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Stochastic methods in population forecasting
International North-Holland
Journal
of Forecasting
521
6 (1990) 521-530
Stochastic methods in population forecasting Juha M. Alho Institute for Environmental Urbana, IL 61801, USA
Studies and Department
of Statistics,
University of Illinois at Urbana-Chainpaign,
Abstract: This paper presents a stochastic version of the demographic cohort-component method of forecasting future population. In this model the sizes of future age-sex groups are non-linear functions of random future vital rates. An approximation to their joint distribution can be obtained using linear approximations or simulation. A stochastic formulation points to the need for new empirical work on both the autocorrelations and the cross-correlations of the vital rates. Problems of forecasting declining mortality and fluctuating fertility are contrasted. A volatility measure for fertility is presented. The model can be used to calculate approximate prediction intervals for births using data from deterministic cohort-component forecasts. The paper compares the use of expert opinion in mortality forecasting with simple extrapolation techniques to see how useful each approach has been in the past. Data from the United States suggest that expert opinion may have caused systematic bias in the forecasts. Keywords: Cohort-component,
Expert opinion, Fertility, Mortality, Volatility.
1. Introduction The cohort-component method of population forecasting typically projects future numbers of annual births, deaths, and migration by one- or five-year age-sex groups, adds them to form a new population vector, and repeats the calculations for each forecast year (Shryock and Siegel, 1976, pp. 443-444). This procedure is usually attributed to P.K. Whelpton and collaborators, who developed it in a sequence of papers beginning in 1928 (Whelpton, 1928; Thompson and Whelpton, 1933, Ch. X; Whelpton, 1936; Whelpton et al., 1947). The equivalent arithmetic had earlier been used by, for example, Cannan (1895) and Bowley (1924). This procedure is not a detailed method of forecasting; rather it provides a general bookkeeping framework within which each age-specific vital rate can be projected as desired. To date, essentially all national population forecasts in countries with the requisite age-sex specific data have been made using the cohort-component method. Unfortunately, the accuracy of population forecasts did not substantially improve due to these 0169-2070/91/$03.50
0 1991 - Elsevier Science Publishers
efforts. For example, Whelpton’s 1947 high forecast for the total fertility rate for 1960 was 2.3 children per woman (Whelpton, 1947, p. 32), whereas the actual rate was 3.5. Changes pf vital rates have been difficult to predict ever since. One reaction to the uncertainty has been to call forecasts “projections”, or “illustrative projections”, as if changing the term would remedy the situation. I propose to accept the observed level of uncertainty for the time being, try to estimate its magnitude, and to analyze its sources. From a statistical point of view, the best forecast is the one with the (in some sense) smallest error. However, there is no substantive reason why the vital processes should be easy to predict, so the smallest possible error may be much larger than the forecasters, the direct users of the forecasts, or the public like or expect (cf. Alho, 1984). One solution to the problem of error estimation is to formulate a stochastic version of the cohortcomponent method, which treats the vital rates as realizations of random processes. This yields, as a by-product, high-low intervals that have a given probability of covering the true size of an age-sex
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group in a given future year (Sykes, 1969; Lee, 1974; McDonald, 1981; Stoto, 1983; Alho and Spencer, 1985; Cohen, 1986). Historically, the first attempt at a stochastic formulation that I am aware of was made by L. Tbrnqvist in Finland in 1949 (HyppGla et al., 1949, pp. 68-74). This early work contained many of the central features of a stochastic forecast. Lack of adequate computing facilities may explain why Tbrnqvist’s lead was not followed in official forecasts until the 1980s (U.S. Bureau of the Census, 1984). In Section 2, I briefly describe a stochastic version of the cohort-component method. In Section 3, I point out how expert opinion and data analysis affect the error of forecasts. In Section 4, I propose a volatility measure of fertility that can be used to calculate approximate prediction intervals for births, based on a deterministic cohortcomponent forecast. In Section 5, I study the bias of simple linear extrapolation techniques in forecasting mortality, and contrast it with the bias that occurs with the use of expert opinion.
2. Calculation
of a stochastic
forecast
Population growth can be described in terms of alineargrowthmodel.Let~(V(t)=(u,(t),...,u,(t))’ be a column vector containing population numbers at (exact) time t, and let an s x s matrix R(t) = (r,,(t)) contain the vital rates (arranged in a suitable way) during the year from t to t + 1. Then, V(t+
1) = R(t)V(t).
(1)
A simple example of this is the classical model of one sex of a closed population considered by Bernardelli (1941) Lewis (1942) and Leslie (1945) (R(t) is sometimes called a “Leslie-matrix”). In this case the only non-zero elements in the R(t) matrices occur on the first row (fertility rates) and on the first sub-diagonal (survival rates). Eq. (1) together with such rate definitions is equivalent to the cohort-component procedure of bookkeeping when there is no migration. As shown, for example, by Rogers (1975), a model of type (1) can be defined to cover both sexes, several geographical regions, or other divisions of the population, such
as type of economic activity. Analogously, the stochastic version can be generalized. Suppose we have observed R(t) for t = 1,. . . , n - 1, have an estimate of V(t) for t = n, and want to forecast V(t) for t=n+l,...,n+m. Suppose also that we have available forecasts of R(t) for t=n,...,n+m-1. Then, the cohortcomponent forecast of V( n + m ) can be written as P(n+m)=k(n+m-l)...k(n)P(n).
(2)
We typically do not know the base population (jump-off population) exactly, so the estimate of V(n) contains error (e.g., Shryock and Siegel, 1976, p. 422, or Spencer and Lee, 1980). To represent this uncertainty we can use a random jump-off population (cf. Alho and Spencer, 1985, p. 310). Interestingly, in very short term forecasts this source of error may dominate. Correlations across forecasts of different components (i.e., different r,,( t)‘s) are also important. We note that the procedure used by most producers of official population forecasts, which combines high fertility assumptions with high life expectancy and high netmigration assumptions, implicitly assumes that these correlations equal 1. This is absurd, since the primary causal factors behind changes in fertility, mortality, and migration are not identical. An assumption of zero correlation is surely much closer to the truth. An analogous problem arises when the autocorrelations of the vital rates are considered. The propagation of error, that is the distribution of the error of the forecast of V(n + m), when the jump-off population and the vital rates are lognormally distributed, is readily handled by approximations provided in Alho and Spencer (1985, p. 309, expressions (3.13)-(3.17)). Simulation (see, e.g., Pflaumer, 1988) could also be used.
