Understanding the Lee-Carter Mortality Forecasting Method1
Federico Girosi2 and Gary King3
September 14, 2007

1

We appreciate the generosity and insight of Ron Lee and Nan Li for help in understanding theirapproach and the demographic literature in general. Many thanks also to John Wilmoth for very
helpful comments.
2
The RAND Corporation (1776 Main Street, P.O. Box 2138, Santa Monica, CA 90407-2138;
girosi@rand.org).
3
David Florence Professor of Government, Harvard University (Center for Basic Research in the Social Sciences, 34 Kirkland Street, Harvard University, Cambridge MA 02138;
http://GKing.Harvard.Edu, King@Harvard.edu, (617) 495-2027).

Abstract
We demonstrate here several previously unrecognized or insufficiently appreciated properties of the Lee-Carter mortality forecasting approach, a method used widely in both theacademic literature and practical applications. We show that this model is a special case of
a considerably simpler, and less often biased, random walk with drift model, and prove thatthe age profile forecast from both approaches will always become less smooth and unrealistic
after a point (when forecasting forward or backwards in time) and will eventually deviate
from any given baseline. We use these and other properties we demonstrate to suggest whenthe model would be most applicable in practice.

The method proposed in Lee and Carter (1992) has become the “leading statistical modelof mortality [forecasting] in the demographic literature” (Deaton and Paxson, 2004). It wasused as a benchmark for recent Census Bureau population forecasts (Hollmann, Mulder
and Kallan, 2000), and two U.S. Social Security Technical Advisory Panels recommended
its use, or the use of a method consistent with it (Lee and Miller, 2001). In the last decade,scholars have “rallied” (White, 2002) to this and closely related approaches, and policy
analysts forecasting all-cause and cause-specific mortality in countries around the world havefollowed suit (Booth, Maindonald and Smith, 2002; Deaton and Paxson, 2004; Haberland
and Bergmann, 1995; Lee, Carter and Tuljapurkar, 1995; Lee and Rofman, 1994; Lee andSkinner, 1999; Miller, 2001; NIPSSR, 2002; Perls et al., 2002; Preston, 1993; Tuljapurkar
and Boe, 1998; Tuljapurkar, Li and Boe, 2000; Wilmoth, 1996, 1998a,b).
Lee and Carter developed their approach specifically for U.S. mortality data, 1933-1987.
However, the method is now being applied to all-cause and cause-specific mortality datafrom many countries and time periods, all well beyond the application for which it was
designed. It thus appears to be a good time to reassess the approach, as the issues these
new applications pose could not have been foreseen by the original authors. See Lee (2000a)
for an earlier effort along these lines.
In this paper, we demonstrate several previously unrecognized or insufficiently appreciated properties of the Lee-Carter model, and use these properties to suggest where andwhen the model would be most applicable. Section 1 describes the forecasting method insome detail. Since the method can be seen as a special case of a principal components
method (Bozik and Bell, 1987; Bell and Monsell, 1991) with a single component , Section 2
summarizes a diverse array of 240 mortality data sets (24 causes of death from 10 countrieseach) to give a sense of where the method has a chance of working well. Then, in Section 3
we show that the Lee-Carter model is equivalent to a special type of multivariate random
walk with drift (RWD) model, in which the covariance matrix depends on the drift vector.
The implication of this special structure is that while the RWD estimator is unbiased fordata generated by the Lee-Carter model and other types of data, the Lee-Carter model
is biased for data generated by the general RWD model with arbitrary covariance matrix.
The Lee-Carter estimator is more efficient when data are known to be drawn from the
Lee-Carter model. These observations suggest that, since the RWD does not make any
assumption about the structure of the covariance matrix, while the Lee-Carter approach
does, the Lee-Carter estimator will be preferable to the RWD only when we have high confidence in its underlying assumptions. The similarity of the two models means that the much1

simpler RWD model and estimator will prove especially useful in elucidating the propertiesof the Lee-Carter approach.
Thus, in Section 4 we illustrate a property common to both the Lee-Carter and the
RWD forecasts that has not previously been highlighted in the literature. We show therethat the age profile of such forecasts will always become less smooth after a point (whenforecasting forward or backwards in time) and this nonsmoothness will continue forever,
and will eventually deviate from any given baseline. Finally, in Section 5, we commenton the Lee and Carter’s second stage reestimation idea and Wilmoth’s (1993) alternative
estimation strategies, based on weighted least squares and maximum likelihood.

1

The Method

1.1

The Model

Let mat denote the log of the mortality rate in age group a (a = 1, . . . , A) and time t(t = 1, . . . , T ) for one country. The first step of the Lee-Carter method consists of modelingthese mortality rates as
mat = αa + βa γt + ²at(1)

where αa , βa and γt are parameters to be estimated and ²at is a set of random disturbances. The parametrization in (1) is not unique, since it is invariant with respect to thetransformations:βa Ã cβa1γt Ã γtc∀c ∈ R, c 6= 0

αa Ã αa − βa cγt Ã γt + c∀c ∈ R.

This is not a conceptual obstacle; it merely means that the likelihood associated with themodel above has an infinite number of equivalent maxima, each of which would produce
identical forecasts. In practice, we merely pick an arbitrary but consistent parameterization
sufficient for identification. This can be done by imposing two constraints. We follow Lee
P
and Carter in adopting the constraint t γt = 0. Unlike Lee and Carter, however, we setPP 2a βa = 1). This last choice is done only to simplify some calculationsa βa = 1 (they set
later on, and has no bearing on empirical applications.
P
The constraintt γt = 0 immediately implies that the parameter αa is simply theempirical average over time of the age profile in age group a, αa = m̄a . We therefore
rewrite the model in terms of the mean centered log-mortality rate, m̃at = mat − m̄a .

2

Since practical uses of the Lee-Carter model implicitly assume that the disturbances ²atare normally distributed, we rewrite Equation 1 as a multiplicative fixed effects model forthe centered age profile:¡¢m̃at ∼ N µ̄at , σ 2(2)E(m̃at ) ≡ µ̄at = βa γt .
In this expression, we use only A + T parameters (βa γt , for all a and t, represented on
the bottom and right margins of the matrix below) to approximate the A × T elements ofthe matrix:

5
10
15
20
m̃ =

25
30
35
..
.
80

1990 1991 1992 1993 1994 m̃5,0 m̃5,1 m̃5,2 m̃5,3 m̃5,4

 m̃10,0 m̃10,1 m̃10,2 m̃10,3 m̃10,4 


 m̃

 15,0 m̃15,1 m̃15,2 m̃15,3 m̃15,4  m̃20,0 m̃20,1 m̃20,2 m̃20,3 m̃20,4  m̃25,0 m̃25,1 m̃25,2 m̃25,3 m̃25,4 


 m̃

 30,0 m̃30,1 m̃30,2 m̃30,3 m̃30,4  m̃35,0 m̃35,1 m̃35,2 m̃35,3 m̃35,4 


 .
..
..
..
.. 
 .

.
.
.
. 
 .
m̃80,0 m̃80,1 m̃80,2 m̃80,3 m̃80,4
γ0
γ1
γ2
γ3
γ4
m̃35,0 m̃35,1 m̃35,2 m̃35,3 m̃35,4


β5
β10
β15
β20
β25
β30

(3)

β35
..
.
β80

For example, Lee-Carter approximates m̃5,0 in the top left cell by the product of the parameters at the end of the first row and column β5 γ0 .
Seen in this framework, the Lee-Carter model can also be thought of as a special caseof log-linear models for contingency tables (Bishop, Fienberg, and Holland, 1975; King,
1989: Ch. 6), where cell values are approximated with estimates of parameters representing
the marginals. Indeed, this model is the most basic version of contingency table models,
where one assumes independence of rows (age groups) and columns (time periods), and the
expected cell value is merely the product of the two parameter values from the respective
marginals: E(m̃at ) = βa γt . In a contingency table model, this assumption would be appropriate if the variable represented as rows in the table were independent of the variable
represented as columns. The same assumption for the log-mortality rate is the absence ofage×time interactions — that βa is fixed over time for all a and γt is fixed over age groups
for all t.
Another way to phrase this independence assumption is that the coefficients β do notvary over time. Lee and Miller (2001) point out that the evidence indicates that most data3

do not fit this assumption. They recommend the simple solution proposed by Tuljapurkar,
Li and Boe (2000), which consists of using as the base forecasts, the year 1950. This isequivalent to excluding the portion of the 20th century where most of the change took
place in infant and child mortality. Although this approach works in some countries, itdoes not work well in others Booth, Maindonald and Smith (2002). In addition, a one-time
fix for β does not solve the problem that the smoothness of the age profiles declines overtime. Carter and Prskawetz (2000) discuss extensions to the Lee-Carter model to allow forstructural shifts and changes in βa over time.

