Econometrics: Choosing the Functional Form of Your Regression Model
In econometrics, the standard estimation procedure for the classical linear regression model, ordinary least squares (OLS), can accommodate complex relationships. Therefore, you have a considerable amount of flexibility in developing the theoretical model. You can estimate linear and nonlinear functions including but not limited to
Polynomial functions (for example, quadratic and cubic functions)
Log functions (log-log, log-linear, and linear-log)
In many cases, the dependent variable in a regression model can be influenced by both quantitative variables and qualitative factors. Aside from keeping track of the units of measurement or converting to a log scale, the use of quantitative variables in regression analysis is usually straightforward. Qualitative variables, however, require conversion to a quantitative scale using dummy variables, which equal 1 when a particular characteristic is present and 0 otherwise. (Note that when more than two qualitative outcomes are possible, the number of dummy variables you need is the number of outcomes minus one.) Utilizing both quantitative and qualitative variables generally results in richer models with more informative results.
Although some experimentation with the exact form of your regression model can be enlightening, take the time to think through specification issues methodically. Be sure you can explain why you’ve chosen specific independent variables for your model. You should also be able to justify the functional form you’ve chosen for the model even if you’ve assumed a simple linear relationship between your variables. Test the assumptions of the classical linear regression model (CLRM) and make changes to the model as necessary. Finally, spend some time examining the sensitivity of your results by making slight modifications to the variables (sometimes influenced by the outcomes of your CLRM tests) included in the model and the functional form of the relationship. If your results are stable to these types of variations, that provides additional justification for your conclusions.