Mathematical demography - The Rockefeller University

MAMEMATICAL DEMOGRAPHY:
RECENT DEVELOPMENTS IN POPUATION PROJECTIONS
Joel E. COHEN
Rockefeller University
New York, U.S.A.
Mathematical demography has flowered in many different gardens during
the past twenty years. Several reviews of mathematical demograph a r e
available: Keyfitz (1968), Feishtinger (1971), Pollard (1973), Ludwig 8978).
Keiding (1975), Menken (1977), Smith and Keyfitz (1977), Kurtz (1981), and
Cohen (1984). This necessarily brief review deals only with stochastic models
for population projection developed since 1965.
POPULATION PROJECTION MODELS
Stochastic models a r e needed for population projection because
deterministic models fail t o account for the variability of historical
demographic data and t o provide probabilistically meaningful estimates of the
uncertainty of demographic predictions. Stochastic population projection models
may include migration, mortality and fertility (Lee, 1978b; McDonald, 1981).
can distinguish, not always
Among stochastic projection models,:one
sharply, between structural models and time-series models. Structural models
reprisent some underlying mechanism of population growth. Time-series models
apply t o demographic data general techniques in which the form of the model
need not be based on demographic theory.
Structural. projection models usually describe either or both of two
sources of random fluctuation: demographic variation and environmental
variation. Demographic variation arises from t h e stochastic operation of
mechanisms with fixed vital rates. Environmental variation arises when the
demographic rates themselves a r e governed by a stochastic process.
Population Projection with Demographic Variation
Deterministic projections (Siegel, 1972) of populations closed t o migration
commonly use the recurrence relation
where y(t) is a vector in which the ith component is the number of females in
age class i a t time t , i = 1,
k; y(0) is a given initial age census of the
...,
female population; and L(t) is a k x k non-negative matrix, conventionally
called a Leslie matrix. See Keyfitz (1968) or Pollard (1973) for details.
Taking L to be independent of t , i.e. constant in time, Pollard (1966)
reinterprets (1) as a multitype branching process. The survival and fertility of
each female are assumed independent of each other and of the survival and
fertility of all other females. Then y(t) in (1) can be interpreted as the
expectation of the age census at time t A linear recurrence relation that uses
the direct or Kronecker product of two matrices describes the variances and
covariances of each census of females. Goodman (1967) computes the
probability i n Pollard's model that the line of descendants of an individual of
any given age will eventually become extinct.
.
Projections of the Norwegian population as a multitype branching process
give estimates of uncertainty that Schweder (1971) considers unrealistically low.
Independently of Pollard (1966), Staroverov (1976) considers exactly the
same model. Because the model variances are implausibly small compared t o
the historical variation i n Soviet birth rates, Staroverov replaces the assumption
that each individual evolves independently with the assumption that groups of
c individuals evolve as units, independently of other groups. As c Increases,
the variance of numbers in each age group increases while the means remain
unaltered. A comparison of observed and projected births from 1960 to 1973
suggests that even c = 100,000 i s too small, and that it is necessary to allow
for temporal variation in the fertility and mortality parameters.
With a different interpretation of L(t) i n (1) from the usual Leslie
matrix, Goodman (1968) describes two sexes; Breev and Staroverov (1977)
describe labor force migration; Wu and Botkin (1980) describe elephants.
Deistler and Feichtinger (1974) show that the multitype branching process
model may be viewed as a special case of a model of additive errors proposed
for population dynamics by Sykes (1969).
Mode (1976) develops population projection models using renewal theory
rather than matrix methods. The continuous-time stochastic theory analogous to
what has just been described is presented by Keiding and Hoem (1976) and
Braun (1978), with extensions to parity-dependent birth rates and multiregional
populations. Cohen (1984) discusses the merits of branching processes as models
of human and nonhuman populations.
Population RojecHon with Environmental Variation
In a large population, the effects of demographic variation are normally
negligible compared to those of apparent changes i n vital rates. Sykes (1969)
supposes that, given L(t+l) , y(t) determines y(t+l) exactly, but that there is
no correlation between L(t) and L(s), s # t Sykes computes the means and
covariances of the age censuses, allowing the means and covariances of the
sequence {(L(t)l to be inhomogeneous i n time. Seneta (1972) pursues the
computation of variances i n the models of Sykes (1969) and Pollard (1966). Lee
(1974) discusses the numerical example that Sykes gives.
.
LeBras (1971, 1974) and Cohen (1976, 1977) develop demographic
applications of products of random matrices (Furstenberg and Kesten, 1960).
Under exactly stated conditions, Furstenberg and Kesten prove theorems that
imply that the census vector y(t) changes asymptotically exponentially and
,
that the elements of y(t) namely the numbers of females in each age class,
are, for large t asymptotically lognormal.
