- Email: [email protected]
Research on forecasting
International North-Holland
Journal
of Forecasting
635
5 (1989) 635-638
Research on forecasting The International Journal of Forecasting provides critiques of papers published elsewhere. The editors try to select recent papers that are likely to be of significant interest to the readers. Our review of each paper is sent to the author for comment prior to publication. Almost all authors respond with suggestions and these typically lead to improvements in the critiques. If you know of interesting papers, or if you have published such a paper, please send a copy to the Editors for possible inclusion in this section of the International Journal of Forecasting. To obtain copies of the papers reviewed in this section, contact the authors of the original papers.
(2) an individual only remains in the class of (3)
(4) (5)
(6) (7)
From the Royal Statistical Society Meeting on AIDS Model& and Forecasting Valerie Isham, “Mathematical modelling of the transmission dynamics of HIV infection and AIDS: A review” (with discussion), Journal of the Royal Statistical Society, Series A 151 (1988) 5-30. Isham’s paper opened the Symposium. The author starts by describing a simple epidemic model where the population can be separated into a group of susceptibles, x(t), and infectives, y(t), where x(t) + y( t) = n. In any short time interval the number of contacts between susceptibles and infectives is proportional to x(t) and y(t) and a fixed proportion of these contacts are assumed to become infected. Then, dx(t)/dt=
-ax(t){n-x(t)}.
This simple model is made progressively more realistic by adding additional assumptions such as: (1) the susceptibles acquire partners at a rate k, the population mixes homogeneously, and the probability that a susceptible catches the infection from an infected is constant; 0169-2070/90/$3.50
0 1990, Elsevier Science Publishers
infected for a finite period of time; an infected individual may join the HIV class with probability p or remain seropositive but no longer infectious. More generally, the mean incubation period for those who subsequently develop AIDS may vary from those infectives who do not; immigration and death from causes other than AIDS can be included; the constant k is likely to be affected by the level of sexual activity and therefore can be assumed to vary across the population; the incubation period can be made to depend (more realistically) on the time since infection; the population can be segmented into heterosexual, bisexual, drug user and homosexual, and models developed for each group and for the contagion between groups.
Including all these seven factors leads to a realistic model for the spread of AIDS. The problem, of course, is that there are only very limited data which can be made to relate to the model’s parameters. In a further contribution to the Symposium, Bailey (pp. 31-34), commenting on Isham’s paper, goes on to estimate a simplified model and then uses it in forecasting, concluding “only the routine testing of large random population samples for HIV antibodies can give the advance warning that is essential to survival”. Additional papers examine the need for data and in particular the concept and estimation of the incubation period. A variant of Isham’s approach is described by Wilkie (pp. 35-39) in “An actuarial model for AIDS’ where he points out the need for age/sex specific modelling for actuarial purposes and based on certain assumptions on the parameters derived from other, detailed studies, he derives population projections. Isham’s paper (and the associated comment) is interesting to the forecaster and diffusion modeller in that it describes ways in which the complexities of a diffusion process can be modelled. It does not attempt to address the issue as to which
B.V. (North-Holland)
636
Research *t2forecasting
of the many simplifying assumptions that a modeller must make are likely to prove important. - Robert [Valerie Isham, Department University College, Gower
of Statistical St., London
Fildes
Science, WC lE,
UK1
M.J.R. Healy and HE. Tillett, “Short-term extrapolation of the AIDS epidemic”, Journal of the Royal Statistical Society, Series A 151 (1988) 5061. The aim of the Healy-Tillett article, the second major article at the symposium, is to produce extrapolative forecasts, needed, they argue, for both scientific and administrative reasons. As the previous discussion made clear, the development of full structural models is impossible with the limited data base available. Nor is it clear that they would offer better forecasts. The authors here examine the data, deciding that the most appropriate variable to forecast is the date of diagnosis. The data are adjusted and two simple models estimated: E(log(x
+ I)) = ty + @t,
X) = a + Pt.
with Poisson
- Robert
Fildes
Referente 1988, Short-Term Prediction ofHIV Infection and AIDS in England and Wales (Department of Health, HMSO).
