Linear model estimated as log-log: What is the bias exactly?

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 $$\boldsymbol{\beta}$$?

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

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 $$(III)$$.
• 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 $\alpha_0$? Commented May 28 at 6:05
• 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 $i \neq 0$. Commented May 28 at 11:07