Maarten Alders and Joop de Beer
2 Methods for Stochastic Forecasting
Population forecasts are calculated by means of the cohort component model. The input of the model is based on assumptions about future changes in fertility, mortality, and migration. In the Dutch population forecasts, assumptions on fertility refer to age-specific rates distinguished by parity, mortality assumptions refer to age- and sex-specific mortality rates, assumptions about immigration refer to absolute numbers distinguished by age, sex and country of birth, and assumptions on emigration are based on a distinction of emigration rates by age, sex and country of birth.
Forecast intervals can be derived from simulations. On the basis of an assessment of the probability of the bandwidth of future values of fertility, mortality, and migration, the probability distribution of the future population size and age structure can be calculated by means of Monte Carlo simulations.
For each year in the forecast period, values of the total fertility rate, life expectancy at birth of men and women, numbers of immigrants and emigration rates are drawn from the probability distributions.
Subsequently, age-sex-specific fertility, mortality and emigration rates, and immigration numbers are specified. Each draw results in a population by age and sex at the end of each year. Thus the simulations provide a distribution of the population by age and sex in each forecast year.
To make the simulations, several assumptions have to be made. First, the type of probability distribution has to be specified. Subsequently, assumptions about the parameter values have to be made. The assumption about the mean or median value can be derived from the medium variant. Next, assumptions about the value of the standard deviation have to be assessed. In the case of asymmetric probability distributions, additional parameters have to be specified. Finally, assumptions about the covariances between the forecast errors across age, between the forecast years, and between the components have to be specified (see e.g. Lee, 1996).
The main assumptions underlying the probability distribution of the future population relate to the variance of the distributions of future fertility, mortality, and migration. The values of the variance can be assessed by using one of the following three methods:
a. an analysis of errors of past forecasts b. model-based estimates of forecast errors c. expert knowledge or judgement.
These methods do not exclude each other; rather they may complement each other. For example, even if the estimate of the variance is based on past errors or on a time-series model, judgement plays an important role. In publications, however, the role of judgement is not always made explicit.
2.1 An Analysis of Errors of Past Forecasts
The probability of a forecast interval can be assessed on the basis of an analysis of the errors of forecasts published in the past. On the assumption that the errors are approximately normally distributed-or can be modelled by some other distribution-and that the future distribution of the
Assumptions on Fertility in Stochastic Population Forecasts 67 errors is the same as the past distribution, these errors can be used to calculate the probability of forecast intervals of new forecasts. Keil man ( 1990) examines the errors of forecasts of fertility, mortality, and migration of Dutch population forecasts published between 1950 and 1980. He finds considerable differences between the errors of the three components. For example, errors in life expectancy grow considerably more slowly than errors in the total fertility rate. Furthermore, he examines to what extent errors vary between periods and whether errors of recent forecasts are smaller than those of older forecasts, taking into account the effect of differences in the length of the forecast period.
In order to assess to what extent errors of past forecasts provide useful information on the degree of uncertainty of new forecasts it has to be examined whether new forecasts are better than older ones. This comparison of errors of forecasts made in different periods requires that we know the reasons why the forecaster chose a specific method for a certain forecast period. This information enables us to conclude whether a certain forecast was accurate, because the forecaster chose the right method for the right period, or whether the forecaster was just more lucky in one period than in another. Thus simply comparing the forecast errors of successive forecasts does not tell us whether recent forecasts are better than preceding forecasts.
Another question related to the analysis of errors of past forecasts is how likely it is that similar developments will occur again. The 1965-based forecasts in the Netherlands were rather poor because forecasters did not anticipate the sharp decline in fertility between 1965 and 197 5. If it is assumed that it is very unlikely that such developments will occur again, one may conclude that errors of new forecasts are likely to be smaller than those of the 1965-based forecasts. For that reason, the degree of uncertainty of new forecasts can be based on errors of forecasts that were made after 1965.
The decision as to which past forecasts are to be included is a matter of judgement. Obviously, one may argue that an 'objective' method would be to include all forecasts made in the past. However, this implies that the results depend on the number of forecasts that were made in different periods. Since more forecasts were made after 1985 than in earlier periods, the errors of more recent forecasts weigh more heavily in calculating the average size of errors. On the other hand, for long-term forecasts, one major problem in using errors of past forecasts for assessing the degree of uncertainty of new forecasts is that the sample of past forecasts tends to be biased towards the older ones. As for recent forecasts the accuracy cannot yet be checked except for the short term (Lutz, Goldstein & Prinz, 1996). Forecast errors for the very long term result from forecasts made a long time ago.
One way of assessing forecast errors in the long term is to extrapolate forecast errors by means of a time-series model (De Beer, 1997). The size of forecast errors for the long term can be projected on the basis of forecast errors of recent forecasts for the short and medium term. Thus, estimates of ex ante forecast errors can be based on an extrapolation of ex post errors. This method combines the analysis of past errors with the use of model-based estimates of errors.
