Chapter 6: Functional Form of the Regression
This chapter shows that the technique of linear regression is an extremely flexible method for describing data. That flexibility derives from the possibility of being able to replace the variables in the regression equation with functions of the original variables. As examples, instead of fitting the equation
Predicted Y = a + bX,
we can fit
Predicted Y = a + bX2,
Predicted ln(Y) = a + bX,
where ln stands for the natural log function. Applying polynomials, multiplying or dividing variables by each other, applying logarithms and exponentials, and taking reciprocals are just a few of the variable transformations available to generate nonlinear fits.
Even though variables may be transformed so that the equation is nonlinear in the original units of the variables, as long as the equation remains in the form of an intercept plus a slope multiplying a (possibly transformed) X variable, it remains a linear regression. In other words, linear regression means linear in the parameters, not the variables. For example, Predicted Y = 1/a + b2X is a nonlinear regression model because the parameters themselves enter into the equation in a nonlinear way. This model cannot be fit using the usual least squares intercept and slope formulas. We will review a specific kind of nonlinear regression model in Chapter 22 but otherwise confine ourselves to linear regression in this book.
This chapter begins with an example of a famous nonlinear equation from the physical sciences. The example will allow us to explore theoretical and practical reasons for using different functional forms. Next, we return to the infant mortality and GDP per capita data set to demonstrate the double-log and reciprocal specifications. The fourth section is devoted to the semilog functional form, which has dominated empirical work in labor economics since it was introduced in the late 1950s. Finally, we explain how elasticities are computed from fitted lines and show how the functional form impacts the elasticity. The appendix to this chapter contains a catalog of functional forms commonly used by economists listing advantages and disadvantages of each specification.