Logging an x-variable

Summary

• In a linear model where one or more explanatory variables are logged we have

\[β_j= \frac{∂E\left( y|x \right)}{∂log x_j}= \frac{∂E\left( y|x \right)}{∂x_j}x_j \]

• for all logged explanatory variables. \(β_j\) /100 measures, approximately, the unit increase in \(E\left( x \right)\) if we increase \(x_j\) by 1% keeping the other explanatory variables constant.