User:Forkandwait/Lee Carter Model

Overview

The Lee-Carter model is a numerical algorithm used in forecasting mortality rates. The input to the method is a matrix of mortality rates sorted monotonically by time, usually with ages in columns and different years in rows. The output is another forecasted matrix of mortality rates. Life expectancy and other life table measures can be derived from this new table. The algorithm (essentially) uses the Singular Value Decomposition (SVD) to find a univariate time series that captures 80-90% of the trend; in the literature, this time series is referred to as "$k_{t}$", with t signifying time. The algorithm also creates a base set of age specific mortality rates ("a_x") which have been transformed into logarithms and centered by the mean, and a vector ("b_x") that describes the amount of mortality change at a give age for a unit of overall mortality change. Future mortality is derived by forecasting k_t with univariate ARIMA methods, then using b_x and a_x to recover a set of logged mortality rates for each forecast year, and finally recovering regular mortality by calculating the exponential of the forecasted log mortality rates. In most implementations, confidence intervals for the forecasts are generated by running simulating multiple k_t's using Monte-Carlo methods. Additionally, many researchers adjust the k_t vector by fitting it to calculated life expectancies for each year, using a_x and b_x just generated; in this approach, changes to k_t are usually small.

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.

History of the model

The 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). See this http://www.soa.org/library/journals/north-american-actuarial-journal/2000/january/naaj0001_5.pdf article in the North American Actuarial Journal for a more contemporary overview, or some of the papers on Prof. Lee's website for extensions and applications.

Extensions and nuances

 Re-fixing e0. Coherent etc. =

Other coolness

SVD.  Linear quality of change.  Idea of regime of mortality change.  

Links

to people Larry Carter, Ron Lee, CEDA, Aussie guy

Papers -- King, SOA, Ron's history of LC