Reciprocal Models

Models of the following type are known as reciprocal models.

Although this model is nonlinear in the variable X because it enters inversely or reciprocally, the model is linear in 0i and 02 and is therefore a linear regression model.i7

This model has these features: As X increases indefinitely, the term 02(l/X) approaches zero (note: 02 is a constant) and Y approaches the limiting or asymptotic value 0i. Therefore, models like (6.7.i) have built in them an asymptote or limit value that the dependent variable will take when the value of the X variable increases indefinitely.i8

Some likely shapes of the curve corresponding to (6.7.i) are shown in Figure 6.6. As an illustration of Figure 6.6a, consider the data given in Table 6.4. These are cross-sectional data for 64 countries on child mortality and a few other variables. For now, concentrate on the variables, child mortality (CM) and per capita GNP, which are plotted in Figure 6.7.

As you can see, this figure resembles Figure 6.6a: As per capita GNP increases, one would expect child mortality to decrease because people can afford to spend more on health care, assuming all other factors remain constant. But the relationship is not a straight line one: As per capita GNP increases, initially there is dramatic drop in CM but the drop tapers off as per capita GNP continues to increase.

i7If we let X? = (i/X;), then (6.7.i) is linear in the parameters as well as the variables Y; and X?.

i8The slope of (6.7.i) is: dY/dX = — 02(i/X2), implying that if 02 is positive, the slope is negative throughout, and if 02 is negative, the slope is positive throughout. See Figures 6.6a and 6.6c, respectively.


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