Mathematical Population Studies, 13:1–18, 2006
Copyright # Taylor & Francis Group, LLC
ISSN: 0889-8480 print=1543-5253 online
DOI: 10.1080/08898480500452109
Extending Lee–Carter Mortality Forecasting
Piet de Jong
Leonie Tickle
Actuarial Studies, Macquarie University, Sydney, Australia
The Lee–Carter method for mortality forecasting is outlined, discussed and
improved utilizing standard time series approaches. The new framework, which
integrates estimation and forecasting, delivers more robust results and permits
more detailed insight into underlying mortality dynamics. An application to
women’s mortality data illustrates the methods.
Keywords: Lee–Carter method; forecasting; Kalman filtering; mortality
1. INTRODUCTION
Figure 1 represents the mortality rates, in logarithm, of Australian
women and female children from ages from 0 to 100 and calendar
years 1921 to 2000. This data set is analyzed in Booth et al. (2002).
The so-called ‘‘Lee–Carter’’ (LC) method (Lee and Carter, 1992) is
often used to forecast mortality rates. We aim to improve on the LC
approach by strengthening both the modeling and estimation framework. Additional previous work related to the LC method includes Bozik
and Bell (1987), Bell and Monsell (1991), Bell (1992), Lee and Miller
(2001) and Bell (1997). Wilmoth (1993) suggested improvements in the
modeling basis and computational methods. Girosi and King (2004), in
a forthcoming book, suggest an alternative Bayesian approach, while a
two dimensional spline approach is developed in Currie et al. (2002).
The Lee–Carter method utilizes singular value decomposition
(SVD) estimation, discussed in Section 3.1. An alternative least
squares estimation method is developed in Section 3.2. The overparametrization of the raw form of the LC model leads to fitted age-related
Address correspondence to Piet de Jong, Department of Actuarial Studies, Division of
Economic and Financial Studies, Macquarie University, NSW 2109, Australia. E-mail:
pdejong@efs.mq.edu.au
1
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P. de Jong and L. Tickle
FIGURE 1 Log-mortality of Australian women and female children,
1921–2000.
functions likely to be too jagged. In Section 4 we introduce a ‘‘smooth’’
version of the LC model, called LC(smooth), which retains the attractive features of LC but with a minimal parametrization. In Section 5,
we discuss maximum likelihood estimation applicable even with more
detailed time series specifications. In Sections 6 and 7, we discuss
extensions and specializations of the LC(smooth) model.
2. THE LEE–CARTER METHOD AND EXTENSIONS
Let yit be the logarithm of the central mortality rate at age
i ¼ 0; 1; . . . ; p and time t ¼ 1; . . . ; n. Put yt ðy0t ; . . . ; ypt Þ0 . Lee and
Carter (1992) proposed to model mortality as:
yt ¼ a þ bat þ Et ;
t ¼ 1; . . . ; n:
ð1Þ
where a and b are ðp þ 1Þ-vectors of unknown parameters, Et is a vector
of disturbance terms, and at is a parameter varying with time such as
a random walk with drift. Here a measures the time invariant age
effect on mortality and b the age response to mortality trends. Parameters a and b and the time series at are to be estimated.
Put Y ðy1 ; . . . ; yn Þ, implying Y is a ðp þ 1Þ n matrix. Then a is
estimated as the vector of row means of Y denoted y with components
Extending Lee–Carter Mortality Forecasting
3
equal to the average log-mortality over the period for each age. The
SVD is then used on the mean corrected log-mortality rates:
Y y10 ’ UDV 0 where 1 denotes a column vector of ones and ’ indicates the retention of a small number of the largest singular values.
Then bb ¼ U, ðb
a1 ; . . . ; b
an Þ ¼ DV 0 , and hats indicate estimates.
Lee and Carter (1992) proposed
only the first singular
P retaining P
value and used the normalization bb ¼ 1 and b
a ¼ 0. Other authors
among those mentioned above have proposed using two or more components to model mortality improvements. In the latter case, at is a
vector and b a matrix with one column for each component of at .
