with mean . Hereto, we maximize the Poisson log-likelihood, conditional on the estimates obtained in step 1:
3.2. The Time Dynamics
3.3. Generating Future Paths of Mortality Rates and Life Expectancies
3.4. The Li and Lee Mortality Model for the Belgian Population
4. Transforming Weekly Mortality Data in Age Buckets to Annual Mortality Data at Individual Ages
4.1. From Weekly to Annual Mortality Data Registered in Age Buckets
4.2. Ungrouping Data from Age Buckets to Individual Ages
4.3. Applying the Protocol to the Multi-Population Data Set to Recalibrate the Li and Lee Mortality Model for the Belgian Population
5. Assessing the Impact of a Pandemic Shock on the Multi-Population Mortality Model
5.1. Limiting the Time Series Likelihood Contribution of the Pandemic Data Point
5.2. Mitigating the Impact of the Pandemic Data Point with a Lee and Miller Inspired Mortality Model
6. Addressing a Mortality Shock in a Mortality Prediction Model: A Literature Review
7. Conclusions and Outlook
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Data Sources
Appendix B. Constructing Virtual Exposure Points
|Age Bucket||Male Exp.||Female Exp.|
Appendix C. Constructing Virtual Death Counts
|Age Bucket||Male Deaths||Female Deaths|
Appendix D. Validation of the Constructed Virtual Death Counts and Exposure Points
Numbers are retrieved from https://www.statista.com/statistics/1102209/coronavirus-cases-development-europe/ (accessed on 13 December 2021) and https://www.statista.com/statistics/1102288/coronavirus-deaths-development-europe/ (accessed on 13 December 2021) and represent the situation at 5 December 2021.
These numbers of COVID-19 deaths come from the COVID-19 Dashboard by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU), see https://www.arcgis.com/apps/opsdashboard/index.html#/bda7594740fd40299423467b48e9ecf6 (accessed on 13 December 2021).
See https://ec.europa.eu/info/live-work-travel-eu/coronavirus-response/safe-covid-19-vaccines-europeans_en (accessed on 21 October 2021) for an overview of the approved, European COVID-19 vaccines and those currently under development, as well as corresponding references.
See https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-working-papers/mortality-projections (accessed on 9 November 2021).
This database is our primary database and can be consulted at https://www.mortality.org/ (accessed on 13 April 2021).
Eurostat is the statistical office of the European Union, see https://ec.europa.eu/eurostat (accessed on 13 April 2021).
Statbel is the Belgian statistical office, see https://statbel.fgov.be/en (accessed on 13 April 2021).
This information can be explored using the visualization toolkit on https://mpidr.shinyapps.io/stmortality/ (accessed on 30 July 2021).
Eurostat provides weekly death statistics at https://ec.europa.eu/eurostat/web/COVID-19/data (accessed on 30 July 2021).
The years 1992, 1998, 2004, 2009, 2015 and 2020 contain 53 weeks instead of the usual 52 weeks (ISO 8601 standard).
The HMD team uses a Lee and Carter model to extrapolate recent exposures. The documentation can be consulted via https://www.mortality.org/Public/STMF_DOC/STMFNote.pdf (accessed on 30 July 2021).
See https://appsso.eurostat.ec.europa.eu/nui/show.do?dataset=demo_r_mwk_05&lang=en (accessed on 30 July 2021).
Eurostat only provides weekly death counts for Germany for age buckets of length 10.
We use the nlminb-function in the stats-package of R.
See https://data.worldbank.org/indicator/NY.GDP.PCAP.CD (accessed on 16 June 2021).
See https://www.iso.org/iso-8601-date-and-time-format.html (accessed on 3 November 2021).
The years 1992, 1998, 2004, 2009, 2015 and 2020 contain 53 weeks.
For all countries except the United Kingdom, we do have the annual exposures at an individual age level in 2019 from either HMD or Eurostat (see Table A1). For Denmark, we already have the annual exposures at an individual age level in the year 2020 available from the HMD.
We use the exposure points and to linearly extrapolate to age 0:
For all countries except the United Kingdom, we do have the annual death counts at individual ages in 2019 from either HMD or Eurostat (see Table A1). For Belgium and Denmark, we even have the annual death counts at an individual age level in 2020 available from HMD and Statbel, respectively. For these two countries, there is no need to create virtual death counts.
For Belgium and Denmark, we already have the death counts at individual ages in 2020 from Statbel and HMD, respectively. We take the average of both ratios.
We only use the weekly death counts collected in age buckets from Eurostat when they match the reported death counts in the larger age buckets from the STMF data series. We do this for safety reasons because some deviations between the weekly death counts on Eurostat and the STMF data series may occur due to for example territorial differences, e.g., France with or without overseas regions.
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|Cohort Life Expectancy in 2020||Males||Females|
|Recalibration 2020 weight = 0||Best. Est.||89.36||19.70||91.42||22.78|
|Recalibration 2020 weight = 0.25||Best. Est.||89.07||19.65||91.11||22.65|
|Recalibration 2020 weight = 0.50||Best. Est.||88.77||19.58||90.78||22.56|
|Recalibration 2020 weight = 0.75||Best. Est.||88.45||19.52||90.46||22.48|
|Recalibration 2020 weight = 1||Best. Est.||88.13||19.45||90.13||22.41|
|IA∣BE 2020||Best. Est.||89.91||20.38||91.54||23.14|
|Cohort Life Expectancy in 2020||Males||Females|
|Recalibration 2020 weight = 0||Best. Est.||89.95||20.31||92.15||23.32|
|Recalibration 2020 weight = 0.25||Best. Est.||89.25||19.89||91.58||22.99|
|Recalibration 2020 weight = 0.50||Best. Est.||88.60||19.57||91.12||22.82|
|Recalibration 2020 weight = 0.75||Best. Est.||88.05||19.37||90.71||22.71|
|Recalibration 2020 weight = 1||Best. Est.||87.57||19.19||90.32||22.58|
|IA∣BE 2020||Best. Est.||89.91||20.38||91.54||23.14|
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Robben, J.; Antonio, K.; Devriendt, S. Assessing the Impact of the COVID-19 Shock on a Stochastic Multi-Population Mortality Model. Risks 2022, 10, 26. https://doi.org/10.3390/risks10020026
Robben J, Antonio K, Devriendt S. Assessing the Impact of the COVID-19 Shock on a Stochastic Multi-Population Mortality Model. Risks. 2022; 10(2):26. https://doi.org/10.3390/risks10020026Chicago/Turabian Style
Robben, Jens, Katrien Antonio, and Sander Devriendt. 2022. "Assessing the Impact of the COVID-19 Shock on a Stochastic Multi-Population Mortality Model" Risks 10, no. 2: 26. https://doi.org/10.3390/risks10020026