3. Expert opinion
and data analysis
Whelpton objected to the sole reliance on mathematical curve-fitting techniques in population forecasting. Referring to the work of Elkanah Watson in about 1815 and Francis Bonynge in 1852, Whelpton noted that such techniques may sometimes be remarkably accurate. For instance, Bonynge’s forecast based on a variable geometric
J.M. Alho / Stochastic methods in population forecasting
growth assumption was only 2.2% off the correct value for the total U.S. population in 1900. The reason for rejecting these methods, and the presumably more sophisticated logistic technique popularized by Pearl and Reed (1920), was that several curves may fit the past data very well but give widely differing forecasts (e.g., Thompson and Whelpton, 1933, pp. 312-313; Whelpton et al., 1947, pp. 10-29). Instead, Whelpton suggested that one should study the past rates, form an opinion of the likely future rates at some target year, and obtain the intermediate values by interpolation. Essentially, the same procedure is still used in official forecasts (see, e.g., U.S. Office of the Actuary, 1987). We can represent Whelpton’s procedure within a statistical model by assuming that the data analysis of the vital rates is done within a parametric statistical model. Subjective input is allowed to influence both the choice of the model and its parameter estimation via Bayesian techniques or, e.g., mixed prediction in which target values are explicitly used (Alho and Spencer, 1985). The statistical procedure gives rise to four different sources of error: (1) model misspecification: the assumed parametric model is only approximately correct; (2) errors in parameter estimates: even if the assumed parametric model would be the correct one, its parameter estimates will be subject to error when only finite data series are available; (3) errors in expert judgement: an outside observer may disagree with our judgements or “prior” beliefs about the parameters of the model, or the weight we give to our beliefs in forecasting; (4) random variation, which would be left unexplained even if the parameters of the process could be specified without any error: since any mathematical model is only an approximation, one would expect there to be residual error. (See Hoem (1973) for further discussion, and an alternative classification.) One should note that the error sources depend conceptually on each other. For example, the presence of (4) gives rise to (2), and errors in (3) may mask or accentuate errors in (l), or vice versa. The next two sections explore some aspects of fertility and mortality forecasting. The primary difference between these series is that fertility has fluctuated wildly during this century, whereas mortality has shown a nearly monotonic decline.
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4. Estimating the volatility of fertility Estimates of the size of prediction error always depend on the statistical model one uses to calculate the forecast (cf. Lee, 1974). They typically depend on the past fit of the model. When expert opinion is used in forecasting, it may also influence error estimates. For example, in the mixed prediction technique introduced in Alho and Spencer (1985) a target value was determined subjectively, and it directly influenced the widths of the prediction intervals. However, for the purpose of setting base-line estimates of the uncertainty of the forecasts, it would be useful to have error estimates that are relatively independent of forecasting methods and subjective assumptions This is especially important if we have available a deterministic cohort-component forecast based on an unknown combination of subjective assumptions and data analysis, and we would like to determine prediction intervals for it. We typically believe that we are able to predict future fertility more accurately than if we merely use the present fertility as a forecast (cf. Keyfitz, 1982, p. 730). For this reason we propose to use the naive forecast of present value as the basis of error estimation. We would expect the errors of naive forecasts to be larger than those of any officially used method, so prediction intervals based on the naive forecasts should be correspondingly wider. In practice, it is first necessary to determine a suitable transformation of the fertility series, and study the change in the transformed fertility series separately for each lag time. More precisely, let w(t) be the transformed fertility rate of year [t, t + 1). Then the volatility of w(t) at lag k > 0 bused on “loss function” d is defined as E[d(w(t)-w(t-k))lw(t-s),
s
(3)
The loss function d is any non-negative function defined on the real line that is symmetric around zero, increases on the positive real line, and vanishes at zero only. For example, we might want to use d(x) = x2 or d(x) = (x I. If the volatility does not change over time, we can estimate it based on the oberved values at t = 1,. . , n by n-k c d(w(t) ,+=I
- w(t - k)),‘(n
- k).
(4)
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In practice, the volatilities would be expected to change over time, so I calculate volatility estimates based on subsets of the observed data, and study their change. My concept of volatility is loosely based on the concept of stock price volatility, which is the basis of the modern theory of stock option pricing developed by Black and Scholes (1973). In that theory one considers the logarithm of the stock price, and uses d(x) = x2. The Black-Scholes theory is based on the assumption that the logarithm of stock price is a Brownian motion. In that case the theoretical volatility at lag = k is k times the volatility at lag = 1. Once the volatility at lag = 1 has been estimated, there is no need to consider other lags separately. In a sense, our concept can be viewed as a nonparametric generalization of the concept used in stock option theory. To illustrate these concepts we shall study the volatility of the Finnish fertility in 1776-1976 (Turpeinen, 1978). We shall use d(x) = 1x 1. Instead of considering all the age-specific fertility rates separately, we concentrate on their average which is equal to the total fertility rate divided by the number of child bearing ages (which we take to be 30, corresponding to ages 15-44). Figure 1 has a plot of the total fertility rate (I). The highest average rate was 0.19710 and the lowest 0.05013. I shall use a logit transformation that forces the average fertility rate between a lower bound rL, and an upper bound ru. Based on the observed minimum and maximum, I shall take rL = 0.025
Fig. 1. Total fertility rate (I), and the summary volatility measure (II) in Finland in 1776-1976.
and ru = 0.25. In other words, letting r(t) denote the average fertility rate, I define w(t) via w(t)=log((r(t)-0.025)/(0.25-r(t))).
Using the whole data period (N = 201) to estimate the lag-specific volatilities, summary statistics of Table 1 are obtained. Figure 1 shows also a plot of the summary volatility measure (II), which I took to be the average of the smoothed lag-specific measures for lags l-15. The procedure RSMOOTH of Minitab, which combines running medians and moving averages, was used to carry out the smoothing. A comparison with the total fertility rate shows that sudden upward or downward spikes (e.g., years 1789, 1809, 1868) in the original series translate
Table 1 Summary statistics of the lag-specific volatilities of the average fertility rate in Finland in 1776-1976.
Lag
N
Mean
Median
SD
Min
MaX
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
200 199 198 197 196 195 194 193 192 191 190 189 188 187 186
0.148 0.170 0.200 0.229 0.237 0.257 0.265 0.285 0.293 0.308 0.318 0.341 0.346 0.363 0.372
0.100 0.141 0.151 0.178 0.184 0.207 0.211 0.241 0.231 0.247 0.250 0.266 0.280 0.273 0.286
0.144 0.157 0.175 0.194 0.205 0.208 0.229 0.224 0.246 0.288 0.262 0.278 0.280 0.286 0.303
0.001 0.003 0.004 0.003 0.002 0.008 0.002 0.005 0.001 0.001 0.000 0.001 0.001 0.003 0.001
0.855 0.876 0.936 1.147 1.069 1.085 1.097 0.998 1.085 1.135 1.126 1.165 1.191 1.206 1.215
525
J.M. Alho / Stochastic methods in population forecasting
into wider peaks in the volatility series. Periods of sustained change, such as the decline from 1900 to 1930, also cause high volatilities. More recently, the dramatic decline from around 1947 to 1973 has created a sizeable peak at the end of the time series. The high volatility observed during this century is in an interesting contrast to the often repeated belief that, after the demographic transition, population growth in industrialized countries should stabilize. We can use our volatility estimates to find out approximately how large are the percentage errors one would expect in birth forecasts using the naive forecast. As noted above, such error bounds ought to be conservative for any forecasting method that is superior to the naive one. Consider the linear growth model (1) for the female population. Let us approximate each age-specific fertility rate by their average; denote r,,.,(t) = r(t). Then the true births are given, approximately, by
where the summation is over ages 15-44. Suppose now we use a naive forecast w for w(r). Then, e(t) = w(t) - w is the prediction error, and we can write i-(t) = 0.025 + 0.225
exp(w + e(t)) + e(f))
1 + exp(w
.
When e(t) = 0, we get the point now that
forecast.