,
Cohen (1977) gives conditions under which the probability distribution of
age structure asymptotically becomes independent of initial conditions. This
weak stochastic ergodic theorem i s the probabilistic analog of the deterministic
weak ergodic theorem of Coole and Lopez (Pollard, 1973, pp. 51-55).
The theory of products of random matrices as models of environmental
variability i n age-structured populations is develo ed by, among others, Cohen
(1980). Lange (1979). Tuljapurkar and Orzack (1 980[ and Tuljapurkar (1984).
An elementary but important observation emerging from these studies is
a distinction between two measures of the long-run rate of growth of a
population i n a stochastic environment. One measure, studied by Furstenberg
and Kesten (1960), is the average of the long-run rates of growth along each
sample path, log A = lirnt+= t-l,E(log y(t)).Another
measure is the long-run rate
of growth of the average population, log p = l i m ,+,,
t-' log E(y 1
... (t)) .
For deterministic models, X = p, but i n general, i n stochastic models,
p with strict inequality i n most examples.
X
Products of random matrices provide a natural model of age-structured
populations of the game fish, striped bass (Cohen e t al., 1983), and can be used
to establish a long-run decline in populations of the striped bass Sam les i n the
1984). Heyde and Cohen 8984) use
Chesapeake Bay (Goodyear et 01.
martingale limit theorems to estimate confidence intervals for demographic
projections based on products of random matrices assuming only ergodicity and
stationarity i n the environment, and apply their methods to the striped bass.
When the models based on products of random matrices are applied to
human data, an updating of the series of Swedish population sizes analyzed by
Saboia (1974), the confidence intervals estimated according to the techniques of
unnecessarily pessimistic, compared
Heyde and Cohen (1984) are brooder, i.e.,
to confidence intervals estimated empirically (Cohen, submitted).
Population Rejection with Demographic and
Emlnmnental Variation
Demographic variation and environmental variation can be modeled
together. I f the probabilities of giving birth and of surviving i n a multitype
branching process are themselves random variables (Pollard, 1968; Bartholomew,
1975), the moments of the number of individuals in each age class can be
computed from a modification of a recurrence relation derived by Pollard
(1966).
For a multitype branching process such that the offspring probability
generating functions at all times are independently and identically distributed,
Weissner (1971) gives some necessary and some sufficient conditions for almost
sure extinction of the population (see also Namkoong, 1972; Athreya and Karlin,
1971). Weissner, Athreya and Karlin do not discuss the application of these
results to age-structured populations.
Time-Series
Models
The application of modern stochastic time-series methods t o demographic
d a t a originates with Lee (1970) and Pollard (1970). L e e (1978a, for summary)
uses long t i m e series, for example, of births and marriages o r of mortality and
wages, t o t e s t alternative historical theories of demographic and economic
dynamics. Pollard (1970) develop a second-order autoregressive model of t h e
growth r a t e of total population size for Australia.
Lee's (1974) analysis of births from 1917 t o 1972 in t h e United States
demonstrates t h a t t h e distinction between structural models and time-series
models is not sharp. Eq. (1) implies t h a t e a c h birth may b e attributed t o t h e
fertility of t h e survivors of some preceding birth cohort. Hence t h e sequence of
births I y (t)} is described by a renewal equation. By a sequence of
approxima\ions t o this renewal equation, Lee transforms t h e residuals of births
from their long-run trend into an autoregressive process for which variations in
the n e t reproduction r a t e a r e t h e error term.
Independently of Lee, Saboia (1974) develops autoregressive moving
average (ARMA) models using Box-Jenkins techniques for t h e t o t a l population
of Sweden. Based on d a t a from 1780 t o 1960 a t 5-year intervals, his projections
for 1965 compare favorably with some standard demographic projections.
Saboia (1977) relates ARMA models t o t h e renewal equation for forecasting births. In these models, t h e age-specific vital r a t e s can vary over
time; migration is recognized. Using female birth time-series for Norway,
1919-1975, he gives forecasts with confidence intervals up t o 2000. However,
Saboia's (1977) models a r e not t h e simplest required t o describe t h e d a t a
(McDonald, 1980).
McDonald (1979) describes t h e relationships among t h e renewal equation
model, with migration added, structural stochastic econometric models, and
ARMA models. H e suggests t h a t exogenous, perhaps economic, variables will
have t o b e invoked t o explain a sharp decline t h a t occurred in t h e number of
Australian births a f t e r 1971. Land (1 980) similarly suggests incorporating
exogenous variables in structural stochastic projection models with
environmental variation.