1.2

Estimation

The parameters βa and γt in model 2 can be estimated via maximum likelihood. However,
the multiple maxima or constraints will make standard optimization programs work poorly.
Fortunately, as Lee and Carter point out, the optima can be found easily via the singularvalue decomposition (SVD) of the matrix of centered age profiles, m̃ = BLU 0 , where theestimate for β is the first column of B, and the estimate for γt is β 0 m̄t .1 If the SVD decomposition of m̃ is not available, one can compute as the normalized eigenvector of the matrix
C ≡ m̃m̃0 corresponding to the largest eigenvalue (we will use this alternative in Section 3,
to perform some analytical calculations). Whether one uses SVD or eignenvalues to findthe optimum, the theoretical justification of the procedure remains maximum likelihood.
In practice, Lee and Carter suggest, after β and γ have been estimated, that the parameter γt be re-estimated using a different criterion. This reestimation step, often called“second stage estimation”, does not always have a unique solution for the criterion outlined
in Lee and Carter (1992). In addition, different criteria have been proposed more recently(Lee and Miller, 2001; Wilmoth, 1993), and some researchers skip this re-estimation stage
altogether. Since it is not a defining feature of the method we skip this step and return tothis issue in Section 5.

1.3

Forecasting

To produce mortality forecasts, Lee and Carter assume that βa remains constant overtime and they use forecasts of γ̂t from a standard univariate time series model. After
testing several ARIMA specifications, they find that a random walk with drift is the mostappropriate model for their data. They make clear that other ARIMA models might be1

We assume that the singular values, the elements on the diagonal of L, are sorted in descending orderand that the columns of B have length one. If not, then the estimate for β is the column of B whichcorresponds to the largest singular value and β should be replaced by β/kβk.

4

preferable for different data sets, but in practice the random walk with drift model for γt
has been used almost exclusively. This model is as follows:γ̂t = γ̂t−1 + θ + ξt2
ξt ∼ N (0, σrw)(4)where θ is known as the drift parameter and its maximum likelihood estimate is simplyθ̂ = (γ̂T − γ̂1 )/(T − 1), which only depends on the first and last of the γ estimates.2 . Then,
to forecast two periods ahead, we plug in the estimate of the drift parameter θ̂ and also
substitute for the definition of γ̂t−1 shifted back in time one period:γ̂t = γ̂t−1 + θ̂ + ξt= (γ̂t−2 + θ̂ + ξt−1 ) + θ̂ + ξt
= γ̂t−2 + 2θ̂ + (ξt−1 + ξt )(5)

To forecast γ̂t at time T + (∆t) with data available up to period T , we follow the sameprocedure iteratively (∆t) times and obtain(∆t)γ̂T +(∆t) = γ̂T + (∆t)θ̂ +X

ξT +l−1l=1= γ̂T + (∆t)θ̂ +p(∆t)ξt ,(6)

where the second line is a simplification made possible by the fact that the random variables
ξt are assumed in this model to be independent with the same variance. The second line
indicates that the conditional standard errors for the forecast increase with the square rootof the distance to the forecast horizon (∆t). These are conditional standard errors andwould be larger if we included estimation uncertainty.
From this model, we can obtain forecast point estimates, which follow a straight line asa function of (∆t), with slope θ̂:E[γ̂T +(∆t) | γ̂1 , . . . , γ̂T ] ≡ µT +(∆t) = γ̂T + (∆t)θ̂(7)

The Lee-Carter model for the γ’s is thus very simple: Extrapolate from a straight line drawn
through the first γ̂1 and last γ̂T points. All other γ̂’s are ignored.
We now plug these expressions into the empirical and vectorized version of Equation 2
to make a point estimate forecast for log-mortality:µT +(∆t) = m̄ + β̂γ̂T +(∆t)= m̄ + β̂[γ̂T + (∆t)θ̂].
2

2
1
The MLE of the variance is σ̂rw
= T −1PT −1

t=1 (γ̂t+1 − γ̂t − θ̂)

5

2

(8)
σ2

rw
with Var[θ̂] = T −1

For example, the Lee-Carter model computes the forecast for year 2030, given data
observed from 1950 to 2000, asµ̂2030 = m̄ + β̂ × [γ̂2000 + 30θ̂]·¸(γ̂2000 − γ̂1950 )= m̄ + β̂ × γ̂2000 + 30
.
50

2

(9)

Information Loss with One Principal Component

As is well known, the estimation stage of Lee-Carter is a special case of principal component
analysis, where the log-mortality data is summarized using only the first single principal
component, with other variation ignored for the purpose of making forecasts.3 The idea ofprincipal components is that a set of data mat (for all a and t) can be decomposed without
error as the sum of a set of basic shapes. The first shape, or principal component, is βa γt in
Equation 1. The full set includes as many as A components with the relationship holding
exactly and so is written without the need for an error term:

mat = αa + βa1 γt1 + βa2 γt2 + · · · + βaA γtA(10)

where the second subscript of each parameter refers to the principal component number
and the first component βa1 γt1 equals βa γt from Equation 1. What was labeled as error
in Equation 1 is now decomposed into the remaining principal components: ²at = βa2 γt2 +· · · + βaA γtA .
One way to understand the entire set of log-mortality data is as a sequence of time
points moving through A-dimensional space. The hypothesis inherent in the Lee-Cartermodel is that the A-dimensional space can be reduced without much loss of information
to one-dimensional space, or in other words that the log-mortality age profiles move alonga straight line in RA . When this assumption is violated for a given dataset, so that one
principal component is insufficient to characterize a large enough proportion of the motionof the age profiles, then the Lee-Carter model would not be expected to forecast as well.
Since comprehending A-dimensional space is difficult for A > 3, we examine a summaryof the data with the first three principal components and study how closely the first component that Lee-Carter uses (i.e., a straight line) fits these three. We do this first graphically3

Principal component analysis was first used in demography by Ledermann and Breas (1959), who usedfactor analysis to analyze life table data from different countries and then Bozik and Bell (1987) and Sivamurthy (1987) for projecting age-specific fertility rates. The method of Bozik and Bell was then extended
by Bell and Monsell (1991) to forecast age-specific mortality rates, but it was not until Lee and Carter’s(1992) simpler formulation that these methods became widely used (see also Lee, 1993, 2000, 2000a; andLee and Tuljapurkar, 1994, 1998, 1998a). .

