Lee–Carter model

The Lee-Carter Model is a numerical algorithm used in mortality and life expectancy forecasting . The model is derived by using the Singular Value Decomposition (SVD) to find a univariate time series "k_t" that captures 80-90% of the mortality trend, along with a vector that determines how fast mortality is changing at specific ages (here the subscript "t" refers to time). Surprisingly, k_t is usually linear, implying that gains to life expectancy are fairly constant year after year in most populations. In addition to k_t, the algorithm creates a base set of age specific mortality rates a_x which have been transformed into logarithms and centered by their age-specific means over time, and a vector b_x that describes the amount of mortality change at a give age for a unit of overall mortality change (here the subscript "x" refers to age). Many researchers also adjust the k_t vector by fitting it to calculated life expectancies for each year, using the a_x and b_x just generated with the SVD; when adjusted using this approach, changes to k_t are usually small.

The input to the method is a matrix of mortality rates sorted monotonically by time, usually with ages in columns and years in rows. The output is another forecasted matrix of mortality rates. Life expectancy and other life table measures can be calculated from this forecasted matrix. In outline, the algorithm is as follows:

2. Create a_x by taking logarithms of the mortality rates and centering the results with the average log mortality at a given age.
3. Derive k_t, a scaling eigenvalue, and b_x from U(:,1), S(1,1), V(1,:), where [U S V] = svd(mort).
4. Forecast k_t with standard univariate ARIMA methods.
5. Use the forecast k_t with the original b_x and a_x to calculate logged mortality rates for each forecast year.
6. Recover regular mortality rates by calculating the exponential of the forecasted log mortality rates.

In most implementations, confidence intervals for the forecasts are generated by simulating multiple mortality forecasts using Monte-Carlo methods; a band of mortality between 5% and 95% percentiles of the simulated results is considered to be a valid forecast. These simulations are done by extending k_t into the future using randomization based on the standard error of k_t derived from the input data.

Without applying SVD or some other method of dimension reduction the table of mortality data is a highly correlated multivariate data series with each age group forming another dimension in addition to time; the complexity of these multidimensional time series makes such them almost impossible to forecast. SVD has become widely used as a method of dimension reduction in many disparate 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). 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 . 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 .

There have been extensions to the Lee-Carter, 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 Hyndman provides an R package for demography that includes routines for creating and forecasting a Lee-Carter Model. Professor German Rodriguez provides code for the Lee-Carter Model using Stata.