Stata Technical Bulletin
15
sbe40 Modeling mortality data using the Lee-Carter model
Duolao Wang, London School of Hygiene and Tropical Medicine, London, UK, [email protected]
Abstract: This article describes the Ieecart command that fits the Lee-Carter model for mortality forecasting. The Lee-Carter
model has been a very successful approach for long-term mortality projection and widely applied in demographic studies,
as well as actuary. A U.S. mortality dataset is used to illustrate the model estimation.
Keywords: Lee-Carter model, mortality forecasting, singular value decomposition.
The Lee-Carter model
Lee and Carter (1992) proposed a model for forecasting mortality based on the past mortality trends. The model can be
written as follows:
fxt = ⅛g(mxt) = ax+bxkt + ext
where mxt is the observed age-specific death rate (ASDR) at age x during time t; ax, bx, and kt are the model’s parameters,
and ext is an error term. ax describes the general age shape of the ASDRs while kt is an index of the general level of mortality.
Z⅛ coefficients describe the tendency of mortality at age x to change when the general level of mortality (kt) changes.
To estimate the model for a given set of ASDRs (mart), ordinary least squares can be applied.
The model evidently is underdetermined, which can be seen as follows. Suppose that a, b, к are one solution. Then for
any c, a — be, b, к + c also must be a solution. It is also clear that if a, b, к are a solution, then a, be, к/c also are a solution.
Therefore, к is determined only up to a linear transformation, b is determined only up to a multiplicative constant, and a is
determined only up to an additive constant. Lee and Carter proposed to normalize the bx to sum to unity and the kt to sum to
0, which implies that ax are simply the averages over time of the log(mart).
The model cannot be fitted by ordinary regression methods because there are no given regressors; on each side of the
equation we have only parameters to be estimated and the unknown index kt. However, the singular value decomposition (SVD)
method can be used to find a least squares solution when applied to the matrix of the logarithms of rates after the averages
over time of the (log) age-specific rates have been subtracted (Good 1969). The first right and left vectors and leading value of
SVD, after normalization described above, provides a unique solution. The Stata matrix function matrix svd is an ideal tool to
estimate the parameters in the Lee-Carter model.
Syntax
Ieecart var~year varage Varmortality [if exp [in range
Description
Ieecart generates three parameter matrices for the Lee-Carter model using the given age-period-specific death rates: ax,
ba, and kt. In addition, it yields a matrix of estimated age-specific mortality rates by year.
Examples
U.S. age-period-specific mortality rates from 1900 to 1995 are used below to demonstrate the use of the Ieecart command
for the estimation of the Lee-Carter model.
. use Ieecart
(US Death Rates by Year: 1900-1995)
. describe
Contains data from leecart.dta
obs : vars: |
: 11,520 |
US Death Rates by Year: 11 Aug 2000 11:11 | ||
: 3 : 184,320 |
(96.4% of memory free) | |||
— 1. |
year |
float |
7.9.0g |
— Year |
2. |
age |
float |
7.9.0g |
Age |
3. |
mort |
float |
7.9.0g |
Mortality |
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