Chapter 3
Experiences from Forecasting Mortality
in Finland
Juha M. Alho
3.1
Modeen and Törnqvist
The first official cohort-component projection of the population of Finland was
prepared by Gunnar Modeen (1934a), an actuary with the Central Statistical Office
of Finland at the time. Modeen’s work had elements of genuine forecasting in that he
commented on past trends in fertility, mortality, and migration, and discussed their
possible long-term implications (Modeen 1934b). On the other hand, the work was
rather schematic in nature. In particular, age-specific mortality was assumed not to
change during the projection period, although Modeen was aware of its declining
trend since the late nineteenth century (Modeen 1934a, 38). Unable to pinpoint the
future rate of decline exactly, Modeen rejected any alternative assumption as
speculative (cf., Modeen and Fougstedt 1938).
Analyses of mortality trends in presumably more advanced countries were used
as leading indicators in the United States by Whelpton et al. (1947), for example. In
Finland, Leo Törnqvist (1949) proposed similar methods. In particular, he used
Swedish mortality as a target towards which he assumed Finnish mortality to
converge. Both series were first transformed via a logistic type transformation.
Then, the curves were aligned, and the Finnish curve was prolonged in accordance
with the Swedish development.
A problem with Modeen’s projection was that it soon became outdated. Fertility
started to increase at the time the projection was published, and mortality continued
to decline. Modeen’s calculations suggested that the Finnish population would never
exceed four million, but this mark had already been crossed by 1950. Törnqvist’s
collected works (Viren et al. 1981) do not mention the error of past forecasts as a
motivation for his own early work. Nevertheless, the future statistics professor, a
J. M. Alho (*)
University of Joensuu, Joensuu, Finland
e-mail: juha.alho@helsinki.fi
© The Author(s) 2019
T. Bengtsson, N. Keilman (eds.), Old and New Perspectives on Mortality
Forecasting, Demographic Research Monographs,
https://doi.org/10.1007/978-3-030-05075-7_3
33
34
J. M. Alho
specialist in time-series analysis (among other fields, cf. Nordberg 1999), was well
aware that forecasts cannot be made without error. He appears to have been the first
to formulate the problem of uncertainty in population forecasting in probabilistic
terms in Törnqvist (1949). Later, Törnqvist also conducted what must be one of the
earliest assessments of the empirical accuracy of Finnish forecasts (including his
own!) in Törnqvist (1967).
In this note, we will outline current developments in Finnish mortality forecasting. In Sect. 3.2, we describe the methods used by official forecasters. These derive
mainly from the tradition of early cohort-component forecasters (cf. DeGans 1999).
In Sect. 3.3, we discuss how uncertainty can be taken into account using probabilistic models and present-day computing facilities. We conclude in Sect. 3.4 by
commenting on some applications for which mortality forecasts are particularly
relevant.
3.2
Official Forecasts1
The arithmetic underlying cohort-component forecasts was understood a hundred
years ago (DeGans 1999). Since the method relies on detailed assumptions
concerning future age-specific rates, the real key to forecast accuracy lies with
those assumptions. One would think that major improvements would have occurred
during the past century, judging from the way the assumptions are formulated. Yet,
the methods were essentially perfected by Whelpton back in the 1940s.
The two producers of official population forecasts in Finland are Statistics
Finland and the Social Insurance Institution of Finland (or KELA, an abbreviation
of the Finnish name). Since the forecasters of the two institutions cooperate on an
informal basis, the forecasts have many similarities.
Both institutions produce forecasts approximately every 3 years. More frequent
updates are made if unexpected developments occur. Both disaggregate the population by sex and single years of age (0, 1, 2, . . ., 99, 100+). Currently both organizations forecast until 2050.
KELA produces a national forecast only, whereas Statistics Finland forecasts the
population of every one of the 448 municipalities of Finland. In the case of mortality,
the country is divided into three relatively homogeneous areas: Northern and Eastern
Finland, which have a high level of mortality (due in particular to cardio-vascular
diseases among males); the Swedish-speaking coastal areas, with low mortality; the
rest of the country, with intermediate mortality. The reason for the low mortality
among the Swedish speakers has not been established, but both socio-economic and
lifestyle factors apparently play a part (Koskinen and Martelin 1995).
1
The author would like to thank Mr. Matti Saari, Statistics Finland, and Mr. Markku Ryynänen,
KELA, for information on the practice of forecasting. Any misunderstandings are the sole responsibility of the author.
