Forecasting using the Lee-Carter model with limited data

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Forecasting using the Lee-Carter model with limited data

M.V. Kruijver

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Master’s Thesis Econometrics, Operations Research & Actuarial Studies Specialization: Actuarial Studies

Faculty of Economics and Business University of Groningen

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Forecasting using the Lee-Carter model with limited data

Maarten Kruijver (s1571575)

University of Groningen

February 27, 2012

Abstract

Life expectancy has risen to historically unprecedented and unforeseen levels. This has led to an increased interest in mortality modeling and forecasting with the Lee Carter (LC) model as a central theme. A broad literature exists discussing application of the LC-model and its variants to developed countries, yet little is known about the application to countries where limited data is available.

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1 Introduction

Life expectancy has recently risen worldwide to historically unprecedented levels. In The Nether-lands, for example, the life expectancy of a a newborn male child rose from 74.6 years in 1995 to 78.8 years in 2010 (CBS Statline). This increase in life expectancy caused by a decrease in aggregate mortality levels has important economical and social implications which have led to an increased interest from academics.

Several models and have been developed to describe as well as predict the development of mor-tality decline. The de facto standard model is the widely used Lee Carter (LC) model. A broad literature exists discussing implementation, limitations and extensions of the model. Practically all literature however focuses on the use of the LC-model on dense and reliable data from developed countries. The majority of the world’s population however lives in (statistically) underdeveloped regions where death registration is incomplete (Mathers et al., 2005). It is questionable to which extent the model is applicable to sparse and possibly unreliable data from developing countries.

This study investigates the extent to which LC-type forecasts can be made when limited data is available. The main research question is: How reliable are Lee-Carter forecasts of aggregate mortality for developing countries where limited data are available? This question is answered in several steps. First, the accuracy of LC-forecasts based on limited data is compared to forecasts based on full data of several countries of which accurate data is available from the Human Mor-tality Database. Fits are made for the period 1950-1980 and forecast error is assessed using data from 1981 to 2006. Second, the quality of LC-forecasts based on limited data from developing countries is assessed. Finally, results are interpreted by drawing conclusions from the literature and simulations.

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2 Modeling and estimating aggregate mortality

This chapter briefly introduces how aggregate mortality is modeled using the life table. Conse-quently, direct (Lexis) and indirect estimation of the life table (using model life tables) is discussed.

2.1 Probabilistic modeling of mortality using the life table

This section reviews some standard actuarial notation, found in for example (Gerber, 1997). For some probability space (Ω, F , P), the future lifetime of a person aged x is a random variable denoted T (x) which has probability distribution

Gx(t) = P(T (x) ≤ t), t ≥ 0.

In actuarial standard notation, the probability that a person aged x dies within t > 0 years is de-noted bytqx, i.e. tqx= P(T (x) ≤ t) = Gx(t). The complementary survival probability is denoted tpx= 1 −tqx. It is a convention to drop the index t when t equals 1. The expectation of T (x) is

usually denoted by ˚ex= E(T (x)).

A life table contains one-year death probabilities qx corresponding to for example the different

sexes. Such a table fully specifies the distribution of the curtate future lifetime K = bT c, but does not fully specify the distribution of T . Under assumptions concerning the distribution of deaths during the year, one can however come up with an approximation to the distribution of T .

2.2 Statistical estimation of death rates

The main ingredient for the life table are the age-specific mortality rates. Under some assumptions these rates can be estimated directly from the observed number of deaths and the size of the population. When these quantities are however not reliably available, one has to rely on indirect estimation techniques.

2.2.1 Direct estimation

Suppose a group of n lives is observed for a certain period. The observation is ended either by death or leaving the sight of the observer (this is common when an insurance policy is terminated). Life number i is observed between the ages of x + ti and x + si (0 ≤ ti ≤ si ≤ 1). Denote by I

the subset of observations i terminated by death. Such a dataset can be visualized using a Lexis diagram; see Figure 1 for an example.

The exposure is then computed as the sum

Ex= (s1− t1) + . . . + (sn− t1),

whence the total length of all line segments in the Lexis diagram equals √2Ex. The classical

estimator for qxis given by (Gerber, 1997)

ˆ qx=

Dx

Ex+P_i∈I(1 − si)

,

which is derived under the assumption that1−uqx+u= (1 − u)qx(known as the Balducci

assump-tion). Further assuming that deaths occur on average at age x + 1/2, the numerator of ˆqx can be

approximated by Ex+1₂DX.

Another estimator given in (Gerber, 1997) is based on the assumption of constant force of mortality during the year, i.e. µx+u = µx+1

2 for 0 < u < 1. This leads to an estimator ˆµx+ 1 2 =

Dx Ex which

allows one to estimate the 1-year probabilities of death by

ˆ

qx= 1 − exp(−ˆµx+1

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Year Age 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 10 20 30 40 50 60 70 80 90 100 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Figure 1: Sample Lexis diagram. The lines denote observed lives. Lines terminated with a dot indicate deaths; no dot indicates termination of observation (by e.g. policy termination)

which can be shown to coincide with the ML-estimate of qxwhen treating P(i ∈ I) as the

param-eter of i.i.d. Bernoulli random variables corresponding to each age x.

In practice, full Lexis diagrams are rarely available at the national level. Therefore a slightly different estimator has to be used. See for example (Wilmoth and Glei, 2007) for details.

2.2.2 Indirect estimation

The direct estimation procedure of the previous section requires availability of accurate data con-cerning deaths (Dx) and number of people exposed to risk of death (Ex). When this kind of data

is not available, one has to rely on indirect estimation techniques. It is then often helpful to model mortality using other countries’ experience (Siegel and Swanson, 2007).

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3 Forecasting using the Lee-Carter model

This chapter discusses mortality forecasting using the Lee-Carter model. First, the original model, properties, applications, forecasting and variants are discussed. Second, a modification by Li, Lee & Tuljapurkar is presented, which makes it possible to use the model when only limited data is available.

3.1 Original formulation of the Lee-Carter model

The Lee-Carter approach is originally developed to forecast U.S. mortality (Lee and Carter, 1992), but has since then be used to model and forecast mortality for developed countries around the world. Several adaptations and extensions of the model have been proposed and performance of the model for developed countries has been extensively studied, yet very little has been published on the use of the model when data availability is limited. Next, the original model specification is given, after which estimation and forecasting is discussed.

