The log-log model

Summary

• If

\[E(y_i|x_i)=β_1x_{i,2}^{β_2}x_{i,3}^{β_3}…x_{i,k}^{β_k} , i=1,…,n\]

• we can define

\[ε_i= \frac{y_i}{E(y_i|x_i)}\]

• and

\[y_i=β_1x_{i,2}^{β_2}x_{i,3}^{β_3}…x_{i,k}^{β_k}ε_i , i=1,…,n\]

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

\[log \left( y_i \right)=log \left( β_1 \right)+β_2log \left( x_{i,2} \right)+β_3log \left( x_{i,3} \right)+…+β_klog \left( x_{i,k} \right)+log \left( ε_i \right) , i=1,…,n\]

• or

\[log \left( y_i \right)=β'_1+β_2log \left( x_{i,2} \right)+β_3log \left( x_{i,3} \right)+…+β_klog \left( x_{i,k} \right)+ε'_i , i=1,…,n\]

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

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

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