Improvements include the specification
yt ¼ Xa þ Xbat þ Et :
ð2Þ
Here X is a known ‘‘design’’ matrix with more rows than columns
unless X ¼ I in which case (2) reduces to (1). As in (1), a and b are
unknown while at is a time series. We call this model the LC(smooth)
model. Model (2) is developed and discussed in Section 4, after the
evaluation of (1) in the next subsections. In models (1) or (2), smoothness across time results from imposing a time series structure on at .
2.1. Forecasting using the LC Approach
In the original formulation, Lee and Carter (1992) suggested an
adjustment to b
at so that the model reproduced the observed total
annual deaths. Alternative criteria for this ‘‘second stage estimation’’
have also been proposed (Lee and Miller, 2001; Booth et al., 2002).
However b
at is derived, a time series model is fitted to the estimates.
Lee and Carter (1992) found that a simple random walk with drift
was an appropriate model for the U.S. data they studied, and although
they highlighted the possibility of more general models, the random
walk with drift is typically used in applications.
In the general case where more than one singular value is retained,
time series models are fitted to each component of b
at . The components
are then forecast together with appropriate error bounds. Multiplying
the forecast components b
anþs by the estimated age effects bb and adding
ab gives the forecast age specific log-mortality. Thus the forecast logmortality rates out to year n þ m are
anþ1 ; . . . ; b
anþm Þ;
ab10 þ bbðb
ð3Þ
where n is the latest observed time point. Cohort aged k at year t ¼ n is
thus forecast to experience future log-mortality displayed along subdiagonal k þ 1 of (3). Cohorts to be born at t ¼ n þ k are predicted to
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P. de Jong and L. Tickle
experience mortality corresponding to superdiagonal k 1 of (3). And
the approximate proportional change in mortality rate at the different
ages out to year n þ m are forecast as bbðb
anþm b
an Þ.
2.2. Application to Women’s Mortality Data
The Australian women’s mortality data are now used to illustrate the
LC approach, thereby providing a basis for comparison. Only one
singular value is retained, and b
at is re-estimated as in the original
Lee and Carter (1992) formulation. The top left panel of Figure 2
shows y, termed the base rates, which is the estimate of a. The bottom
left panel shows the time series plot of b
at corresponding to the first
singular value. This is close to a straight line, which suggests an
exponential decline in mortality over the period. The top right panel
shows the estimate of b constructed using the SVD method and hence
the response by age to the general improvement in mortality over
time. The younger ages appear the most responsive to the improving
trend in mortality. The downward bump around age 17 indicates
decreased response to the improving trend on account of increased
traffic related mortality. The bottom right panel shows fitted year
2000 log-mortalities and forecast log-mortalities at 10-year intervals
from 2010 to 2040. Forecast log-mortalities for 2050 have also been
shown to enable comparison with the results in Figure 8.
FIGURE 2 Results from a Lee–Carter analysis of women’s mortality data.
Extending Lee–Carter Mortality Forecasting
5
For Australian women’s mortality data, time series analysis shows
that b
at is adequately described by a random walk with drift 2.34 and
standard deviation 5.04. The average age effect is 0.0099, which translates to an average improvement in mortality, averaging across all
ages, of 0:0099 2:34 ¼ 2:32% per year.
3. IMPROVING THE LC APPROACH: ESTIMATION
This article suggests improvements to the LC method. Suggested
improvements are the LC(smooth) model (2) and improving the estimation and forecasting basis. This section begins by examining potential shortcomings of the SVD estimation technique usually applied in
the context of the LC model (1), and second suggests a least squares
improvement. This least squares improvement is a first step towards
likelihood based estimation as outlined in Section 5.
