Chapter 11 Functional Forms

\[Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \varepsilon_i\]

The regression model we have studied thus far has two main features. First, the model is linear in the coefficients. This important property allows us to estimate the model using OLS. Second the model is linear in the variables. This property imposes a linear relationship between the dependent and independent variables. In other words, the relationship is a straight line.

A linear relationship between the dependent and an independent variable results in a slope that is constant.

\[\beta_1 = \frac{\Delta Y_i}{\Delta X_{1i}}\]

This means that (holding \(X_{2i}\) constant, of course) the expected value of \(Y\) will increase by \(\beta_1\) units in response to any unit-increase in \(X_{1i}\). The same increase occurs on average no matter where in the range of the independent variable you are. Sometimes this assumption is valid if the range of the independent variable is small enough such that a constant slope is appropriate. Sometimes it isn’t. If this assumption is not valid, then we are committing a specification error even if we have included all of the necessary independent variables. The specification error involves the assumption of a linear model.

The types of models we will consider here are non-linear in the variables but still linear in the coefficients. This means we can estimate the models using OLS, but we will be extending the model to uncover some highly non-linear relationships between the dependent and independent variables. We will do this by transforming the variables prior to estimation, and the names of these models are given by the types of transformations we perform. We then run a traditional OLS estimation on the transformed variables, and back-out the non-linear relationships afterwards. This last bit is what will be new to us, but you will see that it only involves a brief refresher of… calculus.