Lee and Carter (1992) proposed a simple methodfor forecasting future mortality, consisting of a basemodel of age-specific death rates with a dominanttime component and a fixed relative age com-ponent, and a time series model (autoregressiveintegrated moving average (ARIMA)) of the timecomponent. Using US mortality data for the period1900–89, they employed a matrix decompositionmethod to identify the two components, and thenprojected the time component to 2065, using alinear ARIMA forecasting model. The base modeland ARIMA constitute the main features of theLee-Carter method. Several advantages are claimed:a parsimonious demographic model is combinedwith standard statistical time-series methods; nosubjective judgements are involved; forecasting isbased on persistent long-term trends; and proba-bilistic confidence intervals are provided for theforecasts (Lee and Carter 1992).

The Lee-Carter method has been widely adopted(see for instance, North American ActuarialJournal 2(4) 1998; Hollmann, Mulder and Kallan2000; Lee 2000). Tuljapurkar, Li and Boe (2000)applied the method to data for the period 1950–94from the seven most economically developedcountries (G7). They identified a ‘universal pattern’(p.789) of mortality decline characterized byroughly linear decline in the dominant time compo-nent for each country, and used this pattern toproject the declines to 2050 using a linear fore-casting model. This linearity contributes to themethod’s simplicity and hence its popularity (Alho2000; Lee and Miller 2001; Carter and Prskawetz2001).

A major problem with the Lee-Carter method is

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I N T RO D U C T I O N

The populations of the industrialized worldexperienced a major mortality transition over thecourse of the twentieth century. Mortality ratesdeclined dramatically. In Australia, for example, lifeexpectancy increased from 57 years in the period1901–10 to 80 years in 1999. Two trends dominatedthe mortality decline. The first was a reduction inmortality caused by infectious diseases affectingmainly young ages, particularly in the first half ofthe century. The second was increasing and then,from about 1970, decreasing mortality from chronicdiseases affecting older ages. In Australia, thebalance of these two trends resulted in a period ofalmost constant mortality in the 1960s. Othertrends also emerged: an accident ‘hump’ after theSecond World War and, recently, decreasing morta-lity at very old ages. The current pace of decline inlow-mortality populations shows no sign of decel-eration, leading to international debate aboutwhether there are limits to human longevity andhow soon these might be reached (Wachter andFinch 1997).

In order to forecast mortality it must be ade-quately modelled, but although many differentmodels have been developed (Tabeau 2001), not allare suitable for forecasting. For instance, the 8-parameter model developed by Heligman andPollard (1980) has been found to fit a wide range ofmortality schedules, but the large number of para-meters limits the model’s potential for projection(McNown and Rogers 1989).

Applying Lee-Carter under conditions of variablemortality decline

HEATHER BOOTH, JOHN MAINDONALD, A N D LEN SMITH

Abstract. The Lee-Carter method of mortality forecasting assumes an invariant age com-ponent and most applications have adopted a linear time component. The use of the methodwith Australian data is compromised by significant departures from linearity in the timecomponent and changes over time in the age component. We modify the method to adjust thetime component to reproduce the age distribution of deaths, rather than total deaths, and todetermine the optimal fitting period in order to address non-linearity in the time component.In the Australian case the modification has the added advantage that the assumption ofinvariance is better met. For Australian data, the modifications result in higher forecast lifeexpectancy than the original Lee-Carter method and official projections, and a 50 per centreduction in forecast error. The model is also expanded to take account of age-time inter-actions by incorporating additional terms, but these are not readily incorporated intoforecasts.

Population Studies, 56 (2002), 325–336Printed in Great Britain

Submitted: February 2002Final version accepted: August 2002

the assumption that the age component is invariantover time (Lee and Miller 2001). In fact, themortality experience of the industrialized worldover the course of the twentieth century wouldsuggest substantial age-time interaction: the twodominant trends affected different age groups atdifferent times. Lee and Miller (2001, p.545) suggestthat a ‘simple and satisfactory solution’ to thisproblem is to base the forecast on data since 1950,as Tuljapurkar et al. (2000) had done, so that theassumption that the age component is invariantapplies to only half the century. However, in theAustralian case, whether the entire century or onlythe second half is taken into account, it is clear thatthe assumption of invariance in the age componentwill be violated (AIHW 2000, ch.8). Further, pre-liminary exploration of the application of themethod to Australian data for the period 1950–94for comparison with Tuljapurkar et al. (2000) indi-cated a marked departure from linearity in thedominant time component, casting doubt on the‘universal pattern’ of decline and hence the generalapplicability of the linear forecasting model. Theseproblems detract from the validity and reliability ofthe Lee-Carter method with linear temporaldecline.

The aim of the study presented here was toimprove the validity and reliability of the Lee-Carter method by improving the fit of the basemodel and by addressing departures from linearityin the dominant time component and the violationof the invariance assumption in the age component.Specifically, this involved developing methods forimproving the estimation of the time component,for identifying the most appropriate period towhich to apply the Lee-Carter method with a linearARIMA, and for incorporating age-time inter-actions into the base model.

T H E L E E-C A RT E R M E T H O D

The Lee-Carter method (Lee and Carter 1992)comprises a base model of mortality, a method forfitting the model including adjustment of the timecomponent, and a time series model of the timecomponent which is used for forecasting. Thisdescription of the method does not take intoaccount subsequent refinements by Lee and others;these are discussed as appropriate in later sections.The basic Lee-Carter model is:

ln[m(x,t)] = a(x) + b(x)k(t) + ε(x,t)

where

m(x,t) is the central death rate at age x in year t

k(t) is an index of the level of mortality a(x) is a set of age-specific constants describing the general pattern of mortality by age

b(x) is a set of age-specific constants describing the relative speed of change at each age

ε(x,t) is the residual at age x in year t.