The forecasts of t h e time-series models have very wide confidence
intervals (e.g., McDonald, 1979; McNeil, 1974). In view of t h e uncertainty of
t h e demographic future, policy t h a t depends on population size and structure
should be flexible enough t o allow for different possible futures.
In addition t o spectral methods and Box-Jenkins techniques, other recent
approaches t o population time-series modelling include a stochastic version of
t h e logistic equation (McNeil, 1974) a s a model of United States Census total
population counts; the Karhunen-LoBve procedure (Basilevsky and Hum, 1979)
for quarterly records of births in t w o Jamaican parishes, 1880 t o 1938; and a n
age- and density-dependent structural model, estimated by use of t h e
Kalman-Bucy filter (Brillinger et al., 1980), for age-aggregated counts of t h e
sheep blow-fly.
ASSESSMENT AND PROSPECTS
Hajnal (1 957) raises profound doubts about t h e possibility of projecting
t h e future of populations (also bee Hoem, 1973). There is even a joke, not told
by demographers (Heaven forbid!), but by consumers of demographic projections.
Questlon: What is t h e difference between
mathematical demographer ?
a demographer
and
a
Answer: A demographer is somebody who guesses wrong about t h e future
of populations. A mathematical demographer is somebody who uses mathematics
and computers t o guess wrong about t h e future of populations.
Greater e f f o r t needs t o b e made t o evaluate quantitatively t h e m e r i t s
and demerits of population projection techniques and t h e underlying models on
which they rest, following t h e path of Henry and Gutierrez (1977), Ascher
(1978), Keyfitz (1 982), Stoto (1983), Stoto and Schrier (1982), and Smith (1984).
If t h e s e e f f o r t s succeed, demographers c a n replace t h e old joke with a new one.
Questlon: What is t h e
demographer and a good one ?
difference between
a
bad
mathematical
Answer: A bad mathematical demographer uses mathematics and
computers t o guess wrong about t h e f u t u r e of populations. A good mathematical
demographer uses mathematics and computers t o guess wrong about t h e future
of populations, but tells you reliably how f a r wrong you c a n expect him t o be.
SUMMARY
Mathematical demography has flowered recently in more a r e a s than can
b e reviewed here. This review deals only with stochastic models f o r population
projection developed since 1965.
Stochastic models a r e needed for population projection because
deterministic models fail t o account for t h e variability of historical
demographic -data. Deterministic models also fail t o provide probabilistically
meaningful e s t i m a t e s of t h e uncertainty of demographic predictions.
Among stochastic projection models,. one can distinguish, not always
sharply, between structural models and time-series models. Structural models
represent some supposed underlying mechanism of population growth.
Time-series models apply t o demographic d a t a general techniques in which t h e
form of t h e model need not b e based on demographic theory.
Structural. projection models usually describe e i t h e r o r both of two
sources of random fluctuation: demographic variation and environmental
variation. Demographic variation arises from t h e stochastic operation of
mechanisms with fixed vital rates. Environmental variation arises when t h e
demographic r a t e s themselves a r e governed by a stochastic process.
Deterministic projections of po ulations closed t o migration commonly
use t h e recurrence relation y ( t + l ) = LF+l)y(t), t = 0, 1, 2,
where y(t) is a
vector in which t h e ith component is t h e number of females in a g e class i
a t t i m e t , i = 1,
k; y(0) is a given Initial a g e census of t h e female
population; and L(t) is a k x k non-negative matrix, conventionally called a
Leslie matrix. To model demographic variation, this recurrence relation has
been interpreted in t e r m s of multitype branching processes.
...,
...,
However, the effects of demographic variation are normally negligible
compared t o those of apparent chon es in vital rates. To model environmental
variation, the projection matrix L(t7 may be considered t o be random. Under
reasonable conditions, the census vector
y(t) changes asymptotically
exponentially and the elements of y(t) namely, the numbers of females i n each
age class, are, for large t , asymptotically lognormal. Moreover, the
probability distribution of age structure asymptotically becomes independent of
initial conditions. This weak stochastic ergodic theorem is the probabilistic
analog of the deterministic weak ergodic theorem of Coale and Lopez.
Statistical methods for models with environmental variability exist.
,
The application of modern stochastic time-series methods to demographic
data began i n 1970. Some projections based on time-series methods compare
favorably with some standard demographic projections. The forecasts of the
time-series models have very wide confidence intervals. In view o f the
uncertainty of the demographic future, policy that depends on demographic
forecasts should be flexible enough t o allow for different possible futures.
Greater e f f o r t needs t o be made to evaluate quantitatively the merits
and demerits of population projection techniques and the underlying models on
which they rest.