6

for a small number of data sets and then summarize each graph with the percentage ofvariance explained by 1, 2, and 3 principal components, so that we can display the resultsof these analyses for a much larger and more diverse group of data sets.4
In each of the four graphs in Figure 1, we plot the first three principal components forone specific population and cause of death for annual data from 1950 to 2000. The first,
second, and third principal components are plotted on the depth, horizontal, and verticalaxes of the graph. Each data point (summarized by the three principal components rather
than the raw data so that we can fit this in only three dimensions) appears as a circle. We
colored the time series of circles (and vertical lines we added for graphical clarity) in theorder of the rainbow (red, orange, yellow, green, blue, indigo, and violet). A solid black line
represents where all the circles should closely cluster around if the Lee-Carter assumptionof one principal component is adequate.
For example, all cause mortality (the top left graph), and to some degree cardiovasculardisease (the top right graph) show that most of the variation in the three dimensions lies
close to the direction of the first principal component (the depth dimension in the graph).
Cardiovascular disease shows some significant systematic nonlinear variation along the second principal component (the horizontal dimension), and a smaller amount of variation
along the third (the vertical dimension). Note that the Lee-Carter assumption of a single
principal component does not require that the time series move in one direction along theline, and indeed the time series of both top graphs jump back and forth along the line
to some degree (see especially the red circles appearing in both graphs near the front andback).
The three numerical summaries (appearing in the title of each graph) indicates thepercentage of variance explained by the first, second, and third principal components, respectively. Thus, for all-cause mortality (the top left graph), if we try to project the age
profiles on a one-dimensional subspace (which corresponds to the Lee-Carter model) we can
explain 93% of the variance in the data. Using a two-dimensional subspace we can explain
97% of the variance, while adding a third dimension takes us to 99%. Since the scale on thethree axes in each graph is the same, these numbers summarize the figures reasonably well.
(k)

(k)

4
Denote by ²t the error associated with a specification using k principal components: ²t ≡ m̃t −(β1 γ1t + β2 γ2t + · · · + βk γkt ). Averaging over time, a specification with k principal components leads to an
P (k). In order to obtain a measure of relative error, which is easier to understand, we divide
error of t k²t k2P
this expression by t km̃t k2 , which makes it independent of the scale of m̃t . This ratio, which measures how
bad the k-component approximation is, is always between 0 and 1. Since it is customary to report goodness
P

(k) 2

k²

k

t
rather than badness of fit, the standard measure is ∆Ek ≡ 1 − Pt km̃
2 , which is known as the percentagetktof the variance which is “explained” (or linearly accounted for) by the first k principal components: It is 1
when the specification with k components has no error, that is explains 100% of the variance.

7

Cardiovascular Disease (m) USA ( 90 95 99 )

0.0

0.5

1.0

4

0

1st Princ. Comp−0.5

1st Princ. Comp

−1.0

−2

−1
−2

−3

−2.0

−1.5

2
1
0

−0.5
−1.0
−1.5
−2.0

−2.0

−1

1st Princ. Comp

−1.0

0.0

3rd Princ. Comp

0.5
0.0
−0.5

1.5
1.0
0.5

−1.5

3rd Princ. Comp

1

1.0

2

1.5

All Causes (m) USA ( 93 97 99 )

1.5

−3
−3

−2

2nd Princ. Comp

−1

0

1

2

2nd Princ. Comp

−2

1
0

3
2

−1

0

3rd Princ. Comp1

1st Princ. Comp

0

2

−1

1
0

−1

−2

3rd Princ. Comp

2

1

3

4

Homicide (m) Sweden ( 37 50 57 )2

Lung Cancer (m) Sweden ( 38 56 69 )

−1
−2

−3

−3

−2
−3
−3

−2

−1

0

1

2

−3
−3

2nd Princ. Comp

−2

−1

0

1

2

3

4

2nd Princ. Comp

Figure 1: Principal Components of Log-Mortality Age Profiles in Three Dimensions: for
U.S. males for all-causes (top left) and cardiovascular disease (top right), and for males
from Sweden for lung cancer (bottom left) and homicide (bottom right). The black line
in each graph traces out the best approximating first principal component, and the threenumbers at the top of each graph indicate the percentage of variance explained by one, two,and three principal components, respectively.

8While in the top graphs of Figure 1 the low dimensional approximation performs quite
well, we show in the bottom graphs of the same figure two cases at the opposite end of thespectrum. The data correspond to lung cancer and homicide mortality for Sweden, andvary far from the black line in each graph representing the first principal component. The
percentage of variance explained (in the title of the graphs) is low, with one component
explaining only 38% and 37% of the variation in the two graphs, respectively. The LeeCarter method should not be counted on to perform as well in these causes of death, since
it would ignore almost two-thirds of the variation in the data. Unless two-thirds of thedata represent measurement error or other forms of random variation this means that thesemortality rates across different age groups received shocks which are either uncorrelated, orare correlated in a way that is not captured in one-dimensional linear model.
In addition to providing general intuition about PCA, Figure 1 shows the close correspondence between the simple numerical summary of the explanatory power of each dimension and the physical path followed by each of the dimensions. In particular, when thepercentage of variance accounted for by the first dimension is very high, the physical pathfollowed by the profile in three dimensions stays close to the line. We now take advantageof the simplicity of this numerical summary to study a much larger number of data sets.
Thus, Table 1 presents the percentage of the variance accounted for by the first principal
component for all the countries in our data base with more than 40 years of observations,for each of ten causes of death (see White, 2002, for a related analysis). A few observations
emerge from these tables:
1. For some causes of death (like all-cause, TB, and, to a lesser extent, cardiovasculardisease), the motion of the age profiles seem to be confined to a low-dimensional
space fairly consistently across countries, which suggests that there is some underlying
reason for this behavior.
2. Since we expect smaller countries to have larger amounts of measurement or sampling
error we expect to observe a lack of low-dimensionality in smaller countries. While
this may play a role in Iceland there is no obvious pattern relating dimensionality tocountry population, so that other factors are likely to be at work.
3. For some other causes of death (like digestive diseases and lung cancer) we see awide range of behaviors: In some countries, the motion of the age profiles is lowdimensional, while in other countries one principal component is nowhere near enough.
4. For some causes of death (like homicide, respiratory infections, stomach cancer, and9suicide) little evidence exists that the age profiles can be reduced to this lower dimensional summary (indeed even three principal components, which we do not presenthere) is insufficient in most situations.
5. For any given country, the percentage of the variance accounted for varies considerablyacross causes of deaths.

Japan
Chile
CanadaAustralia
U.K
MexicoUSA
Austria
Italy
Finland
Belgium
Portugal
Switzerland
FranceSwedenSpain
Netherlands
Venezuela
Hungary
NorwayIreland
New Zealand
Denmark
Iceland

All
cause
98
97
95
95
94
93
93
93
92
92
92
91
91
90
90
89
89
86
85
84
83
82
70
57

Cardiovascular
87
92
77
79
72
81
90
49
74
56
70
62
36
89
41
91
40
64
64
29
57
57
39
53

Lung
cancer
96
48
84
64
77
77
81
44
88
49
78
80
45
93
55
93
78
46
86
64
68
40
58
42

Homicide
74
52
38
16
60
64
70
16
43
21
40
21
22
44
37
66
41
50
30
24
28
19
31
40

TB
97
96
89
84
91
97
94
86
93
88
85
91
83
94
82
94
81
93
79
82
86
79
67
54

Respiratory
69
52
50
37
80
82
78
35
61
37
47
34
24
78
29
74
39
37
39
24
37
40
36
20

Stomachcancer
66
71
64
61
70
32
63
61
46
75
57
34
64
56
69
54
61
52
52
66
59
40
56
55

Other
cancer
62
55
80
67
87
71
95
69
70
56
67
46
53
74
62
74
70
43
62
64
43
34
51
18

Suicide
65
79
85
66
87
63
91
34
60
52
80
45
33
80
38
73
72
23
73
66
70
59
48
18

Digestive
97
87
79
79
83
97
85
76
90
72
70
87
75
84
64
94
70
93
79
56
71
61
55
32

Table 1: Percentage of Variance Explained in Male Mortality by the First Principal Component. Bold entries, indicating 90% or higher, highlight where Lee-Carter is more likely
to work well.
The main point of this table is the absence of any simple story that could explain thepatterns. As a result, one needs to be careful when analyzing the age profiles of different
countries, groups, and causes: Methods that rely on the assumption of low-dimensionality,
such as Lee-Carter, will usually be appropriate only for idiosyncratic choices of countries
and causes of death, and most importantly, there is no way to know ex ante which thoseare. Across many countries, one principal component fits all-cause mortality well, evenoutside the few developed countries for which the method was designed and where Lee andCarter intended the method to be applied, although for other countries the fit is poor. One
principal component fits cause-specific data more rarely, which supports Lee and Carter’sviews that the method should not routinely be used in these applications.

