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Table 34 shows the relationship using selected values of x. Notice that this relationship is essentially linear due to the small secondorder coefficient. It is not necessary to solve both the linear and curvilinear models, because the secondorder equation can be used in the linear form when this situation occurs. Transforming Nonlinear to Linear Two curves that are often used to forecast prices are logarithmic (power) and exponential relationships (see Figure 3"). The exponential, curving up, is used to scale price data that become more volatile at higher levels. Each of these forms can be solved with unique equations. however, both can be easily transformed into linear relationships and solved using the method of least squares. This will allow you to fool the computer into solving a nonlinear problem using a linear regression tool. The significant difference in the transformations is that the value of x is not scaled for the exponential. Taking the original data and performing the appropriate natural log ftmctions. In results in the linear form that can be solved using least squares. Selected results from the computer program in Appendix 2 are shown in Table 35. EVALUATION OF 2VARIABLE TECHNIQUES Of the three curvefitting techniques, the curvilinear and exponential results are very similar, both curving upward and passing through the main cluster of data points at about the same incline. The log spproximation curves downward after passing through the main group of data points at about the same place as the other approximations. To evaluate objectively whether any of the nonlinear methods are a better fit than the linear spproximation. find the standard deviation of the errors, which gives a statiatical measurement of how close the fitted line comes to the original data points. The results show that the curvilinear is best: the logarithmic, which curves downward, is noticeably the worst (see Figure 38). In the case of the cornsoybean relationship, the model with the smallest vanance makes the most sense. Prices will usually remain within the range tested, and the realities of the productionprice relationship support the selection of the linear (or curvilinear) values. The use of regression analjsis for forecasting the price of soybeans alone has a different conclusion. Although 27 years were used, the last 10 showed a noticeable increase in soybean prices. This rising pattem is best fit by, the curvilinear and exponential models. However, forecasts using these formulas show prices continuing to rise at an increasing rate. Had inflation maintained its doubledigit rate, these forecasts would atill lead to unrealiatically high prices. The logarithmic model, which leveled off after the rise, tumed out to FIGURE 37 Logarithmic and exponential. Logarithmic (Power)
urce ?utiibertDaiiiel andFredS lod. Fitting Fijuations to Data nicuter Aualvs 20,21) Eefnntedw ih tifact.TD..K ;nded..lTewYork J..Li,W,le«&Son be the best for the actual situation. This shows that the problem of forecasting is more complex than this naive solution. The logarithmic model, which showed the worst statiatical results, provided the best forecast. Major factors that cause significant price shifts, such as interest rates, inflation, and the value of the U.S. dollar, must be monitored carefully. The model must be reestimated whenever these factors change. The section Approximations" will discuss this in more detail. Direct Relationships The price at which a maitet trades is limited by the cost of production and dependent upon the prices of other maitets that provide a subatitute product. Arbitrage is based on FIGURE 38 Leastsquares approximation for soybeans using linear, curvilinear, logarithmic, and exponential models. the ability to substitute one product for another after transaction costs, which can include carrjing chaises, shipping, inspection, and commissions. These relationships are carefully monitored and arbitrage opportunities are quickly acted upon by traders. This process keeps futures and cadi prices together and prevents the price of gold in New York London, and Hong Kong from drifting apart any further than the cost of bujinggoldin one location and delivering it to another. For interest rate maitets. strips prevent the large pool of 3month vehicles, notes, and bonds from offering widely different rdums for the same maturity. The Interbant maitet provides the same stability for foreign exchange maitets, while in agriculture the soybean crush and other processing maiins do not stay outofline for long. The
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