Suppose
w = log&( (0.06 - 0.025),‘0.225)
= - 1.69168
(corresponding to the total fertility rate of 1.8, a reasonable forecast in Finland at this time). The ratio of the number of true births to the point forecast is approximately 0.025+0.225exp(w+e(t))/(l+exp(w+e(t))) 0.025+0.225exp(w)/(l+exp(w))
Table 2 Upper and lower 67% and 95% prediction limits divided by the point forecast for births, based on the Finnish birth volatility data from 1776-1976. Forecast year
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Upper end point of interval
Lower end point of interval
95%
67%
67%
95%
1.272 1.306 1.339 1.371 1.400 1.427 1.451 1.472 1.489 1.505 1.524 1.545 1.569 1.593 1.615
1.076 1.096 1.114 1.130 1.142 1.155 1.168 1.181 1.193 1.206 1.217 1.226 1.238 1.256 1.279
0.931 0.916 0.902 0.890 0.881 0.873 0.864 0.856 0.848 0.840 0.833 0.828 0.822 0.812 0.799
0.803 0.786 0.770 0.756 0.743 0.734 0.723 0.716 0.710 0.704 0.698 0.691 0.683 0.676 0.669
For example, the upper end point of the 67% interval for the first forecast year is about 8% above the point forecast. The same analysis was repeated with U.S. white female fertility data from 1920 to 1978. One would expect the resulting error estimates for births to be of the same order of magnitude as in Finland, because both countries experienced a similar increase from-the depression years’ low fertility to a peak after the Second World War, and a subsequent decline. Figure 2 shows that this is, indeed, the case. The prediction intervals based on the I
”
(6)
The volatility estimates give us directly estimates of the distribution of 1e(t) 1 for each lag k = 1, 2,. . .,15. I smooth the empirical distributions first to reduce random error. Then I determine the location of, for instance, the 67th percentile. Replacing e(t) in eq. (6) by + the 67th percentile yields a (conservative) 67% prediction interval for the births. Table 2 gives the width of the resulting prediction intervals in terms of the point forecast.
Fig. 2. Ratio of upper end point of a 67% prediction interval and the point forecast for births based on Finnish volatility in 1776-1976 (I), Finnish volatility in 1920-1976 (II), U.S. volatility in 1920-1978 (III), and the mixed prediction forecast of U.S. births (IV).
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J.M. Alho / Stochastic methods in popdatton
1920-1976 Finnish volatility (II) are very nearly equal to those based on the 1920-1978 U.S. volatility (III). These intervals may be compared to the ones reported in Alho and Spencer (1985) for the U.S. births (IV) that were calibrated so they would have been adequate during the baby-boom years. Overall, the latter intervals are narrower. This is to be expected because the latter intervals were based on a better method of forecasting than the naive method. However, the fact that the volatility based intervals are initially narrower suggests that there may be room for improvement in the mixed prediction methodology. The intervals based on the Finnish volatility from 1776 to 1976 (I) are the narrowest of all. This agrees well with the increase in volatility shown in Fig. 1. It may well be that fertility forecasting has seldom been more difficult than during this century.
5. Uncertainty of mortality decline Figure 3 shows the time series of age-standardized mortality rates for males and females in the U.S. in the period 1900-1986, with the 1980 total population as standard (U.S. Office of the Actuary, 1987, p.7). The nearly steady decline is in contrast to the fluctuations of the fertility series in Fig. 1. Due to biological bounds of longevity, mortality decline cannot continue indefinitely. Therefore, a major source of uncertainty in mortality forecasts comes from the timing of the levelling off of mortality. In 1954-1968, male mortality rose 0.2% per year. At the end of the period it may have seemed that the levelling off had already occurred. The subsequent decline
Fig. 3. Age-standardized mortality rate in the United States in 1900-1986 for males (I) and females (II), per 100,000.
forecasting
shows how uncertain such conclusions may be. Depending on their magnitude, epidemics or war may also affect mortality trends. In view of Fig. 3, an analysis of uncertainty using the present value as a “naive forecast” for all future times does not make sense in mortality forecasting. A more meaningful baseline estimate is obtained by assuming that future changes in mortality are similar to the past changes. We may for example, use the first differences of a suitably transformed mortality series, estimate their mean, and model the deviations from the mean in terms of a simple ARIMA-process. In terms of the original time series, this means that we would use either an ARIMA(1, 1, 0), ARIMA(0, 1, 1). or ARIMA(l, 1, 1) model with a constant term to calculate a baseline forecast. The forecasts of the three models are in this case virtually indistinguishable, and the relatively short-data periods do not allow us to choose between them (or identify other, possibly better fitting, models) on statistical grounds alone. Consequently, I shall determine, a priori, which model to use, estimate its parameters, and calculate a baseline forecast using the estimated model. The crucial assumption in this procedure is the decision to use a constant term in the model. The largest standardized death rate in Fig. 3 is about 0.025 and the smallest is about 0.006. Possible upper and lower bounds for a logit transformation are ru = 0.03 and r,_ = 0.002. I used these values in a logit-transformation to get w(r)
= log((r(t)
- 0.002)/~0.03
-r(t))).
An ARIMA(0, 1, 1) model was used with a 30-year data period, and 15-year forecasts were calculated with the jump-off years 1930-1981 for both male and female mortality rates. Table 3 gives summary data for the prediction error of these forecasts for 5, lo-, and 1%year lead times. To put the figures of Table 3 into perspective, we may note that the 1986 age-standardized mortality rate for males was 1093 (per 100,000) and for females 645. These figures show that the simple ARIMA-extrapolation technique would have given close to unbiased results over this time period. Nearly identical predictions for male mortality are obtained if the same technique is applied to the untransformed mortality series. For females the forecasts based on untransformed rates
J.M. Alho / Stochastic methods in population forecasting
Table 3 Summary statistics of the prediction error of 5-, lo-, and 15-year ahead forecasts of male and female mortality (per 100,000) for the years 1936-1986 in the United States using an ARIMA(O,l,l) model with a constant term.
Males
Females
Lead time
Mean
Median
SD
Min
Max
5 10 15
- 12 -25 -57
-22 -60
6
14 119 141
-158 -262 -317
107 149 151
5 10 15
13 18 17
4 17 -8
62 103 136
-100 -159 -174
178 208 275
are slightly worse. Hence, in these series the effect of data transformation is small. Figure 4 contains a plot of the prediction errors for females. The appearance of male prediction errors is very similar, with somewhat lower average values as indicated by Table 3. For both sexes the forecasts for the beginning of the 1950s were too high, for the end of the 1960s they were too low, and for the early 1980s the male forecasts appear to have been too high. We note that the errors of simple ARIMA extrapolations can serve as a basis for the estimation of the volatility of mortality. Official forecasts typically use other sources of information besides the past mortality time series to forecast future mortality. Therefore, they should be more accurate than the simple trend forecasts. Suppose we are willing to believe that this is the case. Then, we can repeat the volatility estimation procedure described for fertility, using ARIMA extrapolations as the naive forecast, to get error bounds for
I
I
1
i’i
II
Fig. 4. Five-year (I), ten-year (II), and fifteen-year (III) prediction errors of simple ARIMA-forecasts of the U.S. age-standardized mortality rate for females for the years 1935-1986.