3.1.1 Model specification

The LC-method starts by reducing the multi-variate time series of age-specific death rates (see Figure 2 for an example) to a single one dimensional time series (kt) describing overall mortality

improvement. Structure is imposed by modeling the logarithm of the central death rate for age (group) x = 1 . . . p at year t = 1 . . . T using a bilinear specification:

log mxt = ax+ bxkt+ xt,

where axand bx are p-vectors and ktis a univariate time series. The error term xt is assumed to

have mean 0 and constant variance σ2.

1950 1960 1970 1980 1990 2000 0 20 40 60 80 100 −8 −6 −4 −2

Figure 2: Log mortality rates of males between 1950 and 2000 (USA)

Note that restrictions have to be imposed to identify the model, since whenever {ax, bx, kt} forms

a solution, it is seen that {ax− λbx, bx, kt+ λ} also forms a solution (λ ∈ R). For λ 6= 0, another

equivalent solution is given by {ax, λbx, λ−1kt}. Hence, the model is only specified up to a linear

transformation. Lee and Carter therefore impose the restrictionsP bx = 1 and P kt= 0 which

fully identify the model.

Under these restrictions, axis seen to be the average log-mortality over the time interval. This is

often called the baseline mortality. The time series kt captures the mortality improvement over

time, while the vector bx is used to distribute the mortality improvement (kt) over the different

ages. Lee and Carter noticed that kt behaves roughly linear over time for US mortality rates,

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3.1.2 Statistical estimation of the parameters

Lee and Carter estimate ax, bx and kt by finding a least squares fit to the log mortality matrix.

Hence, they minimize the residual sum of squares

RSS =X x,t log mxt− ˆax− ˆbxˆkt 2 ,

under the restrictions P bx = 1 and P kt= 0. It is immediately seen that an unbiased

estima-tor is given by ˆax = log mxt, being the baseline mortality. Now bx and kt are chosen such that

bxkt ≈ log mxt− ˆax. The least square solution to this problem follows from the Singular Value

Decomposition (SVD).

Writing M for the p × T matrix containing the logarithms of mxt minus the row means (ˆax), the

SVD theorem (see for example (Friedberg and Insel, 2003)) guarantees that (as long as M has full rank) M can be written as

M = U ΣV0,

with Σ a uniquely defined diagonal p × T matrix containing the singular values of M . U and V are respectively unitary p × p and p × T matrices. Effectively, this restates M as the sum of p rank 1 matrices. The SVD theorem guarantees that the partial sums of the first k < p matrices are the best (in least square sense) rank k approximations to the matrix M . The special case in which all but the first singular values are dropped is used in the LC-model. An obvious extension to the model is therefore constructed by including additional singular values, which is discussed in section 3.1.5.

To summarize, the original LC-estimation consists of the following steps. First, ˆax is computed

as the average age structure of log mortality over t. Next, the matrix M containing the log death rates minus the row means is decomposed into its singular values. Now the estimate for bxis the

first column of U , while kt is found by computing σ1v1. Finally, the estimates have to be

nor-malized to ensureP bx= 1 andP kt= 0. Hence ˆbxis divided byPˆbxand ktis multiplied byPˆbx.

Note that Lee and Carter propose to do a second stage estimation of ktusing a different criterion.

Since the death rates in the estimated model usually do not lead to the actual number of deaths, they reestimate kt taking ˆax and ˆbx as given. The merits and flaws of the re estimation step are

discussed in (Girosi and King, 2007).

0 20 40 60 80 100 −8 −6 −4 −2 Age ax 0 20 40 60 80 100 0.00 0.01 0.02 0.03 Age bx 1950 1970 1990 −30 −10 10 Year kt

Figure 3: ax, bxand ktfrom application of LC to log mortality rates of males between 1950 and

2000 (USA)

Figure 3 shows an example of the results of the estimation procedure for log mortality rates of males from the USA between 1950 and 2000, which gives some intuition to the meaning of ax,

bx and kt. The baseline mortality ax is the average age pattern of mortality, which is easily seen

when comparing to Figure 2. The age pattern of mortality improvement (bx) shows that relatively

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which may be caused by data defects. The overall mortality improvement (kt) behaves roughly

linear except for a possible break around 1970. Section 6.1.2 will return to the issue of structural breaks.

3.1.3 Forecasting mortality rates

Having computed the estimates for ax, bx and kt, Lee and Carter proceed by forecasting future

mortality rates. They keep axand bxconstant and forecast ktusing the Box-Jenkins approach, the

conclusion being that the random walk with drift is the most appropriate time series specification for the US mortality data. The ARIMA(0,1,0) (random walk with drift) specification of the kt

time series is as follows:

kt= kt−1+ d + ηt,

where d ∈ R and ηt is an i.i.d. error term having finite variance σ2η. Point forecasts are easily

constructed:

Eˆkt+∆t|ˆkt

= ˆkt+ ∆t ˆd,

and a confidence interval is constructed easily when normality of ηtis assumed.

Lee and Carter suggest that other ARIMA specifications might be more suitable for different datasets, but the random walk with drift is used almost exclusively in the literature (Girosi and King, 2007).

3.1.4 Applications of the model

Although Lee and Carter developed their method originally to forecast US mortality rates, its use has stretched far beyond this original purpose. The method has been used directly to forecast mortality rates and related quantities like life expectancy. Indirectly, the method has been used in for example stochastic population forecasts.

Direct use of the LC-method to forecast mortality rates has mainly concentrated on developed countries. Some examples are applications to Australia (Booth et al., 2002), Belgium (Brouhns et al., 2002), Spain (Debón et al., 2008), Sweden (Lundström and Qvist, 2004) and the G7 coun-tries (Tuljapurkar et al., 2000). Applications to developing councoun-tries are more rare. An old one is the application to Chile by Lee in (Lee and Rofman, 1994). More recent applications are to Kuwait (Al-Jarallah and Brooks, 2009), Mexico (Pérez and Guzmán, 2007) and Taiwan (Huang et al., 2008).

