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Say you want to estimate return to education.

Model 1 (no log): assume the DGP is:

$$w_{i} = \alpha + \beta x_{i} + \zeta_{i} + e_{i}$$

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?

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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.

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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 $\beta$.

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