521
each age-specific mortality rate and to derive conservative error bounds for the number of deaths in each age-group. These error bounds would not necessarily equal the theoretical ARIMA bounds that assume the stochastic process has been correctly identified. Rather, they would reflect the empirical prediction error of the naive forecast, as in the case of fertility. Figures 1 and 2 of Olshansky (1988) compare the forecasts of life expectancy of the U.S. Office of the Actuary to the observed development, including jump-off years 1950, 1955, 1965, and 1977, for males and females. The female life expectancy was consistently underestimated. The first three forecasts for males were more accurate than for females. They overestimated the life expectancy in the late 1960s and early 1970s. All forecasts were below the actual life expectancy in the 1980s. These results parallel our results for standarized mortality (Fig. 4) based on ARIMA extrapolations, so the question becomes, which procedure would have been more accurate through the 1960s and 1980s. Using the assumptions of the official forecasts (U.S. Office of the Actuary, Actuarial Studies nos. 33, 46, 62, and 77) I recalculated their forecast for the age-standardized mortality rates of Fig. 3. Figure 5(a)-(d) show the 30-year observation period used in the ARIMA forecast with jump-off years 1950, 1955, 1965, and 1977; the subsequent development of mortality until the year 1986; the ARIMA forecast; and the high and low official forecasts for females (for the last forecast only one variant was used). Only the second official forecast appears to be clearly better than the ARIMA extrapolation. A similar figure for males shows that the first two official forecasts were clearly better than the ARIMA extrapolations. The reverse is true for the latter two. Overall, the official forecasts tended to overshoot the actual mortality, whereas the extrapolations have been usually below it. Ex post facto, we may note that a combination of the “objective” extrapolation technique with the “subjective” expert forecast would have been superior to either technique alone (cf. Winkler, 1984, p. 294). The rate of past mortality decline has, of course, been an important factor when the assumptions of the official forecasts have been made. The results of Fig. 5 indicate that putting somewhat less weight on the expert opinion would have resulted in
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J.M. Alho / Stochastic methods in populatlon forecastmg
YeCi
Fig. 5. Age-standardized
I.
U.S. female mortality rate, ARIMA forecast (I), and high and low official forecasts (a) 1950, (b) 1955, (c) 1965, (d) 1977 (only one official forecast variant).
better forecasts. In this sense, too heavy reliance on expert opinion seems to have biased the forecasts toward high mortality in the past two or three decades. When ARIMA forecasts are continued further from 1986, and compared to a recent U.S. Office of the Actuary (1987) forecast, the difference is that the official forecast of mortality is initially slightly lower, but becomes higher than the extrapolation forecast in about 15 years’ time. This is due to the assumed levelling-off of mortality decline. One is left wondering whether the use of expert judgement will bring better results this time than in the past.
6. Prospects The availability of meaningful error bounds can be of major interest to the users of forecasts, as they indicate the value of past data in decisionmaking. In view of the wide error bounds of birth and death forecasts, urban, educational, or finan-
r7
(II) with Jump-off
year
cial planning should not rely too heavily on current forecasts. The official forecasts may well be the most accurate available but because they are uncertain, it becomes important that one prepare for the possibility that they will be grossly in error. Stoto (1988) reaches a similar conclusion in discussing the role of statistical uncertainty in policy. A full implementation of a stochastic forecast is a considerable task requiring a detailed analysis of the uncertainties of the jump-off population, each of the vital rates, and their joint correlation structure. Much of the work relating to the expectations of these variables is done now under the headings of population estimation and estimation of the vital rates. The analysis of the second moments and the derivation of the full error distribution is a novel task that requires considerable statistical input, in addition to demographic expertise. There are several intermediate stages between a classical deterministic forecas! and a full stochastic forecast. Land (1986) reviews several possibili-
J.M. Alho / Stochastic methods in population forecastmg
ties. In this paper we proposed a measure of the volatility of vital rates. When applied to fertility, it can be used to derive conservative prediction intervals based on a deterministic point forecast of births. The credibility of these intervals hinges only on the fact that the actual forecasting method used is not less accurate than the use of today’s value of fertility for all future times. A similar analysis of mortality can be effected by using the past rate of decline rather than the current value as the naive forecast. All forecasting methods, deterministic and stochastic alike, involve in varying degrees the use of expert judgement. For example, past and current forecasts of mortality have all assumed that the rate of decline of mortality will slow down considerably within one or two decades after the jump-off year. A comparison with simple trend extrapolations shows that putting less weight on expert opinion would have made the official mortality forecasts more accurate in the United States during the past three decades. Over the course of the 20th century the simple extrapolations would have been unbiased, on the average, for forecast periods of 5-15 years. However, the errors of the extrapolations have been highly correlated. It is possible that such short term deviations from the overall trend can be explained by using epidemiologic, health survey, or other data. This suggests that a useful role for expert opinion may be in relatively short term forecasting, in addition to determining where and when the “ultimate” levelling-off of mortality takes place. Expert opinion could cause temporary delays or accelerations in the overall trend forecast. Perhaps the AIDS epidemic will necessitate such an adjustment.
Acknowledgement This research was partially supported by National Institute on Aging Grant AG06996. The author would like to thank Dr. Bruce D. Spencer for several comments on an earlier version of the manuscript, which was presented at the Eighth International Symposium on Forecasting, in Amsterdam, the Netherlands, in June 1988. Useful comments from three anonymous reviewers helped to clarify the presentation.
529
References Alho, J.M., 1984, “Probabilistic forecasts. The case of population projections”, Scandinavian Housing and Planning Research, 1, 99-105. Alho, J.M. and B. Spencer, 1985, “Uncertain population forecasting”, Journal of the American Statistical Association, 80, 306-314. Bemardelli, H., 1941, “Population waves”, Journal of Burma Research Society. 31, 1-18. Black, I. and M. Scholes, 1973, “The pricing of options and corporate liabilities”, Journal of Politrcal Economy, 81, 637-653. Bowley, A.L., 1924, “Births and population of Great Britain”, Journal of the Royal Economic Society, 34, 1888192. Cannan, E., 1895, “The probability of cessation of the growth of population in England and Wales during the next century”, The Economic Journal, 5, 505-515. Cohen, J.E., 1986, “Population forecasts and confidence intervals for Sweden: A comparison of model-based and empirical approaches”, Demography, 23, 105-126. Hoem, J.M., 1973, “Levels of error in population forecasts”, Artikler 61 (Central Bureau of Statistics of Norway, Oslo). HyppGIa, J., A, Tunkelo and L. Tornquist, 1949, “Calculations on the population of Finland, its renewal, and its future development”, Tilastollisia tiedonantoja, 38 (Central Statistical Office of Finland, Helsinki) (in Finnish and Swedish, with a French summary). Keyfitz, N., 1982, “Can knowledge improve forecasts?“, Population and Development Review, 8, 719-751. Land. K.C., 1986, “Methods for national population forecasts: A review”, Journal of the American Statistical Association, 81, 888-901. Lee, R.D., 1974, “Forecasting births in post-transition populations: Stochastic renewal with serially correlated fertility”, Journal of the American Statistical Association, 69. 601-617. Leslie, P.H., 1945, “On the use of matrices in certain population mathematics”, Biometrika, 33, 183-212. Lewis, E.G., 1942, “On the generation and growth of a population”, Sankhya, 16, 93-96. McDonald, J., 1981, “Modeling demographic relationships: An analysis of forecast functions for Australian births”, Journal of the American Statistical Association. 76, 782-792. Olshansky, J., 1988, “On forecasting mortality”, Millbank Quarterly, 66, 482-530. Pearl, R. and L.J. Reed, 1920, “On the rate of growth of the population of the United States since 1970 and its mathematical representation”, Proceedings of the National Academy of Science, 6. 275-288. Pflaumer, P., 1988, “Confidence intervals for population projections based on Monte Carlo methods”, International Journal of Forecastrng, 4, 135-142. Rogers, A., 1975, An Introduction to Multi-Regional Mathematical Demography (Wiley, New York). Shryock, H.S., J.S. Siegel et al., Condensed edition by E.G. Stockwell, 1976, The Methods and Materials of Demography (Academic Press, New York). Spencer, B.D. and C. Lee, 1980, “Postcensal population estimation methods of the Census Bureau”, in: Panel on Small-Area Estimates of Population and Income, Estimating
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Populatron and Income of Small Areas (National Academy Press, Washington, DC) 131-187. Stoto, M., 1983, “The accuracy of population projections”, Journal of the American Statistical Association, 78, 13-20. Stoto, M., 1988, “Dealing with uncertainty: Statistics for an aging population”, The American Statistician, 42, 103-110. Sykes, Z., 1969, “Some stochastic versions of the matrix model for population dynamics”, Journal of the American Statlstical Association, 64, 117-130. Thompson, W.S. and P.K. Whelpton, 1933, Population Trends m the United States (McGraw-Hill, New York). Turpeinen, O., 1978, “Fertility and mortality in Finland since 1750”, Population Studies, 33, 101-14. U.S. Bureau of the Census, 1984, “Projections of the population of the United States, by age, sex, and race: 1983 to 2080”. Current Population Reports, Ser. P-25. No. 952 (U.S. Bureau of the Census, Washington, DC). U.S. Office of the Actuary, 1987, “Social security area population projections: 1987”, Actuurial Study No. 99 (U.S. Department of Health and Human Services, Washington, DC). Whelpton, P.K., 1928, “Population of the United States, 1925 to 1975”, American Journal of Sociology, 343, 253-270.