103

The Lee-Carter Model and the Random Walk with Drift

Although the Lee-Carter model can be estimated with any time series process applied toforecast γt , the random walk with drift specification in Equation 4 accounts for nearly all
real applications. This specification implies that the functional form of the forecast is nodifferent from the one obtained by a multivariate random walk with drift (RWD), althoughwith a different value for the drift parameter. In this section, we analyze the full two-stage
Lee-Carter model and estimator and discuss the following observations:
1. When considered together, the two stage Lee-Carter approach is shown to be equivalent to a particular type of random walk with drift (RWD) model. The crucial
difference between the two is that in the Lee-Carter model the covariance matrix ofthe error term is explicitly dependent on the parameter vector β, while in the RWD
the covariance matrix is not restricted.
2. If data are generated from the Lee-Carter model, then the Lee-Carter estimator andthe RWD estimator are both unbiased.
3. If the data are generated by the RWD model, then the two-stage Lee-Carter estimatoris biased, but the RWD estimator is unbiased.
These results thus pose the question of why or when one would prefer to use the LeeCarter estimator over the simpler RWD estimator. Empirically the Lee-Carter estimator
does not appear to perform better than the RWD (Bell, 1997) in short-term U.S. mortality
forecasts, but unfortunately for long-term forecast and more general cases no evidence
exists about which method performs better. We find that posing this question contributes
to understanding, since the RWD model is much easier to understand and apply.
It seems likely that the answer to this question depends on the framework in which the
Lee-Carter is applied. Researchers whose only interest is to produce a forecast may findit more convenient to use the random walk with drift. However, researchers interested indeveloping new models will find in the Lee-Carter method a solid starting point, and cantake advantage of the large body of knowledge already available on the performance of thismodel.
After demonstrating these points in this section, we then use these results in the next
section to highlight some important properties of the Lee-Carter and RWD model that havenot been noted in the literature previously.

11

3.1

The Multivariate Random Walk with Drift Model

We begin with the pure mulitivariate random walk with drift (RWD) model on a multivariate
time series for the A × 1 vector mt (for t = 1, . . . , T ), which we write as
mt = mt−1 + ψ + ηt

(11)where ηt is normal random error with mean zero and A × A variance matrix Σ, and ψ is an
A × 1 vector of drift parameters.
The maximum likelihood estimate depends only on the first and last data points:ψ̂ =1(mT − m1 )
T −1

(12)

with estimation uncertainty Var[ψ̂] = Σ/(T − 1). Proceeding similarly to Section 1.3 (seeEquation 6, Page 5), the stochastic forecast at time T + (∆t) with data observed up to time
T ism̂T +(∆t) = mT + (∆t)ψ̂ +p(∆t)ηtand its conditional point estimate is a linear function of time:E[m̂T +(∆t) | m1 , . . . , mT ] = mT + (∆t)ψ̂.

(13)

Thus, to forecast under the multivariate random walk with drift model, we merely draw astraight line through the first and last data points and extrapolate.

3.2

Lee-Carter as a Random Walk with Drift

The Lee-Carter model was originally specified in two stages, described by the following
equations:
mt = m̄ + βγt + ²t
γt = γt−1 + θ + ξt

(14)

where the vector ²t and the scalar ξt are independent normal random error terms with
2 , respectively. The two equations in 14 can be combined in one single
variances σ 2 I and σrw

expression, so that the specification of the model then becomes:
mt = mt−1 + θβ + (βξt + ²t − ²t−1 ),(15)

where the parentheses denote the error term. We now equalize this expression with theconditional point estimate from the pure random walk with drift model in Equation 11 by
12noting that the drift parameter vector is θβ here and ψ in the more general random walk.
ψ
If we add to this the Lee-Carter normalization, we get θ = kψk and β = kψk, and thusθβ = ψ. We can now rewrite Equation 15 as¶µψ
mt = mt−1 + ψ +ξt + ²t − ²t−1 .
kψk

(16)

Thus, the only difference between the Lee-Carter model in Equation 16 and the RWD in
Equation 11 is the special restricted error term of Lee-Carter. In the standard RWD model
no structure is imposed on how shocks to mortality are correlated across age groups: The
covariance of the noise is an arbitrary matrix Σ, which is unrelated to the drift vector ψand not restricted. In the Lee-Carter model the error term (denoted by parentheses) is afunction of the drift vector ψ and therefore in the covariance matrix of the noise, which canbe written as follows:2ΣLC = σrwψψ 0
+ 2σ 2 I
kψk2

(17)

This shows that in the Lee-Carter model shocks to mortality can be of only two kinds: The
term (²t −²t−1 ), with variance 2σ 2 I, describes shocks that are uncorrelated across age groups.
ψ
2 ψψ 0 /kψk2 , describes shocks that are perfectlyξt ), with variance σrw
In contrast, the term ( kψkcorrelated across age groups. In addition, the relative size of the perfectly correlated LeeCarter shocks is restricted to be equal to the relative size of the rate of decline of mortalityψover time (since β = kψk). (For example, the ratio between the size of shock in age group 70
70
and in age group 60 is always equal to ββ60.) Therefore, Lee-Carter assumes that age groups

whose mortality rates have been declining faster than others are assumed to receive larger
shocks. If the observed rate of change in mortality in a certain age group is indicative of its“susceptibility to change,” we might expect that future mortality shocks to have the largest
affect on age groups that have already been declining the fastest. A problem is that it is not
immediately clear what these shocks represent. They must have zero mean, so they cannot
be epidemics or other negative health events (many of which also would target specific age
groups), but they could relate for example to weather patterns, changes in pollution levels,or perhaps funding of health care facilities. The difficulty with the Lee-Carter specification,that does not afflict the RWD model, is that shocks to mortality other than those that are
perfectly correlated or uncorrelated across age groups will be missed by the model.55Another way to look at Equation 16 is to think of it as a random walk mt = mt−1 + ψt + (²t − ²t−1 )ψψwith stochastic drift ψt = kψk(kψk + ξt ) that has fixed direction kψk
but whose length (kψk + ξt ) is
normally distributed around the mean. While this particular model is not well known, it is reminiscent of
the commonly used “local linear trend” (LLT) model, in which the drift vector follows a random walk, rather
than being stationary (Harvey, 1991). Again, the distinctive feature here is that shocks are constrained:
the drift vector is only allowed to vary in length, and not in direction, fixing the relative sizes of the shocks
across age groups.

13

The observations above clarify the difference between the Lee-Carter model and the
RWD model. To summarize, the difference lies in the nature of the shocks to mortality,
which in the Lee-Carter model are restricted in a way which explicitly depends on the drift
vector. In other words, Lee-Carter can be thought of as a RWD model with an uninformative
prior on ψ but an extremely strong prior on the covariance matrix Σ.6 Priors of course canbe very useful, but if the wrong prior is selected results could be biased, and one may bebetter off without prior. An example of this bias can be seen if we study the output of one
model when the data are generated from another, the issue to which we now turn.
3.2.1

Using the Random Walk with Drift Model on Data Generated by the
Lee-Carter Model

Let us assume now that mortality data are generated according to the Lee-Carter model.
What would happen if we attempt to forecast them using the random walk with drift
model? It is trivial to see that the RWD estimate for the drift parameter 12 is unbiased,
and therefore it will recover correctly the value of ψ (on average). Arguably this estimateof ψ may be less efficient than the one obtained by the Lee-Carter model, since we expect
the generating model to be optimal in estimating its own parameters.
3.2.2

Using Lee-Carter Model on Data Generated by the Random Walk with
Drift Model

We now show that if data are generated according to a RWD model with an arbitrary
covariance matrix, then the Lee-Carter model will be biased: on average it will not recover
the correct drift parameter. Exceptions will be when the noise in the RWD has sphericalsymmetry, and/or shocks occur along the drift vector. The intuition behind this behaviorof the Lee-Carter model is very simple, and can be best understood with the use of figure
2, which refers to a hypothetical case in which there are only two age groups, and therefore
log-mortality in a given year is a point in a plane. In the left panel of the figure we show a
trajectory of log-mortality generated by a RWD with spherical disturbances (in red). The
starting point is at the top right, and mortality decreases over time following a drift vector
whose direction is shown by the dashed line (in black). The continous straight line (ingreen) is the projection of log-mortality according to the Lee-Carter model. Therefore thedirection of this line coincides with the direction of the first principal component, whichis almost parallel to the direction of the drift vector. In this case the Lee-Carter model
is not biased. In the right panel we use non-spherical disturbances, and set the ratio6şť
2 ψψ 0To be specific, the prior implied by Lee-Carter is P(ψ, Σ) ∝ P(Σ | ψ)P(ψ) = δ Σ − σrw− 2σ 2 Ikψk2

where δ(·) is the point-mass measure.