The time component, k(t), or the ‘dominanttemporal signal’ (Tuljapurkar et al. 2000, p.789)captures the overall time trend in ln[m(x,t)] at allages. The base model makes no assumptions aboutthe functional form of the trend in k(t). The agecomponent, b(x), modifies the main time trendaccording to whether change at a particular age isfaster or slower than the main trend (and in thesame or opposite direction). The model assumesthat b(x) is invariant over time.

Lee and Carter used the singular value decom-position (SVD) to find a least squares solution inthe estimation of b(x) and k(t). The SVD (Trefethenand Bau 1997) decomposes the matrix of ln[m(x,t)]into the product of three matrices, representing theage component, the singular values, and the timecomponent. In order to obtain a unique solution,a(x) is set equal to the means over time of ln[m(x,t)]and b(x) is constrained to sum to unity. The k(t)values sum to zero. Under this normalization, b(x)is the proportion of the change in overall logmortality attributable to age x.

In the rank 1 approximation used by Lee andCarter, k(t) is derived from the first vector of thetime-component matrix weighted by the firstsingular value, and b(x) is derived from the firstvector of the age-component matrix. The secondand higher terms of the SVD together comprise theresidual. Typically for low-mortality populations,the rank 1 approximation accounts for about 95 percent of the variance in ln[m(x,t)].

Since the base model estimates the logarithms ofrates, thus giving equal weight to ratios of rates, Leeand Carter adjusted each k(t) to reproduce annualtotal deaths, D(t), while leaving a(x) and b(x)unchanged. This adjustment gives greater weight toages at which numbers of deaths are large. Anappropriate ARIMA model (Diggle 1990) is thenused to obtain a time series fit to adjusted k(t). Inprinciple, this could be linear or non-linear, but Leeand Carter (1992) and almost all others applyingthe method, including Tuljapurkar et al. (2000),have adopted a random walk with drift. The model,which is linear in t, is

k′(t) = k′(t-1) + d + e(t) (1)

where k′(t) denotes adjusted k(t), d is ‘drift’ (averageannual change in k′(t)), and e(t) are uncorrelatedinnovations. Forecast age-specific death rates and

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life expectancies are obtained by incorporatingextrapolated k′(t) in the base model. The variancesof e(t) and of the estimate of d together estimatethe uncertainty associated with a one-year forecast,which is used to produce probabilistic confidenceintervals for the forecast values of k′(t) and lifeexpectancy. For a forecast p years into the future,the forecast error variance is the sum of p.var(e(t))and p2.var(d). In presenting forecast confidence inter-vals, Lee and Carter (1992, p.665) generally ignore theuncertainty in the estimate of d; the relatively minoruncertainty in the estimates of a(x) and b(x) is alsoignored (Lee and Carter 1992, Appendix B).

M O D I F I C AT I O N S O F T H E L E E-C A RT E R

M E T H O D

Three modifications are made to the Lee-Cartermethod. First, the fit of the base model to observeddeaths is improved by adjusting k(t) by fitting it tothe age distribution of deaths rather than to totaldeaths. The second modification addresses thedeparture from linearity in k′(t) by developing amethod for identifying the most appropriate fittingperiod. Third, the base model is expanded toinclude higher terms.

In the original Lee-Carter method, adjustment ofk(t) to reproduce D(t) renders the minimizationcriterion unclear. Adjustment to reproduce lifeexpectancy, as suggested by Lee and Miller (2001,p.540), in order to avoid the need for populationdata, renders the minimization criterion more com-plex and even less clear. In the present study, k(t) isadjusted by fitting a Poisson regression model to theannual number of deaths at each age, D(x,t)(Brillinger 1986). The Poisson model is

ln[D(x,t)] = ln[N(x,t)] + ln[m′(x,t)] + ε′(x,t)

where N(x,t) denotes the population aged x at timet, ln[m′(x,t)] = a(x) + b(x)k′(t), k′(t) now refers toadjusted k(t) under the present adjustment andε′(x,t) refers to the residuals after adjustment ofk(t). The minimization criterion is the deviance:

deviance(t) = 2 Σx {D(x,t) ln[D(x,t) / D′(x,t)] – (D(x,t) – D′(x,t))} (2)

where D′(x,t) denotes fitted deaths (Maindonald1984, p.243). Fitted deaths are obtained by

D′(x,t) = N(x,t){exp[a(x) + b(x)k′(t)]}.

This modification has several technical advan-tages. First, the use of the deviance as the minimiza-tion criterion is a standard statistical practice,which is useful in comparisons, in particular indetermining the fitting period (see below). Second,

adjusting k(t) by fitting to D(x,t) gives greaterweight to ages at which numbers of deaths are large,using weights that are known, whereas in the orig-inal Lee-Carter adjustment, although the weightsare generally greater at these ages, the weighting iscomplex and does not conform to standard statis-tical criteria. While the total number of annualdeaths may not be reproduced exactly, this loss isoutweighed by the greater accuracy in reproducingthe age distribution of deaths. Third, this is a (con-ditional) maximum likelihood procedure assumingquasi-Poisson errors, the statistical properties ofwhich are well understood. Fourth, it is easier tocalculate than the Lee-Carter minimization becauseit does not require finding the root of a non-linearfunction.

Wilmoth (1993) also examined alternativeapproaches to the estimation of the Lee-Carterparameters. He used weighted least squares andmaximum likelihood estimation, which were shownin the case of Japan to give almost identical results.In his maximum likelihood approach, all three Lee-Carter parameters are estimated simultaneously.The method presented in this paper, which wasdeveloped independently, is an alternative way ofderiving maximum likelihood estimates of k(t),given the initial parameter estimates obtained bySVD, and may be considered as a first iteration ofthe full method used by Wilmoth.