Whelpton, P.K., 1936, “An empirical method of calculating future population”, Journal of the American Statistical Association, 31, 457-473. Whelpton, P.K., H.T. Eldridge and J.S. Siegel, 1947, Forecasts of the Population of the United States 194% 1975 (U.S. Bureau of the Census, Washington, DC). Winkler, R.L., 1984, “Combining forecasts”, in: S. Makridakis et al., The Forecasting Accuracy of Major Time Series Methods (Wiley, New York).
Biography: Juha M. ALHO is Assistant Professor at the Institute for Environmental Studies and Department of Statistics at the University of Illinois at Urbana-Champaign. He was previously Head of Registry Section at the Institute of Occupational Health in Helsinki, Finland. He received a Ph.D. in Statistics from Northwestern University. His primary fields of interest are statistical modelhng in demography and environmental risk analysis. His publications include articles in Biometrics, Biometrika, Demography, the Journal of the American Statistical Association and the American Journal of Industrial Medicine.
Journal
of Forecasting
521
6 (1990) 521-530
Stochastic methods in population forecasting Juha M. Alho Institute for Environmental Urbana, IL 61801, USA
Studies and Department
of Statistics,
University of Illinois at Urbana-Chainpaign,
Abstract: This paper presents a stochastic version of the demographic cohort-component method of forecasting future population. In this model the sizes of future age-sex groups are non-linear functions of random future vital rates. An approximation to their joint distribution can be obtained using linear approximations or simulation. A stochastic formulation points to the need for new empirical work on both the autocorrelations and the cross-correlations of the vital rates. Problems of forecasting declining mortality and fluctuating fertility are contrasted. A volatility measure for fertility is presented. The model can be used to calculate approximate prediction intervals for births using data from deterministic cohort-component forecasts. The paper compares the use of expert opinion in mortality forecasting with simple extrapolation techniques to see how useful each approach has been in the past. Data from the United States suggest that expert opinion may have caused systematic bias in the forecasts. Keywords: Cohort-component,
Expert opinion, Fertility, Mortality, Volatility.
1. Introduction The cohort-component method of population forecasting typically projects future numbers of annual births, deaths, and migration by one- or five-year age-sex groups, adds them to form a new population vector, and repeats the calculations for each forecast year (Shryock and Siegel, 1976, pp. 443-444). This procedure is usually attributed to P.K. Whelpton and collaborators, who developed it in a sequence of papers beginning in 1928 (Whelpton, 1928; Thompson and Whelpton, 1933, Ch. X; Whelpton, 1936; Whelpton et al., 1947). The equivalent arithmetic had earlier been used by, for example, Cannan (1895) and Bowley (1924). This procedure is not a detailed method of forecasting; rather it provides a general bookkeeping framework within which each age-specific vital rate can be projected as desired. To date, essentially all national population forecasts in countries with the requisite age-sex specific data have been made using the cohort-component method. Unfortunately, the accuracy of population forecasts did not substantially improve due to these 0169-2070/91/$03.50
0 1991 - Elsevier Science Publishers
efforts. For example, Whelpton’s 1947 high forecast for the total fertility rate for 1960 was 2.3 children per woman (Whelpton, 1947, p. 32), whereas the actual rate was 3.5. Changes pf vital rates have been difficult to predict ever since. One reaction to the uncertainty has been to call forecasts “projections”, or “illustrative projections”, as if changing the term would remedy the situation. I propose to accept the observed level of uncertainty for the time being, try to estimate its magnitude, and to analyze its sources. From a statistical point of view, the best forecast is the one with the (in some sense) smallest error. However, there is no substantive reason why the vital processes should be easy to predict, so the smallest possible error may be much larger than the forecasters, the direct users of the forecasts, or the public like or expect (cf. Alho, 1984). One solution to the problem of error estimation is to formulate a stochastic version of the cohortcomponent method, which treats the vital rates as realizations of random processes. This yields, as a by-product, high-low intervals that have a given probability of covering the true size of an age-sex
B.V. (North-Holland)
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group in a given future year (Sykes, 1969; Lee, 1974; McDonald, 1981; Stoto, 1983; Alho and Spencer, 1985; Cohen, 1986). Historically, the first attempt at a stochastic formulation that I am aware of was made by L. Tbrnqvist in Finland in 1949 (HyppGla et al., 1949, pp. 68-74). This early work contained many of the central features of a stochastic forecast. Lack of adequate computing facilities may explain why Tbrnqvist’s lead was not followed in official forecasts until the 1980s (U.S. Bureau of the Census, 1984). In Section 2, I briefly describe a stochastic version of the cohort-component method. In Section 3, I point out how expert opinion and data analysis affect the error of forecasts. In Section 4, I propose a volatility measure of fertility that can be used to calculate approximate prediction intervals for births, based on a deterministic cohortcomponent forecast. In Section 5, I study the bias of simple linear extrapolation techniques in forecasting mortality, and contrast it with the bias that occurs with the use of expert opinion.
2. Calculation
of a stochastic
forecast
Population growth can be described in terms of alineargrowthmodel.Let~(V(t)=(u,(t),...,u,(t))’ be a column vector containing population numbers at (exact) time t, and let an s x s matrix R(t) = (r,,(t)) contain the vital rates (arranged in a suitable way) during the year from t to t + 1. Then, V(t+
1) = R(t)V(t).
(1)
A simple example of this is the classical model of one sex of a closed population considered by Bernardelli (1941) Lewis (1942) and Leslie (1945) (R(t) is sometimes called a “Leslie-matrix”). In this case the only non-zero elements in the R(t) matrices occur on the first row (fertility rates) and on the first sub-diagonal (survival rates). Eq. (1) together with such rate definitions is equivalent to the cohort-component procedure of bookkeeping when there is no migration. As shown, for example, by Rogers (1975), a model of type (1) can be defined to cover both sexes, several geographical regions, or other divisions of the population, such
as type of economic activity. Analogously, the stochastic version can be generalized. Suppose we have observed R(t) for t = 1,. . . , n - 1, have an estimate of V(t) for t = n, and want to forecast V(t) for t=n+l,...,n+m. Suppose also that we have available forecasts of R(t) for t=n,...,n+m-1. Then, the cohortcomponent forecast of V( n + m ) can be written as P(n+m)=k(n+m-l)...k(n)P(n).
