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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 $\boldsymbol{\beta}$?

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    $\begingroup$ 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. $\endgroup$ Commented May 27 at 11:16
  • $\begingroup$ Thank you, I will consider it. I posted it here, because these typical economic models usually get better answers here. $\endgroup$
    – Athaeneus
    Commented May 27 at 11:19

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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 $(III)$.

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  • $\begingroup$ 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 $\alpha_0$? $\endgroup$
    – Athaeneus
    Commented May 28 at 6:05
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    $\begingroup$ 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 $i \neq 0$. $\endgroup$ Commented May 28 at 11:07

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