Cox,David,
[M.J.R. Healy, London School of Hygiene and Tropical Medicine, Keppel St., London WC lE7 HT, UK]
Nigel Meade, “‘Forecasting with growth curves: The effect of error structure”, Journal of Forecasting 7 (1988) 235-244.
and E(log
and behavioural aspects of the epidemic, and some observations on the statistical requirements to further our understanding of AIDS The papers at the symposium show how far there is to go before a fully specified structural model of the AIDS epidemic can be useful for forecasting and policy. As a final solution, trend curve analysis was chosen as the forecasting tool. As the epidemic progresses and more data become available, it is to be hoped that comparative testing of the various approaches will be carried out (see, for example Cox, 1988).
errors.
The models produce similar forecasts but if they are extended by including quadratic terms, the two models diverge; the former by predicting increasing incidence of AIDS, the latter suggesting a levelling off (here is yet another example of the classic problem of trend curve forecasting). The authors analyse this key difference by first arguing that more recent data should be given additional weight in forecasting, so that they give geometric declining weights (a weight of 0.8) to the older data. They then calculate rolling horizon trend estimates by adding an additional data point and re-estimating. The analysis suggests a fall in trend in early 1985 which has since levelled off. Based on this conclusion, the two models lead to similar forecasts, when the data are weighted. Other papers in the symposium are concerned with the epidemiology of HIV infection, social
Nigel Meade, “A modified logistic model applied to human populations” (with discussion), Journal of the Royal Stutist~~a~ Society, Series A 151 (1988) 491-498. Robert Raeside, “The use of sigmoids in modelling and forecasting human populations”, Journnf of the Royaf Statistical Society, Series A 151 (1988) 499-513. In the first two of these papers Meade two extensions of the logistic model: X(t)
= S/(1
+ c exp(-bt))
develops
+ e(t),
where S is the saturation level and h is a measure of how fast the population grows. In his earlier survey paper, Meade (1984) considered how such curves could be used in modell-
Research on forecasting
ing the diffusion of innovations, the growth of consumer durables markets and the increase in human populations. These various phenomena (sometimes supported by a detailed mathematical argument) lead to S-shaped growth curves as suitable models. In the first of these papers, Meade suggests a local alternative to the global model given above. Suppose, both S and b are assumed adaptive with their behaviour modelled as a random walk, for example S, = S,_ , + E,. Suppose further that the prediction equation for X(t) is constrained to pass through the last observed value of X(t). The resulting model is:
X(t + 1) =
4% x,+
(St-x,)
exp(-b,)
+“+I’
where a,+, - N(O, <+i>. In the first of these two papers, exploring the implications of this model, Meade concentrates on the effect of the error assumptions. Two cases are examined: the first, apparently more suited to modelling consumer durables markets, assumes y = a*.$( S, - Xr)a, where the variance is zero close to either the beginning of the adoption/ diffusion process or to the market saturation. The second assumption Meade considers is y = u*xp, and is more suited to population growth and describes the situation where the error variance increases with population size. In a careful empirical study of a number of time series including consumer durables and populations using one and five step ahead ex ante iterative forecast-error-analysis, the local logistic with modified error variance assumptions provided generally better forecasts. Meade concludes by offering some guidelines on how to decide whether to adopt the more complicated error assumptions he has proposed. The second paper uses the local logistic to analyse human population increase in some detail. Rigorous error analysis of three populations shows the local logistic outperforming the global logistic model. The two papers taken together offer an intuitively plausible extension of a well-established approach to forecasting. It seems that for many time series the latest observation provides a fixed point
631
from which forecasts should be made. Unusually, the empirical evidence also supports the use of more complex error assumptions than those usually imposed a priori. The third paper, by Raeside, makes a broader point than Meade’s, arguing that S-shaped curves have a forecasting record superior to the more accepted component forecasting approach, in which births, deaths and migrations are separately forecast through age-specific analysis. Raeside illustrates his criticisms by using a number of examples. He then describes an extension in which the population at t is linked to the births at time t - a and life expectancy at a (there is a misprint in the key equation). Using a logistic model for life expectancy, combined with an ARIMA model, Raeside derives substantially better forecasts than those from the UK Office of Population Censuses and Surveys. No clear moral emerges from these three papers; in the examples discussed here, aggregate methods outperform disaggregate and there are major improvements to be gained from adopting more complex statistical assumptions (though perhaps they also offer a more intuitive model of the processes under analysis). - Robert Fildes
Reference Meade, Nigel, 1984, “The use of growth curves in forecasting market development”, Journal of Forecastmg, 3, 429-451.