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3.1.5 Variants of the model and other approaches

The widely applied LC-method is regarded as among the best currently available mortality fore-casting method and is often taken as the point of reference (Booth et al., 2006). It is not the only stochastic mortality forecasting method though. Several variants and extensions of the LC-method have been proposed and some scholars pursue different routes. Some of the research is summarized next.

The two-factor (age and time) LC-method was a significant departure from previous approaches to mortality forecasting. Many attemps have been made to come up with parametric functions to describe the log mortality pattern (Girosi and King, 2008), after which time series methods could be used to forecast the parameters (Alho and Spencer, 2005). Attempts range from the early work of Gompertz (Gompertz, 1825) and Makeham (Makeham, 1860) to more recent functional forms such as the 8-parameter function proposed by Heligman and Pollard (Heligman and Pollard, 1980) and the multi-exponential forms used by Rogers (Rogers and Little, 1994). These functional methods have several drawbacks (Girosi and King, 2008). Simpler functional forms generally fail to capture the mortality profile well, while more sophisticated multi-parameter functions are hard to fit as multiple optima may lie in the high-dimensional parameter space. Moreover, forecasting becomes increasingly complex when more than one parameter is involved. The LC-method to the contrary is exceptionally easy to fit, forecast and interpret.

Since the original publication by Lee and Carter, several modifications to the model have been proposed. (Booth et al., 2006) compare five variants of the model:

• The original LC-model; • The Lee-Miller variant;

• The Booth-Maindonald-Smith (BMS) variant; • The Hyndman-Ullah extension;

• The De Jong-Tickle extension.

These variants are not further discussed here. See for example (Booth et al., 2006) for a review of mortality forecasting or (Booth, 2006) for a comprehensive review of demographic forecasting in general.

Recently, (Coelho and Nunes, 2011) pointed out that the assumption of the constant pace of decline of ktis often violated. In a study of post-1950 data of European countries, they show that

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3.2 The case of limited data

The previous section introduced the original formulation and estimation procedure of the LC-model. A general property of time series models is that a large amount of historical data is needed to obtain a reliable fit. Recently, (Lee et al., 2004) discussed how the LC-method can be applied to countries for which only limited mortality data is available at possible non-equidistant time points. This section restates the main conclusions.

The LC-model requires four pieces of information to be estimated. These are the baseline mor-tality (ax), the improvement by age (bx), the overall improvement (d in kt) and the variability

represented by the variance of the innovation term (σ2

η). Hence there are 2p + 2 parameters to

be estimated. The paper points out when age-specific mortality rates are available for at least two years, it is possible to estimate ax, bx and kt using the SVD of M when the time points are

equidistant. Estimation of the variance σ2

η requires the data to be available at least three time

points, but the LS-estimator has a relatively high variance when the number of data points is so low.

Lee et al. discuss two issues. First, the fitting procedure is modified to deal with the availability of data at not equally distanced time points. Second, they argue that a mean forecast based on so few data points can be just as reliable as one based on many data points.

3.2.1 Data at unequally spaced time points

Let mortality data be available at T not necessarily equally distanced time points τ1, . . . , τT. For

example, τ1= 1, τ2= 10, τ3= 16. The baseline mortality axcan then still be estimated as the row

means of the p × T log mortality matrix:

ˆ ax= 1 T T X t=1 log mx,t.

Writing M for the log mortality matrix minus the row means ˆax, the SVD can be computed as

M = U ΣV0. The estimate for bx is then, like in the normal LC-procedure, given by ˆbx= u1, but

kt is no longer estimated directly. Instead,

(kτ1, . . . , kτT) = σ1v T 1.

Differences of consecutive elements of the k vector are not i.i.d like those obtained in the regular LC-procedure. Instead, kτt+1− kτt = d τt+1− τt +ητt+1+ · · · + ητt+1 .

An estimator for the drift term d is given by

ˆ d = T −1 X t=1 kτt+1− kτt T −1 X t=1 τt+1− τt =kτT − kτ1 τT − τ1 ,

which is seen to be unbiased. The innovation variance ση2is estimated by

ˆ σ_η2= T X t=1 kτt+1− kτt− d(τt+1− τt) 2 τT − τ1− PT −1 t=1 (τt+1−τt)2 τT−τ1 .

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3.2.2 Theoretical performance of forecasts based on limited data

Lee et al. show that the mean forecasts based on only a few time points can be just as accurate as these based on yearly data, as long as the first and the last time points are far enough apart. Moreover, the authors claim that mean forecasts for countries with limited data can be just as accurate as those for G7 countries. The probability intervals on the other hand, can be highly inaccurate when limited data is available. These two phenomena are discussed next.

The claim made concerning the mean forecasts is justified by the following. First note that error in forecasting the mortality index dominates the error in fitting the mortality matrix (Lee and Carter, 1992). Recall that under the assumption that ktfollows the RWD specification, an

unbi-ased estimator for the drift parameter is given by ˆd = kτT−kτ1

τT−τ1 . Since the estimate only depends

on the first and the last value of kt, the mean forecast depends mainly on the death rates at the

start and end of the fitting period, while mortality in between does not matter much. Hence, the LC-approach applied to data from only a few time points can be just as accurate as long as these time points are far enough apart.

The inaccuracy of the probability interval stems from the following. Since only the uncertainty from predicting kt is considered, this probability intervals depends directly on the error in estimating

ση. (Lee et al., 2004) show that the standard deviation of the estimator ˆση equals approximately

the following: s.e.(ˆση) ≈ 2(τT − τ1− PT t=2(τt− τt−1)2 τT − τ1 ) −1/2 ˆ ση,

which is generally large when few time points are available.

The authors also show that the standard error in estimating the drift parameter equals

s.e.( ˆd) = σ 2 η τT − τ1 ≈ σˆ 2 η τT− τ1 ,

which only depends on the length of the interval.

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4 Empirical forecasting performance with limited data from

developed countries

This chapter evaluates the claim that mean forecasts obtained from the LC-model applied to data from only a few time points can be just as accurate as those from a full dataset. The forecasting performance is systematically assessed by fitting, forecasting and back testing the LC-model on various subsets of data from twelve countries for which data is available in the Human Mortality Database (HMD). A systematic evaluation of forecast accuracy for this kind of limited data has not been made before (Booth et al., 2006). Next, the HMD is introduced as a data source, after which the procedure is outlined and illustrative results for the USA are presented. What follows is a repetition of the procedure for several other countries found in the HMD.