3.1. Features of the SVD Estimation Method
In model (1), ai is estimated as the average, over the observed period,
of the log-mortality rate at age i: abi yi , i ¼ 0; . . . ; p where yi is the
mean of row i of Y. We shall show simple relationships for the single
component SVD estimates of bi and at , namely
corðbbi ; si Þ 1;
corðb
at ; yt Þ 1;
ð4Þ
where yt is the mean of column t of Y and si is the standard deviation
of row i of Y and cor denotes correlation.
Insight into the accuracy of (4) is gained from the two panels of
Figure 3 which show, for the women’s mortality data, bbi against si
for i ¼ 0; . . . ; 100 and b
at (without adjustment) against yt for
t ¼ 1; . . . ; 80. Both show a strong linear relationship justifying (4). Thus
the SVD estimates reduce to, in essence, simple means and standard
deviations. These estimates are insensitive to the more subtle aspects
FIGURE 3 Plots of ðsi ; bbi Þ and ðyt ; b
at Þ for women’s mortality data.
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P. de Jong and L. Tickle
of mortality trends and form a crude basis for mortality projection. This
argues for more refined estimation and forecasting methods.
To explain the approximations in (4), put M Y y1 ’ UDV 0
where UDV 0 is a one component singular value decomposition. Then
bb ¼ U and b
a ðb
a1 ; . . . ; b
an Þ0 ¼ VD are eigenvectors of MM 0 and M 0 M,
respectively, corresponding to the largest eigenvalue, D2 . These eigenvectors maximize b0 MM 0 b and aM 0 Ma with respect to b and a, respectively, subject to appropriate normalization. Thus bb is the normalized
linear combination of age-specific mortality which has maximum variance across the calendar years. Since M has mean zero along each row,
diagonal entries of MM 0 are proportional to the estimated variances s2i .
Component bbi tends to be large with si , leading to the first relationship in (4). Thus the age-response estimates bbi are essentially a measure of standard deviation in log-mortality for age i across time.
For the second approximation in (4), the columns of M are not mean
corrected. If m is the vector of column means then
m ðy1 y; . . . ; yn yÞ0 ;
a0 M 0 Ma ¼ a0 ðM 1m0 Þ0 ðM 1m0 Þa þ ðp þ 1Þða0 mÞ2 ;
ð5Þ
where y is the mean of the entries in Y. Maximizing the second
expression with respect to the components of a yields the (unnormalized)
eigenvector of M 0 M. Its components with large mt yt y are heavily
represented, leading to the second approximation in (4). Thus b
at is
essentially the average log-mortality over all ages in calendar year t.
3.2. Least Squares Estimation of the LC Model
One improvement to the ‘‘raw’’ implementation of the LC model as discussed in Section 2 is to integrate the estimation of the time series structure with the estimation of the age-related parameters. This section
displays a least squares approach. This sets the stage for generalized
and improved estimation methods outlined in subsequent sections.
Consider a one component model consisting of a random walk with
drift:
Et
yt ¼ a þ bat þ Et ;
atþ1 ¼ d þ at þ kgt ;
ð6Þ
ð0; r2 IÞ
gt
Multiplying b by a constant and dividing at by the same constant
leaves the model unchanged so that we assume b0 b ¼ 1. Replacing at
by c þ at , t ¼ 1; . . . ; n and a by a cb leads to the same model, so that
we assume a0 ¼ 0. k is a ‘‘signal-to-noise ratio’’ statistic.
