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Population forecasts summarize existing knowledge about the change in the size and structure of populations. They support decision making by providing broad perspectives for future development. In other cases they may provide detailed estimates for speciﬁc planning tasks. The methods that are currently in use were mostly developed in the 1920s, 1930s, and 1940s, before modern computers were available. Although the forecasts provide us with important insight into the possible future development of the populations, a major deﬁciency in those methods is that they provide the forecast users with inadequate assessment of the uncertainty to be expected. The reasons for the unpredictability of demographic processes are only poorly understood. However, the development of new statistical models and the availability of fast computing allows a realistic description of forecast uncertainty in terms that are understandable to nonspecialist users.

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## 1. Development Of The Cohort-Component Method

### 1.1 Curve Fitting Approach

Early calculations concerning the renewal of populations were based on simple mathematical models. Deﬁne V(t) as the total population at time t. If the rate of growth is constant, or V (t)/V(t) = r, then the population grows exponentially, V(t) = V(0)e^{rt}, from the starting or jump-oﬀ population V(0). If the rate of growth decreases with increasing population according to V (t)/V(t) = r(M – V(t))/M where M is a limiting population size, then the growth is logistic, V(t) = Me^{rt} /(1 +e^{rt}). A discrete time version of the exponential model was used, for example, by T. Malthus in his Essay on the Principle of Population in 1798. The logistic model was introduced by F. Verhulst in 1838.

Such models were widely criticized because (a) they do not give age detail; (b) they do not allow a change from increase to decrease; (c) they are too mechanistic for human populations; (d) even though the exponential and logistic models are empirically indistinguishable when V(t) is small relative to M, they give completely diﬀerent long-term forecasts.

### 1.2 Incorporation Of Age Structure

Methods for describing age-structured populations were developed around the turn of the nineteenth century in many countries, ﬁrst in England by E. Cannan in 1895. By the end of the 1920s, forecasts by age and sex had also been made for the Soviet Union, The Netherlands, Sweden, Italy, Germany, France, and the United States (de Gans 1999).

Deﬁne V(x,t) as the size of the female population at age x at the beginning of year t. Then, for ages x > 0, we have the bookkeeping equation V(x +1, t +1) = V(x, t)p(x, t) + N(x + 1, t +1), where p(x, t) is the probability of surviving from age x to age x + 1 during year t, and N(x + 1, t +1) is the net number of migrants during year t who are at age x +1 at the beginning of year t + 1. For age x = 0 we have V(0, t + 1) = Σ_{x} f(x, t)V(x, t) + N(0, t + 1), where the summation is over the childbearing ages (often x = 15,…, 49), and f(x, t) is the expected number of children at age 0 at the beginning of year t + 1 that were borne by women at age x one year earlier. Corresponding formulas can be written for the male population, and they can be extended to comprise several regions. This system of calculation is called the cohort-component (c-c) method.

## 1.3 Advantages Of The Cohort-Component Method

The c-c method can be viewed as a response to the criticisms of Sect. 1.1 in the following sense. (a) Since the c-c method treats the population by age groups, it provides direct estimates of the future population by age and sex. That this is useful for the planning of housing, schools, hospitals etc., was understood from the beginning. (b) The c-c method makes no assumption about the growth rate. It shows that, other things being equal, the growth rate increases with the proportion of women of childbearing ages. In 1925 L. Dublin and A. Lotka introduced the concept of intrinsic rate of growth. For a closed population with unchanging fertility and mortality, it gives the asymptotic rate of growth. As such it provides important insight into the renewal of populations that cannot be detected from current growth rates. Many of the early cohort-component forecasts assumed unchanging fertility and mortality, and no migration, and showed that the size of the childbearing population, and with it the total population, would cease to grow, even though the populations had been growing fast during the early 1900s. This contradicted the forecasts by exponential and logistic models. (c) The reason why the c-c forecasts are less mechanistic than those based on exponential or logistic curves is that they utilize the available information or assumptions about the change of the vital rates (of fertility, mortality, and migration). The forecasters of the 1920s knew that fertility and mortality had been declining for decades. How far down would they go and how fast? Examples from other countries or from privileged social groups were used as leading indicators for future mortality. The spreading of the urban lifestyle was seen to lead to a decrease in fertility. Opinion surveys were developed around 1940 to collect data on desired family size (Whelpton et al. 1947). (d) The early forecasters knew that the exponential, logistic, and other curves often gave similar ﬁts to past data, but led to very diﬀerent forecasts. Since statistical model choice depended on judgment but had a drastic eﬀect on the results, why could one not use judgment in other ways as well? The solution by Whelpton to this problem was to set up targets for future vital rates, and to connect the most recently observed rates to the targets by suitable mathematical curves. Dutch forecasters had also used similar methods. This practice has been widely followed, and most national population forecasts use judgmental methods for specifying future vital rates, often via explicit targets (e.g., Andrews and Beekman 1987).

