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In this section we add a "best-fitting" exponential curve to the data plot and interpret this exponential trend curve.
For CPI data from 1913 through 2002, the equation of the fitted curve is . Your equation may be slightly different if your table includes different data. Here, x is the number of years since 1913, and y is the value of the Consumer Price Index. The statistical term R ^{2} is the coefficient of determination, which in this case is 0.9101. This statistic varies between 0 and 1 and measures how well a trend curve explains the data. An R ^{2} greater than 0.9 suggests a good fit, but the picture shows some obvious problems with the fit. In the next two paragraphs -- for which a little background in statistics is helpful -- I explain why the high R ^{2} does not necessarily mean a good fit.
As we will see shortly, the exponential fit is based on fitting a regression line to something. In regression analysis we fit a straight line to points . (The variables x and y here are "generic" -- they are not necessarily our time and CPI variables.) For each value of the independent variable x and the dependent variable y, we may write , where m and b are the regression coefficients, and (called the residual) is the difference between the actual and fitted values of y. The assumptions of regression analysis are that the values of for all values of x are independent and normally distributed with mean 0 and variance the same for every x. If these assumptions are correct, we should see the data points cluster about the regression line, bouncing erratically above and below it.
For fitting an exponential curve to data, we make corresponding assumptions about the logarithm of y. We fit a regression line to the points , which should then cluster and bounce about the line, if our assumptions are correct. Then we expect the points to cluster and bounce about the corresponding exponential curve. But the CPI points are above the exponential curve until year 18 (1931) and again after year 66 (1979). In between, they are below the curve. This pattern displays a strong dependence of values of on nearby values, not the random fluctuations that we expect. It follows that the assumptions for a regression fit are not satisfied by the CPI data.
Nevertheless, the exponential trend curve does give us a general picture of the growth in CPI over time, and that's the significance of the relatively large R ^{2}. In the next two sections, we will see some algebraic reasons for not placing too much confidence in the trend curve -- in particular, we will see that there is no way to predict inflation accurately. An economist (Haimowitz) notes that governments often manipulate prices. This is certainly the case in controlled economies, but it happens in the U.S. too. This is one of the economic reasons why inflation has often not been exponential. However, learning to fit a curve to data is useful, if only for discerning general trends, and of course you can apply this skill in other situations as well.
Elizabeth B. Appelbaum, "The Consumer Price Index and Inflation - Insert an Exponential Trend Curve," Loci (December 2004)
Journal of Online Mathematics and its Applications