4.1 Data source (HMD)

The HMD is a collaborative project sponsored by the University of California at Berkeley (United States) and the Max Planck Institute for Demographic Research (Rostock, Germany). The database is created with the purpose to provide researchers with easy access to detailed and comparable national mortality data. The methods protocol is described in detail in (Wilmoth and Glei, 2007). An R (R Development Core Team, 2010) package called demography (Hyndman and Maindonald, 2011) offers easy access to the database.

4.2 Procedure

The claim that mean mortality forecasts based on a few time points can be accurate is assessed by systematically disregarding subsets of the dataset. As a benchmark, the LC-model is fit to mortality data covering full data from 1950 to 1980, after which a forecast is made for the year 1981-2005. The fitting and forecasting is repeated for subsets containing k = 2, 3, . . . , 31 (approxi-mately) equally spaced time points. Hence, the model is first fit using data from only the year 1950 and 1980, second to data from the three years 1950, 1965 and 1980 and finally to data covering the full 31-year range 1950-1980. As a start, this procedure is applied to combined (male and female) mortality of the USA for which the results are shown next.

Although the HMD contains age-specific mortality rates for single ages up to and including age 110 for most countries, an open-end age group is constructed for ages 90 and up. Using an open-end age group is a natural decision, since mortality rates at the highest age are generally unreliable and may cause numerical problems.

4.3 Results (USA)

To start, the LC-model is estimated on combined mortality rates from the full dataset, i.e. 1950-1980. A forecast is made for the years 1981-2006. Figure 4 shows ktwith forecast on the left hand

side, while some log mx curves are drawn on the right.

1950 1970 1990 −50 −20 10 Year kt ● 0 20 40 60 80 100 −8 −4 Age log

(

mx

)

Figure 4: Left: ktand forecast for the years 1981-2005. Right: log mxcurves for the years (1950,

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The APE (average prediction error) of the log mortality rates is computed as the average (over x) of:

PEx,T +h= log mx,T +h− log ˆmx,T +h.

Figure 5 shows some results of applying the LC-model to combined mortality in the USA for the years 1950-1980 and forecast years 1981-2006. The conclusion is well known: the LC-model performs very well on predicting life expectancy for the US for the last 30 years, but the predicted log mortality rates have a small negative bias (Lee and Miller, 2001). The mean (over time) of the APE visualized on the left equals -0.0176.

● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ●● ● ● 1985 1995 2005 −0.04 0.00 0.02 Year a v er

age prediction error of log mor

tality r ates 1950 1970 1990 66 68 70 72 74 76 78 Year e0

Figure 5: Left: average (over x) prediction error of log mortality rates during forecasting period. Right: observed life expectancy at birth (HMD) with forecast (dashed)

Next, the whole procedure is repeated with limited data. Instead of estimating using the full data from the years 1950-1980, subsets are used of k = 2, 3, . . . , 31 (approximately) equally spaced time points. The claim that mean forecasts can be just as accurate as those ones computed using the full dataset is assessed by computing the APE and AAPE for all subsets of the data.

Figure 6 shows the mean APE and mean AAPE for the USA for all different subsets of the data. The picture clearly confirms the claim that the accuracy of mean forecasts does not decrease when less time points are used. The story is possibly different when time points are used that are less far apart. This is put to the test by repeating the procedure for k = 2, . . . , 31 time points preceding and including 1980, instead of equally spaced time points in the interval 1950-1980. Hence, the model is first fit to 1979-1980, second to 1978-1980, et cetera.

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5 10 15 20 25 30

−0.4

0.0

0.2

0.4

number of time points used

mean APE 5 10 15 20 25 30 −0.4 0.0 0.2 0.4

mean AAPE

Figure 6: APE and AAPE (of forecasting total mortality rates for the USA in 1981-2005) are practically unaffected by number of time points used in estimation. Time points used in estimation are chosen approximately equidistant from the interval 1950-1980

5 10 15 20 25 30

−0.4

0.0

0.2

0.4

mean APE 5 10 15 20 25 30 −0.4 0.0 0.2 0.4

mean AAPE

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4.4 Results (other countries)

The previous section confirmed for the USA the claim that mean forecasts based on limited data can be accurate as long as time points are far enough apart. Moreover, it was found that the model performed reasonably well when as few as three trailing years were used. To test whether the con-clusion is the same for other countries, the procedure is now repeated for eleven other countries: Australia, Belgium, Canada, Denmark, France, Japan, Netherlands, Norway, Spain, Sweden and Switzerland. The USA is included for comparability with the previous section. Figures 8-11 show the results. 5 10 15 20 25 30 35 −0.4 −0.2 0.0 0.2 0.4

APE AUT BEL CAN CHE DNK ESP FRATNP JPN NLD NOR SWE USA

Figure 8: Mean APE (1981-2006) versus time points (equally spaced over 1950-1980)

5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0

AAPE AUT BEL CAN CHE DNK ESP FRATNP JPN NLD NOR SWE USA

Figure 9: Mean AAPE (1981-2006) versus time points (equally spaced over 1950-1980)

5 10 15 20 25 30 35 −0.4 −0.2 0.0 0.2 0.4

APE AUT BEL CAN CHE DNK ESP FRATNP JPN NLD NOR SWE USA

Figure 10: Mean APE (1981-2006) versus time points (years preceding 1980)

5 10 15 20 25 30 35 0.0 0.2 0.4 0.6 0.8 1.0

AAPE AUT BEL CAN CHE DNK ESP FRATNP JPN NLD NOR SWE USA

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The plots show that behavior of the APE and AAPE is consistent for all countries considered in this study. Figures 8 and 9 confirm that for these countries the prediction accuracy does not decrease when a smaller number of time points are chosen equidistant from the interval 1950-1980. On the other hand, Figures 10 and 11 point out that the prediction accuracy is much lower when few trailing time points are used. Somewhat surprisingly though, for most countries a relatively small mean AAPE of about 0.3 is already achieved when as few as three trailing time points are used.