3 Experiences from Forecasting Mortality in Finland
35
Neither organization uses cause-specific mortality data in the preparation of their
assumptions. This is contrast with the U.S. Office of the Actuary, for example (e.g.,
Wade 1987). However, we have argued elsewhere that cause-specific information
cannot be expected to increase forecast accuracy unless one of two conditions are
met: either leading indicators can be identified in the preparation of forecasts, or
structural changes can be anticipated based on other available information (as in the
case of AIDS, for example) (Alho 1991).
Both organizations use trend extrapolation as a basis for their mortality forecast.
Starting from a target value for life expectancy at birth, e0, Statistics Finland adjusts
future age-specific mortality rates so that the implied increase in life expectancy
gradually slows down until the target of e0 is reached. Age-dependent proportional
adjustment is used to modify the jump-off rates. In KELA the starting point is a
classification of individual ages into aggregates with similar mortality levels.
Regression analysis is used in the log-scale to estimate rates of decline that gradually
decelerate. The assumption, made by both organizations, that the rate of decline
eventually falls off, is far from self-evident. In fact, we have used U.S. data to show
that such an assumption has historically made the U.S. mortality forecasts worse
than simpler trend extrapolations (Alho 1990).
Neither organization formulates their targets on a cohort basis although both
occasionally examine cohort trends to see whether there are any irregularities. A
current example of such an irregularity was reported by KELA: the female cohorts
born in the 1950s appear to have higher mortality than cohorts born earlier,
during WWII.
The methods of trend extrapolation used by the organizations blend judgment and
empirical analysis. Neither organization has experimented with the method proposed
by Lee and Carter (1992). Its performance in regard to ages 65+ in Finland, was
investigated in a University of Joensuu pro gradu thesis by Eklund (1995), who
found that a one-dimensional singular-value decomposition produced a good fit to
the data. Because of random variation, however, the resulting forecast was not
always an increasing function of age.
In addition to the trend forecast, KELA produces another mortality variant in
which it is assumed (as Modeen did) that mortality will remain at the jump-off level.
Statistics Finland limits itself to a single variant even though high and low variants
have previously been used in national forecasts.
3.3
Predictive Distribution of Mortality
A major contribution by Törnqvist (1949) was that he was apparently the first to
maintain that since the future values of a vital rate cannot be totally known, they
must be treated as random variables. The actual future values are then “samples”
from their distributions. In modern terminology, the uncertainty of the future value is
36
J. M. Alho
expressed in terms of a predictive distribution that represents both our best guess and
its uncertainty. The distribution is conditioned on all information available at the
jump-off time of the forecast (e.g., Gelman et al. 1995, 9).
Törnqvist’s contribution may have been ahead of its time. In particular, correct
formal treatment of the predictive distribution would have been difficult before the
availability of high-speed computing. In recent years, the potential usefulness of a
probabilistic approach to uncertainty has been noted on several occasions.2 At the
University of Joensuu, we have written a computer program, PEP (Program for Error
Propagation), which is capable of simulation samples from a wide range of predictive distributions.
The main concept of PEP is that it allows us to describe the uncertainty connected
with a forecast at the time it is being made. All sources of uncertainty – age-specific
fertility and age and sex-specific mortality and migration – are taken into account
and propagated throughout to derive the predictive distribution of the population. In
this sense, PEP is merely a stochastic version of the cohort-component bookkeeping
system. The usefulness of the results depends on the assumptions underlying the
calculations. The user of PEP must specify a point forecast for each of the vital rates
for all future years, just as in ordinary cohort-component forecasting. An additional
step is required in the form of specifying the uncertainty surrounding the forecast.
Suppose R(j,t) is the mortality rate for age j ¼ 0, 1, . . ., ω in a future year t ¼ 1,
2, . . ., T. PEP assumes that
Rðj; tÞ ¼ expð^r ðj; tÞ þ Xðj; tÞÞ,
where ^r (j, t) is the point forecast of the log-rate, and X(j,t) is a random error with a
mean of E[X(j,t)] ¼ 0. The random error can always be written in the form
Xðj; tÞ ¼ εð1; tÞ þ þ εðj; tÞ:
In PEP, the error increments ε(j,t) are assumed to be of the form
εðj; tÞ ¼ Sðj; tÞ η j þ δðj; tÞ ,
where the S(j,t)’s are known scale factors that can be chosen to match any sequence
of error variances Var(X(j,t)) that increases with t. Fixing j, we can think of the terms
ηj as representing errors in forecasted trends. In the case of mortality, the trend
corresponds to the rate of decline, for example. Since the terms δ(j,t) are independent
for any fixed j, they represent unpredictable random variation. The relative roles of
the two types of uncertainties derive from the assumption ηj~N(0, κj), and δ(j, t)~N
2
Review of Land (1986); “Special Section on Statistical Analysis of Errors in Population Forecasting and Its Implications on Policy,” Journal of Official Statistics, September 1997; “Frontiers of
Population Forecasting,” Population and Development Review, 1998 Supplement; review of
U.N. forecasts by the National Research Council (2000).