A consequence of this modification is that thejump-off value of k′(t) (that is k′(t) for the final yearof the fitting period (the jump-off year)) will bedifferent from that in the original Lee-Cartermethod. In fact, the Lee-Carter method has beencriticized for the fact that discontinuity is possiblebetween observed mortality rates and life expectan-cies for the jump-off year and forecast values for thefirst year of the forecast period (Bell 1997). The biasarising from this discontinuity persists throughoutthe forecast. In a recent modification of the Lee-Carter method, Lee and Miller (2001) usedobserved death rates for the jump-off year as thebasis for the forecast, which made a difference of0.6 years in the life expectancy for the jump-offyear. The use of observed rates has the advantage ofavoiding discontinuity but has the disadvantage ofextrapolating any idiosyncrasies that may bepresent in the data for the jump-off year (Lee andMiller 2001, p.539). When k′(t) is based on the fit toD(x,t) rather than D(t), the jump-off rates are likelyto be closer to observed rates so that thediscontinuity is likely to be less while the effect ofidiosyncrasies is reduced.

The second modification introduces a method foridentifying the most appropriate period to which to

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apply the Lee-Carter method with linear ARIMA.In doing so, it is assumed a priori that the trend ink′(t) is linear, based on the ‘universal pattern’ ofdecline identified by Tuljapurkar et al. (2000).When k′(t) departs from linearity (as for example inFigure 1), this assumption may be better met byappropriately restricting the fitting period. Theending year of the fitting period is determined bythe latest available data. The optimal choice ofperiod is thus determined by the starting year,denoted by S. In this analysis, the choice of S isbased on statistical measures of goodness of fit.

The total lack of fit of the model is composed oftwo parts: the base lack of fit from the basic Lee-Carter model after adjustment of k(t), and theadditional lack of fit from the imposition ofthe linear ARIMA model on k′(t). The base lack of fit for the period from S to 1999 is measured by deviancebase(S) = Σtdeviancebase(t) where theD′(x,t) in equation (2) are derived from the k′(t).The total lack of fit is measured by deviancetotal(S)= Σtdeviancetotal(t) where the D′(x,t) are derivedfrom the linear ‘fit’. The linear ‘fit’ to k′(t), which is used only in the determination of the fittingperiod, is taken to have gradient d (see equation (1))and to pass through the mean of k′(t) at mid-period. In order to compare deviancetotal(S) withdeviancebase(S) and to achieve comparabilitybetween periods, they are each divided by therelevant degrees of freedom (df) to produce meandeviance statistics. For n age categories and m yearsin the fitting period (m = 1999-S+1), the df fordeviancebase(S) is (n-1)(m-2) and the df fordeviancetotal(S) is n(m-2) (see Booth, Maindonaldand Smith 2001).

The choice of S is determined by the extent of theadditional lack of fit relative to the total. If the fit ofthe linear model is good, the additional lack of fitwill be small. The extent of the additional lack of fitis indicated by the ratio of total to base lack of fit,

R(S) = {deviancetotal(S)/[n(m-2)]} / {deviancebase(S)/[(n-1)(m-2)]}.

The criterion adopted for choice of S is that R(S)is substantially smaller than the correspondingstatistic for preceding years. A change-point testcould be used to formalize this criterion (Kuan andHornik 1995) but was not considered necessary inthe present application. In situations where there isno clear advantage in restricting the fitting period tothe more recent past (such as would have been thecase in the early 1970s), the longer fitting periodshould be used. However, such situations are notcurrently relevant to the mortality experience ofmost developed countries.

This modification is designed to increase thevalidity and reliability of the forecast. Restricting thefitting period to the longest recent period for whichk′(t) does not deviate markedly from linearity hasseveral advantages. First, since systematic changes inthe trend in k′(t) are avoided, the uncertainty in theforecast will be reduced. Second, b(x) is likely tobetter satisfy the assumption of invariance. Third,the estimate of the drift will more closely reflectrecent experience, resulting in a smoother transitionbetween actual and forecast rates of change.

Having established the optimal fitting periodbased on the additional lack of fit stemming fromthe linear model, both the applicability and perfor-mance of the base model (after adjustment of k(t))need to be examined for this period. The applica-bility of the model in terms of the assumption ofinvariant b(x) is assessed by comparing b(x) distri-butions for a series of subperiods within the fittingperiod. (Note that since b(x) sum to 1, values ofb(x) and k(t) are not directly comparable betweenfitting periods. To assist interpretation and achievecomparability between fitting periods, b(x) may berescaled to describe the average annual rate ofchange at each age (Booth, Maindonald and Smith2001)). Model performance is assessed in terms ofthe randomness of the residuals. A lack of random-ness would indicate the presence of systematic vari-ation, such as age-time interactions, not modelledby b(x)k(t). Further, the adjustment of k(t) mayhave introduced systematic changes to the residuals.The examination of model performance is in factbased on ε′(x,t) calculated using k′(t).

In order to gain a better understanding of age-time interactions, a third modification is intro-duced. The original method makes use of only thefirst terms of the SVD, but in principle the secondand higher order terms could be incorporated in thebase model. The full expanded model is:

ln[m(x,t)] = a(x) + b1(x)k1(t) + b2(x)k2(t) + ...+ bn(x)kn(t)

In this model, bi(x)ki(t) is referred to as the ithterm of the rank n approximation, where i = 1 to n(the number of age categories) (Golub and VanLoan 1983). Any systematic variation in the resid-uals from fitting only the first term would becaptured by the second and higher terms. An itera-tive procedure is used to refine progressively the fitto ln[m(x,t)] by adding successive bi(x)ki(t) afteradjusting ki(t) to minimize the lack of fit to D(x,t).SVD is used to obtain each fit, with each additionalterm taken as the rank 1 approximation to theresiduals from the previous iteration (see Booth,Maindonald and Smith 2001).