(2)
We typically do not know the base population (jump-off population) exactly, so the estimate of V(n) contains error (e.g., Shryock and Siegel, 1976, p. 422, or Spencer and Lee, 1980). To represent this uncertainty we can use a random jump-off population (cf. Alho and Spencer, 1985, p. 310). Interestingly, in very short term forecasts this source of error may dominate. Correlations across forecasts of different components (i.e., different r,,( t)‘s) are also important. We note that the procedure used by most producers of official population forecasts, which combines high fertility assumptions with high life expectancy and high netmigration assumptions, implicitly assumes that these correlations equal 1. This is absurd, since the primary causal factors behind changes in fertility, mortality, and migration are not identical. An assumption of zero correlation is surely much closer to the truth. An analogous problem arises when the autocorrelations of the vital rates are considered. The propagation of error, that is the distribution of the error of the forecast of V(n + m), when the jump-off population and the vital rates are lognormally distributed, is readily handled by approximations provided in Alho and Spencer (1985, p. 309, expressions (3.13)-(3.17)). Simulation (see, e.g., Pflaumer, 1988) could also be used.
3. Expert opinion
and data analysis
Whelpton objected to the sole reliance on mathematical curve-fitting techniques in population forecasting. Referring to the work of Elkanah Watson in about 1815 and Francis Bonynge in 1852, Whelpton noted that such techniques may sometimes be remarkably accurate. For instance, Bonynge’s forecast based on a variable geometric
J.M. Alho / Stochastic methods in population forecasting
growth assumption was only 2.2% off the correct value for the total U.S. population in 1900. The reason for rejecting these methods, and the presumably more sophisticated logistic technique popularized by Pearl and Reed (1920), was that several curves may fit the past data very well but give widely differing forecasts (e.g., Thompson and Whelpton, 1933, pp. 312-313; Whelpton et al., 1947, pp. 10-29). Instead, Whelpton suggested that one should study the past rates, form an opinion of the likely future rates at some target year, and obtain the intermediate values by interpolation. Essentially, the same procedure is still used in official forecasts (see, e.g., U.S. Office of the Actuary, 1987). We can represent Whelpton’s procedure within a statistical model by assuming that the data analysis of the vital rates is done within a parametric statistical model. Subjective input is allowed to influence both the choice of the model and its parameter estimation via Bayesian techniques or, e.g., mixed prediction in which target values are explicitly used (Alho and Spencer, 1985). The statistical procedure gives rise to four different sources of error: (1) model misspecification: the assumed parametric model is only approximately correct; (2) errors in parameter estimates: even if the assumed parametric model would be the correct one, its parameter estimates will be subject to error when only finite data series are available; (3) errors in expert judgement: an outside observer may disagree with our judgements or “prior” beliefs about the parameters of the model, or the weight we give to our beliefs in forecasting; (4) random variation, which would be left unexplained even if the parameters of the process could be specified without any error: since any mathematical model is only an approximation, one would expect there to be residual error. (See Hoem (1973) for further discussion, and an alternative classification.) One should note that the error sources depend conceptually on each other. For example, the presence of (4) gives rise to (2), and errors in (3) may mask or accentuate errors in (l), or vice versa. The next two sections explore some aspects of fertility and mortality forecasting. The primary difference between these series is that fertility has fluctuated wildly during this century, whereas mortality has shown a nearly monotonic decline.
523
4. Estimating the volatility of fertility Estimates of the size of prediction error always depend on the statistical model one uses to calculate the forecast (cf. Lee, 1974). They typically depend on the past fit of the model. When expert opinion is used in forecasting, it may also influence error estimates. For example, in the mixed prediction technique introduced in Alho and Spencer (1985) a target value was determined subjectively, and it directly influenced the widths of the prediction intervals. However, for the purpose of setting base-line estimates of the uncertainty of the forecasts, it would be useful to have error estimates that are relatively independent of forecasting methods and subjective assumptions This is especially important if we have available a deterministic cohort-component forecast based on an unknown combination of subjective assumptions and data analysis, and we would like to determine prediction intervals for it. We typically believe that we are able to predict future fertility more accurately than if we merely use the present fertility as a forecast (cf. Keyfitz, 1982, p. 730). For this reason we propose to use the naive forecast of present value as the basis of error estimation. We would expect the errors of naive forecasts to be larger than those of any officially used method, so prediction intervals based on the naive forecasts should be correspondingly wider. In practice, it is first necessary to determine a suitable transformation of the fertility series, and study the change in the transformed fertility series separately for each lag time. More precisely, let w(t) be the transformed fertility rate of year [t, t + 1). Then the volatility of w(t) at lag k > 0 bused on “loss function” d is defined as E[d(w(t)-w(t-k))lw(t-s),
s
(3)
The loss function d is any non-negative function defined on the real line that is symmetric around zero, increases on the positive real line, and vanishes at zero only. For example, we might want to use d(x) = x2 or d(x) = (x I. If the volatility does not change over time, we can estimate it based on the oberved values at t = 1,. . , n by n-k c d(w(t) ,+=I
- w(t - k)),‘(n
- k).
(4)
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J.M. Alho / Stochastic methods in population forecasting
In practice, the volatilities would be expected to change over time, so I calculate volatility estimates based on subsets of the observed data, and study their change. My concept of volatility is loosely based on the concept of stock price volatility, which is the basis of the modern theory of stock option pricing developed by Black and Scholes (1973). In that theory one considers the logarithm of the stock price, and uses d(x) = x2. The Black-Scholes theory is based on the assumption that the logarithm of stock price is a Brownian motion. In that case the theoretical volatility at lag = k is k times the volatility at lag = 1. Once the volatility at lag = 1 has been estimated, there is no need to consider other lags separately. In a sense, our concept can be viewed as a nonparametric generalization of the concept used in stock option theory. To illustrate these concepts we shall study the volatility of the Finnish fertility in 1776-1976 (Turpeinen, 1978). We shall use d(x) = 1x 1. Instead of considering all the age-specific fertility rates separately, we concentrate on their average which is equal to the total fertility rate divided by the number of child bearing ages (which we take to be 30, corresponding to ages 15-44). Figure 1 has a plot of the total fertility rate (I). The highest average rate was 0.19710 and the lowest 0.05013. I shall use a logit transformation that forces the average fertility rate between a lower bound rL, and an upper bound ru. Based on the observed minimum and maximum, I shall take rL = 0.025
Fig. 1. Total fertility rate (I), and the summary volatility measure (II) in Finland in 1776-1976.
and ru = 0.25. In other words, letting r(t) denote the average fertility rate, I define w(t) via w(t)=log((r(t)-0.025)/(0.25-r(t))).
Using the whole data period (N = 201) to estimate the lag-specific volatilities, summary statistics of Table 1 are obtained. Figure 1 shows also a plot of the summary volatility measure (II), which I took to be the average of the smoothed lag-specific measures for lags l-15. The procedure RSMOOTH of Minitab, which combines running medians and moving averages, was used to carry out the smoothing. A comparison with the total fertility rate shows that sudden upward or downward spikes (e.g., years 1789, 1809, 1868) in the original series translate
Table 1 Summary statistics of the lag-specific volatilities of the average fertility rate in Finland in 1776-1976.