These results give empirical evidence for the claim by Li et al. that the mean forecasts based on limited data can be just as accurate as those based on the full dataset. Two important questions remain however. The first is the question to what extent this result applies to limited data from developing countries. The second is why the prediction error differs between the countries.

The first question is empirically investigated in the next chapter. The just given results about using LC with limited data might make one optimistic about the application of the LC-model to developing countries where mortality data is scarce. Whether or not the model works in practice is investigated next.

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5 Empirical forecasting performance for developing

coun-tries

The previous chapters showed that LC can be applied successfully even with limited data from developed countries. This chapter investigates how well LC-forecasts would have performed, had they been made in 1982 for developing countries. A forecast is made based on life tables published in 1986 by the United Nations. Forecasts accuracy is assessed by comparing predicted life expectancy at birth with UN estimates published in 2010. The next sections introduce the data sources followed by details of the procedure and results.

5.1 Data sources

Comparable historical life tables are found in a book published in 1986 by the United Nations. Contemporary estimates are obtained from the United Nations website. Some details are given next.

5.1.1 Historical UN life tables for developing countries

Back in 1986, the UN published a handbook (United Nations, 1986) containing 78 national life tables from 43 developing countries covering a period from the mid 1940s to the early 1980s. All life tables are sex-specific and abridged into 5-year groups with an open end of ages 85 and up. The number of time points at which the life tables are available varies per country. Table 1 shows the distribution of countries in the database by region and number of life tables included.

Region 1 2 3+ Total

Africa 6 2 1 9

Americas 6 9 4 19

Asia 7 5 3 15

Total 19 16 8 43

Table 1: Distribution of countries in the UN database of life tables for developing countries by region and number of life tables included

Life tables are available at three or more time points for the following countries: Argentina, Hong Kong, Martinique, Singapore and Sri Lanka. There are 16 countries for which there are life tables available at exactly two time points. See Table 5 and 6 in Appendix B for more details.

The life tables are a product of the UN’s best effort to come up with reliable life tables for developing countries. All tables are based on record deaths by age and sex and the population at risk by age and sex. Hence, no model life tables have been used which could possibly distort the conclusions in this study. Data have been adjusted for incompleteness of death registration though. Tables of questionable reliability have been excluded. The effort has led to a set of tables that is ’to the fullest extent feasible, [...] reliable, consistent and comparable’ (United Nations, 1986).

5.1.2 Contemporary UN estimates of life expectancy at birth

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5.2 Procedure

Following the approach of Lee et al. outlined in Section 3.2, the LC-model is fit to the few available historical UN life tables corresponding to the 23 countries. Forecasts are made up to and including the year 2000. The accuracy of this forecast is assessed by comparing the life expectancy at birth corresponding to the forecast with contemporary estimates of historical life expectancy at birth by the UN. Since the UN life tables are abridged into 5-year groups, an interpolation scheme has to be used to compute life expectancy. Following (United Nations, 1986), the method of Greville is used; see Appendix A for details.

5.3 Results

As was to be expected, results differ greatly for the 23 countries considered in this study. The findings are illustrated by describing the application of the model to three countries: Argentina, Bangladesh and Israel. These three countries are chosen because of the typical results found. Argentina is one of the few countries for which the LC-approach would have worked well. Results for Bangladesh are very poor, which is no wonder considering the instability in the country’s development and low data quality. Results for Israel are also poor, although this can not be explained by data quality alone.

5.3.1 Argentina

Of the 23 developing countries to which LC is applied, Argentina is the one with the smallest prediction error. Life tables for Argentina are available for three periods: 1959-1961, 1969-1970 and 1979. The model was fit to mortality rates from these tables, after which a prediction was made up to and including the year 2000. Figure 12 shows the results. The triangles show the life expectancy at birth corresponding to the three life tables where blue indicates male; pink female. The LC-forecast is given by the dotted line. Both the forecast and the data (triangles) are compared with the contemporary estimates found in (United Nations, 2010) and visualized by the solid lines. 1960 1970 1980 1990 2000 55 60 65 70 75 80 85 year e0 Argentina

Figure 12: LC-forecasts for Argentina

The solid lines (contemporary estimates) do not cross the triangles, indicating that there is a minor difference between historical and contemporary estimates of e0. The forecasts (dotted lines) almost

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5.3.2 Bangladesh

The high forecast accuracy just demonstrated for Argentina is the exception rather than the rule for developing countries. Being an extrapolative model, the LC-model relies heavily on two facts: there should be a trend that continues to hold (1) and data quality should be adequate (2). Obviously, a violation of this first assumption would render the forecast useless. Moreover, any bias in the data is increased due to the extrapolative nature of the model. Both assumptions are problematic for Bangladesh. Nevertheless a forecast is made and shown in Figure 13.

1965 1970 1975 1980 1985 1990 1995 2000 30 40 50 60 70 year e0 Bangladesh

Figure 13: LC-forecasts for Bangladesh

Figure 13 shows several issues concerning the application of LC. First, there is a mortality crisis in the Seventies which may or may not bias the forecasts. Second, there is a problem with data quality, since contemporary estimates of e0 differ severely from the value computed from (United

Nations, 1986). Third, there is not much of a trend to extrapolate as is seen by the very small increase in life expectancy between the first and the second life table. Fourth, there are only two life tables available. Given these four issues, it is not a wise idea to apply the model to Bangladesh. This is confirmed by the forecasts which are far off.

5.3.3 Israel 1960 1970 1980 1990 2000 55 60 65 70 75 80 85 year e0 Israel

Figure 14: LC-forecasts for Israel

Israel is - just like Argentina - a country where mortality was relatively low in the Sixties. Much of an increase in life expectancy was however not visible in the life tables published in (United Nations, 1986); see Figure 14. The extrapolative LC-model therefore fails to predict the steep increase in e0since the Seventies. The forecasts of the UN World Population Prospects 1984 also

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5.3.4 Other countries

The previous section sketched the general picture of applying the LC-model to developing coun-tries. Although a small number of data points is theoretically not much a problem, the method generally fails when so few data points are available. While the model produced credible forecasts for Argentina, this was not the case for the other two examples given in the previous section. Table 2 shows for the results obtained for the other countries.