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A P P L I C AT I O N T O AU S T R A L I A N DATA

Data

The database for this study is a set of annual age-specific death rates for Australia from 1907 to 1999.Following the approach used by Lee and Carter(1992) and Tuljapurkar et al. (2000), data for bothsexes combined are used. The age categories are 0,1–4, 5–9,…, 80–84, 85+.

Model fit

The Lee-Carter method with adjustment of k(t) byfitting to the age distribution of deaths was initiallyapplied to the entire data series for the period1907–99. Figure 1 shows k(t) before adjustment,after adjustment by fitting to D(x,t) and, for com-parative purposes, after adjustment by fitting toD(t) as in the original Lee-Carter method. Sinceboth methods give greater weight to ages at whichdeaths are numerous, the two adjustments are gen-erally in the same direction. It is seen that adjust-ment has a substantial effect and (as expected) ismore pronounced for the original Lee-Cartermethod. The pattern in the adjustment is due to theextent of change, relative to overall change, inm(x,t) at ages where D(x,t) comprises a large pro-portion of total deaths. In the early part of theperiod the distribution of deaths was fairly evenlyspread over age (with the exception of age 0) so thatreweighting has a minor effect on k(t). In the 1960sthe majority of deaths occurred at ages 50–79, andat these ages m(x,t) was virtually constant, resultingin an upward adjustment of k(t). From about 1980,deaths at ages 80+ increased proportionately whiledeath rates at these ages decreased more rapidlythan overall, resulting in a downward adjustment.

Figure 1 also shows that k′(t) departs sub-stantially from linearity. This is also seen in Figure 2from the large additional lack of fit (the difference

between total and base lack of fit) for periodsstarting before the late 1960s. Examination of thecorresponding ratios, R(S), shown in Figure 3identifies 1968 as the pivotal year: the addition ofthe preceding year marks the beginning of a periodof substantial and consistent increase.

Figure 2 shows that, compared with the period1968–99 or shorter fitting periods, the period1907–99 results in a relatively poor fit of both thebasic Lee-Carter model and the linear model.Violation of the assumption of invariant b(x)contributes to the lack of fit of the base model forthis longer period. As seen in Figure 4, the b(x)distributions differ substantially when differentsubperiods are used in fitting. Figure 2 also showsthat the fit of the base model is better for fittingperiods starting in later years, particularly after themid-1940s. This suggests less variation in b(x) overtime for these later fitting periods. Thus, the use of1950 as the starting year as proposed by Lee andMiller (2001) would indeed improve the fit of thebase model (though significant variation in b(x)clearly remains (Figure 4)). However, the lack of fitattributable to the linear model for this starting yearremains almost unchanged in comparison with theperiod 1907–99, and constitutes a relatively largeproportion of the total lack of fit (see Figure 3).

For the optimal fitting period, 1968–99, the

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Figure 1. k(t), k′(t) when fitted to D(x,t), and k′(t) when fitted toD(t), Australia 1907–99Source: Based on data from Australian Bureau of Statistics

Figure 2. Mean deviance statistics of total and base lack of fitfor periods 1907–99 to 1997–99, AustraliaSource: As for Figure 1

Figure 3. Ratio of total to base lack of fit for periods 1907–99 to1997–99, AustraliaSource: As for Figure 1

assumption of invariant b(x) holds fairly well, asseen in Figure 5 which shows b(x) values forsubperiods of the optimal period. The rank 1approximation accounts for 97 per cent of thevariance in ln[m(x,t)]. Values of k(t) and k′(t) areshown in Figure 6. As is to be expected, theassumption of linearity is clearly better met thanfor the period 1907–99 (Figure 1).

Figure 7 shows b(x)k′(t) as a surface of age bytime effects. Since k′(t) is roughly balanced aboutzero, positive effects in the earlier part of the periodare balanced by negative effects in the later part. Alarge b(x) results in greater reductions at age x (forexample at age 0), and a small b(x) results in smallerreductions (for example at age 25–29).

Forecasts to 2050

Forecasts were made to 2050, following Lee andCarter and Tuljapurkar et al. For the optimal fittingperiod, the jump-off rates obtained by fitting toD(x,t) imply a life expectancy of 79.7, which is 0.1years greater than the actual life expectancy in 1999of 79.6 years. By 2050, the forecast value is 92.0years with a 95 per cent confidence interval of 85.1to 99.8 years. The asymmetry in this confidenceinterval, which changes over time, is due to the non-linearity of the transformation from k′(t) to lifeexpectancy arising from the combined effects of the

entropy of the life table (Keyfitz 1985) and theshape of b(x).

Since incorporating choice of fitting period intothe Lee-Carter method is designed to improve thefit of the linear model with a view to increasing thevalidity and reliability of the forecast, it is ofinterest to compare the above forecast with similarforecasts based on longer fitting periods. Theperiods 1950–99 and 1907–99 are used for compari-son, the first following Tuljapurkar et al. (2000) andLee and Miller (2001) and the second using allavailable data as implicit in the assumptions of theoriginal Lee-Carter method. Table 1 shows forecastcharacteristics for the three fitting periods. Since thescale of k(t) is arbitrary and different for eachfitting period, the drift and standard errors cannotbe compared directly between periods. The coeffi-cients of variation of the innovation term showincreasing variation with length of fitting period.Since ordinarily the reverse would be expected, thisunderlines the magnitude of the improvement in thefit. Comparison between forecast life expectanciesin 2050, which are also shown in Figure 8, showsthat the most rapid increase occurs in the forecastbased on 1968–99, reflecting the more rapid rate ofmortality decline in recent decades.