Lag
N
Mean
Median
SD
Min
MaX
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
200 199 198 197 196 195 194 193 192 191 190 189 188 187 186
0.148 0.170 0.200 0.229 0.237 0.257 0.265 0.285 0.293 0.308 0.318 0.341 0.346 0.363 0.372
0.100 0.141 0.151 0.178 0.184 0.207 0.211 0.241 0.231 0.247 0.250 0.266 0.280 0.273 0.286
0.144 0.157 0.175 0.194 0.205 0.208 0.229 0.224 0.246 0.288 0.262 0.278 0.280 0.286 0.303
0.001 0.003 0.004 0.003 0.002 0.008 0.002 0.005 0.001 0.001 0.000 0.001 0.001 0.003 0.001
0.855 0.876 0.936 1.147 1.069 1.085 1.097 0.998 1.085 1.135 1.126 1.165 1.191 1.206 1.215
525
J.M. Alho / Stochastic methods in population forecasting
into wider peaks in the volatility series. Periods of sustained change, such as the decline from 1900 to 1930, also cause high volatilities. More recently, the dramatic decline from around 1947 to 1973 has created a sizeable peak at the end of the time series. The high volatility observed during this century is in an interesting contrast to the often repeated belief that, after the demographic transition, population growth in industrialized countries should stabilize. We can use our volatility estimates to find out approximately how large are the percentage errors one would expect in birth forecasts using the naive forecast. As noted above, such error bounds ought to be conservative for any forecasting method that is superior to the naive one. Consider the linear growth model (1) for the female population. Let us approximate each age-specific fertility rate by their average; denote r,,.,(t) = r(t). Then the true births are given, approximately, by
where the summation is over ages 15-44. Suppose now we use a naive forecast w for w(r). Then, e(t) = w(t) - w is the prediction error, and we can write i-(t) = 0.025 + 0.225
exp(w + e(t)) + e(f))
1 + exp(w
.
When e(t) = 0, we get the point now that
forecast.
Suppose
w = log&( (0.06 - 0.025),‘0.225)
= - 1.69168
(corresponding to the total fertility rate of 1.8, a reasonable forecast in Finland at this time). The ratio of the number of true births to the point forecast is approximately 0.025+0.225exp(w+e(t))/(l+exp(w+e(t))) 0.025+0.225exp(w)/(l+exp(w))
Table 2 Upper and lower 67% and 95% prediction limits divided by the point forecast for births, based on the Finnish birth volatility data from 1776-1976. Forecast year
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Upper end point of interval
Lower end point of interval
95%
67%
67%
95%
1.272 1.306 1.339 1.371 1.400 1.427 1.451 1.472 1.489 1.505 1.524 1.545 1.569 1.593 1.615
1.076 1.096 1.114 1.130 1.142 1.155 1.168 1.181 1.193 1.206 1.217 1.226 1.238 1.256 1.279
0.931 0.916 0.902 0.890 0.881 0.873 0.864 0.856 0.848 0.840 0.833 0.828 0.822 0.812 0.799
0.803 0.786 0.770 0.756 0.743 0.734 0.723 0.716 0.710 0.704 0.698 0.691 0.683 0.676 0.669
For example, the upper end point of the 67% interval for the first forecast year is about 8% above the point forecast. The same analysis was repeated with U.S. white female fertility data from 1920 to 1978. One would expect the resulting error estimates for births to be of the same order of magnitude as in Finland, because both countries experienced a similar increase from-the depression years’ low fertility to a peak after the Second World War, and a subsequent decline. Figure 2 shows that this is, indeed, the case. The prediction intervals based on the I
”
(6)
The volatility estimates give us directly estimates of the distribution of 1e(t) 1 for each lag k = 1, 2,. . .,15. I smooth the empirical distributions first to reduce random error. Then I determine the location of, for instance, the 67th percentile. Replacing e(t) in eq. (6) by + the 67th percentile yields a (conservative) 67% prediction interval for the births. Table 2 gives the width of the resulting prediction intervals in terms of the point forecast.
Fig. 2. Ratio of upper end point of a 67% prediction interval and the point forecast for births based on Finnish volatility in 1776-1976 (I), Finnish volatility in 1920-1976 (II), U.S. volatility in 1920-1978 (III), and the mixed prediction forecast of U.S. births (IV).
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J.M. Alho / Stochastic methods in popdatton
1920-1976 Finnish volatility (II) are very nearly equal to those based on the 1920-1978 U.S. volatility (III). These intervals may be compared to the ones reported in Alho and Spencer (1985) for the U.S. births (IV) that were calibrated so they would have been adequate during the baby-boom years. Overall, the latter intervals are narrower. This is to be expected because the latter intervals were based on a better method of forecasting than the naive method. However, the fact that the volatility based intervals are initially narrower suggests that there may be room for improvement in the mixed prediction methodology. The intervals based on the Finnish volatility from 1776 to 1976 (I) are the narrowest of all. This agrees well with the increase in volatility shown in Fig. 1. It may well be that fertility forecasting has seldom been more difficult than during this century.
5. Uncertainty of mortality decline Figure 3 shows the time series of age-standardized mortality rates for males and females in the U.S. in the period 1900-1986, with the 1980 total population as standard (U.S. Office of the Actuary, 1987, p.7). The nearly steady decline is in contrast to the fluctuations of the fertility series in Fig. 1. Due to biological bounds of longevity, mortality decline cannot continue indefinitely. Therefore, a major source of uncertainty in mortality forecasts comes from the timing of the levelling off of mortality. In 1954-1968, male mortality rose 0.2% per year. At the end of the period it may have seemed that the levelling off had already occurred. The subsequent decline
Fig. 3. Age-standardized mortality rate in the United States in 1900-1986 for males (I) and females (II), per 100,000.
forecasting
shows how uncertain such conclusions may be. Depending on their magnitude, epidemics or war may also affect mortality trends. In view of Fig. 3, an analysis of uncertainty using the present value as a “naive forecast” for all future times does not make sense in mortality forecasting. A more meaningful baseline estimate is obtained by assuming that future changes in mortality are similar to the past changes. We may for example, use the first differences of a suitably transformed mortality series, estimate their mean, and model the deviations from the mean in terms of a simple ARIMA-process. In terms of the original time series, this means that we would use either an ARIMA(1, 1, 0), ARIMA(0, 1, 1). or ARIMA(l, 1, 1) model with a constant term to calculate a baseline forecast. The forecasts of the three models are in this case virtually indistinguishable, and the relatively short-data periods do not allow us to choose between them (or identify other, possibly better fitting, models) on statistical grounds alone. Consequently, I shall determine, a priori, which model to use, estimate its parameters, and calculate a baseline forecast using the estimated model. The crucial assumption in this procedure is the decision to use a constant term in the model. The largest standardized death rate in Fig. 3 is about 0.025 and the smallest is about 0.006. Possible upper and lower bounds for a logit transformation are ru = 0.03 and r,_ = 0.002. I used these values in a logit-transformation to get w(r)
= log((r(t)
- 0.002)/~0.03
-r(t))).
An ARIMA(0, 1, 1) model was used with a 30-year data period, and 15-year forecasts were calculated with the jump-off years 1930-1981 for both male and female mortality rates. Table 3 gives summary data for the prediction error of these forecasts for 5, lo-, and 1%year lead times. To put the figures of Table 3 into perspective, we may note that the 1986 age-standardized mortality rate for males was 1093 (per 100,000) and for females 645. These figures show that the simple ARIMA-extrapolation technique would have given close to unbiased results over this time period. Nearly identical predictions for male mortality are obtained if the same technique is applied to the untransformed mortality series. For females the forecasts based on untransformed rates
J.M. Alho / Stochastic methods in population forecasting
Table 3 Summary statistics of the prediction error of 5-, lo-, and 15-year ahead forecasts of male and female mortality (per 100,000) for the years 1936-1986 in the United States using an ARIMA(O,l,l) model with a constant term.
