Lee–Carter model

The algorithm seeks to find the least squares solution to the equation:

${\displaystyle \ln {(\mathbf {m} _{x,t})}=\mathbf {a} _{x}+\mathbf {k} _{t}\mathbf {b} _{x}+\epsilon _{x,t}}$

where ${\displaystyle \mathbf {m} _{x,t}}$ is a matrix of mortality rate for each age ${\displaystyle x}$ in each year ${\displaystyle t}$.

1. Compute ${\displaystyle \mathbf {a} _{x}}$ which is the average over time of ${\displaystyle \ln {(\mathbf {m} _{x,t})}}$ for each age:

   ${\displaystyle \mathbf {a} _{x}={\frac {\sum _{t=1}^{T}{\ln {(\mathbf {m} _{x,t})}}}{T}}}$

2. Compute ${\displaystyle \mathbf {A} _{x,t}}$ which will be used in SVD:

   ${\displaystyle \mathbf {A} _{x,t}=\ln {(\mathbf {m} _{x,t})}-\mathbf {a} _{x}}$

3. Compute the singular value decomposition of ${\displaystyle \mathbf {A} _{x,t}}$:

   ${\displaystyle \mathbf {U} \mathbf {S} \mathbf {V^{*}} ={\text{svd}}(\mathbf {A} _{x,t})}$

4. Derive ${\displaystyle \mathbf {k} _{t}}$, ${\displaystyle s_{1}}$ (the scaling eigenvalue), and ${\displaystyle \mathbf {b} _{x}}$ from ${\displaystyle \mathbf {U} }$, ${\displaystyle \mathbf {S} }$, and ${\displaystyle \mathbf {V^{*}} }$:

   ${\displaystyle \mathbf {k} _{t}=(u_{1,1},u_{2,1},...,u_{t,1})}$

   ${\displaystyle \mathbf {b} _{x}=(v_{1,1},v_{1,2},...,v_{1,x})}$

5. Forecast ${\displaystyle \mathbf {k} _{t}}$ using a standard univariate ARIMA model to ${\displaystyle n}$ additional years:

   ${\displaystyle \mathbf {k} _{t+n}={\text{ARIMA}}(\mathbf {k} _{t},n)}$

6. Use the forecasted ${\displaystyle \mathbf {k} _{t+n}}$, with the original ${\displaystyle \mathbf {b} _{x}}$, and ${\displaystyle \mathbf {a} _{x}}$ to calculate the forecasted mortality rate for each age:

   ${\displaystyle \mathbf {m} _{x,t+n}=\exp(\mathbf {a} _{x}+s_{1}\mathbf {k} _{t+n}\mathbf {b} _{x})}$

Without applying SVD or some other method of dimension reduction the table of mortality data is a highly correlated multivariate data series, and the complexity of these multidimensional time series makes them difficult to forecast. SVD has become widely used as a method of dimension reduction in many different fields, including by Google in their page rank algorithm.

The Lee–Carter model was introduced by Ronald D. Lee and Lawrence Carter in 1992 with the article "Modeling and Forecasting the Time Series of U.S. Mortality," (Journal of the American Statistical Association 87 (September): 659–671).[2] The model grew out of their work in the late 1980s and early 1990s attempting to use inverse projection to infer rates in historical demography.[3] The model has been used by the United States Social Security Administration, the US Census Bureau, and the United Nations. It has become the most widely used mortality forecasting technique in the world today.[4]

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.

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 Toolbox for parameter estimation.