14

between the standard deviation of the noise along the vertical and horizontal directions
to 20 (we choose such a large number in order to make the effect clear, not to portray a
common empirical pattern). Notice that as the trajectory moves along the drift direction,
on average, it experiences large variations on the vertical direction. Therefore the principal
component for this pattern of log-mortality is “fooled”, because it picks up this variationand incorporates it in the estimate for β. As a result the Lee-Carter forecast takes off in
the wrong direction, and bias results.

Figure 2: (Left) A two dimensional random walk with drift with spherical disturbances.
(Right) A two dimensional random walk with drift with non-spherical disturbances: thestandard deviation of the noise along the vertical axis is 20 times larger than the one on
the horizontal axis. The dashed line denotes the direction of the drift vector, while thecontinuous straight line (in green) denotes the Lee-Carter forecast.
In order to see this phenomenon more formally it is convenient to notice that the vector
β in the Lee-Carter model can be estimated in a slightly different way from we have used
so far. Evaluating the Lee-Carter specification in the first line of Equation 14 at two points
in time, t and t − 1, and taking the difference, we write:
mt − mt−1 = (γt − γt−1 )β + ²t − ²t−1 .

(18)

The least square estimate for β under this model is the same as the one obtained in thestandard way, albeit more noisy, as the first eigenvector of the matrix
T1X(mt − mt−1 )(mt − mt−1 )0 .
C=
T
t=1

This approach is used here only to simplify the calculations, which are longer if we proceed
in the standard way. To see what happens when mt follows the more general random walkprocess like in Equation 11, we replace mt − mt−1 with its equivalent under Equation 11,
ψ + ηt , in the expression for C, obtaining
15

T

T

T

t=1

t=1

t=1

1X
1X
1X
(ψ + ηt )(ψ + ηt )0 = ψψ 0 +ηt ηt0 +(ψηt0 + ηt ψ 0 ).
C=
T
TT
We rewrite this expression as:
C = ψψ 0 + Σ + Z(19)

where Z is a random matrix with zero mean. The parameter vector β is now the firstprincipal component associated to the matrix C in Equation 19: in order for the Lee-Carterψmethod to be unbiased on average this vector should equal to kψk. This clearly cannot

happen in general, because the matrix Σ is arbitrary. The only exceptions occurr when Σis the identity matrix, or when the the first principal component of Σ happens, by chance,ψto be kψk.

We now turn to the demographic implications of these implicit assumptions of the LeeCarter model.

4

Smoothness of Age Profiles Under Lee-Carter and RWD

The forecasts produced by Lee-Carter and RWD have identical functional forms: They area line in the multidimensional space of age profiles, and we refer to them as linear. Linear
forecasts are unlikely to generate unusual mortality patterns in the short run, although inthe long run some unlikely patterns will often develop. Some of these, such as inversions
in rank order of the age groups’ mortality rates, may or not may not occurr depending on
the data: even if they may look unlikely, they are not intrinisc features of these models.
However, there is one feature which is common to all forecasts of this type no matter whatpatterns are present in the data: They will always produce a mortality age profile that
becomes less smooth, and more deviant from any given fixed baseline, over time, after apoint. A key point is that this phenomenon is data independent: If the data were generated
by a model in which the age profiles become smoother over time, these models will not beable to capture this feature, and will always produce a forecast in which age profiles will
become less smooth overtime after a certain year.
The fact that age profiles may evolve in implausible ways under the Lee-Carter model
has been noted in the context of particular data sets (Alho, 1992). We extend this point
here and demonstrate that it does not depend on the particular time series extrapolation
process used or any idiosyncracies of the data set chosen. This point would also seem toresolve a major point of contention in the published debates between Lee and Carter (1992)
16

and McNown (1992) about whether the Lee-Carter model sufficiently constrains the outof-sample age profile of mortality: Almost no matter what one’s prior is for a reasonable
age profile, Lee-Carter forecasts although they may be reasonable over the short run will
eventually violate it as time passes. Notice that we refer to the Lee-Carter model in thefollowing, but our arguments apply to the RWD model as well, since they originate fromthe assumption of linearity, which is shared by the Lee-Carter and the RWD models.

4.1

Raw Patterns

We begin by examining raw patterns in the data that highlight the consequences of linearity
and then move to successively more general results and analyses. Figures 3 and 4 offer
examples of six datasets, one in each row. The left graph in each row is a time series plotof the log-mortality rate for each age group (color-coded by age group and labeled on theright), and the right graphs include the age profiles (color coded by year). For each, thedata are plotted up to year 2000 and the Lee-Carter forecasts are plotted for subsequent
years. We do not plot the RWD forecasts here, but they are easy to imagine: they aresimply a straight line drawn through the first and last observed data point and continued
into the future. The Lee-Carter does not necessarily goes through the first and last data
point, but comes close to these points. In any case the forecast is linear over time, and theconsequence of this is that, except in the knife-edged case where all the lines happen to be
exactly parallel, the time series plots of age groups will always fan out after a point. In
other words the age profiles of log-mortality will always eventually become less smooth overtime, since the distance between log-mortality in adjacent age groups can only increase.
The fanning out of the Lee-Carter forecasts can be seen clearly in all-cause male mortality
in New Zealand and Hungary (the left graph in the first two rows of Figure 3) and female
mortality from digestive disease (the left graph in the second row of Figure 4). Age groupforecasts that fan out have age profiles that become progressively less smooth over time, as
can be seen in the increasingly exaggerated right graphs in each of these examples. These
patterns account for the vast majority of the cross-sections in our data set.
In other data sets, the forecast lines converge for a period, but after converging they
cross and then from then on they too fan out — as in male suicide in the U.S. (Figure 4,
row 1, left graph). For data like these, the age profile pattern (in the right graph) inverts,
with the forecasted pattern the opposite of that indicated in the observed data. In mostcircumstances, this inversion would be judged to be highly implausible.
The knife-edged case of exactly parallel time series forecasts is very rare, but we found

17

−2

All Causes (m) New Zealand−2

All Causes (m) New Zealand
1950

80

2060

0

−4
−6

−6

50
25
20
45
15 40
35 30

−8

60
55

Data and Forecasts

−4

70
65

−8

Data and Forecasts

75

5

1980

2000

2020

2040

2060

0

25
20
−8

15

2000

2020

2040

2060

−6

−4

1955

80

−8

−2
−4
−6

30

−2

Age

5

0
10

2060

0

20

40

60

80

2060

−9.0

−8.5

−8.0

−7.5

1955

−10.0

−9.5

10
0

−9.5

−9.0

5

Data and Forecasts

−8.5

−8.0

−7.5

−7.0

80
75
65 70
20
60
55
50
25
15
45
30
35 40

−7.0

−6.5

Age

Transportation Accidents (m) Portugal

−6.5

Time

Transportation Accidents (m) Portugal−10.0

Data and Forecasts

60

All Causes (m) Hungary

35

1980

40

Time

80
75
70
65
55 60
50
45
40

1960

20

All Causes (m) Hungary

Data and Forecasts

1960

Data and Forecasts

−10

−10

10

1960

1980

2000

2020

2040

2060

0

Time

20

40

60

80

Age

Figure 3: Data and, after year 2000, Lee-Carter Forecasts by Age and Time, Part I

