# Temporal extrapolation methods

*The text on this page is taken from an equivalent page of the IEHIAS-project.*

One of the easiest ways of obtaining approximate estimates of future (or past) conditions is by extrapolating the data we do have over time. Thus, if we want to make a 'rough-and-ready' assessment of future levels of contaminants in drinking waters for a period 20 years ahead, we can extrapolate the trend from the last 20 years or so forward. Similarly, if we need to estimate the background prevalence of smoking, or lung cancer, in the years ahead, we can do so by calculating the trend over recent years and projecting this into the future.

Extrapolations of this type can be done using more of less sophisticated methods. In many cases (especially in areas of marketing and business management), it has traditionally been done using judgemental methods - for example by looking at recent data and intuitively estimating what this implies for the future. Rule-based methods can also be used, by applying a set of predefined principles or expectations based on prior understanding of the system, together with recent data, to interpret future developments. In addition, and more objectively, trends can be derived statistically, using regression or related (e.g. autoregressive or Bayesian) methods.

Whatever method is used, care is essential in extrapolation because of the numerous uncertainties involved. Any extrapolation procedure, for example, is based on the assumption that there is valid information in past data and knowledge, and thus that the future is conditioned by the same (or similar) factors to those that have operated previously. While this may be true for business-as-usual scenarios, it clearly is less valid if we are trying to estimate future conditions in the context of new policies or technologies. The amount of information available in the historic record also depends on the length of that record and the accuracy of the data. In some cases it has been suggested that greater weight should be attached to more recent data, both because this is likely to be more accurate and more relevant to current conditions. Nevertheless this can have dangers, especially if there are large, random, short-term variations in the data, which mean that any apparent short-term trend is relatively uninformative. Likewise, the possibility of non-linear (including cyclic or seasonal) variations needs to be recognised, especially when making long-term extrapolations; these are often detectable only with relatively long runs of data.

In the light of these considerations, many analysts have used a combination of autoregressive and moving average functions, in what are known as ARMA (or ARIMA) models. In order to ensure rigour in the way these are applied, the so-called Box-Jenkins methodology is often followed; this defines an explicit set of methods and checks to deal with the uncertainties involved. Nevertheless, there is considerable research in the field of forecasting to show that sophisticated statistical methods do not always out-perform simpler approaches. In the end it seems apparent that the best extrapolations tend to be achieved when prior knowledge about the causal processes operating within the system is combined with robust statistical techniques - and when these are supported by equally robust time series data.

In making extrapolations, therefore, we should alwaysd bear in mind that:

*A trend is a trend is a trend,**But the question is, will it bend?**Will it alter its course**Through some unforeseen force**And come to a premature end?*

- Cairncross (1969), quoted in Armstrong (2001).

## References

- Armstrong, J.S. 2001 Extrapolation for time-series and cross-sectional data. Principles of Forecasting: A Handbook for Researchers and Practitioners (J.S. Armstrong, ed.), Norwell, MA: Kluwer Academic Publishers.
- Armstrong, J.S., Adya, M. and Collopy, F. 2001 Rule-based forecasting: using judgement in time-series extrapolation. Principles of Forecasting: A Handbook for Researchers and Practitioners (J.S. Armstrong, ed.), Norwell, MA: Kluwer Academic Publishers.
- Cairncross, A. 1969 Economic forecasting. Economic Journal 79, 797-812.
- Makrides, S. and Hibron, N. 1997 ARMA models and the Box-Jenkins methodology. Journal of Forecasting 16, 147-163.