How do I interpret a regression coefficient when a variable is in logs?
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The question
My model regresses log wages on years of education and log firm size. I get coefficients of 0.08 and 0.05 but I am not sure how to write up what they mean.
Short answer
When the dependent variable is logged and the regressor is not, multiply the coefficient by 100 for an approximate percentage change: one more year of education is associated with about 8% higher wages. When both are logged, the coefficient is an elasticity: a 1% larger firm is associated with 0.05% higher wages.
Full expert answer
Econometrics tutor
MSc Econometrics, LSE
The interpretation depends on which side of the equation is logged. There are four combinations and each one reads differently.
- Level-level: a one unit increase in x changes y by β units
- Log-level: a one unit increase in x changes y by approximately 100 × β percent
- Level-log: a 1% increase in x changes y by approximately β / 100 units
- Log-log: a 1% increase in x changes y by approximately β percent, which is an elasticity
Applying it to your model
Education is log-level, so 0.08 means each extra year is associated with roughly 8% higher wages, holding firm size constant. Firm size is log-log, so 0.05 is an elasticity: wages rise by about 0.05% for each 1% increase in firm size.
Use 'associated with' unless your identification strategy supports a causal reading. Examiners notice when observational estimates are described as effects.
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