Building and Testing a Multiple Linear Regression Model 91
Let’s see how a simple transformation can work as explained in Chap-
ter 2. Suppose that the true relationship of interest is exponential, that is,
y=βεαx (4.11)
Taking the natural logarithms of both sides of equation (4.11) will result in
lnlnyx=+βα (4.12)
which is again linear.
Now consider that the fit in equation (4.12) is not exact; that is, there is
some random deviation by some residual. Then we obtain
lnlnyx=+βα+ε (4.13)
If we let z = ln y and adjust the table of observations for y accordingly and
let λ = ln β, we can then rewrite equation (4.13) as
z = λ + αx + ε (4.14)
This regression model is now linear with the parameters to be estimated
being λ and α.
Now we transform equation (4.14) back into the shape of equation
(4.11) ending up with
y==βεαεxx⋅⋅εβεξα (4.15)
where in equation (4.15) the deviation is multiplicative rather than additive
as would be the case in a linear model. This would be a possible explanation
of the nonlinear function relationship observed for the residuals.
However, not every functional form that one might be interested in esti-
mating can be transformed or modified so as to create a linear regression.
For example, consider the following relationship:
y = (b 1 x)/(b 2 + x) + ε (4.16)
Admittedly, this is an odd looking functional form. What is important
here is that the regression parameters to be estimated (b 1 and b 2 ) cannot
be transformed to create a linear regression model. A regression such as
equation (4.16) is referred to as a nonlinear regression and the estimation
of nonlinear regressions is far more complex than that of a linear regres-
sion because they have no closed-form formulas for the parameters to be