Keyfitz N., Caswell H. Applied Mathematical Demography
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12.4. The Search for Constancies 285
A
B
C
a
t
0
Death rate
Time
A
B
C
b
t
0
Birth rate
Time
Figure 12.1. Demographic transition for three provinces or other groups
designated as A, B, and C.
12.4.2 Are Longitudinal Relations Demonstrated by
Cross-Sectional Data?
In seeking a constancy of some function of the population variables
through time one is tempted to transfer a relation found in cross-sectional
data. Constancy from place to place at a given time ought to be evidence
of constancy through time. The provinces of India whose mortality is low
seem also to be the ones whose fertility is low; does it not follow that, as
mortality falls in the years ahead for the whole of India, so also will fertil-
ity? And—only a short step further—the low mortality may be argued to
cause the low fertility, among other reasons because parents aim to have
a certain number of children who survive to maturity. These statements
may be true, and if they are they would be very helpful in forecasting. But
other evidence than the cross-sectional relation is required to prove them,
as can be seen from the very simple cases of Figures 12.1 to 12.3.
Figure 12.1 is a stylized illustration of the problem for three provinces
or other population groups, A, B,andC, with deaths in Figure 12.1a and
births in Figure 12.1b. From the birth and death rates of the provinces at
t
0
, i.e., taking vertical cross sections of the two families of curves at t
0
,we
can obtain a cross-sectional correlation of birth rates and death rates, and
it would be positive; the provinces are in the same sequence A, B,andC.
On the other hand, we obtain a longitudinal correlation by comparing birth
and death rates through time for any particular province. In Figure 12.1
the longitudinal, like the cross-sectional, correlations of birth and death
rates are positive.
But we could draw the curves differently and obtain the opposite result,
as has been done in Figure 12.2, where deaths are declining and births are
rising. The correlation of birth and death rates through time for any of the
286 12. Projection and Forecasting
A
B
C
A
B
C
a
t
0
Death rate
Time
b
t
0
Birth rate
Time
Figure 12.2. Opposite to demographic transition for three provinces A, B, and
C.
A
B
C
C
B
A
a
t
0
Death rate
Time
b
t
0
Birth rate
Time
Figure 12.3. Demographic transition for three provinces with sequence reversed.
provinces A, B,orC would be negative, though that for the cross section
at any given time would again be positive.
Is the cross-sectional correlation even necessary to the correlation in
time? Examine Figure 12.3, drawn in the same way as Figure 12.1, ex-
ceptthatinFigure12.3b the births have been labeled differently. In this
diagram the cross-sectional correlation is negative, while the correlation
through time is positive. There is nothing impossible a priori about this
arrangement. The cross-sectional correlation is neither necessary nor suffi-
cient for the longitudinal one (Janowitz 1971). This is part of the elusiveness
of the inference of future trends from past conditions.
12.5. Features of Forecasting and Forecasting Error 287
12.5 Features of Forecasting and Forecasting Error
No one expects botanists, geographers, or anthropologists to make declara-
tions regarding the future.
∗
Physicists do make predictions, but within an
experimental situation closely protected against interference from outside
events in the real world. Demographers, on the other hand, are expected
to predict future population as it will actually occur. And they respond as
best they can; much of the published literature in demography, and even
more of the unpublished, whether written by amateurs or by professionals,
consist in statements about the future.
We saw that a projection over t units of time can be written as A
t
n,
where A is a projection matrix and n the initial population vector. If we
know the elements of A and of n, the estimate of the future consists in the
multiplication. We may incorporate in A and n several regions, rural and
urban populations, ethnic and racial groups, or other divisions (Chapter
7). We may admit that the transition probabilities are a function of time
and say (equation 12.2.5) that at time t the population is
n(t)=A
t−1
A
t−2
···A
0
n(0).