Forecast standard errors and confidence intervalsfor forecast life expectancies, obtained by adjustingk(t) by fitting to D(x,t), are shown in Table 1 on twodifferent bases. The first combines uncertainty inthe estimates of innovation and drift (used in thepresent analysis) and the second is based on uncer-tainty in the innovation term only (following Leeand Carter). The fact that uncertainty in the drift isproportional to the square of the forecast length,p, means that the more complete standard error of the forecast increases at a faster rate than thatbased on innovation alone, the relative factor being(1 + p/(m-1))1/2. Thus, the relative increase is greater for a shorter fitting period, m. This largelyexplains why the more complete confidence interval

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Figure 4. b(x) values for selected subperiods, Australia 1907–99Source: As for Figure 1

Figure 5. b(x) values for selected subperiods, Australia 1968–99Source: As for Figure 1

Figure 6. k(t) and k′(t), Australia 1968–99Source: As for Figure 1

for forecast life expectancy based on the optimalperiod is wider than those based on longer fittingperiods.

Table 1 also compares the results of the adjust-ment of k(t) by fitting to D(x,t) rather than to D(t)as in the original Lee-Carter method. It is seen thatthe effect on the drift and jump-off bias is negligiblefor the periods 1968–99 and 1950–99, resulting in adifference of less than 0.2 years in life expectancy in2050, but that for the period 1907–99 this differenceis 0.8 years, owing largely to a reduced jump-offbias. The effect on the standard errors (of innova-tion and drift) is one of substantial reduction for allfitting periods. The standard error of the forecastbased on innovation only is reduced by 12 to 18 percent. In terms of life expectancy, this translates intoa reduction in the width of the confidence intervalof 13 to 20 per cent by 2050.

In the short term, this reduction in forecastuncertainty achieved by fitting to D(x,t) is greaterthan the increase caused by taking account ofuncertainty in the drift. As the length of theforecast period increases, however, this advantage islost. The period for which the advantage ismaintained depends on the ratio of the standarderrors of the innovation term and the length of thefitting period. For forecasts based on the periods1968–99 and 1950–99, the advantage in thestandard error of the forecast is maintained for onlythe first 12 and 14 years respectively, equivalent to11 and 13 years for the width of the confidenceinterval for life expectancy. For 1907–99, theadvantage is maintained for 46 years, or 47 years for

the width of the confidence interval; and the asym-metry in the confidence interval, noted above,results in advantage for 44 years for the lowerbound and for over 51 years for the upper bound.

Comparison of the confidence limits, shown inTable 1, also depends on differences in the drift andjump-off bias. As already noted, these have negli-gible effects for the periods 1968–99 and 1950–99.For the period 1907–99, however, the difference inthe jump-off bias is 0.54 years, sufficient to placethe upper confidence limit based on innovation anddrift below that based on the original Lee-Cartermethod.

The total effect of the modified method, that iswith adjustment of k(t) by fitting to D(x,t) andoptimizing the fitting period (when forecast error isbased on innovation only), is seen by comparison ofthe second and last columns of data in Table 1.Jump-off bias is reduced from 0.40 to 0.11 years,and forecast life expectancy in 2050 is increasedfrom 87.0 to 92.0 years. The standard error of theforecast is reduced by 50 per cent, resulting in thewidth of the confidence interval in 2050 beingreduced by 28 per cent, from 12.4 to 8.9 years.

Figure 8 also shows the most recent officialprojection of life expectancy for Australia, whichembodies an implicit asymptotic limit (ABS 2000).It is seen that the modified Lee-Carter forecastexpectancy based on the period 1968–99 exceedsthe official projection, though the official projectionfalls within the lower confidence interval for almostthe entire forecast period. In 2027, forecast lifeexpectancy is 86.5 years compared with the official

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Figure 7. Surface of age by time effects, b(x)k′(t), Australia 1968–99Source: As for Figure 1

projection of 83.3; and in 2050 these figures are 92.0and 84.9 years. The officially projected figure is alsolower than the forecast expectancy based on theperiod 1950–99, and while it exceeds that based onthe period 1907–99 for the first half of the period byup to 0.5 years, by 2050 it is lower by 1.3 years.

Model performance

The above forecasts are based on the rank 1approximation of the Lee-Carter model whichincludes only the first term, b1(x)k1(t), of the rank napproximation or expanded model. Figure 9 shows

the residual death rates from this rank 1approximation after adjustment of k(t). Theseexhibit a systematic pattern that clearly demon-strates the presence of age-time interaction. Thepattern remains the same if unadjusted k(t) is usedand an examination of the difference betweenresiduals based on k(t) and k′(t) indicates that age-time interaction is not an artefact of the adjustment.The diagonal pattern in the residuals is rather toosteep to be purely a cohort effect and might better bedescribed as a cohort-period effect. After the secondterm was incorporated, the residuals appeared muchless systematic, suggesting that the second termlargely models the cohort-period effect (though itaccounts for only 1.3 per cent of the variance inln[m(x,t)]). However, marked clustering remained,and this was reduced by the addition of the thirdterm (not shown; see Booth, Maindonald andSmith 2001, Figures 20–23). Further terms were notadded because they accounted for a negligibleproportion of the variance.

D I S C U S S I O N

This analysis has introduced three modifications tothe Lee-Carter method, namely an improved fit ofthe base model, the identification of the optimalfitting period for the linear ARIMA and theincorporation of age-time interactions in the basemodel.