2.2 Model-based Estimates of Forecast Errors
Instead of assuming that future forecast errors will be similar to errors of past forecasts, one may attempt to estimate the size of future forecast errors on the basis of the assumptions underlying the methods used in making new forecasts. If the forecasts are based on an extrapolation of observed trends, ex ante forecast uncertainty can be assessed on the basis of the time-series model used for producing the extrapolations. If the forecasts are based on a stochastic time-series model, the model produces not only the point forecast, but also the probability distribution. For example, ARIMA (Autoregressive Integrated Moving Average)-models are stochastic univariate time-series models that can be used for calculating the probability distribution of a forecast (Box & Jenkins, 1970).
Alternatively, a structural time-series model can be used for this purpose (Harvey, 1989). The latter model is based on a Bayesian approach: the probability distribution may change as new observations
68 M. ALDERS & J. DE BEER
become available. The Kalman filter is used for updating the estimates of the parameters.
One problem in using stochastic models for assessing the probability of a forecast is that the proba-bility depends on the assumption that the model is correct. Obviously, the validity of this assumption is uncertain, particularly in the long run. If the point forecast of the time-series model does not correspond with the medium variant, the forecaster apparently does not regard the time-series model as correct. Moreover, time-series forecasting models were developed for short term horizons, and they are not generally suitable for long term forecasts (Lee, 1996). Usually, stochastic time-series models are identified on the basis of autocorrelations for short time intervals only. Alternatively, the form of the time-series model can be based on a judgement to constrain the long-run behaviour of the point forecasts such that they are in line with the medium variant of the official forecast (Tuljapurkar, 1996). However, one should be careful in using such a model for calculating the variance of ex ante forecast errors, because of the uncertainty of the validity of the constraint imposed on the model. In assessing the degree of uncertainty of the projections of the model one should take into account the uncertainty of the constraint, which is based on judgement.
Rather than identifying an appropriate time-series model and analytically deriving forecast in-tervals from the selected model, empirical forecast errors can be assessed by means of calculating the forecast errors of simple baseline projections. Alho (1998) notes that the point forecasts of the official Finnish population forecasts are similar to projections of simple baseline projections, such as assuming a constant rate of change of age-specific mortality rates. If these baseline projections are applied to past observations, forecast errors can be calculated. The relationship between these forecast errors and the length of the forecast period can be used to assess forecast intervals for new forecasts.
2.3 Expert Knowledge or Judgement
In assessing the probability of forecast intervals on the basis of either an analysis of errors of past forecasts or an estimate of the size of model-based errors, it is assumed that the future will be like the past. Instead, the probability of forecasts can be assessed on the basis of experts' opinions about the possibility of changes in trends. For example, fertility forecasts are usually based on the assumption that trends like the increase of female labour force participation will continue. Even though a reversal of this trend may not be assumed to occur in the most likely variant, an assessment of the probability of such an event is needed to determine the uncertainty of the forecast. More generally, an assessment of ex ante uncertainty requires assumptions about the probability that the future will be different from the past. If a forecast is based on an extrapolation of past trends, the assessment of the probability of structural changes which may cause a reversal of trends cannot be derived directly from an analysis of historical data and therefore requires the judgement of the forecaster. Lutz, Goldstein & Prinz ( 1996) argue that subjective distributions are to be preferred to a time-series approach, because "structural changes and unexpected events are likely to happen". Lutz, Sanderson, Scherbov & Goujon ( 1996) assess the probability of forecasts on the basis of opinions of a group of experts. The experts are asked to indicate the upper and lower boundaries of 90 percent forecast intervals for the total fertility rate, life expectancy, and net migration up to the year 2030.
Subjective probability distributions of a number of experts are combined in order to diminish the danger of individual bias.
In the Dutch population forecasts, an assessment of the degree of uncertainty of fertility forecasts is primarily based on expert judgement, taking into account errors of past forecasts and model-based estimates of the forecast errors.
Assumptions on Fertility in Stochastic Population Forecasts 69 3 Assumptions on Fertility
In the Dutch population forecasts the assumptions about future changes in fertility are based on a distinction by parity. Accordingly, the assumptions about the standard deviation of forecast errors are based on a distinction by parity. The assumptions relate to the upper and lower limits of a 95%
interval. This implies that it is assumed that it is very unlikely that a forecast that is higher than the upper limit or lower than the lower limits will come true.
The assumptions underlying the medium variant refer to cohort fertility. In the long run, age-specific fertility rates are assumed to remain constant. As a consequence, the total fertility rate in 2050 equals the average number of children for women born around 2020. The 2000 Dutch population forecast assumes a level of 1.75 children per woman for the TFR in 2050 (De Jong, 2001).
3.1 An Analysis of Errors of Past Forecasts of Fertility
The first population forecast that was published by Statistics Netherlands dates from 1950. In total 23 forecasts of the TFR are published. Between the mid-eighties and mid-nineties the population forecasts were revised every year. Since 1996 the forecasts for the long run are revised every other year, and for the short run (5 years ahead) in the intermediate years. Figure 1 shows some of the historic forecasts of the TFR. As is clear from figure 1, the 1965-based forecast was rather poor since it did not anticipate the sharp decline in fertility during the late sixties and early seventies.