Extending Lee–Carter Mortality Forecasting
7
From (6):
Eðyt ja0 Þ ¼ a þ bða0 þ dtÞ;
EðDyt Þ ¼ db;
ð7Þ
where Dyt yt yt1 . Straightforward calculations show
r2 fI þ k2 tbb0 g;
t ¼ s;
covðyt ; ys ja0 Þ ¼ 2 2
r k minðt; sÞbb0 ; t 6¼ s;
8
<r2 ð2I þ k2 bb0 Þ; t ¼ s;
covðDyt ; Dys Þ ¼ r2 I;
t ¼ s 1;
:
0;
jt sj > 1:
ð8Þ
ð9Þ
Unbiased estimators of db and covðDyt Þ are
c¼
db
n
1 X
1
ðyn y1 Þ;
Dyt ¼
n 1 t¼2
n1
n
1 X
c
c 0:
ðDyt db
dbÞðDy
dbÞ
t db
n 1 t¼2
Using the normalization b0 b ¼ 1 yields the estimates
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
b
ðyn y1 Þ0 ðyn y1 Þ;
d¼
n1
yn y1
bb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
ðyn y1 Þ0 ðyn y1 Þ
ð10Þ
ð11Þ
Furthermore, y a þ bfa0 þ dðn þ 1Þ=2g given bb, and hence0 if a0 is
normalized such that a0 ¼ b
dðn þ 1Þ=2, then ab ¼ y and b
at ¼ bb ðyt yÞ
are reasonable estimates of a and at . The normalization on a0 facilitates direct comparison to the estimates displayed in Section 2.
From (8) we find
(
)
2
2
k
r2 k2
0
c ¼ r2
bb
bb0 :
I
þ
ð12Þ
covðdb
dbÞ
n1
n1
ðn 1Þ2
From (9), given b0 b ¼ 1, it follows that covðb0 Dyt Þ ¼ r2 ð2 þ k2 Þ and
covðb0 Dyt ; b0 Dyt1 Þ ¼ r2 suggesting the moment estimates
n
0
0
1 X
ðbb Dyt b
dÞðbb Dyt1 b
dÞ;
n 1 t¼2
(
)
Pn b 0
2
b
2
ð
b
Dy
d
Þ
t
t¼2
b
k ¼ 2þ
:
ðn 1Þb
r2
b
r2 ¼
ð13Þ
If these estimates are negative, the model appears inappropriate.
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P. de Jong and L. Tickle
3.3. Least Squares Estimates for Women’s Mortality Data
Figure 4 illustrates the use of the least squares method applied to
women’s mortality data. The estimated base rates are, by construction,
the same as those displayed in Figure 2. The estimated trend b
at displayed in the bottom left panel is similar to the one estimated using
the LC method. (The different normalization results in a difference
of scale.) The response profile bb in the top right panel appears more
regular than the LC estimate. However, it is quite jagged arguing
for appropriate smoothing. The final panel shows fitted year 2000
log-mortalities and forecast log-mortalities at 10-year intervals from
2010 to 2040. These forecast rates also display a jaggedness which
would not be expected of actual rates. Estimates are b
d ¼ 0:220,
b
r ¼ 0:155 and b
k ¼ 1:911. The standard deviation associated with the
random walk b
kb
r ¼ 0:296 is almost twice that associated with each
observed log-mortality.
The least squares and LC estimates of the random walk are similar
after allowing for scaling. This shows, from Section 3.1, that the least
squares estimate of the random walk is essentially the scaled column
means of Y. One suggestion for improvement is to regress yt on ð1; tÞ,
leading to estimates of a and db. However, if the covariance matrix of
FIGURE 4 Results from least squares analysis for women’s mortality data.
Extending Lee–Carter Mortality Forecasting
9
yt , as implied by the random walk model, is factored in, then the
regression leads to the same estimates displayed in (11). Further, it
appears worthwhile to iterate the estimates, improving efficiency. At
each iteration, estimates of b and k are used to define the estimate
of covfvecðDYÞg as in (9) and covariances are used to improve estimates. A unified approach is discussed in Section 5.
4. THE LC(SMOOTH) MODEL
The LC(smooth) model (2) builds in the expected smooth behavior of
mortality by age. In practice X has fewer columns than rows, and
hence a and b in (2) have few parameters as compared to a and b in
the LC model (1). The effect of X is to impose smoothness and robustness. In contrast to (1), the effects of the at time series on each age are
not independent across age but are constrained by the variables
making up X.
An example of X is with entry ði; jÞ as ði þ 1Þj1 , i ¼ 0; . . . ; p and j
indicates the order of polynomial. Then X interpolates low order polynomials and at impacts mortality smoothly at each t across ages. A
variant is when X contains splines or b-splines (De Boor, 1978), yielding polynomial smoothness across age with knots at critical ages corresponding to break points in the momentum of age-related mortality.
