# A New Fourier Approach under the Lee-Carter Model for Incorporating Time-Varying Age Patterns of Structural Changes

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## Abstract

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## 1. Introduction

## 2. Methodology

**splines**”. Again, we modeled the cubic-spline parameters as time series processes to predict future age sensitivity. In matrix notation, the age sensitivity factor under the cubic Lee-Carter model could be written as follows,

**ns**” in R. Then the $\left(4\times 1\right)$ vector ${c}_{t}={\left({c}_{1,t},{c}_{2,t},{c}_{3,t},{c}_{4,t}\right)}^{\prime}$ containing the time-varying cubic-spline coefficients was estimated via linear regression.

## 3. Projections

## 4. Backtesting

#### 4.1. Rolling-Window Backtesting

#### 4.2. Backtesting with Simulation

## 5. Integrating the Age and Period Effects of Structural Changes

**R**package ‘

**strucchange**’ to identify the potential structural change for male mortality in the United Kingdom between 1950 and 2001. The structural change is detected in 1979, which is in accordance with the graphical observation. For more details of this modelling strategy, one may refer to Section 3 of van Berkum et al. (2013) who investigated the period effects under various mortality models. The estimated value of $\widehat{\mu}$ based on the whole fitting period is −1.35, and the ${\widehat{\mu}}_{2}$ calculated using the latest period is −2.25.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Age sensitivity factors under the Lee-Carter model estimated from sequential subperiods with different fixed lengths. The legend $b\left[x,t\right]$ refers to the age response ${\beta}_{x}$ calibrated on a subperiod ending in year $t$, $tend$ refers to the ending year of the entire sample period.

## Notes

1 | The mortality data for New Zealand are available up to 2013 at the time of collection. |

2 | In the following, we use the terms age sensitivity and age response interchangeably. |

3 | The general Fourier series contains sinusoidal waves in the time dimension. We borrow the functional form and apply it to depict the shape of the age sensitivity parameter in the Lee-Carter model. |

4 | Choosing overlapping subperiods is a common choice in modelling structural changes (e.g., Goyal and Welch 2003; Inoue et al. 2017). We use overlapping rather than distinct subperiods, as the latter would not provide sufficient data points for reliable estimation and prediction of the time-varying age sensitivity factors. For instance, the whole sample period of 67 years (1950–2016) can only be split into fewer than 4 non-overlapping subsamples if the length of each subperiod is required to be at least 20 years. There are then at most only 4 data points available to calibrate the time series models for the time-varying Fourier (cubic-spline) parameters. |

5 | The original Lee-Carter model (Lee and Carter 1992) was estimated via singular vector decomposition with an assumption that the error terms are homoscedastic, which can be unrealistic (Alho 2000). Brouhns et al. (2002) improved the fitting approach by assuming a Poisson distribution for the age-specific number of deaths in each year. The model parameters are then estimated using maximum likelihood method. We use the Poisson assumption in this article, although other distributions can be employed based on particular datasets (e.g., Pitt et al. 2018; Wong et al. 2018; Awad et al. 2022). To avoid the identification problem when fitting the Lee-Carter model to the entire sample, the constraints ${{\displaystyle \sum}}_{t}{k}_{t}=0$ and ${{\displaystyle \sum}}_{x}{\beta}_{x}=1$ are adopted, where ${\beta}_{x}$ refers to the age response estimated from the entire sample. |

6 | We also repeat the rolling-window backtesting using different lengths of 20 and 30 years. The results (not presented in the article) are comparable to those presented in Section 4.1. |

7 | Although not presented here, the differences in heights between the waves for younger and older ages are rather significant when a shorter sample period is employed. |

8 | As illustrated in Figure 5, one set of Fourier parameters for each region appears to be adequate in portraying the pattern of age sensitivity factors. |

9 | Our purpose is to demonstrate one way of incorporating the time-varying age patterns of mortality development. Other time series processes can be adopted if one sets different assumptions about the future age patterns or if the data exhibit different trends. |

10 | We apply the Coale-Kisker method (Coale and Kisker 1990) to extend mortality predictions up to a predetermined maximum age of 110 with an ultimate mortality rate of 0.7 (Gampe 2010). |

11 | For simplicity we consider the difference stationary model with one breakpoint. For a test of multiple breakpoints, one may refer to van Berkum et al. (2016). |

