Abstract

This paper compares several methods of generating confidence intervals for forecasts of population size. Two rest on a demographic model for age-structured populations with stochastic fluctuations in vital rates. Two rest on empirical analyses of past forecasts of population sizes of Sweden at five-year intervals from 1780 to 1980 inclusive. Confidence intervals produced by the different methods vary substantially. The relative sizes differ in the various historical periods. The narrowest intervals offer a lower bound on uncertainty about the future. Procedures for estimating a range of confidence intervals are tentatively recommended. A major lesson is that finitely many observations of the past and incomplete theoretical understanding of the present and future can justify at best a range of confidence intervals for population projections. Uncertainty attaches not only to the point forecasts of future population, but also to the estimates of those forecasts’ uncertainty.

An erratum to this article is available at http://dx.doi.org/10.2307/2061297.

References

Alho, J. M., & Spencer, B. D. (
1985
).
Uncertain population forecasting
.
Journal of the American Statistical Association
,
80
,
306
314
. 10.2307/2287887
Ascher, W. (
1978
).
Forecasting: An Appraisal for Policy-Makers and Planners
.
Baltimore
:
Johns Hopkins University Press
.
Charlesworth, B. (
1980
).
Evolution in Age-Structured Populations
.
Cambridge
:
Cambridge University Press
.
Coale, A. J. (
1972
).
The Growth and Structure of Human Populations: A Mathematical Investigation
.
Princeton
:
Princeton University Press
.
Cohen, J. E. (
1976
).
Ergodicity of age structure in populations with Markovian vital rates, I: Countable states
.
Journal of the American Statistical Association
,
71
,
335
339
. 10.2307/2285308
Cohen, J. E. (
1976
).
Irreproducible results and the breeding of pigs; or nondegenerate limit random variables in biology
.
BioScience
,
26
,
391
394
. 10.2307/1297412
Cohen, J. E. (
1977
).
Ergodicity of age structure in populations with Markovian vital rates, II: General states
.
Advances in Applied Probability
,
9
,
18
37
. 10.2307/1425814
Cohen, J. E. (
1977
).
Ergodicity of age structure in populations with Markovian vital rates, III: Finite-state moments and growth rates; illustration
.
Advances in Applied Probability
,
9
,
462
475
. 10.2307/1426109
Cohen, J. E. (
1979
).
Long-run growth rates of discrete multiplicative processes in Markovian environments
.
Journal of Mathematical Analysis and Applications
,
69
,
243
251
. 10.1016/0022-247X(79)90191-4
Cohen, J. E. (
1979
).
Ergodic theorems of demography
.
Bulletin of the American Mathematical Society N.S.
,
1
,
275
295
. 10.1090/S0273-0979-1979-14594-4
Cohen, J. E. (
1979
).
Contractive inhomogeneous products of non-negative matrices
.
Mathematical Proceedings of the Cambridge Philosophical Society
,
86
,
351
364
. 10.1017/S0305004100056176
Cohen, J. E. (
1979
).
Comparative statics and stochastic dynamics of age-structured populations
.
Theoretical Population Biology
,
16
,
159
171
. 10.1016/0040-5809(79)90011-X
Cohen, J. E. (
1980
).
Convexity properties of products of random non-negative matrices
.
Proceedings of the National Academy of Sciences
,
77
,
3749
3752
. 10.1073/pnas.77.7.3749
Cohen, J. E. (
1982
).
Multiregional age-structured populations with changing rates: weak and stochastic ergodic theorems
. In Land, K. C., & Rogers, A. (Eds.),
Multidimensional Mathematical Demography
(pp.
477
503
).
New York
:
Academic Press
.
Cohen, J. E. 1983. How is the past related to the future? Pp. 59–71 in Center for Advanced Study of the Behavioral Sciences Annual Report 1982. Stanford, California.
Cohen, J. E. (
1985
).
Stochastic demography
. In Kotz, S., & Johnson, N. L. (Eds.),
Encyclopedia of Statistical Sciences
.
New York
:
John Wiley and Sons
.
Cohen, J. E. In press. An uncertainty principle in demography and the unisex issue. American Statistician.
Cohen, J. E., Christensen, S. W., & Goodyear, C. P. (
1983
).
A stochastic age-structured population model of striped bass (Morone saxatilis) in the Potomac River
.
Canadian Journal of Fisheries and Aquatic Sciences
,
40
,
2170
2183
. 10.1139/f83-251
Furstenberg, H., & Kesten, H. (
1960
).
Products of random matrices
.
Annals of Mathematical Statistics
,
31
,
457
469
. 10.1214/aoms/1177705909
Goodyear, C. P., Cohen, J. E., & Christensen, S. W. (
1985
).
Maryland striped bass: recruitment declining below replacement
.
