Beyond Six Billion Forecasting the World's Population (2000) / Chapter Skim
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Appendix F: Estimating Expected Errors from Past Errors
Appendix F: Estimating Expected Errors from Past Errors
Pages 326-348
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From page 326... ...
First we work through the necessary equations, and then we present estimates of prediction intervals by region and for selected countries. The reader who is primarily interested in the results might focus on the latter 326
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Therefore, the estimated average growth rate during [m - 5, m)
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The error in population estimation may change over time both due to systematic factors (such as improved coverage of a census) and due to random factors (such as sampling variability, if survey techniques form part of the estimated population total; changes in population inclusion criteria; lost data; etc.)
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+ ((m - 1~/5. cam- 1~/5, Estimated Error in the Average Growth Rate Forecast For data analysis we need a formula for the estimated average growth rate, the forecast growth rate, and the estimated error of the forecast growth rate.
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This implies that, for data analysis, systematic linear changes in the errors of demographic data do not matter. Note, however, that both or and ~ will continue to influence future forecast accuracy, because they both influence the accuracy of the estimate of jump-off population V(k)
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parameter c.. such that C,~ij = cijc,~i, C,~j = cc, and C,~ij = cc, where the region-specific variance components are identified via the normalizing condition c,~i2 + cons + c,~,i2 = 1.
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c,2~,i+ c, 2~;/5. ~1 This estimate depends crucially on the estimated normalized variance components via Si.
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Instead we have used geographic proximity and, in borderline cases, the average past forecast error to define the regions. The world was divided into 10 regions: Western and Middle Africa; North, Eastern, and Southern Africa; the Middle East; South Asia and China; East Asia, excluding China; the Pacific Islands; Latin America and the Caribbean; Northern America and Australia; Western Europe; and Eastern Europe and the former Soviet Union (Table F-1~.
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Rep. Benin Burkina Faso C6te d'Ivoire Cape Verde Ghana Guinea Gambia, The Guinea-Bissau Liberia Mali Mauritania Niger Nigeria Senegal Sierra Leone Togo South Asia and China Algeria Botswana Burundi Djibouti Egypt Eritrea Ethiopia Kenya Lesotho Libya Madagascar Malawi Maldives Mauritius Morocco Mozambique Namibia Reunion Rwanda Somalia South Africa Sudan Swaziland .
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FYR Moldova Poland Romania Russia Slovakia Slovenia 'To · · 1 1 a~l~ |
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Prudence dictates that we not think of such events as being less frequent in the future than in the past. To arrive at a plausible specification of the correlations we must keep in mind that r meaTABLE F-2 Estimates of within-region correlations of errors Region Correlation Western and Middle Africa North, Eastern, and Southern Africa Middle East South Asia and China East Asia, excluding China Pacific Islands Latin America and Caribbean Northern America and Australia Western Europe Eastern Europe and former Soviet Union Mean 0.08 0.13 0.01 0.22 0.05 0.05 0.14 0.21 0.07 0.50 0.15
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PREDICTIVE DISTRIBUTIONS Prediction Intervals for Individual Countries via Bootstrap The probability model we have defined in earlier sections is a pure random-effects model with four variance parameters for each country j in region i: O~i2, ~,~2, CIti2, and cij2. In addition, there are correlation parameters Pi for within-region correlation.
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In this way, uncertainty concerning the relative shares of the three normalized variance components becomes a part of the estimated uncertainty of the forecast. The middle three columns of Table F-3 give the upper endpoints of the resulting bootstrap prediction intervals obtained for selected countries.
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339 ~ a, lo be o o V)
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The last column gives the multipliers that can be applied to the country-specific scales to get the average regional scale. Using the multipliers, the reader can easily calculate the widths of intervals corresponding to any composite estimator.
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Prediction Intervals for World Population We parametrize the models for future population size in terms of the scale parameter At= 1.0, 0.85, 0.70) , and the intraregional correlation parameter (p = 0.15, 0.375, 0.50~.
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For developing regions, the U.N. high/ medium ratios are clearly lower than the ratios derived from the 95-percent prediction intervals, even though for the world as a whole the U.N.
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high-low projections for the world are very close to the 95-percent prediction intervals that one can reasonably infer from our empirical analysis based on past errors. In Table F-6, we present quartiles of the predictive distribution of the world population in 2010, 2030, and 2050, under the assumption that the population in the year 2000 is known without error.
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Adjustments for fump-Off Error and Interregional Correlations We have derived predictive distributions for future population from past errors of U.N. forecasts with jump-off years in 1970-1990.
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The data provide little information on interregional correlations because the data period is short and the observations are highly correlated over time. One might infer that various factors influence correlations in opposite directions.
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A reader who believes that forecasts are more difficult now than in the recent past may assume that the relative error in a 50-year forecast could reach 20 percent. For total world population, the uncertainty estimates are more reliable than those given for individual countries, because of the cancellation of error in the estimates of the scales.
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Lee, R.D., and S Tuljapurkar 1994 Stochastic population forecasts for the United States: Beyond high, medium and low.
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