## References

- Alho, Juha M. 2000. “The Lee-Carter Method for Forecasting Mortality, with Various Extensions and Applications”, Ronald Lee, January 2000. North American Actuarial Journal 4: 91–93. [Google Scholar] [CrossRef]
- Amaral, Luiz Felipe, Reinaldo Castro Souza, and Maxwell Stevenson. 2008. A smooth transition periodic autoregressive (STPAR) model for short-term load forecasting. International Journal of Forecasting 24: 603–15. [Google Scholar] [CrossRef]
- Armstrong, Ben. 2006. Models for the Relationship between Ambient Temperature and Daily Mortality. Epidemiology 17: 624–31. Available online: http://www.jstor.org.simsrad.net.ocs.mq.edu.au/stable/20486290 (accessed on 14 March 2020). [CrossRef] [PubMed]
- Awad, Yaser, Shaul K. Bar-Lev, and Udi Makov. 2022. A New Class of Counting Distributions Embedded in the Lee–Carter Model for Mortality Projections: A Bayesian Approach. Risks 10: 111. [Google Scholar] [CrossRef]
- Booth, Heather, Rob Hyndman, Leonie Tickle, and Piet de Jong. 2006. Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research 15: 289–310. Available online: https://EconPapers.repec.org/RePEc:dem:demres:v:15:y:2006:i:9 (accessed on 30 April 2020). [CrossRef]
- Bracewell, Ronald N. 1978. The Fourier Transform and Its Applications, 2nd ed. New York: McGraw-Hill. [Google Scholar]
- Brouhns, Natacha, Michel Denuit, and Jeroen K. Vermunt. 2002. A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics 31: 373–93. Available online: https://EconPapers.repec.org/RePEc:eee:insuma:v:31:y:2002:i:3:p:373-393 (accessed on 15 November 2019). [CrossRef] [Green Version]
- Cairns, Andrew J. G., David Blake, and Kevin Dowd. 2006. A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of Risk and Insurance 73: 687–718. [Google Scholar] [CrossRef]
- Cairns, Andrew J. G., David Blake, Kevin Dowd, Guy D. Coughlan, David Epstein, Alen Ong, and Igor Balevich. 2009. A Quantitative Comparison of Stochastic Mortality Models Using Data From England and Wales and the United States. North American Actuarial Journal 13: 1–35. [Google Scholar] [CrossRef]
- Canisius, Francis, Hugh Turral, and David J. Molden. 2007. Fourier analysis of historical NOAA time series data to estimate bimodal agriculture. International Journal of Remote Sensing 28: 5503–22. [Google Scholar] [CrossRef]
- Carter, Lawrence R., and Alexia Prskawetz. 2001. Examining structural shifts in mortality using the Lee-Carter method. Methoden und Ziele 39. Available online: https://www.demogr.mpg.de/Papers/Working/wp-2001-007.pdf (accessed on 3 June 2022).
- Coale, Aj, and Ee Kisker. 1990. Defects in data on old-age mortality in the United States: New procedures for calculating mortality schedules and life tables at the highest ages. In Asian and Pacific Population Forum. Honolulu: East-West Population Institute, vol. 4. [Google Scholar]
- Coelho, Edviges, and Luis C. Nunes. 2011. Forecasting mortality in the event of a structural change. Journal of the Royal Statistical Society: Series A (Statistics in Society) 174: 713–36. [Google Scholar] [CrossRef]
- Currie, Iain D., Maria Durban, and Paul H. Eilers. 2004. Smoothing and forecasting mortality rates. Statal Modelling An International Journal 4: 279–98. [Google Scholar] [CrossRef]
- De Jong, Piet, and Leonie Tickle. 2006. Extending Lee–Carter Mortality Forecasting. Mathematical Population Studies 13: 1–18. [Google Scholar] [CrossRef]
- Debón, Ana, Francisco Montes, and Ramón Sala. 2006. A Comparison of Nonparametric Methods in the Graduation of Mortality: Application to Data from the Valencia Region (Spain). International Statistical Review/Revue Internationale de Statistique 74: 215–33. Available online: http://www.jstor.org/stable/25472704 (accessed on 2 January 2020).
- Dowd, Kevin, Andrew J. G. Cairns, David Blake, Guy D. Coughlan, David Epstein, and Marwa Khalaf-Allah. 2010. Backtesting Stochastic Mortality Models. North American Actuarial Journal 14: 281–98. [Google Scholar] [CrossRef]
- Gampe, Jutta. 2010. Human mortality beyond age 110. In Supercentenarians. Edited by Heiner Maier, Jutta Gampe, Bernard Jeune, Jean-Marie Robine and James W. Vaupel. Berlin and Heidelberg: Springer, pp. 219–30. [Google Scholar]
- Goyal, Amit, and Ivo Welch. 2003. Predicting the equity premium with dividend ratios. Management Science 49: 639–54. [Google Scholar] [CrossRef] [Green Version]
- Gysen, Bart L. J., Esin Ilhan, Koen J. Meessen, Johannes J. H. Paulides, and Elena A. Lomonova. 2010. Modeling of Flux Switching Permanent Magnet Machines With Fourier Analysis. IEEE Transactions on Magnetics 46: 1499–502. [Google Scholar] [CrossRef] [Green Version]