18

20 15

−7.5
−8.0
−8.5
−10.5 −10.0

1980

2000

2020

2040

2060

20

30

40

50

60

70

Time

Age

Digestive Disease (f) Hungary

Digestive Disease (f) Hungary

−12

25
20

−6

2060

−8

10

−14

0

−16

−14

15

1955

−10

−10

−8

30

Data and Forecasts

−6

80
55
45 50
40
60
75
35
65 70

80

−12

1960

Data and Forecasts

2060

−9.0

−8.5
−9.5

−9.0

45
65
50 55
60

Data and Forecasts

−8.0

75
35
70
40

1950

−9.5

−7.5

25
80
30

−10.5 −10.0

Data and Forecasts−7.0

Suicide (m) USA−7.0

Suicide (m) USA

−16

5

1980

2000

2020

2040

2060

0

20

60

80

Age

Cervix Cancer (f) United Kingdom
−8

Time

−11

40
60

−9
−10
−12

−12

55
50 45

2060

−11

−10

80 75
70 25
30 65
35

Data and Forecasts

−9

1950
Data and Forecasts

40

Cervix Cancer (f) United Kingdom
−8

1960

1960

1980

2000

2020

2040

2060

30

Time

40

50

60

70

80

Age

Figure 4: Data and, after year 2000, Lee-Carter Forecasts by Age and Time, Part II

19

one that was close: male transportation accidents in Portugal (Figure 3, row 3, left graph).
The Lee-Carter forecast lines do not fan out (much) in this data set, and so the age profile
stays relatively constant. Coincidentally, however, this example also dramatically illustrates
the consequences of a forecasting method that ignores all but the first and last data points.
Except in the knife-edged case, the linearity of the forecasts frequently produce implausible changes in the out of sample age profiles. We have already seen the dramatic example
of suicides in U.S. males. For another example, consider forecasts of all-cause male mortality
in New Zealand (Figure 3, first row). In these forecasts, the lines crossing in the left graphproduce implausible patterns in the age profiles, which can be seen in the right graph, with
20-year-olds dying at a higher rate than 40-year-olds. Mortality from cervix cancer in the
United Kingdom (Figure 4, last row) is another exmaple with implausible out-of-sample
age profiles. Cervix cancer is a disease known biologically to increase with age, with therate usually slowing after menopause. Although this familiar pattern can be seen in theraw data in the right graph, the forecasts have lost any biological plausibility.

4.2

Formalizations

We now formalize and generalize the insights illustrated by the empirical analyses in Figures
3 and 4. To do this, we first note that in real data, including all-cause mortality for which themodel was originally designed, the Lee-Carter mortality index γt is typically a monotonic,
although not necessarily linear, function of time. The result we describe here indicates thatin this situation, Lee-Carter age profile forecasts always become less smooth. They may
become more smooth for a period, but they will always become less smooth eventually, andat that point will continue to become less smooth for every subsequent year. This property,
which would seem to violate one of most well-known relationships in classical demography,
holds regardless of the patterns in the empirical data and for an extraordinarily large rangeof different definitions of smoothness.
We now discuss this point in some detail, describing the problem in three distinct ways.
Ultimately, however, the culprit is the following elementary observation: When a one dimensional function, such as an age profile, is “stretched”, due to multiplication by a scale
factor, small differences between neighboring values become “amplified”. The one dimensional function we have in mind, of course, is the age profile of the parameters βa , where
the scale factor that stretches the age profile at any point in time is γt .
We begin with the differences in expected log-mortality at any two points in time, t andthe “base period” t0 :20µat − µat0 = (γt − γt0 )βa(20)

To fix ideas, take all the βa positive and γt to be a decreasing function of time, so thatthe (γt − γt0 ) < 0 and hence the expected log-mortality rate µat decreases over time. We
see from Equation 20 that, as time passes, expected log-mortality decreases, relative to thebase period, but at different rates in different age groups: Age groups with βa small will seevery little decrease and age groups with larger βa will see a larger decrease. This illustrates
the universal increasing nonsmoothness property of Lee-Carter forecasts.
An interesting point is that the forecast age profile will always becomes less smooth,no matter whether we project forward or backward. Or put differently, if the Lee-Carter
forecasts fan out, and we reverse the time index on the data (so that 1950 is switched with
2000, 1951 is switched with 1999, etc.), the forecasts on the altered data set will still fan out.
This demonstrates that universal increasing nonsmoothness is a property of the model, notof the data. Every difference in βa is thus amplified as we forecast farther into the future(or past). Any values of βa that are not constant over a, in combination with the particularparametric form of the Lee-Carter model, are the cause of the problem.7
To clarify and generalize these points, we now compute a direct measure of the smoothness of the log-mortality age profiles generated by the Lee-Carter model, and observe how
it evolves over time. Thus, one simple measure of nonsmoothness is to take the average ofthe squared differences between adjacent age groups:
A

Nonsmoothness(µt ) ≡1 X(µat − µa−1,t )2
A−1

(21)

a=2

where µt ∈ RA is the age profile of mortality at time t. (This measure is also closely related
to the even simpler variance of the age profiles at any point in time, which also increases in
Lee-Carter forecasts.) In Lee-Carter forecasts, this measure of nonsmoothness will always
increase after a point, and then will continue to increase forever thenceforth. (Proving
this point is slightly more convenient in a more general context, and so delay presenting
the proof for a moment.) The point past which Lee-Carter forecasts become increasingly
nonsmooth without limit will frequently occur before the end of the observed data (as in7∂ ∂µatWe can also study this result by computing the mixed derivatives with respect to age and time: ∂t=∂a∂γt ∂βa
.
The
left
side
of
this
expression
is
the
time
variation
of
the
slope
of
the
age
profile.
Ifγ
ismonotonict∂t ∂awith time then its time derivative, on the right side, always has the same sign. Therefore the sign of theleft hand side is determined by the sign of the derivative of βa with respect to age. When the profile ofβa has a “bump,” its derivative changes sign, implying that there will be nearby age groups whose slopes
move in different directions over time, decreasing the smoothness of the age profile. This problem would be
attenuated if the profile of βa were a monotonic function of age, but unfortunately this is rarely the case
(we have not been able to find a single instance of this in any of the countries in our data base).

21

most of the examples in Figures 3 and 4, resulting in the time series plots of age groupforecasts fanning out) but will sometimes occur after the last observed point (in which casethe lines will converge until a point in time, and then diverge from then on, as in malesuicide in the U.S. in Figure 4, row 1). The point here is that the pattern seen in the figures
is perfectly general, when formalized by the expression in Equation 21.

4.3

Empirical Illustrations

To illustrate this point, we offer Figure 5, which plots the nonsmoothness measure in Equation 21 applied to observed data (in red dashed lines) and Lee-Carter forecasts (in solid
black lines) up to 2060 and backwards (or “backcast”) to 1900, for four data sets observedfor 1950–2000. (The solid line is thus the smoothness of the Lee-Carter forecast age profile
rather than a forecast of the in-sample smoothness line.) For example, the top left graphplots nonsmoothness for female cardiovascular disease in the U.S., which has been increasing throughout the entire observed period. The nonsmoothness of the Lee-Carter forecasts
continues this trend into the future, which is obviously plausible. However, the backcasts
turn upward too, which cannot be seen as plausibly related to any empirical trend in thedata.
The top right graph, portraying nonsmoothness for all-cause male mortality in Hungary,shows almost the opposite pattern. The backcasts seem to extend the trend in the data
upward, which may seem plausible, but the forecasts implausibly turn upward in about
2040, and if extended beyond where the graph ends in 2060 would increase for the rest oftime. Although short term forecasts may not be too severly affected by this property, themethod does not seem appropriate for most term forecasts unless one had some reason to
impose this property on the forecasts, since the patterns produced clearly have nothing to
do with empirical patterns generated by the data.
The two graphs at the bottom of Figure 5 offer examples with no clear in-sample trend innonsmoothness, but with nonsmoothness of forecasts and backcasts both increasing fast andextending far above the range in the data. The bottom right graph, for female cardiovasculardisease in Norway, is especially dispositive, since in these data the first prinicple component
used by Lee-Carter explains only 29% of the variation (see Table 1, Page 10), and so wemight expect that forecasts from the model to “oversmooth.” Yet the nonsmoothness of
the Lee-Carter forecasts (and backcasts) are still rapidly increasing way above that in theobserved data. Taken together, the results from all four figures illustrate the main point:
Lee-Carter forecasts of mortality produce age profiles that are less and less smooth over22

All Causes (m) Hungary

1.2
1.0
0.6

0.8

Lack of Smoothness

1.2
1.0
0.8
0.6
0.4

Lack of Smoothness

1.4

Cardiovascular Disease (f) USA

1900

1950

2000

2050

1900

2000

2050

Time
Cardiovascular Disease (f) Norway

1.4
1.2
1.0
0.6

0.8

Lack of Smoothness

0.70
0.65
0.60

0.4

0.55

Lack of Smoothness

1.6

0.75

1.8

Time
All Causes (m) USA

1950

1900

1950

2000

2050

Time

1900

1950

2000

2050

Time

Figure 5: Nonsmoothness, as defined in Equation 21, plotted as a function of time. The
red dashed line refers to the in-sample data and the continuous line corresponds to LeeCarter forecasts fit to the in-sample data (not fit to the nonsmoothness) and forecast toyear 2060 and backcast to year 1900. The parabolic shape would continue to increase inboth directions if extended.

