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Stata Technical Bulletin

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(m_xt) = a_x+b_xk_t + e_xt

where m_xt is the observed age-specific death rate (ASDR) at age x during time t; a_x, b_x, and k_t are the model’s parameters,
and e_xt is an error term. a_x describes the general age shape of the ASDRs while k_t 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 (k_t) changes.

To estimate the model for a given set of ASDRs (m_art), 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 b_x to sum to unity and the k_t to sum to
0, which implies that a_x are simply the averages over time of the log(m_art).

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 k_t. 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: a_x,
ba, and k_t. 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

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