- Heligman, L., and J. H. Pollard. 1980. The age pattern of mortality. Journal of the Institute of Actuaries 107: 49–80. [Google Scholar] [CrossRef]
- Human Mortality Database. 2019. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available online: www.mortality.org (accessed on 10 December 2019).
- Inoue, Atsushi, Lu Jin, and Barbara Rossi. 2017. Rolling window selection for out-of-sample forecasting with time-varying parameters. Journal of econometrics 196: 55–67. [Google Scholar] [CrossRef] [Green Version]
- Lee, R. Ronald, and Lawrence R. Carter. 1992. Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association 87: 659–71. [Google Scholar] [CrossRef]
- Lee, Ronald, and Timothy Miller. 2000. Assessing the Performance of the Lee-Carter Approach to Modeling and Forecasting Mortality. Los Angeles: Annual Meeting of the Population Association of America. [Google Scholar]
- Li, Jackie. 2013. A Poisson common factor model for projecting mortality and life expectancy jointly for females and males. Population Studies 67: 111–26. [Google Scholar] [CrossRef]
- Li, Nan, and Ronald Lee. 2005. Coherent Mortality Forecasts for a Group of Populations: An Extension of the Lee-Carter Method. Demography 42: 575–94. Available online: http://www.jstor.org/stable/4147363 (accessed on 5 November 2019). [CrossRef] [PubMed] [Green Version]
- Li, Jackie, and Kenneth Wong. 2020. Incorporating structural changes in mortality improvements for mortality forecasting. Scandinavian Actuarial Journal 2020: 776–91. [Google Scholar] [CrossRef]
- Li, Johnny S.-H., Wai-Sum Chan, and Siu-Hung Cheung. 2011. Structural Changes in the Lee-Carter Mortality Indexes. North American Actuarial Journal 15: 13–31. [Google Scholar] [CrossRef]
- Li, Nan, Ronald Lee, and Patrick Gerland. 2013. Extending the Lee-Carter method to model the rotation of age patterns of mortality decline for long-term projections. Demography 50: 2037–51. [Google Scholar] [CrossRef] [Green Version]
- Lombardet, Benoît, L. Andrea Dunbar, Rolando Ferrini, and Romuald Houdré. 2005. Fourier analysis of Bloch wave propagation in photonic crystals. Journal of the Optical Society of America B 22: 1179–90. [Google Scholar] [CrossRef]
- Mei, Liang, and Sune Svanberg. 2015. Wavelength modulation spectroscopy—Digital detection of gas absorption harmonics based on Fourier analysis. Applied Optics 54: 2234–43. [Google Scholar] [CrossRef]
- Milidonis, Andreas, Yijia Lin, and Samuel H. Cox. 2011. Mortality Regimes and Pricing. North American Actuarial Journal 15: 266–89. [Google Scholar] [CrossRef]
- O’Hare, Colin, and Youwei Li. 2015. Identifying Structural Breaks in Stochastic Mortality Models. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering 1: 021001. [Google Scholar] [CrossRef]
- Perron, Pierre. 1989. The great crash, the oil price shock, and the unit root hypothesis. Econometrica: Journal of the Econometric Society 57: 1361–401. [Google Scholar] [CrossRef]
- Pitt, David, Jackie Li, and Tian Kang Lim. 2018. Smoothing Poisson common factor model for projecting mortality jointly for both sexes. ASTIN Bulletin: The Journal of the IAA 48: 509–41. [Google Scholar] [CrossRef]
- Powers, Michael R., and Thomas Y. Powers. 2015. Fourier-analytic measures for heavy-tailed insurance losses. Scandinavian Actuarial Journal 2015: 527–47. [Google Scholar] [CrossRef]
- Sweeting, Paul J. 2011. A Trend-Change Extension of the Cairns-Blake-Dowd Model. Annals of Actuarial Science 5: 143–62. [Google Scholar] [CrossRef]
- van Berkum, Frans, Katrien Antonio, and Michel Vellekoop. 2013. Structural Changes in Mortality Rates with an Application to Dutch and Belgian Data. FEB Research Report AFI-1379: 1–27. [Google Scholar] [CrossRef]
- van Berkum, Frans, Katrien Antonio, and Michel Vellekoop. 2016. The impact of multiple structural changes on mortality predictions. Scandinavian Actuarial Journal 2016: 581–603. [Google Scholar] [CrossRef]
- Vyas, Seema, and Lilani Kumaranayake. 2006. Constructing socio-economic status indices: How to use principal components analysis. Health Policy and Planning 21: 459–68. [Google Scholar] [CrossRef] [Green Version]
- Wong, Jackie S. T., Jonathan J. Forster, and Peter W. F. Smith. 2018. Bayesian mortality forecasting with overdispersion. Insurance: Mathematics and Economics 83: 206–21. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Age sensitivity factors under the Lee-Carter model estimated from sequential subsamples with a fixed sample size.