23

time, no matter what trends exist in the empirical data.
A different, more substantive way to make the same methodological point is to suppose
we were interested in forecasting the degree to which the forecast age profile flattens out
over time, an empirical pattern observed in some countries by Lee and Miller (2001). The
point of the study would be to quantify how much flattening there is in various countries,
causes, or groups. The implication of our results is that Lee-Carter would not be of much
use in a study like this because its forecasts will (except sometimes temporarily) neverindicate that the age profile is flattening. Under Lee-Carter, forecast age profile will always
be less and less flat after a point.

4.4

Generalizations

To generalize this result, suppose we (now) recognize that the flattness of the empirical orforecast age profile (or equivalently the degree of fanning out of the time series age plots)
cannot be reasonably modeled by Lee-Carter in the long run, and we want to move on toother issues. One way to do this is to use a measure of nonsmoothness that is indifferent tolevels and to slope shifts (or “rotations”) in the age profile. For example, as the standard allcause mortality “swoosh” shape rotates clockwise, the degree of flattness increases. Such asmeasure is easy to construct by switching from the average first difference used in Equation
21 (a measure that is indifferent only to the level of log-mortality) to the average second
difference, that is differences between each pair of log-mortality differences of adjacent age
groups. This new measure indicates how close adjacent age groups are while ignoring theabsolute level of log-mortality for the entire age profile as well as any slope shifts in the
profile. Put differently, it measures how much log-mortality oscillates over the age groups
around its basic shape, ignoring any drop in average mortality or rotation in the age profile.
Our result indicates that Lee-Carter (or the more general random walk with drift) forecasts
guarantee that, after a point, this oscillation will increase, effectively indicating that eventhis adjusted shape will fit less well over time.
To continue generalizing, suppose furthermore that one hypothesized that the shape ofthe log-mortality age profile in some theoretical model or model life table is the place towhich log-mortality will eventually converge. To do this, we merely subtract the predicted
age profile from the observed age profile data, and then apply Equation 21, or a version
based on second differences. Our result is that Lee-Carter will still not be able to model theconvergence since it will always forecast that such predictions will become more and moreincorrect over time.

24

We now provide a simple proof of these claims. We begin with the same function in
Equation 21 applied to the mean adjusted age profiles, µ̃at = γt βa (which define the vectorµ̃t ), where the adjustment accounts for the overall average shape or predicted pattern of thelog-mortality age profile. (Subtracting out some predicted age profile other than based on
the empirical mean would only slightly complicate this expression.) We express our resultin three ways:
Nonsmoothness(µ̃t ) = µ̃0t W µ̃t(22)= γt2 β 0 W β

(23)

∝ γt2

(24)

where, for first differences corresponding to Equation 21, W is a tridiagonal matrix withones on the diagonal and negative ones on the two adjacent bands. The result, in thelast line of this expression, combined with the empirical result that γt is monotonicallyincreasing, is enough to show that nonsmoothness will eventually increase as time passes.
To be specific, the lack of smoothness will increase as soon as γt is positive. If γt starts
out negative, the lack of smoothness will drop for a time, until γt is zero, and it will thenincrease inexorably as all subsequent years go by. Nonsmoothness bottoms out at a pointthat is a function of the predicted age profile. The increase in nonsmoothness after thispoint will not stop, and no feature of the data or forecasting method for projecting the γt ’s
can make it stop. The U-shaped solid line in Figure 5 is merely a plot Equation 23.
By generalizing the definition of W , so that it can be any semi-positive definite matrix,
Equation 22 can be used to examine the consequences of using any of a very broad classof nonsmoothness definitions. (Many more sophisticated definitions of smoothness can bederived, but most lead to quadratic forms like the one above, with different definitions forW .) The key result of Equation 24 is that no matter what matrix is used for W , and thus nomatter what definition of smoothness one adopts, Lee-Carter forecasts still have the same
universal increasing nonsmoothness property. To be more precise, Equation 24 shows that
nonsmoothness, by any of these definitions, will always increase after a point.
These generalizations extend the substantive areas to which these results apply. They
imply that the same universal increasing nonsmoothness result applies if we consider nonsmoothness only after say age 20 and not before, or we penalize for deviations from smoothness less for younger ages. The weights would merely change the definition of W . Suppose
furthermore that we recognize that the average squared differences in adjacent age groups inEquation 21 reflects a random walk with drift over age groups and so might not be smooth25

enough: If we adjust W so this is not an issue, nonsmoothness by this alternative will still
be universally increasing. Alternatively, if instead of using the first or second difference,
we adjust W so that it measures nonsmoothness by the n-th derivative, or the n-th derivative combined with the m-th derivative, or we adjusted for discreteness or edge effects, or
any of a variety of other patterns and issues, this definition would produce a measure ofnonsmoothness for Lee-Carter forecasts that would still increase after a point and continue
increasing for the rest of time. The cause of these issues is solely because of the assumptionof linearity and would occur regardless of patterns in the data.8
Therefore, forecasts from models such as Lee-Carter and RWD will be appropriate only
if we have external knowledge that the log-mortality age profile will eventually become lesssmooth in this way. When mortality forecasts are used to allocate spending to reduce healthrisks among chosen age groups, declining smoothness will unrealistically exaggerate allocation differentials. Insurance companies, which often impose a form of smoothness (or what
they call “graduation”) on the mortality age profiles to keep their insurance schedules looking simple, would presumably find this aspect of the Lee-Carter model to be problematic.
Politically, forecasts that are insufficiently smooth could lead to unnecessarily adversarial
relationships among interest groups supporting research funding for the health problems ofdifferent age groups.

5

Properties of Alternative Estimation Approaches

In this section, we discuss two alternative approaches to estimating the parameters of the
Lee-Carter model.

5.1

Second Stage Reestimation

In their original paper, Lee and Carter (1992) suggest an alternative method of estimating
the mortality index γt . We outline this method here and then point out several previously
unrecognized problems with this approach. Lee and Carter notice that once βa and γt havePbeen estimated, the observed total number of deaths dt ≡ a dat is not guaranteed to equal
to the total number of deaths dˆt predicted by the model:
dˆt =

X

pat em̄a +β̂a γ̂t .

a
8We can also put a lower bound on the degree of smoothness in Equation 23: Nonsmoothness(µ̃t ) => λ1 γt2 , where λ1 is the first non-zero eigenvalue of W . This inequality is derived by taking
advantage of the normalization kβk = 1 and of the Raileigh variational characterization of the eigenvaluesof a matrix (Strang, 1988). This relationship is useful because it provides a lower bound for nonsmoothness
independent of the profile of βa , which is another way of showing that this property is an intrinsic featureof the model and not of the data.
γt2 β 0 W β

26

As a result, they propose computing a new estimate of γt , for each year t, by searchingfor the value that makes the observed number of deaths equal to the predicted number ofdeaths. Defining this new estimate of γt as γ̂t∗ , Lee and Carter suggest setting:
dt =

X

∗

pat em̄a +β̂a γ̂t .