4.1. Least Squares Estimation of the LC(Smooth) Model
To estimate the LC(smooth) model (2), a least squares estimate is
obtained by transforming the model to a lower dimension based on
X. The suggested transformation is based on X ¼ UDV 0 , the SVD of
X, where V and D are square corresponding to the nonzero singular
values of X. Multiplying the LC(smooth) model (2) by U 0 yields the
reduced dimension LC model (1)
yt ¼ a þ b at þ Et ;
ð14Þ
where yt ¼ U 0 yt , a U 0 Xa ¼ DV 0 a, b U 0 Xb ¼ DV 0 b, Et ¼ U 0 Et with
covðEt Þ ¼ r2 I if covðEt Þ ¼ r2 I. The number of components in yt is much
less than yt since X has fewer columns than rows. The least squares
method of Section 3.2 is applied to yt ensuring smoothness and robustness in the age direction. Estimates of a and b are arrived at by multiplying the estimates of a and b by ðU 0 XÞ1 ¼ VD1 . Further,
estimates of Xa and Xb are arrived at by multiplying the estimates
of a and b by U.
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P. de Jong and L. Tickle
4.2. Application of the LC(Smooth) Model to Women’s
Mortality Data
Figure 6 presents the results of fitting the LC(smooth) model (2) to the
women’s mortality data of Figure 1 using the least squares method of
Section 3.2 applied to (14). The matrix X is based on the b-splines displayed in Figure 5. Between knots at ages 0.5, 9.5, 19.5, 60.5, 90.5 and
105, log-mortality is assumed quadratic. At knots the quadratics join
up in such a way that the first derivative is continuous but the second
derivative is discontinuous. Knots correspond to apparent changes in
the momentum of age-related mortality. At each age there are at most
three nonzero splines, thus each row of X in (2) has at most three nonzero entries. The selected knots imply there are eight age parameters
associated with each time series component. This is an improvement
on the p þ 1 ¼ 101 parameters, for each time series component,
implicit in the Lee–Carter model of Section 2.
Figure 6 displays the fit to women’s mortality data. Both the base
rates and the age response profile are smooth. The estimated historical trend is jagged, largely because the coefficients of the b-splines
associated with the oldest ages are erratic. This suggests the need to
factor in standard errors associated with different ages.
FIGURE 5 b-splines used for LC(smooth) applied to women’s mortality data.
Extending Lee–Carter Mortality Forecasting
11
FIGURE 6 Results from LC(smooth) least squares analysis for women’s mortality data.
4.3. Testing Model Adequacy
Consider yt ¼ U 0 yt and N 0 yt where the columns of N span the null
space of the columns of X. If (2) holds, each of the time series in yt
is a common random walk plus noise. The time series in N 0 yt , provided
covðEt Þ ¼ r2 I and N 0 N ¼ I, are zero mean noise because N 0 X ¼ 0. For
example, serial correlation in N 0 yt indicates not all the time dependence in log-mortality is picked up through the smooth age dependent
behavior modeled with X.
The top left panel in Figure 7 plots the yt for women’s mortality
data. The series in U 0 yt are standardized such that U 0 y1 ¼ 0 and each
load negatively (as determined by least squares) on at . The increase of
the series over time indicates mortality improvement at all ages, with
lesser improvements experienced by b-splines associated with older
ages. The top right panel shows the autocorrelation function of the series in yt up to lag 20. Five of the autocorrelation functions are consistent with that of an autoregressive process with a unit root and
consistent with a random walk. The last two autocorrelation functions,
corresponding to the oldest ages, are not consistent with a random
walk specification. The partial autocorrelation functions plotted in
the bottom left panel of Figure 7 also suggest AR(1) models.
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P. de Jong and L. Tickle
FIGURE 7 Time series properties of yt ¼ U 0 yt and N 0 yt .