**Figure 3.**Age sensitivity factors under the Lee-Carter model estimated from sequential subperiods (using the same set of mortality level and mortality index parameters). The legend $b\left[x,t\right]$ refers to the age response ${\beta}_{x}$ calibrated on a subperiod ending in year $t$.

**Figure 4.**Curves of $\mathrm{sin}\left(x+{P}_{t}^{\left(i\right)}\right)$ for different phase values. The left (right) column indicates possible shapes for the Fourier series before (after) the splitting age. The parts on the left of the two black dashed (vertical) lines in the left and right plots denote the shape of a quarter and half of a complete cycle, respectively.

**Figure 5.**Estimated (solid lines) and fitted (dashed lines) age sensitivity factor under the Lee-Carter model estimated from the latest 24-year subperiod. The blue (red) lines refer to fitted values using the Fourier (cubic-spline) method.

**Figure 6.**Observed (blue lines) and projected (black lines) life expectancy at birth (${e}_{0}$), calibrated on mortality data from the whole sample period. The solid lines correspond to forecasts under the original Lee-Carter model. The dashed and dotted lines refer to projections from the Fourier Lee-Carter model and cubic Lee-Carter model, respectively.

**Figure 7.**Observed values (solid lines), projected values (dotted lines) and 95% prediction intervals (dashed lines) of life expectancies at birth (top panel) and age 60 (bottom panel), calibrated on years 1950–2001. The blue lines correspond to the projections from the original Lee-Carter model, and the red (grey) lines refer to the projections from the Fourier (cubic) Lee-Carter model.

**Figure 8.**Parameter estimates of mortality index ${k}_{t}$ under the original Lee-Carter model for the United Kingdom, calibrated on years 1950–2001. The red dashed line is based on two random walk processes connected at a breakpoint. The blue dashed line is based on a single random walk process over the whole fitting period.

**Figure 9.**Observed values (solid lines), projected values (dotted lines), and 95% prediction intervals (dashed lines) of life expectancies at ages 0 (panel (

**a**)) and 60 (panel (

**b**)) for the United Kingdom, calibrated on years 1950–2001.

**Figure 10.**Observed values (solid lines), projected values (dotted lines) and 95% prediction intervals (dashed lines) of life expectancies at birth (top panel) and age 60 (bottom panel), calibrated on years 1950-2001. The blue lines correspond to the projections from the original Lee-Carter model, and the red (grey) lines refer to the projections from the Fourier (cubic) Lee-Carter model. The mortality index is modelled as piecewise RWD.

**Table 1.**MAPE values (%) of projected log central death rates of all ages and 30-year age groups (rolling-window basis). The minimum MAPE values for each age group and each country are given in bold.

