<p>There is a very simple relationship between these two models (i.e. <span class="math-container">$(II)$</span> and <span class="math-container">$(I)$</span>):
<span class="math-container">$$\beta_{i}=\frac{\partial{y}}{\partial{x_{i}}}\quad \forall i \in \{1,2,3,\dots\}$$</span>
<span class="math-container">$$\alpha_{i}=\frac{\partial{\ln y}}{\partial{\ln x_{i}}}\quad \forall i \in \{1,2,3,\dots\}\\
\Rightarrow \alpha_{i}= \frac{\frac{\partial{y}}{y}}{\frac{\partial{x_{i}}}{x_{i}}}\quad\forall i \in \{1,2,3,\dots\}\\
\Rightarrow \frac{\partial{y}}{\partial{x_{i}}}=\alpha_{i}\frac{x_{i}}{y}\quad\forall i \in \{1,2,3,\dots\}\\
\Rightarrow \beta_{i}=\alpha_{i}\frac{x_{i}}{y}\quad\forall i \in \{1,2,3,\dots\}$$</span></p>
<p>For an estimate you can take the respective mean values i.e.
<span class="math-container">$$\beta_{i}=\alpha_{i}\frac{\bar{x_{i}}}{\bar{y}}.$$</span></p>
<p>For <span class="math-container">$\beta_{0}$</span> and <span class="math-container">$\alpha_{0}$</span> I think one can do manipulations to find the relation, but it is not very simple and elegant.</p>
<p>I don't see an point in finding the relationship between models <span class="math-container">$(II)$</span> and <span class="math-container">$(III)$</span>.</p>

CC BY-SA 4.0
2024-05-28T06:05:23.877Z
Thank you very much. This was great answer! Additionally, just for curiosity, what kind of manipulations, roughly, could you do to interconnect $\beta_0$ and $\alpha_0$?
Athaeneus
CC BY-SA 4.0
2024-05-28T11:07:15.710Z
You could use the relation $\beta_{0}=\bar{y}-\beta_{1}\bar{x_{1}}-\beta_{2}\bar{x_{2}}-\beta_{3}\bar{x_{3}}$ and manipulate to substitute $\alpha_{i}$ in place of $\beta_{i}$ for $i \neq 0$.
Pallak Goyal
CC BY-SA 4.0
2024-05-27T14:08:00.363Z
2024-05-27T16:38:37.650Z
2024-05-27T16:38:37.650Z
Pallak Goyal
<p>Consider a causal model:</p>
<p><span class="math-container">$$ \text{(I)} \qquad y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon$$</span></p>
<p>Now, assume we estimate it by OLS in the form:</p>
<p><span class="math-container">$$ \text{(II)} \qquad \ln(y) = \alpha_0 + \alpha_1 \ln(x_1) + \alpha_2 \ln(x_2) + \alpha_3 \ln(x_3) + u$$</span></p>
<p>We can easily identify that this does not correspond to:</p>
<p><span class="math-container">$$ \text{(III)} \qquad \ln(y) = \ln(\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \epsilon)$$</span></p>
<p><strong>Can we, though, identify what precisely we estimate by model (II) and how does this relate to model (I)?</strong></p>
<p>For instance, how do coefficient <span class="math-container">$\boldsymbol{\alpha}$</span> relate to coefficients <span class="math-container">$\boldsymbol{\beta}$</span>?</p>

CC BY-SA 4.0
2024-05-27T11:16:27.893Z
This seems like a good question for Cross Validated (the statistics site of Stack Exchange). Posting on multiple SE sites is discouraged, but if you do not get an answer here over, say, 3 days, then I think you could post there, too.
Richard Hardy
CC BY-SA 4.0
2024-05-27T11:19:27.710Z
Thank you, I will consider it. I posted it here, because these typical economic models usually get better answers here.
Athaeneus
<econometrics>
Linear model estimated as log-log: What is the bias exactly?
Athaeneus
# Linear model estimated as log-log: What is the bias exactly?

(Asked by: username_0 on 2024-05-27)

Consider a causal model:

$$ \text{(I)} \qquad y = \beta\_0 + \beta\_1 x\_1 + \beta\_2 x\_2 + \beta\_3 x\_3 + \epsilon$$

Now, assume we estimate it by OLS in the form:

$$ \text{(II)} \qquad \ln(y) = \alpha\_0 + \alpha\_1 \ln(x\_1) + \alpha\_2 \ln(x\_2) + \alpha\_3 \ln(x\_3) + u$$

We can easily identify that this does not correspond to:

$$ \text{(III)} \qquad \ln(y) = \ln(\beta\_0 + \beta\_1 x\_1 + \beta\_2 x\_2 + \beta\_3 x\_3 + \epsilon)$$

**Can we, though, identify what precisely we estimate by model (II) and how does this relate to model (I)?**

For instance, how do coefficient $\boldsymbol{\alpha}$ relate to coefficients $\boldsymbol{\beta}$?

**Tags:** econometrics
> **Comments:**
> **username_1**: This seems like a good question for Cross Validated (the statistics site of Stack Exchange). Posting on multiple SE sites is discouraged, but if you do not get an answer here over, say, 3 days, then I think you could post there, too.  
> **username_0**: Thank you, I will consider it. I posted it here, because these typical economic models usually get better answers here.  

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## 1 answers

### Answer by username_2 on 2024-05-27. Score: 2 | Accepted answer

There is a very simple relationship between these two models (i.e. $(II)$ and $(I)$):
$$\beta\_{i}=\frac{\partial{y}}{\partial{x\_{i}}}\quad \forall i \in \{1,2,3,\dots\}$$
$$\alpha\_{i}=\frac{\partial{\ln y}}{\partial{\ln x\_{i}}}\quad \forall i \in \{1,2,3,\dots\}\\
\Rightarrow \alpha\_{i}= \frac{\frac{\partial{y}}{y}}{\frac{\partial{x\_{i}}}{x\_{i}}}\quad\forall i \in \{1,2,3,\dots\}\\
\Rightarrow \frac{\partial{y}}{\partial{x\_{i}}}=\alpha\_{i}\frac{x\_{i}}{y}\quad\forall i \in \{1,2,3,\dots\}\\
\Rightarrow \beta\_{i}=\alpha\_{i}\frac{x\_{i}}{y}\quad\forall i \in \{1,2,3,\dots\}$$

For an estimate you can take the respective mean values i.e.
$$\beta\_{i}=\alpha\_{i}\frac{\bar{x\_{i}}}{\bar{y}}.$$

For $\beta\_{0}$ and $\alpha\_{0}$ I think one can do manipulations to find the relation, but it is not very simple and elegant.

I don't see an point in finding the relationship between models $(II)$ and $(III)$.

> **Comments:**
> **username_0**: Thank you very much. This was great answer! Additionally, just for curiosity, what kind of manipulations, roughly, could you do to interconnect $\beta_0$ and $\alpha_0$?  
> **username_2**: You could use the relation $\beta_{0}=\bar{y}-\beta_{1}\bar{x_{1}}-\beta_{2}\bar{x_{2}}-\beta_{3}\bar{x_{3}}$ and manipulate to substitute $\alpha_{i}$ in place of $\beta_{i}$ for $i \neq 0$.  

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