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Table 1. Characteristics of Lee-Carter mortality forecasts by fitting period, method of adjustment of k(t) and basis of forecast error,Australia

1968–1999 1950–1999 1907–1999

D(x,t)D(t)

D(x,t)D(t)

D(x,t) D(t)

Forecast Innovation Innovation Innovation Innovation Innovation Innovation Innovation Innovation Innovationcharacteristic +drift only only +drift only only +drift only only

Drift –0.4720 –0.4776 –0.3544 –0.3512 –0.3771 –0.3912se(innovation) 0.5923 0.6991 0.6306 0.7202 0.9587 1.1759coefficient of variation 1.25 1.46 1.78 2.05 2.54 3.01(innovation)

se(drift) 0.1064 0.1256 0.0901 0.1029 0.1000 0.1226se(1-year forecast) 0.6018 0.5923 0.6991 0.6370 0.6306 0.7202 0.9639 0.9587 1.1759se(11-year forecast) 2.2865 1.9644 2.3187 2.3142 2.0914 2.3886 3.3644 3.1796 3.9000se(21-year forecast) 3.5152 2.7142 3.2038 3.4538 2.8897 3.3003 4.8690 4.3933 5.3886se(51-year forecast) 6.8792 4.2297 4.9927 6.4332 4.5032 5.1432 8.5357 6.8465 8.3976

Jump-off bias 0.11 0.07 0.12 0.07 –0.14 0.40Life expectancy in 2050 92.0 92.1 89.3 89.1 86.2 87.0– lower confidence limit 85.1 87.7 87.1 82.5 84.5 83.7 80.3 81.5 81.1– upper confidence limit 99.8 96.6 97.6 96.9 94.5 95.1 92.7 91.3 93.5– confidence interval width 14.7 8.9 10.5 14.4 10.0 11.4 12.4 9.9 12.4

Notes: In the original Lee-Carter method, k(t) is adjusted by fitting to D(t); in the modified method fitting is to D(x,t). Drift andstandard errors are not comparable between fitting periods.Source: Forecasts based on data from Australian Bureau of Statistics.

Figure 8. Actual and forecast values of life expectancy and 95per cent confidence intervals by fitting period, and official pro-jection, Australia 1968–2050Source: Official ABS projection from Australian Bureau ofStatistics (ABS). 2000. Population Projections, Australia,1999–2101. Canberra. Catalogue No. 3222.0.

The improvement of the fit of the base model wasachieved through the adjustment of k(t) by fittingto D(x,t) rather than to D(t). While this maximumlikelihood approach was adopted on grounds oftechnical advantage, in particular to address theneed for a comparable goodness-of-fit statistic tofacilitate determination of the optimal fittingperiod, it has also resulted in substantive benefits,including an improved estimate of the drift andreduced forecast error.

When comparing the modified method with theoriginal Lee-Carter method, both the adjustment ofk(t) and choice of fitting period are taken intoaccount. For Australian data for 1907–99, adoptingthe maximum likelihood approach results in areduction of 0.8 years in life expectancy by 2050and a reduction in forecast error of 18 per cent. Theadditional adoption of the optimal fitting periodresults in a significant change in drift, producing anincrease in life expectancy of 5.8 years by 2050, anda 38 per cent reduction in forecast error. In total,therefore, the modified method results in anincrease in forecast life expectancy of 5.0 years in2050, and a 50 per cent reduction in forecast error,equivalent to a 28 per cent reduction in the width ofthe confidence interval for life expectancy.

The forecast error in the above comparison isbased on innovation only. In fact, the confidenceintervals presented in this paper are based onforecast standard errors that include uncertainty inboth innovation and drift. Lee and Carter (1992,p.665) acknowledge that inclusion of uncertainty inthe drift would increase the standard error of theirforecast based on the period 1900–89 by 25 per centafter 50 years, but choose (without stating a reason)not to take it routinely into account. The inclusionof uncertainty in the drift counterbalances thereduction in forecast uncertainty due to themaximum likelihood adjustment. For Australiandata for the shorter fitting periods, this restricts the

period of advantage to the first 11–13 years of theforecast, while for the longer fitting period its effectis less pronounced and the advantage is maintainedfor 46 years.

The two fundamental issues raised by theapplication of the Lee-Carter method to Australianmortality are the departure from the ‘universalpattern’ of constant exponential decline over theperiod 1950–94 identified by Tuljapurkar et al.(2000, p.789) for the G7 countries and the failure tosatisfy the assumption of a constant age pattern ofmortality decline. These issues are central to theaccuracy of the forecasts. Only when the assump-tion of invariance in the age component is satisfiedwill the time component be an accurate indicator ofthe level of, and hence change in, overall mortality.Further, only when a good fit of the ARIMA (inthis case, linear) model is obtained will the pastfitting period be a reliable guide for the future.

While the focus has been on Australian mortality,these issues are of general relevance. The evidenceprovided by Tuljapurkar et al. for the G7 countriesand by Lee and Carter for the US in fact demon-strates similar but less pronounced patterns ofdeviation from linearity in k′(t) to that found forAustralia. Deviations from linearity are also evidentin the data for Austria for the period 1947–99analyzed by Carter and Prskawetz (2001). The factthat the age pattern of mortality decline varies overtime has been demonstrated for a range of coun-tries (Kannisto, Lauritsen, Thatcher and Vaupel1994; Horiuchi and Wilmoth 1998; Wilmoth 1998;Carter and Prskawetz 2001; Lee and Miller 2001).The method developed in this paper to take accountof these issues is generally applicable to suchpopulations.

Since the two issues relate separately to the baseand ARIMA models in the Lee-Carter method,they have been addressed separately rather thansimultaneously. The issue of linearity in k′(t) wasaddressed first. The approach was to choose thelongest recent fitting period for which the assump-tion of linearity holds good and then to examinevariability in b(x) and the performance of the basemodel over this period. This is more straight-forward than choosing a fitting period to satisfy theassumption of invariant b(x) and then examiningthe performance of the linear model for this period.Choosing the fitting period based on invariance inb(x) is problematic, not least because of the numberof age categories involved.