3 Experiences from Forecasting Mortality in Finland
37
(0, 1 κj), where 0 κj 1. The terms ηj are assumed to be independent of the
terms δ(j,t). Finally, the terms ηj can either have a constant correlation across j, or an
AR(1) type correlation. The same is true for the δ(j,t)’s, when t is fixed. This scaled
model for error was introduced in Alho and Spencer (1997).
In Alho (1998) we provide details of the application of PEP to the population of
Finland for 1999–2050. The point forecasts for each vital rate were as specified by
Statistics Finland. We now present some details on the treatment of uncertainty in the
mortality forecast.
Age-specific mortality data in 5-year age-groups 0–4, 5–9, . . ., 75–79, and 80+
were available for the years 1900–1994. After a preliminary analysis, the data were
aggregated into the broader age groups 0–4, 5–34, 35–59, 60–79, and 80+ by adding
the age-specific rates together. This increased the stability of the trends. The analysis
was carried out in terms of the logarithm of the sum (cf. Alho 1998, Figures 5a–e,
pp. 19–21). The unusual values produced by the civil war in 1918 and WWII in
1939–1944 were smoothed using values from the previous year. For each of the five
broad age groups, we produced baseline forecasts as follows:
• Starting from year y ¼ 1915, we used the data for the previous 15 years (y, y –
1, . . ., y – 15) to calculate a trend forecast for all future years until 1994.
• A linear trend was estimated from the first and the last observation of the 15-year
data period.
• In case the linear trend was positive, it was replaced by a constant value (i.e.,
slope ¼ 0).
For each y ¼ 1915, 1916, we calculated the empirical forecast error for lead times
t ¼ 1, 2, . . ., 50. For each lead time t, we could then estimate the standard deviation
of the error around zero (i.e., assuming that the forecasts are unbiased). This would
give us estimates of Var(X(j,t)) directly, from which the scales S(j,t) could be
deduced. However, it turned out that especially for younger ages the estimates
were somewhat erratic because of the large random (Poisson-type) variation in the
counts. Therefore, final estimates were produced by averaging the estimates from the
six time series corresponding to the three broad age groups of 35–59, 60–79, and 80
+ for males and females. The resulting estimate of the standard deviation of the
relative error starts from approximately 0.06 at t ¼ 1 and increases in a linear fashion
to about 0.6 at t ¼ 50. Otherwise expressed, the relative error one might expect for a
single age group increases from 6% to roughly 60% in 50 years (cf., Alho 1998,
Fig. 6, p. 22). These estimates were used for all ages.
The results were checked by fitting an ARIMA(1,1,0) model to the data series,
and similar results were obtained (Alho 1998, Fig. 6, p. 22).
The parameter κ was estimated by the least-squares method. The single value
κ ¼ 0.149 was applied for all ages.
An AR(1) process was used to model the autocorrelation of the error terms ηj and
δ(j,t) across age j. Otherwise expressed, the correlation was assumed to be ρji jj for
any two single years of age i and j, where the empirical estimate ρ ¼ .945 was used
for ηj’s and ρ ¼ .977 was used for δ(j,t)’s. Finally, a parameter for contemporaneous
38
J. M. Alho
Life Expectancy
87
82
77
72
2000
2010
2020
2030
Year
2040
2050
Fig. 3.1 Predictive distribution of male life expectancy in Finland in 1998–2050
89
88
Life Expectancy
87
86
85
84
83
82
81
80
2000
2010
2020
2030
Year
2040
2050
Fig. 3.2 Predictive distribution of female life expectancy in Finland in 1998–2050
crosscorrelation between the error of male mortality and the female mortality was
estimated as .795.