The preceding paragraph, like Section 12.2, is complicated enough to
have the effect of concealing the logical status of what we are doing. Age
and other distinctions do matter, but their numerical effects are often small
in comparison with the prediction error within any category. Let us drop the
categories and think only of total population projected over one period, say
of 30 years; thus we are back to multiplying scalars rather than matrices and
vectors. For example, the population of the United States was counted in
1970 at 203 million; what will be its number in the year 2000? The ultimate
in simplicity is to say that, if the population increases by 33 percent, it will
number 270 million. Most projections snow us with breakdowns by age, sex,
race, and region, and take a number of time intervals, all of them valuable,
but in their combination causing us to lose perspective on the problem. By
peeling off the breakdown, we arrive at the essence of calculation of future
population in its two aspects of projection and prediction.
Projection is where the 33 percent is hypothetical. All projection con-
sists of such statements as the following: “If (which we do not assert) the
population grows at 33 percent in 30 years, then by the year 2000 it will
have increased to 203 × 1.33 = 270 million.” The projection consists in
performing the multiplication, is conditional on the 33 percent, and is as
unassailable as the laws of arithmetic. No projection risks being in error; it
∗
Actually, this statement has become less and less true over time, as problems in con-
servation and resource management have required ecologists to try harder and harder to
predict the future growth of plant and animal populations. This has led them to confront
exactly the same problems of forecasting and forecasting error as human demographers.
288 12. Projection and Forecasting
cannot be proved or disproved by what the population actually turns out
to be in the year 2000.
Stated in this stripped-down form, the result is wholly uninteresting. At
most it does the reader the service of performing the multiplication, and in
an age of computers no one would pay much to be told the bare arithmetical
fact that 203 × 1.33 = 270. The demographer’s service surely starts with
his comment on the realism of the 1.33, and with that he enters the field
of forecasting.
Forecasting is where the 33 percent is taken as the real prospective in-
crease: “The population will grow 33 percent in 30 years.” Unless this is
to be soothsaying, we now need evidence for the 33 percent. Such evidence
may be one or another kind of extrapolation; in the simplest case the argu-
ment would be that the next 30 years will show an increase of 33 percent
because the last 30 years showed such an increase. A more sophisticated
version would assert, not that the ratio will be constant, but that the rate
of change in the ratio will be constant. In the most general statement the
forecast asserts that some more complex function, possibly involving eco-
nomic, ecological, psychological, or other variables sufficient to determine
the future population numbers, will remain constant into the future, and
the task of forecasting consists in the search for such a functional invariance.
One circumstance that can make prediction interesting is what makes a
poker game interesting: the fact that the cards, first hidden as the future is
hidden, are later turned face upward on the table, and so test the accuracy
of the forecast. In economic forecasting a frequent span is 6 months or 1
year; such a period is not too long for the predictor to see whether she
guessed right, or at least more accurately than someone else, and to learn
from the experience. But 30 years, which is a typical span for demographic
forecasting, is too long a wait for learning to occur. To wait 30 years to see
the cards removes the interest inherent in the rapid alternation of forecast–
verification–forecast–verification that holds us at the poker table.
12.5.1 Extrapolation Versus Mechanism
Any forecasting that takes itself seriously tries to base itself on the un-
derlying mechanisms, rather than merely extrapolating trends. Yet the
distinction is not always sharp. The components projection on which much
stress is laid is indeed a mechanism, but it still depends on extrapolation
of birth, death, and migration rates.
A mechanism that lends itself to prediction is Easterlin’s model described
in Section 14.11, in which age-specific birth rates depend negatively on the
size of the cohort of an age to marry and bear children at the given time.
In effect it predicts waves of twice the length of generation, so that in
the United States birth rates would be low for the next decade or more,
and only in the 1990s would they pick up again. [Of course, even if the
mechanism is operative exactly as described, the intensity of its operation,
12.5. Features of Forecasting and Forecasting Error 289
measured by the constant γ of (14.11.1), would still have to be ascertained
by extrapolation.]