[Dr Nigel Meade, Management School, Imperial College, Exhibition Rd., London SW7 2BX, UK; Dr Robert Raeside, Department of Mathematics, Napier College, Sighthill Court, Edinburgh, EH14 1DJ UK]
Kenneth F. Wallis, “Macroeconomic forecasting: A survey”, Economic Journal 99 (1989) 28-61. Despite its importance to government, industry and commerce, macroeconomic forecasting still receives little attention in the mainstream economic journals. Cairncross (1969) commented in
638
Research on forecasting
his presidential address that he had found no earlier discussion of the subject at the Royal Economic Society. Taking Cairncross as his starting point, Wallis surveys the developments of the last 20 years. He first describes how forecasts from macro models are typically constructed and some of the associated problems. He then charts the learning that took place from the two major recessions of 1974-1975 and 1979-1981, where the major changes, experienced in the economy, exposed failings in most UK models, which in turn led to their revision. For example, the re-evaluation provoked by the 1979-1981 recession led to a more important role for expectations variables in the models, and in particular, expectations extrapolated from past data were typically replaced by forward looking expectations variables, which in their extreme form assumed that economic agents were aware of the model structure and had coincident expectations to the model’s forecasts, this last being an example of the rational expectations hypothesis. Wallis then goes on to discuss the evaluation of macro forecasts, describing some of the literature on the comparison with univariate extrapolative models and VAR models. Research here has concentrated on MSE (or MAPE) comparisons and neglects the possible influence of outliers (important in the small sample comparisons made) and turning point accuracy; issues I believe to be important. Wallis concludes by pointing out that the early success of extrapolative methods compared with macro models only illustrated dynamic misspecification in the macro models. He then considers cross-model comparisons in the UK, and the decomposition of forecast error. Ex ante forecast error can be broken down into errors due to:
(1) model inadequacy, (2) m&-forecasting the exogenous (3) data revisions, (4) parameter estimation, made (5) the adjustments model output.
variables,
by the forecasters
to
Some of these errors are self-cancelling, some consistently improve the overall forecast error and typically, their order of magnitude varies from model to model. Wallis and his colleagues working in the UK Economic and Social Research Council funded Macroeconomic Modelling Bureau have carried out substantial work in trying to understand these various sources. As yet, hypotheses such as ‘modeller adjustments typically help’ and ‘exogenous variable error is less important than model mis-specification’ remain open and, of course, the analysis needs to be supplemented by US evidence such as that given by Dhrymes and Peristiani (1988). Wallis concludes his paper by commenting that “Models and forecasts benefit from public discussion and assessment and in most countries much more could be done.” He and his colleagues at the Warwick Bureau have done much to illuminate UK practice. What seems to me to be needed now, is the specification of more explicit hypotheses concerning propositions in economics, forecasting and model building. Wallis in the UK and McNees in the US have carried out the exploratory work - now we should know what we are looking for in a good model (see, for example, Stekler, 1987). The IJF ‘s proposed special issue on macroeconomic forecasting may perhaps offer some illumination. _ Robert
Fildes
References Cairncross, A., 1969, “Economic forecasting”, Economic Journal, 19, 797-812. Dhrymes, Phoebus J. and Stavros C. Peristiani, 1988. “A comparison of the forecasting performance of WEFA and ARIMA time series methods”, International Journal of Forecartrng, 4, 81-101. Stekler, H.O., 1987, “Who forecasts better?“, Journal of Business and Economic Statistics, 5, 155-158.
