Multiplicative errors for linear model I am trying to figure out the 'standard' way of handling multiplicative error in a linear model, i.e. my model reads:
$$
Y_i = (ax_i + b)\varepsilon_i
, \quad \varepsilon_i\sim\mathcal{N}(1, \sigma^2)
$$
How do I fit this (in R)? My idea was to use a log transform on both sides:
$$
Z_i = \log(Y_i) = \log(ax_i + b) + \log(\varepsilon_i)
$$ 
The additive errors are now lognormally distributed, but how do I fit this? I would say this is a GLM with link function $g=\exp$ and lognormal errors, but R does not seem to know such a model. Am I missing something? This seems to be an extremely 'standard' model...
Edit: I obviously mixed something up here ;) $\varepsilon_i$ should be lognormally distributed, then it is valid to take logs. But still, the link function would be the exponential - or not? So I tried
glm(z ~ x, data = data.frame(x, z = log(y)), family = gaussian(link = 'exp'))

but the link function does not exist...
 A: This is a standard question on econometrics comprehensive exams.  I don't think I've ever seen it used in practice, though.  Here is the standard answer.
First, usually you would want to assume $E\{\epsilon_i|X\}=1$ and $V\{\epsilon_i|X\}=\sigma^2$.  This is as close as you can get to the usual regression modelling assumptions here.  Thinking about the mean of $Y$:
\begin{align}
E\{Y_i|X\} &= E\left\{ \left( ax_i+b\right)\epsilon_i|X\right\}\\
           &= \left( ax_i+b\right)E\left\{ \epsilon_i|X\right\}\\
           &= \left( ax_i+b\right)1\\
           &= \left( ax_i+b\right)
\end{align}
Now, let's think about the variance of $Y$:
\begin{align}
V\{Y_i|X\} &= V\left\{ \left( ax_i+b\right)\epsilon_i|X\right\}\\
           &= \left( ax_i+b\right)^2V\left\{ \epsilon_i|X\right\}\\
           &= \left( ax_i+b\right)^2\sigma^2
\end{align}
Hmmmmm.  That kind of reminds me of a heteroskedastic regression model, like:
\begin{align}
Y_i =& ax_i+b+\nu_i\\
    &\; E\left\{ \nu_i|X\right\}=0\\
    &\; V\left\{ \nu_i|X\right\}=\left( ax_i+b\right)^2\sigma^2
\end{align}
If that were the model, then we know that the BLUE is the GLS estimator.  Also, we know that OLS would be unbiased, consistent, asymptotically normal.  Also, we would know how to calculate the variance of the OLS estimator.
But that model, the linear one with the $\nu$, is not the model we are given.  Here is the trick.  We make the model with the $\nu$ be the model we are given by forcing it to be:
\begin{align}
Y_i=&(ax_i+b)\epsilon_i\\
   =&ax_i+b + \left[(ax_i+b)\epsilon_i - ax_i-b \right]
\end{align}
So, just call the thing in square brackets $\nu_i$.  It's easy to verify now that $\nu_i$, the thing in square brackets, conditional on $x$, has mean zero and variance $\left( ax_i+b\right)^2\sigma^2$.  So, this multiplicative errors model is just a cleverly disguised linear model with heteroskedasticity.
To estimate this model, you would just run OLS and use heteroskedasticity-robust standard errors.