Although the c-c method was an improvement over the curve ﬁtting methods, it was soon understood that it is really a bookkeeping framework, and without an accurate speciﬁcation of the future change in the vital rates improvements in forecast accuracy would not materialize.

As an example we may consider a forecast of Sweden that was crafted by S. Wicksell in 1926. The forecast had 1920 as the jump-oﬀ year with a population of 5.90 million. It had ﬁve fertility variants. For 1940 the most likely forecast was 6.53 million inhabitants. The high forecast was 6.87 and the low forecast was 6.28. The actual population size was 6.37 so the error was 2.5 percent. For 1970 the most likely value was given as 6.67 and the high–low interval was [6.26, 8.44]. The true value was 8.08 so the error was –17.5 percent. Even though many of the errors were large, the high–low intervals were wide and they gave a realistic indication of the uncertainty of the forecast.

## 2. Foundational Aspects In The Forecasting Of The Vital Rates

Most national forecasts of the vital rates are purely demographic, i.e., they do not explicitly incorporate social or economic factors. Only rarely do they take into account the fact that the forecast itself might inﬂuence the future development. Partly because of these factors, and partly because of the poor accuracy of past forecasts, there has been a lengthy debate about whether the results of the calculations would be better interpreted as mere ‘projections’ rather than ‘forecasts.’

### 2.1 The Diﬃculty With Explanatory Models

Reﬁnement in demographic analysis often refers to the partitioning of the population at risk into more meaningful subgroups. For example, instead of relating births and deaths to the total population (which gives the crude birth and death rates), one may relate births to age of mother and consider deaths by age and sex (which gives the age-speciﬁc fertility and mortality rates). The use of crude rates in exponential or logistic models can imply growth forever, whereas the use of the age-speciﬁc rates in a c-c model may imply a future decrease of the same population.

The analysis can be further reﬁned. Age-speciﬁc rates may be analyzed for birth cohorts and by parity; marital and nonmarital fertility may be considered separately; or one may consider fertility by race, for example. Empirical evidence of whether this improves accuracy is mixed.

However, the more fundamental social or economic factors that determine the fertility, mortality, and migration of population subgroups are rarely used explicitly in forecasting. This is because future changes in such factors can be harder to forecast than the vital processes themselves. Keyﬁtz (1982) reviews many of the diﬃculties.

One attempt at solving the problem has been conditioning. By making detailed assumptions about the future course of relevant social and economic factors, forecasters have sought to formulate conditional forecasts, or scenarios, in which the evolution of the vital processes would follow assumed relationships with their determinants. A drawback of such an approach is that the manner in which the conditioning is done can have a major eﬀect on the results. A sensitivity analysis that systematically varies some of the assumptions may then be helpful.

### 2.2 Self-Fulﬁlling Or Self-Negating Prophecies

In some countries such as Germany and France the early population forecasts were motivated by the fear of population decline and loss of political power. In The Netherlands overpopulation was the concern. In both cases the forecasts were seen as a means of inﬂuencing social policies (de Gans 1999). Inasmuch as the forecasts were inﬂuential, they were self-negating.

In other cases such as regional forecasting, a forecast of population growth may cause the local society to develop the infrastructure of an area to allow for future growth. The favorable circumstances created may then turn the forecast into a self-fulﬁlling prophecy.

Although examples of both types of ‘prophecies’ exist, it has been diﬃcult to demonstrate the causality in a historical context. Deliberate government attempts at lowering (or increasing) fertility, or lowering mortality, have seldom had long-lasting eﬀects that could be attributed to the policies as opposed to other concurrent factors.

### 2.3 Debate On Terminology

From the very start forecasters have been aware that the future vital rates cannot be known with certainty. Evaluations of past forecasts have vividly demonstrated the unpredictability of demographic change. For example, the post-World War II baby boom was missed by all national forecasters. Mortality has similarly declined faster in the industrialized countries than the forecasters anticipated.

To safeguard their reputation some forecasters have occasionally tried to communicate the uncertainty by using the terms ‘projection,’ ‘illustrative projection,’ or even ‘horoscope’ for the results of their calculations. This is a reasonable practice, e.g., if the calculation consists of a c-c forecast with constant vital rates, when no one expects the rates to remain constant. However, as pointed out by H. Dorn in 1950, the producers of oﬃcial forecasts typically publish calculations that are based on what they consider as the most likely future vital rates, and users also interpret the results in this way. Indeed, we may deﬁne a calculation based on what is regarded as the most likely assumptions as a forecast.