[Professor Kenneth nomics, University 7AL, UK]
F. Wallis, Department of Ecoof Warwick, Coventry, CV4
Journal
of Forecasting
635
5 (1989) 635-638
Research on forecasting The International Journal of Forecasting provides critiques of papers published elsewhere. The editors try to select recent papers that are likely to be of significant interest to the readers. Our review of each paper is sent to the author for comment prior to publication. Almost all authors respond with suggestions and these typically lead to improvements in the critiques. If you know of interesting papers, or if you have published such a paper, please send a copy to the Editors for possible inclusion in this section of the International Journal of Forecasting. To obtain copies of the papers reviewed in this section, contact the authors of the original papers.
(2) an individual only remains in the class of (3)
(4) (5)
(6) (7)
From the Royal Statistical Society Meeting on AIDS Model& and Forecasting Valerie Isham, “Mathematical modelling of the transmission dynamics of HIV infection and AIDS: A review” (with discussion), Journal of the Royal Statistical Society, Series A 151 (1988) 5-30. Isham’s paper opened the Symposium. The author starts by describing a simple epidemic model where the population can be separated into a group of susceptibles, x(t), and infectives, y(t), where x(t) + y( t) = n. In any short time interval the number of contacts between susceptibles and infectives is proportional to x(t) and y(t) and a fixed proportion of these contacts are assumed to become infected. Then, dx(t)/dt=
-ax(t){n-x(t)}.
This simple model is made progressively more realistic by adding additional assumptions such as: (1) the susceptibles acquire partners at a rate k, the population mixes homogeneously, and the probability that a susceptible catches the infection from an infected is constant; 0169-2070/90/$3.50
0 1990, Elsevier Science Publishers
infected for a finite period of time; an infected individual may join the HIV class with probability p or remain seropositive but no longer infectious. More generally, the mean incubation period for those who subsequently develop AIDS may vary from those infectives who do not; immigration and death from causes other than AIDS can be included; the constant k is likely to be affected by the level of sexual activity and therefore can be assumed to vary across the population; the incubation period can be made to depend (more realistically) on the time since infection; the population can be segmented into heterosexual, bisexual, drug user and homosexual, and models developed for each group and for the contagion between groups.
Including all these seven factors leads to a realistic model for the spread of AIDS. The problem, of course, is that there are only very limited data which can be made to relate to the model’s parameters. In a further contribution to the Symposium, Bailey (pp. 31-34), commenting on Isham’s paper, goes on to estimate a simplified model and then uses it in forecasting, concluding “only the routine testing of large random population samples for HIV antibodies can give the advance warning that is essential to survival”. Additional papers examine the need for data and in particular the concept and estimation of the incubation period. A variant of Isham’s approach is described by Wilkie (pp. 35-39) in “An actuarial model for AIDS’ where he points out the need for age/sex specific modelling for actuarial purposes and based on certain assumptions on the parameters derived from other, detailed studies, he derives population projections. Isham’s paper (and the associated comment) is interesting to the forecaster and diffusion modeller in that it describes ways in which the complexities of a diffusion process can be modelled. It does not attempt to address the issue as to which
B.V. (North-Holland)
636
Research *t2forecasting
of the many simplifying assumptions that a modeller must make are likely to prove important. - Robert [Valerie Isham, Department University College, Gower
of Statistical St., London
Fildes
Science, WC lE,
UK1
M.J.R. Healy and HE. Tillett, “Short-term extrapolation of the AIDS epidemic”, Journal of the Royal Statistical Society, Series A 151 (1988) 5061. The aim of the Healy-Tillett article, the second major article at the symposium, is to produce extrapolative forecasts, needed, they argue, for both scientific and administrative reasons. As the previous discussion made clear, the development of full structural models is impossible with the limited data base available. Nor is it clear that they would offer better forecasts. The authors here examine the data, deciding that the most appropriate variable to forecast is the date of diagnosis. The data are adjusted and two simple models estimated: E(log(x
+ I)) = ty + @t,
X) = a + Pt.
with Poisson
- Robert
Fildes
Referente 1988, Short-Term Prediction ofHIV Infection and AIDS in England and Wales (Department of Health, HMSO).