Other mechanisms are less useful for forecasting. That working women
have fewer children than nonworking women is true without telling us
whether seeking work is primary and lack of children the result, or whether
selection is involved: women who do not have children take jobs. But sup-
pose the first case: that the fall in births during the 1960s was due to more
and more women preferring work outside the home to child raising. Perhaps
they needed the income; perhaps outside work came to carry greater pres-
tige. Certainly women’s liberation will reduce childbearing, at least until
the day when men take over the unpaid work of child raising. To use this
in prediction we would need some way of knowing in advance about the
shifting preferences for work and income versus the satisfactions of moth-
erhood. Will the uptrend in the former continue over the next decade and
so maintain the downtrend in births?
The difficulty of answering this question suggests that the mechanism in
question is an ad hoc explanation of the past, to be retained as long as
women increasingly enter the labor force and the birth rate falls, but to be
quickly dropped and replaced by some other mechanism as soon as births
turn upward. One can imagine, after births do start to rise, articles showing
how inevitable is the reassertion of the durable values of motherhood as
opposed to ephemeral economic interest. Whether such explanations are
true or false, if they come after the fact they are too late to forecast turning
points.
In this difficult situation it is natural to resort to asking women what
their childbearing intentions are—just as people are queried about their
house-buying intentions, and firms their investment plans. Such data help
but only in short-term forecasting. Most of the children born more than 5
years from now will be to mothers who are presently still in their teens,
unmarried, in no position to provide a realistic statement of their futures as
mothers (Ryder and Westoff 1967, Siegel and Akers 1969). For the shorter
term, intentions may predict well; Westoff, Potter, and Sagi (1963) found
that about the best predictor of whether a couple with two children will
have a third is their own statement.
Even for estimating turning points in the birth curve less than 5 years
ahead, however, the statements of wives have not always been borne out.
Women may declare an intention that accords with present rates of child-
bearing, but once the future becomes the present they are influenced by
whatever the fashion is at the time. It is too much to say that current
childbearing influences stated intentions more than intentions anticipate
future childbearing, but some hint of this does appear in the time series of
intentions, on the one hand, and performance, on the other.
We have some evidence (Masnick and McFalls 1976) that women’s at-
titudes toward childbearing are formed (or manifest themselves) early in
their married careers. If they start married life in a time of low fertility
290 12. Projection and Forecasting
when contraception is practiced rigorously, they tend to continue to prac-
tice it. Such a fact, if the manner and degree of its operation could be
established, would facilitate forecasting.
12.5.2 Shape of the Projection Fan
If we know the past, and must make estimates of the future, the population
trajectory starts out as a single, somewhat jagged line, and at the moment
where past and future meet fans out into a set of relatively smooth lines
representing the several possibilities. Disregarding for the moment the er-
rors, incompleteness, and delays in statistics on the past, the question is
how the single line or curve, representing more or less certain past knowl-
edge, fans out into the future to make the horn shape familiar in graphical
representations of projection.
If the lines of the fan were straight, each would require only one
parameter—say, the angle made with the time axis—for its full description.
Such straight lines imply that uncertainty increases in equal increments
with time from the present—that if we know in 1970 that the population
for 1980 will be in a certain range with a certain probability, the population
for 1990 will be in twice that range, both expressed in numbers of persons
(Figure 12.4a).
If, on the other hand, one could say that for all future time the (fixed)
rate of increase r would be within the range r
1
to r
2
, the horn would take
the shape of exponentials as in Figure 12.4b. Drawn on semilog paper, this
would be back to the form of Figure 12.4a.
Only slightly more difficult is thinking that the future rate of increase
would be r(t)attimet, drawn from a specified probability distribution.