(25)

a

This way of estimating γt has several advantages, which are described in detail in Lee andCarter (1992). These can be useful in the life table representation of the data, and especiallyfor cases where only the total, rather than age-specific, death rates are known in certain
years.
A problem with the estimate of γt given by Equation 25 is that this equation has either
0, 1 or 2 solutions for γt∗ , depending on the values of β̂. When it has one solution, all isfine, but with 0 or 2 solutions, the approach can be logically inconsistent and empiricallyunhelpful.
A number of solutions different from 1 arise only when the estimated values β̂a do not all
have the same sign, which fortunately is easy to check. A nonuniform sign for the values ofβ̂a implies that mortality is increasing in some age groups and decreasing in others. This israre when predicting all-cause mortality, which has been decreasing more or less worldwide,with relatively few exceptions such as Hungary. Thus, when used for all-cause mortality
in the U.S., for which Lee-Carter was designed, this issue will only occasionally arise, andin these situations this section only points out what will happen fairly rarely, althoughenough to crash software not designed to detect the problem. In cause-specific mortality,
nonuniform signs in β̂a are much more common, and so reestimation would be ill-advised.
In order to see where this result comes from, we partition the coefficients β̂a in two
groups, those with positive sign, which we denote β̂a+ = |β̂a+ | and those with negative sign,
which we denote β̂a− = −|β̂a− |. Then we rewrite Equation 25 as:
dt =

X

+ ∗

pat em̄a eβ̂a γ̂t +

a

X

− ∗

pat em̄a eβ̂a γ̂t .

aWhen all the β̂a have the same sign then the right side of this equation is a sum of exponentials in γ̂t∗ , either all increasing or all decreasing, and therefore a monotonic function ofγ̂t∗ , with range (0, +∞). This ensures that the equation has only one solution.
When the β̂a have different signs, the right side is the sum of exponentials, some increasing and some decreasing. As is well known, such a function is U -shaped, with its minimum
a strictly positive number. Therefore, if dt is large enough, a horizontal line at height equal
to dt will intersect the U -shaped function in two places, and we have two solutions. If dt is27

small enough, it will pass below the U -shaped function entirely, and the problem will haveno solution.
One alternative strategy is due to Lee and Miller (2001) who propose reestimating
γt so that the predicted and observed value of life expectancy at birth, rather than the
total number of deaths, match in each year. However, this result has the same property
as matching on the number of deaths.9 Of course, in addition, this approach cannot beapplied to cause-specific mortality. Fortunately, reestimation is not a crucial feature of the
Lee-Carter model.

5.2

Wilmoth’s Weighted Least Squares

Two other alternatives to the basic Lee-Carter estimation procedure are described in Wilmoth(1993). This is a often-cited technical report, and since the methods proposed seem to bequite popular we comment on them here.
Wilmoth’s insight was that the reason why the fitted number of deaths differs from theobserved number is that the estimates of γt are computed by minimizing the least squareerror over log-mortality, rather than mortality. As a result, age groups with small numbersof deaths receive the same weight as age groups with large numbers, even though they
contribute very little to the totals. Hence, rather than using the second stage estimationof γt as described in Lee and Carter (1992), Wilmoth proposes two alternative estimation
strategies: a weighted least square (WLS) and a maximum likelihood (MLE) technique.
Both techniques have the significant advantage, over the original Lee-Carter formulation,that they deal naturally with the case in which the observed number of deaths is zero, which
occurs when analyzing cause-specific data and/or when dealing with small countries. While
the MLE technique is statistically sound (if appropriate to the data at hand), we show that
the WLS procedure is not.
The WLS technique is based on the recognition that one could weight the first stage of
Lee-Carter in such a way that observed and predicted deaths are closer to each other. To
be specific, Wilmoth suggests finding the parameters αa , βa and γt of the Lee-Carter modelof Equation 1 as the solution of the weighted least squares (WLS) problemminαa ,βa ,γtXdat (mat − αa − γt βa )2 ,

(26)

at

where dat is the observed number of deaths in age group a at time t, and where γt andβa satisfy the usual constraints of the Lee-Carter model. By weighting the error by the9

Our thanks to Ron Lee and Nan Li for pointing this out to us.

28

observed number of deaths, the expression above gives more weight to those age groups andyears with large numbers of deaths, and the resulting estimates are more likely to fit the
total number of deaths in each year.
This procedure has two additional apparently appealing features. The first is that it
eliminates the problem that log-mortality is not defined when the number of deaths is zero.
Although when dat = 0 the expression mat = ln(dat /pat ) is meaningless, these observations
are discarded in Equation 26, since they receive zero weight. A second appealing featureof Equation 26 is that it is easy to write down the corresponding first order conditions(Wilmoth, 1993: 3).
Unfortunately, the procedure is not statistically sound, and these advantages may be outweighed by the bias induced by the estimator. In particular, Wilmoth interprets Equation
26 as the log-likelihood of a model with normal, heteroskedastic disturbances, and varianceproportional to 1/dat . This claim is incorrect, since the variance of the disturbances cannot
be (inversely) proportional to the observed number of deaths. A valid weighted likelihood,must use exogenous weights, but obviously the number of deaths is a random variable. As
such, estimates resulting from this minimization problem have no known statistical properties.
The other troubling feature of this approach is computational. Despite the apparent
simplicity of Equation 26, its minimization with numerical methods may suffer of a problem
of mulitple local minima. This is easily seen in the non-weighted case, where all the weights
are equal, which corresponds to the standard Lee-Carter model. In this case, one chooses
the value of β that maximizes the quadratic form β 0 Cβ in the unit sphere kβk = 1. Unfortunately, once the quadratic form β 0 Cβ (in A variables) is restricted to the unit sphere,it acquires exactly 2A local extrema, and therefore any numerical maximization algorithm
which relies only on first order conditions may converge to a suboptimal solution. Fortunately, using principal components or SVD to estimate β bypasses this problem. However,when the weights are not all equal SVD cannot be used any longer. It is not obvious in thiscase what happens to the number of local extrema, and we cannot prove easily that it is stillequal to 2A. However, since the function to be maximized in Equation 26 depends linearlyon the weights dat , it is a smooth deformation of the objective function whose weights areconstant. Consequently, we can still expect a certain number of local minima and maxima. It may turn out in practice that some minimization techniques lead to apparently
meaningful results. However, this adds another reason to prefer the MLE over the WLS
method. The MLE is based on the observation that the number of deaths is a counting

29

random variable which can be modeled by a Poisson process. Therefore, instead of the OLS
specification for log-mortality with homoskedastic errors used by Lee and Carter one could
use instead the following Poisson specification:dat∼ Poisson(eµat ) µat = αa + βa γt
pat

(27)

This approach leads to a standard maximum likelihood estimator, with its usual well-knwon
properties: its only difficulty is that many commonly used statistical packages for Poisson
regression will not be able to handle the bilinear form βa γt . However, in a recent paper
Brouhns et al. (2002) report the successful implementation of Wilmoth’s strategy 27 using
LEM, a freely available software package for the analysis of categorical data (Brouhns,
Denuit and Vermunt, 2002). Therefore if choosing between the WLS and MLE estimators,
MLE would be preferable, even if in the analysis of U.S. data little difference between thetwo has been found.

6

Concluding Remarks

The insights in Lee and Carter (1992), and formalized in their model, clearly must be
included or at least addressed in any approach taken to forecasting mortality from U.S. orother closely related data. The key recognition by Lee and Carter is that U.S. nationallog-mortality data have with few exceptions followed a fairly linear path over recent history,
and different age groups are highly correlated over time. Over relatively short time periods
that may be adequate for some purposes, in some data the Lee-Carter forecasting model
continues these patterns. Over longer periods, the model is guaranteed to violate theobserved empirical patterns, and any fixed baseline prediction.
Whether any empirical pattern will continue into the future is of course a key questionthat is the subject of almost every forecasting work. In this paper, we point out some apparently unforseen or under-appreciated properties of the Lee-Carter model that guaranteethat forecasts (or backcasts) from it will not remain consistent with observed U.S. mortality
data and it will deviate in a way that will likely also be inconsistent with many demographers’ priors beliefs about the patterns of future mortality. Less surprisingly, we showthat the model is unlikely to be applicable to many cause- and country-specific mortality
data sets for which it was not designed. Those considering forecasting using the Lee-Cartermodel should consider whether the patterns in their data match those that can be represented by this model, and they should assess whether the properties that all forecasts fromthis model have are consistent with their prior beliefs.

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