The log-linear model

Summary

• If

\[E(y_i|x_i)=β_1exp \left( β_2x_{2i}+β_3x_{3i}+…+β_kx_{ki} \right) , i=1,…,n\]

• we can define

\[ε_i= \frac{y_i}{E(y_i|x_i)}\]

• and

\[y_i=β_1exp \left( β_2x_{2i}+β_3x_{3i}+…+β_kx_{ki} \right)ε_i , i=1,…,n\]

• We can transform this into a linear regression model by taking log of both sides:

\[log \left( y_i \right)=log \left( β_1 \right)+β_2x_{2i}+β_3x_{3i}+…+β_kx_{ki}+log \left( ε_i \right) , i=1,…,n\]

• or

\[log \left( y_i \right)=β'_1+β_2x_{2i}+β_3x_{3i}+…+β_kx_{ki}+ε'_i , i=1,…,n\]

• which is a linear model (linear in parameters, not in data).
• For this model

\[β_j= \frac{∂log \left( E\left( y|x \right) \right)}{∂x_j}= \frac{∂E\left( y|x \right)}{∂x_j} \frac{1}{E\left( y|x \right)} , j=2,…,k\]

• \(β_j\) *100 measures, approximately, the percentage increase in \(E\left(y | x \right)\) if we increase \(x_j\) by 1 unit keeping the other explanatory variables constant.