The details are fairly complex. One way to assess the reasonableness of the
procedures is to consider their implications for life expectancy. Figure 3.1 has a
predictive distribution for male life expectancy at birth, and Fig. 3.2 has a plot for
female life expectancy. The median of the predictive distribution is 82.0 for males
and 85.6 for females in 2050. A 50% prediction interval (or interval between the first
and third quartile) is [79.0, 84.4] for males and [83.9, 87.5] for females. An 80%
prediction interval for males is [76.5, 86.5] and [82.3, 89.0] for females. The
narrower spread for females is probably due to their lower level of mortality.
3 Experiences from Forecasting Mortality in Finland
39
Two concerns can be raised concerning the intervals. First, the long-term point
forecasts are based on an eventual slowdown of the decline in mortality; this may
make the Finnish forecast too conservative, as it did in the U.S. earlier. However, we
may note that the life expectancy implied by the current Swedish forecasts for males
is 82.6 and for females 86.5 years in 2050. In the intermediate variant of the
Norwegian forecast, the corresponding ages assumed are 80.0 and 84.5 years. We
see that despite the assumption of a slow-down in the mortality decline in Finland,
the Finnish forecast is the most optimistic of the three in terms of improvement, since
the current life expectancy in Finland is the lowest. Even though the Finnish point
forecast may be too low at the end of the forecast period, from this perspective the
Finnish forecast appears less conservative.
Second, in view of the vast potential for new medical advances, one could argue
that the range of uncertainty expressed by the widths of the intervals might be overly
narrow. Two arguments seem relevant here. For the U.S. (both sexes combined), Lee
and Carter (1992, p. 660, Fig. 3.1) calculated model-based 95% intervals for life
expectancy 50 years ahead. The width of these intervals was approximately
8.4 years. In a normal model, the corresponding width for an 80% interval would
be approximately 5.5 years. Thus, our intervals are clearly wider. In a discussion of
the paper by Lee and Carter, we noted that by including all sources of variation, the
Lee and Carter intervals would have been approximately one half wider (Alho
1992). This would have resulted in estimates close to ours. (One could also argue
that in a large country with heterogeneous sub-populations there might be some
offsetting variation, resulting in a national average more stable than in a small
homogeneous country. While conceivable, this possibility does not seem to be an
adequate reason for inflating the Finnish intervals, since it does not show up in the
Finnish time series).
A related criticism suggests that future advances in medical knowledge may be so
unprecedented that intervals based on past outcomes are too narrow. However, in
1915, at the start of our observation period, both life expectancy for both males and
females was substantially lower – 43.2 and 49.2 years (Kannisto and Nieminen
1996); the improvements by the end of the twentieth century were 27.5 and
29.5 years, respectively. Our estimates reflect the variation in this turbulent period
of major improvement. The future can be even more volatile, but the advantage of
our intervals is that they correspond to actual past variation, rather than to a
subjective assessment. Of course, a subjective assessment may be used as a basis
for other calculations that would complement our own.
3.4
Applications
Now that we have the analytical capability to produce predictive distributions for
future vital rates and future population, it is of some interest to consider how they
might be applied. There are two aspects to this.
40
J. M. Alho
First, it is critical that we understand how the predictive distribution can be
understood. As noted, e.g., in Alho (1998), predictive distributions can be based
on (1) formal statistical models, (2) errors of past forecasts can be used to estimate
the error of future forecasts, (3) errors of baseline forecasts can be used to estimate
future error, and (4) error specification can be purely subjective. Of course, any
mixture of the four is also a possibility. The results we have shown rely primarily on
(3), although they have elements of (1) and (4), as well. The aim was to provide an
empirical assessment of the difficulty of forecasting (or “forecastability”) of the vital
processes for different times. As such, the results for mortality correspond to the
uncertainty in mortality forecasting during the twentieth century. One can reasonably question whether forecasting is now easier, or more difficult, than in the past,
but at least we now have a quantitative empirical assessment of how things were
before.
Second, the predictive distribution can be used to address numerous social-policy
issues that depend on future population and its age structure. For example, in Alho
et al. (2001) we review an example in which output from PEP was used in
combination with the Finnish overlapping-generations model to devise alternative
pension-funding rules, and another example in which output from PEP was used to
assess the stability of the current rules for state aid to municipalities. In a University
of Joensuu pro gradu thesis, Polvinen (2001) used PEP to form a predictive
distribution of the so-called generational accounts. All these questions are of fundamental concern for the long-term planning of the social-support systems in Finland.
In no case has the effect of uncertain population age-distribution previously been
recognized. Other applications have been presented by Lee and Tuljapurkar (1998)
for the Social Security system of the United States, for example. Further research
opportunities are discussed in Auerbach and Lee (2001).
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