11.3 Functional Forms

Despite the fact that the regression model is linear, non-linear relationships can be measured. For example, the relation between consumption and income might be non linear since a change in consumption due to extra income may decrease with income. Or the relationship between income and education can exhibit a non-linear form because a change in income due to more education may decrease with more education. Consider the following relationship between \(y\) and \(x\): \[y = \beta_0 + \beta_1 x + \beta_2 x^2\] If a nonlinear relation can be expressed as a linear relation by re-defining variables we can estimate that relation using ordinary least square. For the aforementioned equation, we can define new variables \(x_1\) and \(x_2\): \(x_1 = x\) and \(x_2 = x^2\). Note that this will lead to a multivariate relationship.