## 3. Statistical Uncertainty In Forecasting

Although probabilistic terminology is used in many branches of demography (e.g., probability of death in life tables), and experience has shown that forecasts of the future population have been highly uncertain, the forecasting models considered so far have all been deterministic. The recognition of the uncertainties in probabilistic terms opens up new possibilities to enhance the usability of the forecasts.

### 3.1 Uncertainty In The Events And In The Rates

Consider a child-bearing population of size V. Suppose the number of births during a year is B. We can think of B as a sum of V random experiments with an average probability f of a birth. The expected number of births is E = fV. Based on a Poisson approximation one can estimate that in a sequence of B value s the relative variability (coeﬃcient of variation) is E^{-1/2}. In a small regional population we may have E = 100, for example. Then, the births vary by about 10 percent each year due to random variation alone. However, if E = 10,000, they only vary by about 1 percent, and if E = 1,000,000 they vary by 0.1 percent each year. It was pointed out by J. Pollard in 1973 that much larger relative variations are observed in the numbers of births in diﬀerent countries, so this type of Poisson variation does not account for the variability of the observed birth processes.

The situation changes if we allow f to change from year to year with a coeﬃcient of variation c (or by 100c percent). Then the relative variation is at least c irrespective of the expectation E. The conclusion is that for forecasting large populations, the uncertainty lies in the randomness of the rates rather than in the Poisson type variation in the events themselves.

### 3.2 Logical Flaws In The High And Low Variants

Over the years it became customary to produce high and low variants, in addition to the most likely medium variant, to give the forecast users an indication of the plausible range of the future developments. Users have typically ignored the high and low variants. One reason has been the added complexity of having more than one number in a planning context. More fundamentally, an additional reason may have been that the high and low variants have not had a logically coherent interpretation. There are two reasons for this. First, suppose [B_{L}, B_{H}] and [D_{L}, D_{H}] are high–low intervals for births B and deaths D, respectively. In general, there is no guarantee that the two intervals cover an equal range, because no probability content for the intervals has traditionally been given. Any calculation based on intervals with unequal ranges is diﬃcult to interpret. Second, suppose that the intervals are given a probabilistic interpretation, so that the probability is, e.g., 80 percent that each one of them contains the true value. This was ﬁrst suggested by L. Tornqvist in 1949. Suppose there is no migration. Then the increase of the population is I = B–D. On can show that the usual high–low interval [B_{L}–D_{H}, B_{H}–D_{L}] for I has a higher probability than 80 percent to cover the increase. Only if the number of births and deaths have correlation –1 does the high–low interval have probability content 80 percent. But the assumption of perfect negative correlation is known empirically to be false.

In a traditional c-c forecast a very large number of vital rates are given high–low variants, and these are combined in complex ways to produce high–low intervals for population sizes at diﬀerent ages. No uniform interpretation can be given to the latter intervals. For example, a United Nations forecast of the total world population published in 1998 gives a high–low interval of [7.4, 8.8] billions for the year 2030 and [7.3, 10.7] for 2050. Since it is not known what probability contents these intervals are intended to have, it is not clear how they should be interpreted.

### 3.3 Expressing Uncertainty Probabilistically

Modern probability theory provides a language that can be used to describe the uncertainty of the future vital rates and vital events. This leads to a predictive distribution of the future population given what we have observed in the past.

The availability of fast computing allows the use of stochastic simulation to describe numerically the predictive distribution. In practice this is done so that a set of age-speciﬁc vital rates is sampled from their respective predictive distributions. A c-c calculation is then made based on the sampled values to produce a possible sample path of the population in each age and sex group. This is repeated thousands of times and the results are stored. For any age–sex group, or for any larger population aggregate, one can represent the uncertainty of the forecast in terms of graphics (histograms, boxplots, etc.), or in terms of statistical summaries (means, medians, standard deviations, etc.). Such techniques are available for single region populations, and they are being developed for multiregional (or multistate) populations, household forecasts, and other functional forecasts.

The key element in such calculations is the formulation of the predictive distributions of the vital rates based on statistical analysis of past data. This can be based on formal time series methods, the analysis of the accuracy of the past forecasts, and judgment.

Predictive distributions provide the forecast users with a richer view of the possible future developments than the traditional population forecasts. For users needing a point forecast only, one would typically oﬀer the median of the predictive distribution. At the other end, a complete predictive distribution is available to a sophisticated user who wants to use formal loss functions in risk analyses. Stochastic forecasts are well suited to the analysis of the ﬁnancial stability of pension funds and other social security systems, for example.

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