The alternative approach of first addressing theissue of invariance in b(x) was adopted by Carterand Prskawetz (2001). They demonstrated that, inthe case of Austria, significantly different patterns

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Figure 9. Residual death rates from the rank 1 approximationafter adjustment of k(t), Australia 1968–99Source: As for Figure 1Note: Except at the oldest ages, most residuals lie in the interval-0.005 to 0.005.

of b(x) are obtained for different fitting periods andproposed ‘an extension of the original Lee-Cartermethod that takes into account the changing agepattern of mortality ….[and] can be applied tolocate structural changes in the historical pattern ofmortality that could ultimately be used to determinethe time horizon of past mortality rates on whichforecasts should be based’ (p.2). Their extensioninvolved the application of the Lee-Carter methodto the 30 overlapping 24-year subperiods of their 53-year dataset. The issue of linearity in k′(t) was notaddressed and it was implicitly assumed that k′(t)was linear for all fitting periods. They conclude fromtheir heuristic examination of variation in b(x) thattheir analysis ‘should be taken as evidence that thechoice of the base period may not be negligible’(p.12) in forecasting. However, they ‘refrain fromdrawing any definite conclusions about the optimalperiod on which to base forecasts’ (p.12).

In the present paper, an independently-developedmethod for determining the optimal fitting periodfor the linear model of k′(t) has been proposed. Inthe case of Australia, limiting the fitting period tothe period 1968–99 had the additional advantage ofincreasing the applicability of the base model: thevariation in b(x) was significantly reduced. (It isnoted that reduced variation in b(x) does notnecessarily follow from ‘optimizing’ the fittingperiod, but is likely to occur in practice in developedcountries given known changes in mortality patternsover the twentieth century.) By choosing the optimalfitting period, the assumption of invariant b(x) wasbetter met and the problem associated with violationof this assumption was largely avoided. Thus to alarge extent this approach has addressed both issuesof concern: linearity in k′(t) and invariance in b(x).

When considering the adequacy of the Lee-Carter method with linear ARIMA, it is the good-ness of fit of the total model that should be takeninto account. In past applications, the adequacy ofthe fit of the base and linear models has beenconsidered in isolation. This may lead to less thanoptimal choices being made. For example, thoughthe adoption of 1950 as the starting year for thefitting period, as proposed by Lee and Miller,resulted in an improved fit of the base model toAustralian data, the fit of the total model was poor.The additional lack of fit attributable to the linearmodel outweighed that attributable to the basemodel by approximately 3:1. It is likely that asimilar scenario would prevail in other developedcountries since mortality patterns have been subjectto similar influences.

In the approach taken here, it was assumed apriori that k′(t) is linear, i.e. that each age-specific

death rate declines exponentially at a constant rate.This is the paradigm within which the Lee-Cartermethod has been previously applied. While provi-sion is made in the Lee-Carter method for a non-linear ARIMA model for k′(t) (Lee and Carter1992, p.663), it would appear from the literaturethat such a model has never been applied. Lee andNault (1993) found a superior model was obtainedby the addition of a moving average or auto-regressive term, but this made almost no differenceto the forecast. Lee and Miller (2001, p.538) reportthat for at least 10 national datasets ‘k has alwaysbeen highly linear with time’. Further, exponentialdecline is the simplest model to describe mortalitychange and has theoretical merit. A linear trend ink′(t) is also convenient for extrapolation.

As noted above, this linearity in k′(t) contributesto the method’s simplicity and popularity. Indeed,the simplest approach is to assume that k′(t) islinear no matter what the period of available data,in other words without consideration either of non-linear models of k′(t) or of more appropriate fittingperiods. However, the danger is that this approachwill be adopted at the expense of forecastingaccuracy. It is argued here that this computationalsimplicity belies the complex issues involved in thechoices inherent in applying the method, in parti-cular the choice of fitting period.

The process of statistically determining the fittingperiod may be contrasted with the common prac-tice of taking the maximum period for which dataare available, based on a belief that data should notbe discarded and implying that all experience isuseful and relevant. This seems to have been thebasis for the selection of fitting periods by Lee andCarter (1992, p.659) for the US and by Tuljapurkaret al. (2000, p.790) for the G7 countries, though inboth cases it is clear that their underlying approachis to base long-term forecasting on long-term trends(Lee and Carter 1992, p.668; Tuljapurkar et al.2000, pp.790–1). Lee and Carter made forecasts for76 years (to 2065) based on data for 90 years(1900–89), while Tuljapurkar et al. made forecastsfor 56 years (to 2050) based on data for 45 years(1950–94). This is in keeping with convention: it isgenerally the case that long-term trends are used asa basis for long-term forecasts and short-termtrends for short-term forecasts. The inappropri-ateness of basing long-term forecasts on short-termtrends stems from the possibility that the short-term trend may be dominated by temporary devia-tions from a more persistent underlying trend.Similarly, it may be inappropriate to base short-term forecasts on long-term trends.

Whether a long-term or short-term approach is

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adopted, the central issues in relating past experi-ence to future forecasts are theoretical and sub-stantive. How relevant is average past experience tothe future? In taking a long-term approach, it isimplicit that past long-term trends will be ofcontinuing relevance to the long-term future. Ashort-term approach implies that only recentexperience is relevant for the short-term future. Inthe case of Australia, the optimal choice of the 32-year fitting period implies the adoption of arelatively short-term approach. In basing forecastson the period 1968–99, the experience of the earlypart of the century and the mortality stagnation ofthe 1960s are implicitly deemed not relevant forfuture mortality. Rather, only the relatively recentexperience of rapid decline is taken into account.

