Marginal effect for a nonlinear model
Problem
Consider \(y_i=β_1+β_2x_{2i}x_{3i}+ε_i\) where GM applies.
a. Find \(E\left( y \mid x \right)\)
b. Find
\[ \frac{∂E\left( y \mid x \right)}{∂x_2}\]
c. In the linear regression model, \(∂E\left( y \mid x \right)/∂x_2=β_2\) . Will this result hold for this model?
Solution
a.
\[E\left( y \mid x \right)=E\left( β_1+β_2x_2x_3+ε \mid x \right)=E\left( β_1 \mid x \right)+E\left( β_2x_2x_3 \mid x \right)+E\left( ε \mid x \right)=\]
\[β_1+β_2x_2x_3+0=β_1+β_2x_2x_3\]
simply the right hand side of the LRM without the error term. This is a “generic” observation where I have skipped subscript “i”, it applies equally well to all observations.
b. This is the derivative of \(β_1+β_2x_2x_3\) with respect to \(x_2\) which is \(β_2x_3\) . The marginal effect of \(x_2\) is \(β_2x_3\) which here depends on \(x_3\) (not constant as in the linear regression model)
c. No. The beta-parameters in a nonlinear model are no longer the marginal effects. The no longer have any obvious interpretation.