∆e0 in LT’s PE1 (e0 in 2000)

Country # LT’s, time span m f m f

Argentina 3 (21 years: ’59-’79) 3.98 4.94 -0.60 -0.26 Bangladesh 2 (18 years: ’64-’81) 1.40 2.47 -12.95 -13.61 Chile 2 (13 years: ’59-’71) 4.16 4.80 -8.67 -8.65 Costa Rica 2 (13 years: ’62-’74) 6.69 7.28 3.37 1.52 Egypt 2 (13 years: ’65-’77) 3.62 5.21 -9.06 -3.90 Guatemala 2 (11 years: ’63-’73) 7.01 7.64 7.30 2.13 Guadeloupe 2 (18 years: ’66-’83) 7.05 9.12 1.71 2.78 Guyana 2 (13 years: ’59-’71) 2.43 3.23 5.40 5.06 Hong Kong 4 (22 years: ’60-’81) 7.90 5.91 -1.47 -2.94 Honduras 2 (16 years: ’60-’75) 9.67 10.29 -0.97 -3.97 Israel 2 (14 years: ’60-’73) -0.56 0.37 -8.81 -7.85 Jamaica 2 (13 years: ’59-’71) 3.75 3.67 5.72 2.13 Kuwait 2 (8 years: ’74-’81) 3.16 2.79 2.70 5.36 Martinique 3 (24 years: ’60-’83) 8.48 8.85 0.94 -2.26 Mauritius 2 (13 years: ’61-’73) 1.97 3.46 -2.91 -2.85 Pakistan 2 (10 years: ’62-’71) 3.16 1.70 3.53 -4.18 Panama 2 (21 years: ’60-’80)2 3.16 2.17 2.47 -4.02 Philippines 2 (8 years: ’69-’76) -0.54 -1.58 -6.87 -12.33 R´eunion 3 (10 years: ’61-’70) 4.17 5.41 -6.86 -4.31 Singapore 3 (25 years: ’56-’81) 8.35 7.58 -2.72 -3.21 Sri Lanka 4 (28 years: ’45-’72) 19.09 23.71 7.48 4.02 Trinidad & Tobago 3 (27 years: ’45-’71) 10.65 12.02 5.79 2.80 Uruguay 2 (15 years: ’62-’76) 0.01 0.90 -5.39 -3.99

Table 2: Forecast accuracy for 23 developing countries with limited data

Part of the results shown in Table 2 can be explained by two observations:

• Forecasts are never accurate when mortality increases at a very high (e.g. Sri Lanka) or very low (e.g. Uruguay) pace. This is no surprise given the extrapolative nature of the model. • Forecast error is generally large. Possible causes are data quality issues (bias gets inflated)

and non-constant rate of mortality improvement.

The conclusion drawn from the preceding is that it is in general not a wise idea to apply the LC-model to countries with limited data. As the previous chapter pointed out, the data limitations are only part of the problem. A more serious issue is the extrapolative nature of the model which forces one to make strong assumptions about the past and future development of mortality rates. At this point in the study it is hard to assess what exactly these assumptions are and to what extent these are met. To this end, the next chapter further investigates the assumptions incorporated in the LC-model by taking a second look at the HMD data from developed countries.

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6 Model assumptions and forecast error

The previous chapters showed that forecast error of the LC-model varies wildly for developing as well as developed countries. This chapter further investigates when the LC-model is appropriate and forecasts are reliable. First, the assumptions of the model are discussed in detail. Second, the extent to which the assumptions are met is compared with model fit and forecast error for the twelve countries discussed in Chapter 4.

6.1 Model assumptions

Recall from Chapter 3 that in the LC-model the log mortality rate is specified as

log mxt = ax+ bxkt+ xt,

where ax and bx are p-vectors and kt is a univariate time series. The error term xt is assumed

to have mean 0 and constant variance σ2. Lee and Carter write that ktcan be forecast using the

Box-Jenkins approach, but in practice the RWD specification is almost exclusively used. There are several implicit assumptions underlying this model. Three are identified next.

6.1.1 Estimation using one principal component

The most notable assumption is that the log mortality improvement can be described by the uni-variate time series kt and improvement profile bx. It is well known that this type of modeling is

an application of PCA where all but the first principal components are dropped.

A visually appealing way to think about PCA is described in (Girosi and King, 2007). The au-thors point out that the (vector) time series of age-specific mortality rates can be thought of as a sequence of time points moving through an A-dimensional space, where A is the number of age groups. The hypothesis inherent in the LC-model is then that the dimensionality of this space can be reduced to 1 without loss of much information. Geometrically, this means that the log mortality age profiles should move along a straight line in RA_{. Girosi & King add that the LC-model is not}

expected to forecast reliably when this assumption is violated.

−20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 −20 −15 −10 −5 0 5 10 15 20 2nd Princ. Comp. 1st Pr inc. Comp . 3rd Pr inc. Comp . ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●●●●● ● ● ● ● ● ● ● ● USA (88, 95, 97)

Figure 15: First three Principal Compo-nents of USA total mortality (1950-1980)

−8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 2nd Princ. Comp. 1st Pr inc. Comp . 3rd Pr inc. Comp . ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● DNK (32, 45, 51)

Figure 16: First three Principal Compo-nents of Danish male mortality (1950-1980)

The geometrical assumption that the log mortality age profiles move along a straight line in RA

can be visually inspected after plotting the projection of this space on the subspace R3_{. Figures 15}

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first two and first three principal components. It is seen that the one dimensional approximation seems to describe the pattern of the US well, while the Danish male mortality pattern is more complicated. This is in line with the summary measures of percentage of variation explained by the first principal component.

6.1.2 Forecast assumptions

Forecasts based on the LC-model are made by keeping bx constant and forecasting kt using time

series methods. In practice, the RWD model is used almost exclusively in applications. Implicit lie two assumptions. The first assumption is that the age pattern of mortality reduction is and stays constant. The second assumption is that the RWD specification is and stays appropriate for the one dimensional approximation kt. Both are discussed next.