The bottom right panel indicates the average autocorrelation function of N 0 yt (middle line) with 95% confidence intervals. The average
autocorrelation suggests that the series N 0 yt also have a time series
structure. Thus the predictability in yt is only partly captured through
the smooth age behavior modeled by the X matrix.
5. MAXIMUM LIKELIHOOD ESTIMATION OF THE
LC(SMOOTH) MODEL USING KALMAN FILTERING
Maximum likelihood estimates of the LC(smooth) model (2) can be
derived using Kalman filtering and smoothing (Harvey, 1989) and
assuming an appropriate time series model for at . Consider a random
walk with drift: atþ1 ¼ d þ at þ kgt and disturbance terms normally distributed. Then the procedure consists in:
1. Evaluating the normal based likelihood for different values of parameters b and k using Kalman filtering and determining the
maximum likelihood values be and e
k;
2. Given be and e
k, determining the maximum likelihood values of a, d
and r, denoted ae, e
d and e
r, again using Kalman filtering;
Extending Lee–Carter Mortality Forecasting
13
3. Given the maximum likelihood estimates of the parameters, using
Kalman and smoothing filters to determine
e
at Eðat jy1 ; . . . ; yn Þ;
Xðe
a þ bee
at Þ ¼ EðXa þ Xbat Þ;
t ¼ 1; . . . ; n þ s;
ð15Þ
and associated error variances. Here s > 0 indicates the appropriate lead time for forecasting.
Kalman filtering can use either (2) or (14). The latter system is of
lower dimension, but is less efficient because the data Y ðy1 ; . . . ; yn Þ
are summarized into fewer observations U 0 Y ¼ ðy1 ; . . . ; yn Þ. The
numerical maximization in step 1 uses standard numerical routines,
such as the simplex method (Nelder and Mead, 1965) starting from
the least squares estimates discussed in Section 4.1.
Kalman filtering and smoothing can be applied whenever the time
series model for at is of the generic ‘‘state space’’ form, such as the random walk model atþ1 ¼ d þ at þ kgt , ARIMA (Box et al., 1994) or structural models (Harvey, 1989).
5.1. Application to Women’s Mortality Data
Figure 8 presents the maximum likelihood estimates of model (2) combined with a random walk with drift for at . The top left panel shows
the estimated (mle and least squares) and actual 2000 log-mortality
rates (jagged line). The top right panel shows the estimated age
response patterns corresponding to mle and least squares. These are
almost identical. The bottom left panel shows the estimate of at with
95% confidence intervals obtained from the smoothing filter companion to the Kalman filter. The estimate is smoother than the one
obtained from least squares. The bottom right panel displays the latest
observed log-mortalities (jagged line) and forecast log-mortalities for
the year 2050, together with 95% confidence intervals. The interval
is for the expected value of log mortality in 2050 and incorporates
the estimation uncertainty associated with a and d and the process
uncertainty associated with the random walk at . Not incorporated in
the interval is the estimation uncertainty associated with b and r2 .
The detailed confidence intervals on both historical estimates of at
and projected future log-mortalities are an advantage of the utility
of the current maximum likelihood method.
Table 1 compares estimates from least squares, smooth least
squares and mle methods. The expected improvement per year is estimated to be around 0:22 X be by each method, where X be is plotted in
14
P. de Jong and L. Tickle
FIGURE 8 Results from maximum likelihood estimation analysis for
women’s mortality data.
the top right panel of Figure 8. At age 10 for example, the improvement is estimated to be about 0:22 0:15 ¼ 3:3% per year, while at
age 70 it is about 0:22 0:05 ¼ 1:1% per year. These estimates
assume that the fitted model is an appropriate guide to the future.
6. EXTENSIONS OF THE LC(SMOOTH) MODEL
LC(smooth) model (2) reduces to the LC model if X ¼ I. Further specializations and extensions are discussed in the following subsections.