3,5
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1950 1960 1970 1990 1990 2000
Figure 1. Historic forecasts and observations of the TFR in the Netherlands.
To assess the accuracy of the (medium variants of the) historic forecasts, errors are calculated by subtracting the observed values from the forecasted values. An empirical 67% interval was then obtained specifying the lower and upper boundaries such that one sixth of the errors is lower than the lower boundary and one sixth is higher than the upper boundary. Similarly, a 95% interval is calculated. Figure 2 shows the duration specific intervals, together with the mean of the errors. It
70 M. ALDERS & J. DE BEER
should be noted that the number of forecasts on which the intervals are based, decreases with the forecast lead time. The intervals for the duration of 3 years are based on 20 forecasts, and those for the duration of 20 years on only 4 forecasts.
What strikes one's attention from figure 2 is that the intervals are rather asymmetric around zero.
The mean is significantly positive. This is to a large extent due to the 1965-based forecast. If the same analysis is applied to all forecasts except the 1965-based forecast, the mean is close to zero (figure 3). Moreover, the 95% interval is much smaller and more symmetric. As was stated in section 2.1 the question is to what extent this analysis provides useful information about the degree of uncertainty of new forecasts. If it is assumed that it is unlikely that developments that took place in the sixties will occur again, one may assume that errors of new forecasts will be smaller than those of the 1965-based forecast.
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Forecast duration in years
Figure 2. Mean and 67% and 95% interval of errors of past forecasts of the TFR.
The question as to which historic forecast to use can only be based on judgement. Figure 4 compares the analysis in which all forecasts are used with analyses in which the 1965-based forecast and the forecasts made before 197 5 are excluded respectively. The duration is restricted to 10 years.
For this duration at least 14 historic forecasts are available if all forecasts are included.
Two conclusions can be derived from figure 4. First, the differences between the 67% interval and the 95% interval indicate that the errors are not normally distributed. Second, the way in which the errors increase over the forecast lead time is somewhat different than in the latest forecast where it was assumed that the errors increase proportional to the square root of the forecast lead time (like in a random walk model).
One problem in using forecast errors from historic forecasts for assessing intervals of long-term forecasts is that there are no forecast errors for a period of 50 years. Therefore the standard deviation of forecast errors for the long run is assessed on the basis of a projection of forecast errors for shorter time horizons. A random walk model is used for this purpose.
Table 1 shows the 95% interval for the TFR in 2050. If the model is applied to all forecasts since
Assumptions on Fertility in Stochastic Population Forecasts
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Figure 3. Mean and 67% and 95% interval of errors of past forecasts of the TFR, excluding 1965-basedforecast.
Table 1
Random walk model of forecasts errors of the TFR.
95% interval of TFR in 2050 Lower bound Upper bound
All forecasts since 1950 0.89 2.61
All forecasts excL 1965 Forecast 0.97 2.53
All forecasts since 1975 1.23 2.27
2000 Population forecast 1.20 2.30
71
1950 the 95% interval ranges from 0.9 to 2.6 children per woman. If only forecasts made since 1975 are included, the interval is again much smaller: 1.2-2.3 children per woman. The interval assumed in the latest Dutch forecast equals that of the forecasts made since 1975. The decision as to which forecasts one is to include cannot be made on purely statistical grounds. Judgement plays a role.
This choice is based on an analysis of fertility rates by parity. The large forecast errors in the 1960s and 1970s occurred when fertility was at a considerably higher level than at present. Fertility rates for third and fourth births were much larger. Similar forecast errors in the same direction seem very unlikely in the future. Fertility rates of parity 3 and higher can hardly decline to the same extent as in the past, since they are already rather low. This would imply that fertility rates of parity 1 and 2 would have to decline much more than they did in the past. It would imply, for example, that a vast majority of women would remain childless. The Netherlands Fertility and Family Surveys (FFS), however, indicate that only a small minority of women from the younger generations wish to remain childless. Thus, very low fertility levels would imply that a large number of women could not realise their expectations. Similar large errors in an upward direction do not seem likely either, because of the changed position of women. The strong increase in the female labour force participation rates does not make it very likely that the percentage of women having three or four children will increase
72 M. ALDERS & J. DE BEER
67% interval
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Forecast duration in years
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Figure 4. Width of 67% and 95% interval of errors of past forecasts of the TFR.
strongly. For this reason it is assumed that the interval of new forecasts is smaller than that based on the errors of the forecasts made before 197 5.
It should be noted that the 95% interval in 2050 as modelled by the random walk model of the errors of all forecasts (0.89-2.61 children per woman) is small compared to the corresponding empirical 95% interval as shown in figure 4. Since only a few historic forecasts are available, the latter interval is determined to a large extent by outliers (e.g. the 1965-based forecast). Although such outliers cannot be excluded, the probability that such errors will occur again is very small as was explained before.
3.2 Model-based Estimates of Forecast Errors of Fertility
The assessment of the standard deviation of the TFR can also be based on time-series models.
The assessment of the standard deviation of the TFR can also be based on time-series models.