Males
Females
Lead time
Mean
Median
SD
Min
Max
5 10 15
- 12 -25 -57
-22 -60
6
14 119 141
-158 -262 -317
107 149 151
5 10 15
13 18 17
4 17 -8
62 103 136
-100 -159 -174
178 208 275
are slightly worse. Hence, in these series the effect of data transformation is small. Figure 4 contains a plot of the prediction errors for females. The appearance of male prediction errors is very similar, with somewhat lower average values as indicated by Table 3. For both sexes the forecasts for the beginning of the 1950s were too high, for the end of the 1960s they were too low, and for the early 1980s the male forecasts appear to have been too high. We note that the errors of simple ARIMA extrapolations can serve as a basis for the estimation of the volatility of mortality. Official forecasts typically use other sources of information besides the past mortality time series to forecast future mortality. Therefore, they should be more accurate than the simple trend forecasts. Suppose we are willing to believe that this is the case. Then, we can repeat the volatility estimation procedure described for fertility, using ARIMA extrapolations as the naive forecast, to get error bounds for
I
I
1
i’i
II
Fig. 4. Five-year (I), ten-year (II), and fifteen-year (III) prediction errors of simple ARIMA-forecasts of the U.S. age-standardized mortality rate for females for the years 1935-1986.
521
each age-specific mortality rate and to derive conservative error bounds for the number of deaths in each age-group. These error bounds would not necessarily equal the theoretical ARIMA bounds that assume the stochastic process has been correctly identified. Rather, they would reflect the empirical prediction error of the naive forecast, as in the case of fertility. Figures 1 and 2 of Olshansky (1988) compare the forecasts of life expectancy of the U.S. Office of the Actuary to the observed development, including jump-off years 1950, 1955, 1965, and 1977, for males and females. The female life expectancy was consistently underestimated. The first three forecasts for males were more accurate than for females. They overestimated the life expectancy in the late 1960s and early 1970s. All forecasts were below the actual life expectancy in the 1980s. These results parallel our results for standarized mortality (Fig. 4) based on ARIMA extrapolations, so the question becomes, which procedure would have been more accurate through the 1960s and 1980s. Using the assumptions of the official forecasts (U.S. Office of the Actuary, Actuarial Studies nos. 33, 46, 62, and 77) I recalculated their forecast for the age-standardized mortality rates of Fig. 3. Figure 5(a)-(d) show the 30-year observation period used in the ARIMA forecast with jump-off years 1950, 1955, 1965, and 1977; the subsequent development of mortality until the year 1986; the ARIMA forecast; and the high and low official forecasts for females (for the last forecast only one variant was used). Only the second official forecast appears to be clearly better than the ARIMA extrapolation. A similar figure for males shows that the first two official forecasts were clearly better than the ARIMA extrapolations. The reverse is true for the latter two. Overall, the official forecasts tended to overshoot the actual mortality, whereas the extrapolations have been usually below it. Ex post facto, we may note that a combination of the “objective” extrapolation technique with the “subjective” expert forecast would have been superior to either technique alone (cf. Winkler, 1984, p. 294). The rate of past mortality decline has, of course, been an important factor when the assumptions of the official forecasts have been made. The results of Fig. 5 indicate that putting somewhat less weight on the expert opinion would have resulted in
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J.M. Alho / Stochastic methods in populatlon forecastmg
YeCi
Fig. 5. Age-standardized
I.
U.S. female mortality rate, ARIMA forecast (I), and high and low official forecasts (a) 1950, (b) 1955, (c) 1965, (d) 1977 (only one official forecast variant).
better forecasts. In this sense, too heavy reliance on expert opinion seems to have biased the forecasts toward high mortality in the past two or three decades. When ARIMA forecasts are continued further from 1986, and compared to a recent U.S. Office of the Actuary (1987) forecast, the difference is that the official forecast of mortality is initially slightly lower, but becomes higher than the extrapolation forecast in about 15 years’ time. This is due to the assumed levelling-off of mortality decline. One is left wondering whether the use of expert judgement will bring better results this time than in the past.
6. Prospects The availability of meaningful error bounds can be of major interest to the users of forecasts, as they indicate the value of past data in decisionmaking. In view of the wide error bounds of birth and death forecasts, urban, educational, or finan-
r7
(II) with Jump-off
year
cial planning should not rely too heavily on current forecasts. The official forecasts may well be the most accurate available but because they are uncertain, it becomes important that one prepare for the possibility that they will be grossly in error. Stoto (1988) reaches a similar conclusion in discussing the role of statistical uncertainty in policy. A full implementation of a stochastic forecast is a considerable task requiring a detailed analysis of the uncertainties of the jump-off population, each of the vital rates, and their joint correlation structure. Much of the work relating to the expectations of these variables is done now under the headings of population estimation and estimation of the vital rates. The analysis of the second moments and the derivation of the full error distribution is a novel task that requires considerable statistical input, in addition to demographic expertise. There are several intermediate stages between a classical deterministic forecas! and a full stochastic forecast. Land (1986) reviews several possibili-
J.M. Alho / Stochastic methods in population forecastmg
ties. In this paper we proposed a measure of the volatility of vital rates. When applied to fertility, it can be used to derive conservative prediction intervals based on a deterministic point forecast of births. The credibility of these intervals hinges only on the fact that the actual forecasting method used is not less accurate than the use of today’s value of fertility for all future times. A similar analysis of mortality can be effected by using the past rate of decline rather than the current value as the naive forecast. All forecasting methods, deterministic and stochastic alike, involve in varying degrees the use of expert judgement. For example, past and current forecasts of mortality have all assumed that the rate of decline of mortality will slow down considerably within one or two decades after the jump-off year. A comparison with simple trend extrapolations shows that putting less weight on expert opinion would have made the official mortality forecasts more accurate in the United States during the past three decades. Over the course of the 20th century the simple extrapolations would have been unbiased, on the average, for forecast periods of 5-15 years. However, the errors of the extrapolations have been highly correlated. It is possible that such short term deviations from the overall trend can be explained by using epidemiologic, health survey, or other data. This suggests that a useful role for expert opinion may be in relatively short term forecasting, in addition to determining where and when the “ultimate” levelling-off of mortality takes place. Expert opinion could cause temporary delays or accelerations in the overall trend forecast. Perhaps the AIDS epidemic will necessitate such an adjustment.
Acknowledgement This research was partially supported by National Institute on Aging Grant AG06996. The author would like to thank Dr. Bruce D. Spencer for several comments on an earlier version of the manuscript, which was presented at the Eighth International Symposium on Forecasting, in Amsterdam, the Netherlands, in June 1988. Useful comments from three anonymous reviewers helped to clarify the presentation.
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Biography: Juha M. ALHO is Assistant Professor at the Institute for Environmental Studies and Department of Statistics at the University of Illinois at Urbana-Champaign. He was previously Head of Registry Section at the Institute of Occupational Health in Helsinki, Finland. He received a Ph.D. in Statistics from Northwestern University. His primary fields of interest are statistical modelhng in demography and environmental risk analysis. His publications include articles in Biometrics, Biometrika, Demography, the Journal of the American Statistical Association and the American Journal of Industrial Medicine.