The first assumption is put to the test by plotting the age structure of mortality improvement for different intervals. Figures 17 and 18 show the average improvement for respectively the USA and Denmark for different intervals. In line with the findings of (Lee and Miller, 2001), it is seen that the shape of the age profile differs before and after 1950. Moreover, it is seen that after about age 50 the mortality rates for the USA decline roughly at the same rate. On the other hand, it appears that the age pattern of mortality improvement is changing, especially between ages 15 and 50. Hence, the assumption of constant bxis not met here. Lee and Miller argue however that

depending on the purpose of the forecast procedure this might not be a problem, since only a small fraction of life table deaths occurs between age 15 and 40.

0 20 40 60 80 −2 0 2 4 6 8 Age P ercentage decline

Figure 17: Average annual reduction in age specific death rates (USA total mortality): 1935-1949 (dashed), 1950-1964 (solid), 1965-1979 (dotted), 1980-1995 (dotdash)

The changing age pattern of mortality improvement of the USA seems a minor issue when com-pared to the pattern of Denmark shown in Figure 18. The pattern behaves much wilder than the US pattern. Moreover, the assumption of constant bx seems peculiar here. It is not immediately

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0 20 40 60 80 −15 −5 0 5 Age P ercentage decline

Figure 18: Average annual reduction in age specific death rates (Danish total mortality): 1935-1949 (dashed), 1950-1964 (solid), 1965-1979 (dotted), 1980-1995 (dotdash)

The second assumption in forecasting is that kt can be modeled and forecast reliably. Only the

case of using the RWD specification is treated here. Recently, (Coelho and Nunes, 2011) pointed out again that a structural break in kt is often present for post-1950 data. The authors illustrate

how accounting for such a strutural break can have a major effect on mortality and life expectancy forecasts.

6.2 Forecast error for different countries

The previous section identified three (implicit) assumptions made when forecasting using the LC-model:

1. Only one principal component describes the age and time structure of log mortality suffi-ciently;

2. The age pattern of mortality improvement (bx) is and stays constant;

3. The overall mortality improvement (kt) follows the RWD specification.

Compliance with the first assumption is easily tested by computing the percentage of variation ex-plained by the first principal component. Testing for compliance with the other two assumptions is more involved. It is unclear to what extent the forecast accuracy of the LC-model is affected by violations of these three assumptions, although it seems obvious that a violation of the first assumption makes it less likely that the forecasts are accurate (Girosi and King, 2007).

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● ● ● ● ● ● ● ● ● ● ● ● ● 0.5 0.6 0.7 0.8 0.9 0.10 0.15 0.20 0.25 0.30 R2 mean AAPE

Figure 19: Scatter plot of mean (over time) of average (over age groups) absolute prediction error of log mortality rates (total mortality in 1981-2005) versus the percentage of variation explained by the first Principal Component for the twelve countries.

A surprising observation from Table 3 is that there is only a weak relation between the percentage of variation explained by the first principal component and the absolute prediction error. This is even better seen in Figure 19, which shows a scatter plot of the mean AAPE versus R2 _{for the 12}

countries. The prediction error is particularly high for Japan and Spain, which have a very high R2 _{possibly as a result of autocorrelation. These findings contradict the intuition that a bad fit}

implies a bad forecast.

Male mortality Total mortality

Country 1 PC 2 PC 3 PC MAAPE 1 PC 2 PC 3 PC MAAPE

Australia 58% 65% 70% 0.28 73% 79% 82% 0.20 Belgium 60% 66% 71% 0.23 71% 78% 81% 0.20 Canada 78% 83% 85% 0.25 85% 90% 91% 0.14 Denmark 32% 45% 51% 0.26 45% 57% 63% 0.18 France 79% 86% 89% 0.21 86% 93% 94% 0.16 Japan 97% 98% 99% 0.27 98% 99% 99% 0.34 Netherlands 70% 81% 84% 0.26 80% 87% 90% 0.15 Norway 57% 64% 68% 0.28 67% 73% 76% 0.18 Spain 92% 95% 96% 0.26 95% 97% 98% 0.28 Sweden 58% 63% 67% 0.27 69% 75% 78% 0.17 Switzerland 63% 68% 72% 0.22 77% 81% 83% 0.18 USA 81% 91% 95% 0.13 88% 95% 97% 0.08

Table 3: Mean (over time) of average (over age groups) absolute prediction error of log mortality rates (male and total mortality in 1981-2005) by country compared to the percentage of variation explained by the first three principal components.

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Following (Coelho and Nunes, 2011) and assuming ktis I(1), an OLS regression of the first

differ-ences of kt on a dummy variable for a structural break is carried out:

∆kt= β0+ β1χt(b) + t,

where χt(b) = 1 for t > b (the break date) and zero elsewhere. After this regression, a t-test is

carried out for significance of β1, i.e. H0 : β1 = 0 is tested against the alternative Ha : β1 6= 0.

The date of the break (b) is inferred from the data by computing the t-statistic for different values of b and computing the maximal t-value as:

t∗= sup b∈Λ

|tb|,

where Λ is the set of possible break dates. Coelho and Nunes follow (Harris et al., 2009) and ex-clude the first and last 10% of dates, since allowing for structural changes so close to the beginning or end of the sample may lead to erroneous conclusions since there may not be enough observations to support it. In this study, only a small sample of 30 years is used. Therefore the first and last 5 years are excluded from Λ. Results are shown in Table 4.