Transactions of the American Fisheries Society
,
114
,
146
151
. 10.1577/1548-8659(1985)114<146:MSB>2.0.CO;2
Hajnal, J. (
1957
).
Mathematical models in demography
.
Cold Spring Harbor Symposia on Quantitative Biology
,
22
,
97
103
.
Hall, P., & Heyde, C. C. (
1980
).
Martingale Limit Theory and Its Applications
.
New York
:
Academic Press
.
Henry, L., & Gutierrez, H. (
1977
).
Qualité des prévisions démographiques à court terme. Étude de l’extrapolation de la population totale des départements et villes de France, 1821–1975
.
Population (Paris)
,
32
,
625
647
.
Heyde, C. C. (
1985
).
On inference for demographic projection of small populations
. In LeCam, L. M., & Olshen, R. (Eds.),
Proceedings ofthe Berkeley Conference in honor of Jerzy Neyman and J. Kiefer
(pp.
215
223
).
Monterey, Calif.
:
Wadsworth and Hayward: Institute of Mathematical Statistics
.
Heyde, C. C., & Cohen, J. E. (
1985
).
Confidence intervals for demographic projections based on products of random matrices
.
Theoretical Population Biology
,
27
,
120
153
. 10.1016/0040-5809(85)90007-3
Hofsten, E. (
1972
).
The Swedish Population 1750–1970
.
Stockholm
:
National Bureau of Statistics
.
Keyfitz, N. (
1968
).
An Introduction to the Mathematics of Population
.
Reading, Mass.
:
Addison Wesley
.
Keyfitz, N. (
1982
).
The limits of population forecasting. Chapter 13 in Keyfiz, N., 1982. Population Change and Social Policy
.
Cambridge, Mass.
:
Abt Books
.
Keyfitz, N., & Flieger, W. (
1968
).
World Population: An Analysis of Vital Data
.
Chicago
:
University of Chicago Press
.
Keyfitz, N. (
1971
).
Population: Facts and Methods of Demography
.
San Francisco
:
W. H. Freeman
.
Land, K. C. 1985. Methods for National Population Forecasts: A Critical Review. Population Research Center Paper 7.001, University of Texas at Austin.
Lange, K. (
1979
).
On Cohen’s stochastic generalization of the strong ergodic theorem of demography
.
Journal of Applied Probability
,
16
,
496
504
. 10.2307/3213079
Lange, K., & Hargrove, J. (
1980
).
Mean and variance of population size assuming Markovian vital rates
.
Mathematical Biosciences
,
52
,
289
301
. 10.1016/0025-5564(80)90073-5
Lange, K., & Holmes, W. (
1981
).
Stochastic stable population growth
.
Journal of Applied Probability
,
18
,
325
334
. 10.2307/3213280
Lee, R. D. (
1974
).
Forecasting births in post-transition populations: stochastic renewal with serially correlated fertility
.
Journal of the American Statistical Association
,
69
,
607
617
. 10.2307/2285990
Saboia, J. (
1974
).
Modeling and forecasting populations by time series: the Swedish case
.
Demography
,
11
,
483
492
. 10.2307/2060440
Siegel, J. S. (
1972
).
Development and accuracy of projections of population and households in the United States
.
Demography
,
9
,
51
68
. 10.2307/2060545
Slade, N. A., & Levenson, H. (
1982
).
Estimating population growth rates from stochastic Leslie matrices
.
Theoretical Population Biology
,
22
,
299
308
. 10.1016/0040-5809(82)90047-8
Smith, S. K. 1985. Population projections in the real world: tests of forecast accuracy. University of Florida, manuscript.
Stoto, M. A. (
1983
).
The accuracy of population projections
.
Journal of the American Statistical Association
,
78
,
13
20
. 10.2307/2287094
Stoto, M. A., & Schrier, A. P. (
1982
).
The Accuracy of State Population Projections. John F. Kennedy School of Government, Discussion Paper 117D
.
Cambridge, Mass.
:
Harvard University
.
Tuljapurkar, S. D. (
1982
).
Population dynamics in variable environments II. Correlated environments, sensitivity analysis and dynamics
.
Theoretical Population Biology
,
21
,
114
140
. 10.1016/0040-5809(82)90009-0
Tuljapurkar, S. D. Demographic applications of random matrix products. In J. E. Cohen, H. Kesten, and C. M. Newman (eds.), Random Matrices and Their Applications. Contemporary Mathematics no. 50. Providence, RI: American Mathematical Society. In press.
Tuljapurkar, S. D., & Orzack, S. H. (
1980
).
Population dynamics in variable environments. I. Long run growth rates and extinction
.
Theoretical Population Biology
,
18
,
314
342
. 10.1016/0040-5809(80)90057-X
Demographic Yearbook Historical Supplement. ST/ESA/STAT/SER.R/8
. (
1979
).
New York
:
United Nations
.
Demographic Yearbook 1981
. (
1983
). 33d ed.
New York
:
United Nations
.
Williams, W. H., & Goodman, M. L. (
1971
).
A simple method for the construction of empirical confidence limits for economic forecasts
.
Journal of the American Statistical Association
,
66
,
752
754
. 10.2307/2284223