Cox,David,
[M.J.R. Healy, London School of Hygiene and Tropical Medicine, Keppel St., London WC lE7 HT, UK]
Nigel Meade, “‘Forecasting with growth curves: The effect of error structure”, Journal of Forecasting 7 (1988) 235-244.
and E(log
and behavioural aspects of the epidemic, and some observations on the statistical requirements to further our understanding of AIDS The papers at the symposium show how far there is to go before a fully specified structural model of the AIDS epidemic can be useful for forecasting and policy. As a final solution, trend curve analysis was chosen as the forecasting tool. As the epidemic progresses and more data become available, it is to be hoped that comparative testing of the various approaches will be carried out (see, for example Cox, 1988).
errors.
The models produce similar forecasts but if they are extended by including quadratic terms, the two models diverge; the former by predicting increasing incidence of AIDS, the latter suggesting a levelling off (here is yet another example of the classic problem of trend curve forecasting). The authors analyse this key difference by first arguing that more recent data should be given additional weight in forecasting, so that they give geometric declining weights (a weight of 0.8) to the older data. They then calculate rolling horizon trend estimates by adding an additional data point and re-estimating. The analysis suggests a fall in trend in early 1985 which has since levelled off. Based on this conclusion, the two models lead to similar forecasts, when the data are weighted. Other papers in the symposium are concerned with the epidemiology of HIV infection, social
Nigel Meade, “A modified logistic model applied to human populations” (with discussion), Journal of the Royal Stutist~~a~ Society, Series A 151 (1988) 491-498. Robert Raeside, “The use of sigmoids in modelling and forecasting human populations”, Journnf of the Royaf Statistical Society, Series A 151 (1988) 499-513. In the first two of these papers Meade two extensions of the logistic model: X(t)
= S/(1
+ c exp(-bt))
develops
+ e(t),
where S is the saturation level and h is a measure of how fast the population grows. In his earlier survey paper, Meade (1984) considered how such curves could be used in modell-
Research on forecasting
ing the diffusion of innovations, the growth of consumer durables markets and the increase in human populations. These various phenomena (sometimes supported by a detailed mathematical argument) lead to S-shaped growth curves as suitable models. In the first of these papers, Meade suggests a local alternative to the global model given above. Suppose, both S and b are assumed adaptive with their behaviour modelled as a random walk, for example S, = S,_ , + E,. Suppose further that the prediction equation for X(t) is constrained to pass through the last observed value of X(t). The resulting model is:
X(t + 1) =
4% x,+
(St-x,)
exp(-b,)
+“+I’
where a,+, - N(O, <+i>. In the first of these two papers, exploring the implications of this model, Meade concentrates on the effect of the error assumptions. Two cases are examined: the first, apparently more suited to modelling consumer durables markets, assumes y = a*.$( S, - Xr)a, where the variance is zero close to either the beginning of the adoption/ diffusion process or to the market saturation. The second assumption Meade considers is y = u*xp, and is more suited to population growth and describes the situation where the error variance increases with population size. In a careful empirical study of a number of time series including consumer durables and populations using one and five step ahead ex ante iterative forecast-error-analysis, the local logistic with modified error variance assumptions provided generally better forecasts. Meade concludes by offering some guidelines on how to decide whether to adopt the more complicated error assumptions he has proposed. The second paper uses the local logistic to analyse human population increase in some detail. Rigorous error analysis of three populations shows the local logistic outperforming the global logistic model. The two papers taken together offer an intuitively plausible extension of a well-established approach to forecasting. It seems that for many time series the latest observation provides a fixed point
631
from which forecasts should be made. Unusually, the empirical evidence also supports the use of more complex error assumptions than those usually imposed a priori. The third paper, by Raeside, makes a broader point than Meade’s, arguing that S-shaped curves have a forecasting record superior to the more accepted component forecasting approach, in which births, deaths and migrations are separately forecast through age-specific analysis. Raeside illustrates his criticisms by using a number of examples. He then describes an extension in which the population at t is linked to the births at time t - a and life expectancy at a (there is a misprint in the key equation). Using a logistic model for life expectancy, combined with an ARIMA model, Raeside derives substantially better forecasts than those from the UK Office of Population Censuses and Surveys. No clear moral emerges from these three papers; in the examples discussed here, aggregate methods outperform disaggregate and there are major improvements to be gained from adopting more complex statistical assumptions (though perhaps they also offer a more intuitive model of the processes under analysis). - Robert Fildes
Reference Meade, Nigel, 1984, “The use of growth curves in forecasting market development”, Journal of Forecastmg, 3, 429-451.
