I understand that taking the logarithm of the multiplicative error model transforms it into the additive error model. Let $y'$ be the observed response variable, with $y$ being the true response variable and $\alpha$ the multiplicative error and $\log(\alpha)=\alpha'$. We know that $\mathrm{E}(\alpha) = 1$. $$ \log(y') = \log(y \cdot \alpha) = \log(y) + \log(\alpha) = \beta X + \epsilon \\ \Downarrow \\ \log(y) = \beta X + \epsilon - \log(\alpha) = \beta X + \epsilon - \alpha'. $$ I want to know how multiplicative measurement error affects the model, and our assumptions, if we do not take log of response variable. Is it possible to write the following? If so, how do we interpret this, and how does it affect our OLS model assumptions? $$ y \cdot \alpha = \beta X + \epsilon \\ \Downarrow \\ y = \frac{1}{\alpha}\beta X + \frac{1}{\alpha} \epsilon $$
Assume that the standard OLS model $y = \beta X + \epsilon$ meets all assumptions.
Similarly, I want to understand what happens when you take the logarithm of the additive error model. Thus far, I have $$ \log(y') = \log(y+\alpha) = \beta X + \epsilon \\ \Downarrow \\ y = \mathrm{e}^{\beta X + \epsilon} - \alpha $$
This is no longer a linear model, correct? Can this be estimated with nonlinear least squares, or would we need to use nonparametric methods?