Lee–Carter model
The Lee–Carter model is a numerical algorithm used in
The model uses singular value decomposition (SVD) to find:
- A univariate time series vector that captures 80–90% of the mortality trend (here the subscript refers to time),
- A vector that describes the relative mortality at each age (here the subscript refers to age), and
- A scaling constant (referred to here as but unnamed in the literature).
Surprisingly, is usually linear, implying that gains to life expectancy are fairly constant year after year in most populations. Prior to computing SVD, age specific mortality rates are first transformed into , by taking their
Many researchers adjust the vector by fitting it to empirical life expectancies for each year, using the and generated with SVD. When adjusted using this approach, changes to are usually small.
To forecast mortality, (either adjusted or not) is projected into future years using an
Because of the linearity of , it is generally modeled as a random walk with trend. Life expectancy and other life table measures can be calculated from this forecasted matrix after adding back the means and taking exponentials to yield regular mortality rates.
In most implementations,
Algorithm
The algorithm seeks to find the least squares solution to the equation:
where is a matrix of mortality rate for each age in each year .
- Compute which is the average over time of for each age:
- Compute which will be used in SVD:
- Compute the singular value decomposition of :
- Derive , (the scaling eigenvalue), and from , , and :
- Forecast using a standard univariate ARIMAmodel to additional years:
- Use the forecasted , with the original , and to calculate the forecasted mortality rate for each age:
Discussion
Without applying SVD or some other method of
The Lee–Carter model was introduced by
There have been extensions to the Lee–Carter model, most notably to account for missing years, correlated male and female populations, and large scale coherency in populations that share a mortality regime (western Europe, for example). Many related papers can be found on Professor Ronald Lee's website.
Implementations
There are surprisingly few software packages for forecasting with the Lee–Carter model.
- LCFIT is a web-based package with interactive forms.
- Professor Rob J. Hyndman provides an R package for demography that includes routines for creating and forecasting a Lee–Carter model.
- Alternatives in R include the StMoMo package of Villegas, Millossovich and Kaishev (2015).
- Professor German Rodriguez provides code for the Lee–Carter Model using Stata.
- Using Matlab, Professor Eric Jondeau and Professor Michael Rockinger have put together the Longevity Toolboxfor parameter estimation.
References
- ^ "The Lee-Carter Method for Forecasting Mortality, with Various Extensions and Applications | SOA" (PDF). Archived from the original (PDF) on March 7, 2019. Retrieved September 28, 2010.
- doi:10.2307/2290201.
- ^ Lee, Ronald (June 5, 2003). "Reflections on Inverse Projection: Its Origins, Development, Extensions, and Relation to Forecasting".
- ^ Federico Girosi; Gary King. "Understanding the Lee-Carter Mortality Forecasting Method" (PDF). Harvard University. Retrieved April 12, 2023.