For how long will this experience continue to berelevant? While in this paper, a forecast has beenmade for 51 years (to 2050) based on only 32 yearsof experience, this is for illustrative purposes only.In actual applications, convention and cautionwould dictate that the maximum length of theforecast period be approximately the same as thelength of the fitting period. The need for caution indetermining forecasting length is underlined by twoimportant considerations. The first is that in theAustralian case the continuation of the rapiddecline to 2050 produces a forecast life expectancy(92 years) that is beyond what is considered feasibleby some scholars (for example, Olshansky, Carnesand Desesquelles 2001), although others find noevidence for an upper limit (for example, Wilmoth2000, pp.1120–1). Certain age-specific rates, forexample the infant death rate of 0.5 per 1,000forecast for 2050, may be regarded as implausiblylow, though it should be noted that this also appliesto the rate of 0.7 per 1,000 for the forecast based onthe period 1907–99. The second consideration isthat, when uncertainty in the drift is taken intoaccount, forecast uncertainty and hence the width ofthe confidence interval for life expectancy increase ata faster rate for a shorter fitting period. As a rule ofthumb, when length of forecast period equals lengthof fitting period (p = m), the standard error of theforecast will have increased by the factor (2p)1/2.

Compared with the official projection forAustralia, the forecast based on the period 1968–99gives more optimistic results: by 2027 life expectancyis 3.2 years higher than the official projection and7.1 years higher by 2050. This is to be expectedbecause only the recent trend of rapid decline istaken into account in the forecast. However, thisrelative optimism is also true of the forecast basedon the period 1950–99 and true in the longer termof that based on the period 1907–99. This is in

keeping with the results of other research. Theforecasts of life expectancy for the G7 countriesmade by Tuljapurkar et al. (2000), which are basedon the period 1950–94, also exceed the officialprojections for all or the later part of the forecastperiod. Forecasts for Austria of sex-specific lifeexpectancy to 2050 based on the period 1947–99also indicate a longer life expectancy for both sexescombined than official projections throughout theforecast period (Carter and Prskawetz 2001). Forthe US, Lee and Carter (1992) forecast lifeexpectancy in 2065, based on the period 1900–89, tobe 5.6 years above the official projection. Combinedwith historical demonstrations both of the greateraccuracy of the Lee-Carter method over officialmethods and of a systematic tendency of the Lee-Carter method to under-predict long term gains inlife expectancy (Bell 1997; Lee and Miller 2001),these comparatively optimistic results suggest thatcurrent official projections for Australia are highlylikely to underestimate future mortality decline,particularly in the long term.

When the standard error of the forecast is basedon the innovation term only, the forecast based onthe period 1907–99 has the widest confidenceinterval. This reflects in part the fact that the linearARIMA model is not entirely appropriate for thislonger series of data. The relatively large pastdeviations from linearity (see Figure 1) compriserandom error and systematic deviations, both ofwhich are treated as random with the result that theconfidence interval is wider than would be expectedfrom the longer period. The implication is that atany point in the future, random deviation from theforecast trend could be as great as the deviationgenerated by the combined effect of random andsystematic error in the past. Clearly, the relativelylarge deviations in the early part of the century, thestagnation of the 1960s, and the accelerated declinein recent decades have contributed significantly tothe width of the confidence interval. (Lee andCarter (1992) included a dummy variable for the1918 influenza epidemic in their US analysis, signi-ficantly reducing the standard error of the forecastattributable to innovation only. Such an interven-tion does not address systematic error.) The narrowerconfidence interval based on the period 1950–99reflects both the exclusion of the earlier deviationsand a better fit of the linear model for these morerecent decades. In comparison with the confidenceinterval for the period 1968–99 however, the effectof the mortality stagnation in the 1960s is suffi-ciently great to outweigh the effect of the longerfitting period.

The approach adopted for the analysis reported

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here is to restrict the model of k′(t) to a linearprocess, based on the experience of previousapplications. However, for the longer fitting period,the systematic error or changes in trend, seen in thestagnation of the 1960s and subsequent accelerateddecline, would ideally be modelled. This approach isbased on the premise that, while the determinants ofpast changes are unlikely to be repeated, futuredevelopments may lead to comparable changes.Further refinement might be possible by takingaccount of the underlying determinants of changes intrend. An analysis by cause of death would increaseunderstanding of past changes and provide insightthat might allow the incorporation of the possibilityof changes in trend into estimates of uncertainty.

To conclude, improvements in the performance ofthe Lee-Carter method with linear ARIMA havebeen achieved through two modifications. The firstwas to improve the fit of the basic Lee-Carter modelby adopting a maximum likelihood approach to theadjustment of k(t). The second was to base theforecast on the optimal fitting period determined bythe goodness of fit of the total model. These mod-ifications resulted in the assumption of invariancein b(x) being better met, a smoother transitionbetween observed and forecast rates of change, anupward adjustment in forecast life expectancy, anda substantial reduction in forecast error. The thirdmodification addressed the presence of age-timeinteractions in the residuals. Though these werepartially modelled, their incorporation in forecast-ing demands further research.

N OT E S

Heather Booth is at the Demography and Sociology Program,Australian National University, heather.booth@anu.edu.au

John Maindonald is at the Centre for Bioinformation Science,John Curtin School of Medical Research, Australian NationalUniversity, and Mathematical Sciences Institute, AustralianNational University, and Len Smith is at the Australian Centrefor Population Research, Australian National University

An earlier version of this paper was presented at the AnnualMeeting of the Population Association of America, WashingtonDC, 29–31 March 2001. The analysis was undertaken usingpurpose-written code in R. The authors are grateful to RajeshChauhan for research assistance and to Leonie Tickle and twoanonymous referees for valuable comments.

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