## Extrapolation

Extrapolation is the word which describes the continuation of a trend, usually quantitative. As applied to time series, the idea is that, barring unexpected events, the future will be like the past. If the sun has come up every morning for 10 000 years, it seems reasonable to assume it will come up again tomorrow. If the moon has waxed for 14 days and waned for 14 days for 10 000 years, it seems reasonable to assume that the pattern will be repeated next month. If spring, summer, fall and winter have followed each other regularly for 10 000 years, we feel comfortable in assuming that they will do so again in the coming year.

It has been said that man is a pattern-making animal. More accurately, some of us are good at pattern-recognition. The discovery of regularities in the observed motions of the stars and planets by the Babylonians was certainly the first step towards developing the science of astronomy. Copernicus was the first to provide a theory of sorts to explain the observable patterns, while retaining the notion that Earth was the center of the universe. His theory of cycles and hyper-cycles was soon overturned by Kepler and Galileo, who realized that the earth and planets revolve around the sun in elliptical orbits. Newton asked himself what sort of force law would account for an elliptical orbit, and concluded that such a force would have to be inversely proportional to the square of the distance from the point of attraction. From this kind of reasoning came Newton's laws of motion and the law of gravity. Newton's theory of gravitational attraction was accepted for 250 years. But some small but persistent deviations from the Newtonian scheme, notably the precession of the perihelion of the orbit of Mercury, inspired Einstein to formulate his more general theory. And so it goes. From patterns one can postulate and test cause-effect relationships. As competing hypotheses are winnowed, 'laws of nature' may emerge.

Evidently pattern recognition is a basic tool for forecasting. The simplest pattern of all is a geometrically straight line. Motion in a straight line is likely to continue along the same line unless it is deflected by some force. Newton's law of gravity is a more complex extrapolation, but no different in principle. Different 'laws of motion', such as exponential growth (compound interest), the logarithmic, sinusoidal and logistic functions, are commonplace in physics and economics. Quite elaborate statistical methods have been developed to identify superimposed sinusoidal patterns, such as business cycles, from apparently random fluctuations. But the underlying 'force laws' explaining such behaviors remain obscure. However, if a Delphic forecast panel of leading economists were created today and the question to them were 'what average real growth rate do you expect the US to enjoy during the 21st century?' we think the answer would fall between 1.5 percent and 2.5 percent per annum. But that is because virtually all economists assume that economic growth is automatic and costless, and that it is independent of energy price or availability.

Since the US economy has grown at a relatively steady rate for well over a century (nearer two centuries), it seems reasonable to suppose that it will continue to do so. In fact, it is usually reasonable to assume past trends will continue for some time to come, absent definite reasons for expecting a change of direction. A supertanker has enormous inertia. It cannot turn on a proverbial dime. The economy also has a lot of inertia.

We need to digress for a moment, here, to define the term 'trend'. The most common sort of trend in economics is exponential. Extrapolation of an exponential trend is tantamount to extrapolating the logarithms of the appropriate time series as a straight line. Of course there is an enormous gap between simple extrapolations of single variables and elaborate and complex models intended to capture a variety of interacting trends simultaneously. Nevertheless, it happens that most complex economic models depend on several extrapolations, as will be seen.

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