Since, as in (1.6.1), population at time t is N
t
= N
0
exp[
t
0
r(τ) dτ ], what
is operative is the sum of r(t). If the increases at the several times in
the future are drawn independently at random, the variance of
t
0
r(τ) dτ
is proportional to the time t, the standard deviation to
√
t. The range
in this case would be a curve like e
√
t
, in suitable units. Or under other
circumstances the curve might be of the form exp(
√
t) (Figure 12.4c). A
priori the possibilities are unlimited.
But now recall that we can predict survivorship better than births. As
time moves on, more and more of the population will have been born since
the jumping-off point, and a smaller and smaller fraction will consist of
survivors from that point. As a larger fraction passes into the group known
less well, the fan widens more rapidly than is shown in any of the three
diagrams of Figure 12.4.
In symbols, suppose that at a given future time f of the population
will have been born since the jumping-off point and will be known with
uncertainty represented by the standard deviation σ,and1− f of the
population with uncertainty kσ,wherek<1. Then the accuracy with
12.5. Features of Forecasting and Forecasting Error 291
(a)
(b) (c)
Figure 12.4. Shape of the fan under various conditions. (a) Ignorance of popu-
lation increasing with time in equal increments. (b) Ignorance of (fixed) rate of
increase. (c) Future subject to diminishing variation.
which the whole population is known has the standard deviation
σ
∗
= σ
f
2
+(1− f )
2
k
2
.
The spread can be expressed directly in terms of survivors and births.
For ex ante evaluation the population t years hence may be written as
N
t
=
t
0
B(τ)l(t − τ) dτ +
ω
0
p(a)
l(a + t)
l(a)
da + I
t
− E
t
,
292 12. Projection and Forecasting
where
B(τ) = the births at time τ,
l(t − τ) = the probability of survival of a birth to
age t − τ , that is, to time t,
p(a) da = the number of persons age a to a + da
at the jumping-off point where t =0,
I
t
= the cumulative number of immigrants by time t,as
projected from immigration statistics by a
survivorship function,
E
t
= the cumulative number of emigrants, also as projected
to time t.
The quantities in the equation are subject to different amounts of error—
current population p(a) is best known, prospective survivorship l(a+t)/l(a)
next best, future births B(τ) only poorly. Emigrants present more difficul-
ties than immigrants in current statistics as well as in the future. Separate
estimates need to be made of the precision of each of these components.
As time goes on from the jumping-off point, the dependence of the fore-
cast on future births increases, and hence the precision decreases. The fan
of projections will correspondingly open out more widely than in straight
lines or exponentials.
These considerations apply to projections using matrix population mod-
els with arbitrary stage classification. If the projection matrix A
t
at time
t is determined by an stationary ergodic stochastic process, and if the ma-
trices are sufficiently well behaved (a stochastic analogue of the primitivity
condition for convergence), then the log of population size is asymptoti-
cally normally distributed with a variance that increases linearly with time
at a rate we can denote by σ
2
(Furstenberg and Kesten 1960, Tuljapurkar
and Orzack 1980). Thus the projection fan in this case also increases as
exp(σ
√
t), but matrix methods are required to estimate σ
2
(see Tuljapurkar
1990 and MPM Chapter 14).
These results are asymptotic, and apply in stationary stochastic environ-
ments, whereas projections are often short-term and in conditions where
the statistical properties of the vital rates are not constant. For an approach
to such cases, see Lee and Tuljapurkar (1994).
12.6 The Components of Forecasting Error Ex
Ante
The errors that arise in estimates of the future are classifiable under five
headings. Arrangement is in ascending order of difficulty of estimation, and
probably also of magnitude.
12.6. The Components of Forecasting Error Ex Ante 293
1. There is random variation from the deterministic model on which
the estimate was made. Corresponding to the deterministic model
used for forecasting, one can construct many stochastic models, all
with expected values equal to the forecast. The most obvious of these
stochastic models allows the same probability to act on the several
individuals independently.