TABLE 1 Comparison of Estimates
Method
Least squares
Smooth least squares
mle using yt
d
r
k
0.220
0.218
0.215
0.155
0.148
0.359
1.911
1.983
0.661
Extending Lee–Carter Mortality Forecasting
15
6.1. Evolving Mortality Trend
In the random walk with drift model, the approximate annual change
d in mortality is fixed, independent of t. Booth et al. (2002) noted
that application of Lee–Carter using a random walk with drift model
was compromised with Australian data by significant departures
from fixed d. They addressed this by determining an optimal fitting
period. More generally, mortality trends that vary over time can be
modeled by
atþ1
1 1
at
k1 0
g1t
¼
þ
:
ð16Þ
0 k2
dtþ1
dt
g2t
0 1
k2 ¼ 0 corresponds to constant mortality improvement. The ‘‘slope’’
noise g2t and hence k2 > 0 ensures that most recent data prevails in
determining mortality trends (Harvey, 1989, p. 47).
6.2. More Detailed Time Series Modeling
An example is to add an AR(1) component to the random walk:
g1t
d
1 0
k1 0
atþ1 ¼
þ
a þ
:
ð17Þ
0 k2
l
0 / t
g2t
In this case, b is a matrix with two columns indicating the log-mortality response profiles to the random walk and AR(1) components.
6.3. Differential Measurement Error Variances
Variability in log-mortality rates is expected to vary with age. This is
rendered by replacing Et in (2) by GEt where G 6¼ I. In the estimation,
ages with more variable rates weigh less in trends. Wilmoth (1993)
imputes variances according to Poisson assumptions.
6.4. Constraining Fitted Rates
The fitted rates can be constrained so that, for example, the sum of the
observed log rates is equal to the sum of the fitted log rates at each t.
For example define M I n1 ii0 where i ¼ ð1; . . . ; 1Þ0 . Then i0 M ¼ 0
and if
yt ¼ Xa þ Xbat þ MEt ;
ð18Þ
then i0 yt ¼ i0 ðXa þ Xbat Þ. A fit based on (18) leads to fitted rates
at such that i0 yt ¼ i0 yet and hence the fitted ‘‘jump off’’
yet X ae þ X bee
16
P. de Jong and L. Tickle
rates yen X ae þ X bee
an are, on average, equal to the average of the
latest actual rates yn . Smaller groups of fitted rates can also be constrained, by replacing M in (18) by a block diagonal matrix with smaller versions of M on each block. These constraints can also be imposed
on the transformed system yt ¼ a þ b þ MEt where sums of fitted
rates in the transformed system coincide with sums of components
of yt .
7. SEPARATE TRENDS
The LC(smooth) model states that a single common random walk
drives the behavior over time of all age-specific mortality rates. This
can be relaxed in favor of several random walks allowing, for example,
a behavior varying with age:
yt ¼ Xa þ Xat þ Et ;
atþ1 ¼ d þ at þ Hgt
ð19Þ
where at is a vector time series. The scaled versions of a single random
walk bat in LC(smooth) (2) is replaced by a vector of random walks at .
Model (19) is a special case of the LC(smooth) model (2) by constraining d and H in (19) to be scalar multiples of a common vector.
Equations (19) were considered in De Jong and Boyle (1983), who
studied the mortality experience associated with a small pension fund.
In their model, the considerable variations in exposure by age and
time were modeled by heteroskedastic error terms Et . With large sample size, as in for example the women’s mortality data, exposures are
uniformly large and death probabilities are small except for the oldest
ages, so that covðEt Þ ¼ r2 I is a reasonable assumption.
Multiplying the first equation in (19) by U 0 , where X ¼ UDV 0 is the
SVD of X, yields the lower order system
yt ¼ a þ at þ Et ;
atþ1 ¼ d þ at þ H gt ;
ð20Þ
where a ¼ DV 0 a, at DV 0 at , Et ¼ U 0 Et , d ¼ DV 0 d and H ¼ DV 0 H. Each
random walk component has its own error term and if H ¼ diagðk Þ,
then
atþ1 ¼ d þ at þ diagðk Þgt ;
ð21Þ
where gt is a vector of disturbance terms, Ua ¼ Xa and Uat ¼ Xat .