Age Group | Overall | 0–29 | 30–59 | 60–89 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Population | LC | Fourier | Cubic | LC | Fourier | Cubic | LC | Fourier | Cubic | LC | Fourier | Cubic |

Australia | 3.02 | 2.92 | 2.88 | 3.23 | 3.34 | 3.24 | 1.82 | 1.74 | 1.70 | 4.01 | 3.67 | 3.70 |

Canada | 2.66 | 2.31 | 2.53 | 2.54 | 2.33 | 2.31 | 1.39 | 1.36 | 1.32 | 4.05 | 3.23 | 3.95 |

New Zealand | 4.50 | 4.36 | 4.35 | 5.21 | 5.29 | 5.17 | 2.98 | 2.95 | 2.91 | 5.31 | 4.85 | 4.97 |

Sweden | 3.64 | 3.29 | 3.52 | 4.38 | 4.67 | 4.30 | 2.96 | 2.80 | 2.86 | 3.58 | 2.40 | 3.39 |

United Kingdom | 2.71 | 2.51 | 2.57 | 2.25 | 2.25 | 2.21 | 1.26 | 1.23 | 1.13 | 4.62 | 4.06 | 4.38 |

United States | 2.03 | 1.92 | 1.96 | 1.59 | 1.62 | 1.58 | 1.58 | 1.64 | 1.49 | 2.93 | 2.48 | 2.80 |

**Table 2.**MAPE values (%) of projected life expectancies at ages 0 and 60 (rolling-window basis). The minimum MAPE values for each age and each country are given in bold.

Age Group | ${\mathit{e}}_{0}$ | ${\mathit{e}}_{60}$ | ||||
---|---|---|---|---|---|---|

Population | LC | Fourier | Cubic | LC | Fourier | Cubic |

Australia | 1.31 | 1.22 | 1.19 | 4.16 | 3.72 | 3.78 |

Canada | 1.17 | 0.87 | 1.10 | 4.42 | 3.14 | 4.15 |

New Zealand | 1.78 | 1.64 | 1.64 | 4.90 | 4.23 | 4.43 |

Sweden | 1.30 | 0.90 | 1.18 | 3.98 | 2.07 | 3.67 |

United Kingdom | 1.46 | 1.31 | 1.34 | 5.35 | 4.59 | 4.86 |

United States | 0.87 | 0.83 | 0.82 | 3.08 | 2.45 | 2.81 |

**Table 3.**Proportions (%) of observed life expectancies at ages 0 and 60 falling outside 95% prediction intervals, predicted over the last 15 years of the sample.

Age Group | ${\mathit{e}}_{0}$ | ${\mathit{e}}_{60}$ | ||||
---|---|---|---|---|---|---|

Population | LC | Fourier | Cubic | LC | Fourier | Cubic |

Australia | 0 | 0 | 0 | 60 | 20 | 20 |

Canada | 73 | 47 | 73 | 87 | 87 | 87 |

New Zealand | 17 | 0 | 0 | 75 | 0 | 50 |

Sweden | 53 | 0 | 13 | 67 | 0 | 60 |

United Kingdom | 40 | 13 | 20 | 87 | 53 | 67 |

United States | 7 | 0 | 0 | 87 | 67 | 87 |

**Table 4.**Proportions (%) of observed life expectancies at ages 0 and 60 falling outside 95% prediction intervals, predicted over the last 15 years of the sample. The mortality index is modelled by a piecewise RWD with one breakpoint.

Age Group | ${\mathit{e}}_{0}$ | ${\mathit{e}}_{60}$ | ||||
---|---|---|---|---|---|---|

Population | LC | Fourier | Cubic | LC | Fourier | Cubic |

Australia | 0 | 0 | 0 | 0 | 0 | 0 |

Canada | 20 | 0 | 7 | 87 | 27 | 87 |

New Zealand | 0 | 0 | 0 | 0 | 0 | 0 |

Sweden | 0 | 0 | 0 | 0 | 0 | 0 |

United Kingdom | 0 | 0 | 0 | 33 | 0 | 0 |

United States | 0 | 0 | 0 | 80 | 13 | 40 |

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**MDPI and ACS Style**

Tang, S.; Li, J.; Tickle, L.
A New Fourier Approach under the Lee-Carter Model for Incorporating Time-Varying Age Patterns of Structural Changes. *Risks* **2022**, *10*, 147.
https://doi.org/10.3390/risks10080147

**AMA Style**

Tang S, Li J, Tickle L.
A New Fourier Approach under the Lee-Carter Model for Incorporating Time-Varying Age Patterns of Structural Changes. *Risks*. 2022; 10(8):147.
https://doi.org/10.3390/risks10080147

**Chicago/Turabian Style**

Tang, Sixian, Jackie Li, and Leonie Tickle.
2022. "A New Fourier Approach under the Lee-Carter Model for Incorporating Time-Varying Age Patterns of Structural Changes" *Risks* 10, no. 8: 147.
https://doi.org/10.3390/risks10080147