<p>The questions of what the real DGP is and how you interpret the coefficients of the model you estimate are entirely unrelated.</p>

<p>(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.</p>

<p>(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.</p>

CC BY-SA 3.0
2016-12-17T20:16:36.290Z
2016-12-17T20:16:36.290Z
Tobias
<p>A possible answer if it's not too late.</p>

<p>Q1: Yes, if you mean "per years of schooling" at the end of your question. So, the unit of $\beta x_i$ is $\$/hr$.</p>

<p>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 $\beta$.</p>

CC BY-SA 3.0
2016-10-18T13:17:59.563Z
2016-10-18T13:17:59.563Z
chan1142
<p>Say you want to estimate return to education. </p>

<p><strong>Model 1 (no log)</strong>: assume the DGP is:</p>

<p>$$w_{i} = \alpha + \beta x_{i} + \zeta_{i} + e_{i}$$</p>

<p>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. </p>

<p><strong>Question 1</strong>: 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.</p>

<p><strong>Model 2 (log)</strong>: assume the DGP is:</p>

<p>$$\ln{w_{i}} = \alpha + \beta x_{i} + \zeta_{i} + e_{i}$$</p>

<p>where wage, $x$, $\zeta_{i}$ and $e$ are as defined above.</p>

<p>According to answers to <a href="https://math.stackexchange.com/questions/238390/units-of-a-log-of-a-physical-quantity">this question</a>, the log transformation of a variable is dimensionless. In consequence, $\ln{w_{i}}$ is dimensionless. 
<strong>Question 2</strong>: 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? </p>

<econometrics><applied-econometrics><wages>
Interpretation of regression coefficients with and without logs
luchonacho
# Interpretation of regression coefficients with and without logs

(Asked by: username_0 on 2016-07-04)

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](https://math.stackexchange.com/questions/238390/units-of-a-log-of-a-physical-quantity), 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?

**Tags:** econometrics, applied-econometrics, wages


---

## 2 answers

### Answer by username_1 on 2016-12-17. Score: 1 | Accepted answer

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.


---

### Answer by username_2 on 2016-10-18. Score: 2

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 $\beta$.


---