[Dr Nigel Meade, Management School, Imperial College, Exhibition Rd., London SW7 2BX, UK; Dr Robert Raeside, Department of Mathematics, Napier College, Sighthill Court, Edinburgh, EH14 1DJ UK]
Kenneth F. Wallis, “Macroeconomic forecasting: A survey”, Economic Journal 99 (1989) 28-61. Despite its importance to government, industry and commerce, macroeconomic forecasting still receives little attention in the mainstream economic journals. Cairncross (1969) commented in
638
Research on forecasting
his presidential address that he had found no earlier discussion of the subject at the Royal Economic Society. Taking Cairncross as his starting point, Wallis surveys the developments of the last 20 years. He first describes how forecasts from macro models are typically constructed and some of the associated problems. He then charts the learning that took place from the two major recessions of 1974-1975 and 1979-1981, where the major changes, experienced in the economy, exposed failings in most UK models, which in turn led to their revision. For example, the re-evaluation provoked by the 1979-1981 recession led to a more important role for expectations variables in the models, and in particular, expectations extrapolated from past data were typically replaced by forward looking expectations variables, which in their extreme form assumed that economic agents were aware of the model structure and had coincident expectations to the model’s forecasts, this last being an example of the rational expectations hypothesis. Wallis then goes on to discuss the evaluation of macro forecasts, describing some of the literature on the comparison with univariate extrapolative models and VAR models. Research here has concentrated on MSE (or MAPE) comparisons and neglects the possible influence of outliers (important in the small sample comparisons made) and turning point accuracy; issues I believe to be important. Wallis concludes by pointing out that the early success of extrapolative methods compared with macro models only illustrated dynamic misspecification in the macro models. He then considers cross-model comparisons in the UK, and the decomposition of forecast error. Ex ante forecast error can be broken down into errors due to:
(1) model inadequacy, (2) m&-forecasting the exogenous (3) data revisions, (4) parameter estimation, made (5) the adjustments model output.
variables,
by the forecasters
to
Some of these errors are self-cancelling, some consistently improve the overall forecast error and typically, their order of magnitude varies from model to model. Wallis and his colleagues working in the UK Economic and Social Research Council funded Macroeconomic Modelling Bureau have carried out substantial work in trying to understand these various sources. As yet, hypotheses such as ‘modeller adjustments typically help’ and ‘exogenous variable error is less important than model mis-specification’ remain open and, of course, the analysis needs to be supplemented by US evidence such as that given by Dhrymes and Peristiani (1988). Wallis concludes his paper by commenting that “Models and forecasts benefit from public discussion and assessment and in most countries much more could be done.” He and his colleagues at the Warwick Bureau have done much to illuminate UK practice. What seems to me to be needed now, is the specification of more explicit hypotheses concerning propositions in economics, forecasting and model building. Wallis in the UK and McNees in the US have carried out the exploratory work - now we should know what we are looking for in a good model (see, for example, Stekler, 1987). The IJF ‘s proposed special issue on macroeconomic forecasting may perhaps offer some illumination. _ Robert
Fildes
References Cairncross, A., 1969, “Economic forecasting”, Economic Journal, 19, 797-812. Dhrymes, Phoebus J. and Stavros C. Peristiani, 1988. “A comparison of the forecasting performance of WEFA and ARIMA time series methods”, International Journal of Forecartrng, 4, 81-101. Stekler, H.O., 1987, “Who forecasts better?“, Journal of Business and Economic Statistics, 5, 155-158.
[Professor Kenneth nomics, University 7AL, UK]
F. Wallis, Department of Ecoof Warwick, Coventry, CV4