Forecasting is involved with trends of variables in time and the belief that trends can be extrapolated. The more chaotic a system is, the less confidence can be attached to extrapolations. A famous saying is "A trend is a trend is a trend until it will bend". So extrapolations, without a proper assessment of the uncertainties can be dangerous depending on the context. However, certain processes are reasonably predictable, such as demographics and transport. Any change in such processes is slow enough to have confidence in their continuance. In
Various methods for forecasting as "trend extrapolation" have been proposed. I have encountered only a few and attempted their use:
Examples of forecasts are oil price, OPEC production, exploration effort in number of wildcats per year, etc.. A single linear regression using past data can be used to extrapolate into the future. The statistical significance of the observed trend may be exagerated because of autocorrelation in the sequence. On the other hand, such autocorrelation may show the structure of the process, such as a random walk, a moving average process, or other regularities which can be used to refine the extrapolation.
An example of forecasting with a single variable is an estimate of the total ultimate oil resources of the world (Odell & Rosing, 1973). Historical estimates of this total were plotted against time and a linear regression vs, time calculated. Estimates increased with time and a significant trend seen. However extrapolating this would not make much sense. In the first place: the estimates comprise ever increasing parts of the globe, because more areas can be explored technically. So the data for the regression are not at all "iid", (independent). And second: there is only one globe. This latter condition suggests a logistic approach, rather than a linear one.
Examples, involving two variables in this website are Groningen earthquakes, using one year l lag'. In the early 80ies I tried to make a forecast of oil production in the non-OPEC countries, based on oil price. The correlations suggested that production increases were significantly linked to the oil price two and three years before. Quite logical, as it takes some time between realizing that oil prices increase, have confidence in this increase for the near future and invest in production. It appeared that an important factor was the grear number of "stripper wells" in the USA that could react to oil price reasonably fast. So, a lagged regression gave forecasts under various oil price scenarios. But then came the 1986 price collapse and this change was too much for the model I had constructed.
Here the understanding of the process is used to extrapolate into the future. It will normally involve a number of variables, that themselves can be extrapolated in time. An example is
creaming, a way of forecasting future rates of discovery, hence "exploration process modeling".
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Single variable time trend extrapolation
Multivatiate time trend extrapolation
Process modeling