# Interpretation of regression coefficients with and without logs

$$w_{i} = \alpha + \beta x_{i} + \zeta_{i} + e_{i}$$
where $x$ is years of schooling, $\zeta_{i}$ is unobserved heterogeneity (e.g. capturing ability), and $e$ is an idiosyncratic error. Furthermore, wage is measured in \$per hour. Question 1: since the dependent variable is in wage per hour, does this imply that$\alpha, \zeta_{i}$,$e_{i}$are also in wage per hour? Same for the composite$\beta x_{i}$? If this is the case, then$\beta$-- the return to education -- is defined as wage per hour, per year of experience. Model 2 (log): assume the DGP is: $$\ln{w_{i}} = \alpha + \beta x_{i} + \zeta_{i} + e_{i}$$ where wage,$x$,$\zeta_{i}$and$e$are as defined above. According to answers to this question, the log transformation of a variable is dimensionless. In consequence,$\ln{w_{i}}$is dimensionless. Question 2: does this imply that$\alpha, \zeta_{i}$and$e_{i}$are also dimensionless? Is the composite$\beta x_{i}$dimensionless? If so, the dimension of$\beta$is "1 over years of experience"(!). How it is then that we can interpret$\beta$as the % change in wage following a extra year of education? ## 2 Answers The questions of what the real DGP is and how you interpret the coefficients of the model you estimate are entirely unrelated. (1) For your first model$\frac{\partial w}{\partial x} = \beta$. The interpretation is that the predicted w will increase by$\beta$units of whatever unit w is in when you increase x by 1 unit of whatever unit x is in. (2) For your second model$\frac{\partial w}{\partial x}\frac{x}{w} = \beta$. That means$\beta$is an elasticity and if you increase x by 1%, then the w predicted by your model will increase by$\beta$%. Since proportional increases are independent of units, you may say that your model/coefficient is dimensionless, but I think the term you are looking for is elasticity. A possible answer if it's not too late. Q1: Yes, if you mean "per years of schooling" at the end of your question. So, the unit of$\beta x_i$is$\$/hr$.
Q2: The unit of $\beta$ is $\log(\$/hr)/years$literally. Then the unit of$\Delta (\beta x_i)$is$\Delta \log(\$/hr)$, nothing strange, which is approximately the % change times 0.01. This is "dimensionless" if you like to say so, or would "dimension-irrelevant" be better? We interpret $\Delta (\beta x)$ as % change, not $\beta$.