In application to a single item, say births numbering B, the coefficient
of variation (standard deviation divided by expectation) can be calcu-
lated as a function of the numbers involved, for example the number
of births B. For large closed populations, assuming statistical inde-
pendence between individuals gives a coefficient of variation equal to
1/
√
B. When births are in millions, the error is trifling; when even as
few as 10,000 births the coefficient of variation is only 1 percent. This
component of forecasting error is important only for populations as
small as a neighborhood or a village.
This type of variation is called demographic stochasticity,todis-
tinguish it from environmental stochasticity caused by stochastic
temporal fluctuations in the vital rates. Including demographic
stochasticity in matrix population models leads to a multi-type
branching process, an extension of the model described in Section
16.4. The connection between matrix population models and multi-
type branching processes was first studied by Pollard (1966, 1973) for
age-classified models, and was generalized to stage-classified models
in Chapter 15 of MPM.
2. The constants in the models are not known exactly. The projection
may suppose the age-specific rates of birth and death, first differences
of these rates over time, ultimate family size, or something else to be
fixed into the future. If whatever is taken as constant really is constant
but is improperly measured or assessed, the future will to that extent
be incorrectly estimated.
The extent of this uncertainty can be evaluated by associating confi-
dence intervals with projections, based on the sampling distributions
of the parameters on which they are based. This was once difficult
or impossible, but the development of bootstrap resampling methods
(Efron and Tibshirani 1993) now make it a fairly straightforward task
in even complicated matrix population models (MPM Chapter 12).
Estimation errors may be amplified in a forecast, but they still add
little to the other, deeper errors inherent in forecasting and considered
below.
3. The quantities supposed constant, which constitute the parameters
of the model, really vary in time. Only a finite past record, a short
sample of time periods, is available for estimating their paths. Of-
ten the sample of time periods is a single past year or decade. It is
294 12. Projection and Forecasting
hardly a properly drawn sample, yet the forecaster has no other way
of estimating the future values of the needed parameters.
4. Probably more important than any of the foregoing, the model itself
is incorrectly specified. The main fault is not the bad estimates of
parameters, but an altogether inappropriate function supposed to
hold for all time. For example, the period age-specific rates of birth are
held constant, when what is really constant is their first differences,
or else not period rates at all but the average number of children
born to a cohort. The finite past experience that can be brought to
bear has little power to discriminate among models, as can be seen
by experimenting with the logistic and explosion models of Section
1.7. It is not easy to compel the past to inform us which of several
models that fit indifferently well will give the best forecast, yet the
futures given by these models can be very different.
Recent statistical developments, combining information theory and
maximum likelihood estimation, make it possible to objectively as-
sign weights to different competing models, and take account of the
uncertainty of model specification in projections (see Burnham and
Anderson 2002 for a particularly clear account, with many appli-
cations to population biology). This approach is rapidly becoming
standard in some branches of population ecology, and has great
potential for human demography as well.
5. Finally, even perfect use of exact facts regarding a homogeneous past
may be frustrated by the future being genuinely different. The uncer-
tainty about the future is superimposed on projections of past data
at the disposal of the demographer, and such uncertainty does not
lend itself to estimation in advance. In some countries and epochs this
fifth component of error will be small, but the twentieth century is
not such an epoch. This component is wholly distinct from sampling
error and is the characteristic special difficulty of prediction.
Of the five components of error the first four are, at least in principle,
accessible to statistical analysis, and the fifth is not accessible by any means.
A rough technique may be devised for handling the fourth component or,
more strictly, the first four combined. This provides a lower bound for the
overall error ex ante, that is to say, made at the same time as the estimate
itself. This is contrasted below with the error ex post, calculable after the
event predicted, the predictand is known, and which does include all five
sources of error.
In the course of estimating the future many separate decisions have to
be made, and none of these follows uniquely from any accepted principles.
Extrapolation of death rates can proceed from a shorter or longer expe-
rience; it can be done separately for the several age–sex groups or for all
groups together, supposing that mortality will improve according to the