RESUME
DEMOGRAPHIE MA THEMATIQUE: APPLICA TIONS RECENTES
DANS LE DOMAINE DES PR03ECTIONS DEMOGRAPHIQUES
La ddrnographle rnathdrnatlque s'est ddveloppde rckernrnent dans
tellernent de dornalnes qufll est lrnposslble de les passer en revue, seuls les
rnod6les stochastlques pour les profectlons ddrnographlques dlabordes depuls
1965 seront exarnlndes lcl.
Il est ndcessalre dfavolr recours aux rnod6les stochastlques pour
lf61aboratlon des profectlons ddrnographlques parce que les rnod6les
ddterrnlnlstes ne parvlennent pas 3 rendre cornpte de la varlabllltd des donndes
ddrnographlques dans le temps. Les rnod6les ddterrnlnlstes ne parvlennent pas
non plus 3 fournlr des estlrnatlons de lflncertltude des pr6dlctlons
ddrnographlques qul ont un sens du polnt de vue probablllste.
Parrnl ces rnod6les stochastlques, ll est posslble de dlstlnguer, quolque
pas toufours de rnanl6re nette, les rnod6les structurels des rnod6les basds sur
des s k l e s chronologlques. Les rnod6les structurels reprdsentent l'un ou lfautre
rndcanlsrne qul est supposd rdglr la crolssance ddrnographlque. Les rnod6les
bas& sur des sdrfes chronologlques appllquent aux donndes ddrnographlques des
techniques gdndrales qul n'exlgent pas que la forrne du rnod6le solt basde sur la
th6orle ddrnographlque.
Les rnod6les structurels de profectlon ddcrlvent gdndralernent slnon les
deux sources d e varlatlon a l b t o l r e du rnolns lfune dlentre elles, 2 savolr: la
varlatlon ddrnographlque e t la varlatlon de lfenvlronnernent. La varlatlon
dimographlque rQulte du jeu de l'actlon stochastlque des micanlsmes assoclis
d des taux dimographlques f l x b au prhlable. La varlatlon de l'envlronnement
lntervlent lorsque les taux dimographlques eux-mimes sont gouvernis par un
processus stochastlque.
Les projectlons ditermlnlstes de populatlons fermies utlllsent
communiment la relatlon de ricurrence y(t+l) = L(t+l)y(t), t = 0, 1, 2,
OG
y(t) est un vecteur sur lequel la l&re composante est le nombre de femmes
k; y(o) est la
appartenant d la classe dra^ge I au temps t , I = 1,
distrlbutlon par a^ge lnltlale de la populatlon fimlnlne obtenue lors dlun
recensement donni; e t L(t) est une matrlce kxk non-n&gatlve, appelie par
conventlon la matrlce de Leslle. Pour rendre la varlatlon dkmographlque sous
forme de modgles, ll faut lnterpriter la relatlon de ricurrence en termes de
processus d dklvatlons multlples.
...,
...,
Normalement, les e f f e t s de la varlatlon dimographlque restent cependant
nigllgeables comparatlvement 2 ceux prodults par les changements apparents
des taux dimographlques. Pour rendre compte de la varlatlon de
l'envlronnement sous forme de modgles, la matrlce de projection L(t) sera
consldirie comme aliatolre. Sous certalnes condltlons ralsonnables, le vecteur
du recensement y(t) change (de manlgre asymptotlque e t exponentlelle) e t les
iliments de y(t) , d savolr les nombres de femmes appartenant d chaque classe
dla^ges, sont asymptotlquement Iognormaux pour une valeur de t
.
De plus, la dlstrlbutlon probabillste de la structure par Sges devlent
asymptotlquement Indipendante des condltlons lnltlales. Ce thiorgme de falble
ergodlclti stochastlque est le correspondant probablllste du thgorgrne de falble
ergodlclti ditermlnlste. Enonci par Coale e t Lopez, ll exlste des mithodes
statlstlques qul s'appllquent aux modgles Incorporant la varlablllti de
l'envlronnement.
C'est en 1970 que l'appllcatlon aux donnies dimographlques, des
mithodes stochastlques modernes basies sur des s k l e s chronologlques a
commenci. Certalnes projectlons calculies selon des mkthodes basies sur des
sirles chronologlques se comparent avantageusement d certalnes projectlons
dimographlques courantes. Les prkvlslons de ces modgles basis sur des sirles
chronologlques ont de trgs grands lntervalles de conflance. Etant donnd
l'lncertltude de l'avenlr dimographlque, toute polltlque qul se fonde sur des
privlslons dimographlques devralt i t r e sufflsamment flexible pour s'adapter d
diffirents avenlrs posslble.
Une attention plus grande devralt i t r e accordie 2 l'ivaluatlon
quantltatlve du pour e t du contre des dlffirentes techniques dimographlques
alnsl que des modgles qul les sous-tendent.
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