Country b∗ β1 |t∗| P(T > |t∗|) Total mortality Australia 1961 -2.21 1.38 0.18 Belgium 1958 -2.33 1.82 0.08 Canada 1975 -1.26 1.51 0.14 Denmark 1956 -1.91 1.34 0.19 France 1958 -2.56 1.56 0.13 Japan 1955 -7.25 4.03 0.00 Netherlands 1972 -0.86 1.11 0.28 Norway 1956 -2.23 2.00 0.06 Spain 1958 -5.46 2.93 0.01 Sweden 1955 -2.10 0.80 0.43 Switzerland 1971 -1.40 0.82 0.42 USA 1973 -0.87 2.13 0.04 Male mortality Australia 1962 -1.34 1.09 0.28 Belgium 1958 -1.66 1.52 0.14 Canada 1975 -0.81 1.36 0.18 Denmark 1975 -0.27 1.25 0.22 France 1958 -1.62 1.39 0.18 Japan 1955 -6.23 3.77 0.00 Netherlands 1972 -0.43 1.01 0.32 Norway 1956 -1.05 1.14 0.26 Spain 1958 -4.98 3.12 0.00 Sweden 1975 -0.71 0.75 0.46 Switzerland 1969 -0.95 0.85 0.40 USA 1973 -0.58 2.52 0.02

Table 4: Test for a structural break in ktcorresponding to total and male mortality (1950-1980)

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The bold entries in Table 4 indicate significance of the structural break at the 5% level. It is seen that a break is found for the countries Japan, Spain and the USA for male as well as total mortality. This explains part of the discrepancies in forecast error found previously. Indeed, a relatively high MAPE was found for Japan and Spain. The USA on the other hand is a country for which the model performs very well.

The conclusion drawn from the preceding is that there is only little relation seen between forecast accuracy and compliance with the model assumptions. Three assumptions were defined: one principal components explains enough of the variation, bx is constant and ktbehaves linearly. It

was shown that the magnitude of R2 is not a good indicator of prediction error, except for the peculiar case of R2_{> 0.9. It is well known that non-constancy of b}

xis a problem (Lee and Miller,

2001) and this is confirmed visually. Stuctural breaks in ktwere however not found to be significant

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7 Conclusion

The main research question is: How reliable are Lee-Carter forecasts of aggregate mortality for developing countries where limited data is available? This question is answered in several steps. Chapters 2 and 3 gave an introduction to respectively mortality modelling using the life table and forecasting using the Lee Carter model.

Chapter 4 discussed how LC-forecasts made with limited data performed for twelve developed countries. The model was fit to mortality rates from 1950 to 1980 and a forecast was made for the years 1981 to 2005. Subsequently the procedure was repeated for several subsets of the data. Somewhat surprisingly, it was found that forecast accuracy was just as accurate when only 1950 and 1980 were used to fit the model instead of the full dataset. When the model was fit to data from a number of years preceding 1980 on the other hand, it was found that forecast accuracy increased rapidly when more data points were added. These result might make one optimistic about the use of LC on developed countries.

Chapter 5 showed however that it is in general not a wise idea to apply LC to limited data from developing countries. The model was fit to historical life tables from 23 developing countries and forecasts were compared with contemporary UN estimates of life expectancy at birth. Forecast ac-curacy was found to be very low for most developing countries. The low forecast acac-curacy can not be explained by the lack of data points. Instead, the development of mortality rates of developing countries do not seem to follow the strong pattern imposed by the LC-model.

Therefore Chapter 6 investigated the assumptions behind the model and their effect on forecast accuracy for the 12 developed countries discussed earlier. Three assumptions were identified. It was found that there is only little relation between model fit and forecast accuracy. For a few countries a significant structural break in mortality improvement was found. This explains part of the differences in forecast accuracy for different countries.

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Deb´on, A., F. Montes, and F. Puig (2008). Modelling and forecasting mortality in Spain. European Journal of Operational Research 189 (3), 624–637.

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Girosi, F. and G. King (2007). Understanding the Lee-Carter mortality forecasting method. Har-vard University.

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A

Computing life expectancy from abridged life tables

In the construction of life tables in (United Nations, 1986), a method described in (Greville, 1943) is used to convert central death rates (nmx) to probabilities of death nqx by use of the formula

(Siegel and Swanson, 2007):

nqx= nmx 1 n +nmx 1 2+ n 12(nmx− log c) ,

where c is assumed to be equal to e0.095. The next columns of the life table are calculated in the usual way, as described in Chapter 2, where the life expectancy at birth is computed as e0= T_l00.

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B

Historical life tables for developing countries

Country no. of life tables Periods

Bangladesh 2 1964-1965 1981 Chile 2 1959-1961 1969-1971 Costa Rica 2 1962-1964 1972-1974 Egypt 2 1965-1967 1975-1977 Guadeloupe 2 1966-1968 1981-1983 Guatemala 2 1963-1965 1972-1973 Guyana 2 1959-1961 1969-1971 Honduras 2 1960-1962 1973-1975 Israel 2 1960-1962 1971-1973 Jamaica 2 1959-1961 1969-1971 Kuwait 2 1974-1976 1979-1981 Mauritius 2 1961-1963 1971-1973 Pakistan 2 1962-1965 1968-1971 Panama 2 1960-1970 1970-1980 Philippines 2 1969-1971 1974-1976 Uruguay 2 1962-1964 1974-1976

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Country no. of life tables Periods Argentina 3 1959-1961 1969-1970 1979 Hong Kong 4 1960-1962 1970-1972 1976 1981 Martinique 3 1960-1962 1966-1968 1981-1983 Puerto Rico 4 1949-1951 1959-1961 1969-1971 1979-1981 R´eunion 3 1961-1962 1967-1968 1970 Singapore 3 1956-1958 1969-1971 1979-1981 Sri Lanka 4 1945-1947 1952-1954 1962-1964 1970-1972 Trinidad and Tobago 3 1945-1947 1959-1961 1969-1971

Forecasting using the Lee-Carter model with limited data

Forecasting using the Lee-Carter model with limited data

M.V. Kruijver

Forecasting using the Lee-Carter model with limited data

Maarten Kruijver (s1571575)

University of Groningen

February 27, 2012

Contents

1

Introduction

2

Modeling and estimating aggregate mortality

2.1

Probabilistic modeling of mortality using the life table

2.2

Statistical estimation of death rates

3

Forecasting using the Lee-Carter model

3.1

Original formulation of the Lee-Carter model

3.2

The case of limited data

4

Empirical forecasting performance with limited data from

developed countries

4.1

Data source (HMD)

4.2

Procedure

4.3

Results (USA)

(

)

4.4

Results (other countries)

5

Empirical forecasting performance for developing

coun-tries

5.1

Data sources

5.2

Procedure

5.3

Results

6

Model assumptions and forecast error

6.1

Model assumptions

6.2

Forecast error for different countries

7

Conclusion

References

A

Computing life expectancy from abridged life tables

B

Historical life tables for developing countries