The system can be estimated using maximum likelihood and the
Kalman filter. Again, estimation from (20) rather than from (19) is less
efficient because the data Y ðy1 ; . . . ; yn Þ are summarized into fewer
observations Y U 0 Y. r2 and the entries of H of (20) are unknown.
The mle of k yields the mle’s of a , d and r2 . As Xa ¼ Ua and
Extending Lee–Carter Mortality Forecasting
17
Xat ¼ Uat , the fitted values of yt are determined directly from the estimates. Forecast log-mortality rates at time t ¼ n þ m are
Uðe
a þ e
an þ me
d Þ UU 0 yn þ mU e
d yn þ mU e
d :
ð22Þ
The latest observed rates yn are used as jump off rates for the random
walk projections rather than the latest smoothed rates Uðe
a þ e
an Þ.
8. CONCLUSION
We have discussed the Lee–Carter framework of mortality forecasting
and suggested improvements. The suggested improvements are based
on standard time series methods. The improvements integrate estimation and forecasting. Additionally, the improved framework is
amenable to extension.
9. ACKNOWLEDGEMENTS
Our thanks to Len Smith for providing data from the Australian
Demographic DataBank produced by the Australian Centre for Population Research.
REFERENCES
Bell, W. (1992). Arima and principal component models in forecasting age-specific
fertility. In N. Keilman and H. Cruijsen (Eds.), National Population Forecasting
in Industrialized Countries. Amsterdam: Swets and Zeitlinger, pp. 159–177.
Bell, W. (1997). Comparing and assessing time series methods for forecasting agespecific fertility and mortality rates. Journal of Official Statistics 13: 279–303.
Bell, W. and Monsell, B. (1991). Using principal components in time series modeling and
forecasting age-specific mortality rates. In Proceedings of the American Statistical
Association, Social Statistics Section, Alexandria, VA, USA, pp. 154–159.
Booth, H., Maindonald, J., and Smith, L. (2002). Applying Lee–Carter under conditions
of variable mortality decline. Population Studies 56: 325–336.
Box, G.E.P., Jenkins, G.M., and Reinsel, G.C. (1994). Time Series Analysis: Forecasting
and Control (3rd ed.). New Jersey: Prentice-Hall, Englewood Cliffs.
Bozik, J. and Bell, W. (1987). Forecasting age-specific fertility using principal components. In Proceedings of the American Statistical Association, Social Statistics
Section, pp. 396–401.
Currie, I., Durban, M., and Eilers, P. (2002). Using p-splines to smooth two-dimensional
Poisson data. In Proceedings of the 17th International Workshop on Statistical Modelling, Crete, pp. 127–134.
De Boor, C. (1978). A Practical Guide to Splines. New York: Springer.
De Jong, P. and Boyle, P. (1983). Monitoring mortality: A state-space approach. Journal
of Econometrics 23: 131–146.
Girosi, F. and King, G. (2004). Demographic Forecasting. Forthcoming: available at
http:==gking.harvard.edu=files=smooth.pdf.
18
P. de Jong and L. Tickle
Harvey, A.C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter.
Cambridge: Cambridge University Press, UK.
Lee, R.D. and Carter, L.W. (1992). Modeling and forecasting U.S. mortality (with discussion). Journal of the American Statistical Association 87(419): 659–675.
Lee, R.D. and Miller, T. (2001). Evaluating the performance of the Lee–Carter method
for forecasting mortality. Demography 38: 537–549.
Nelder, J. and Mead, R. (1965). A simplex method for function minimization. The
Computer Journal 7: 308–313.
Wilmoth, J.R. (1993). Computational methods for fitting and extrapolating the Lee–
Carter model of mortality change. Technical report, Department of Demography,
Berkeley: University of California.