Linear regression models which are nonlinear in data
Summary
Main points:
- If a regression model is nonlinear in data then we can create new variables transforming it into a linear regression model which is linear in data.
- The result
\[ \frac{∂E\left( y|x \right)}{∂x_j}=β_j , j=2,…,k\]
- is no longer necessarily true.
- We often use squares of \(x\) -variable to capture nonlinear effects in the data.
- We often use interactive terms to allow the marginal effect of one variable to depend on another variable.
Example 1:
\[y_i=β_1+β_2x_{i,2}+β_3x_{i,2}^2+ε_i i=1,…,n\]
- Create \(x_{i,3}=x_{i,2}^2\) and estimate the model
\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+ε_i i=1,…,n\]
- For this model,
\[ \frac{∂E\left(y| x \right)}{∂x_2}=β_2+2β_3x_2\]
- which depends on \(x_2\) .
- \(β_2+2β_3x_2\) is estimated by \(b_2+2b_3x_2\) .
Example 2:
\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+β_4x_{i,2}x_{i,3}+ε_i i=1,…,n\]
- Create \(x_{i,4}=x_{i,2}x_{i,3}\) (the interactive term) and estimate the model
\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+β_4x_{i,4}+ε_i i=1,…,n\]
- For this model,
\[ \frac{∂E\left( y|x \right)}{∂x_2}=β_2+β_4x_3\]
- which depends on \(x_3\) .
- \(β_2+β_4x_3\) is estimated by \(b_2+{2b}_4x_3\) .