interview questIon: what's the bias of y = $a(x+\epsilon) + b$ linear regression It is an interview question, I simplify the question as dim = 1, it is better if the solution is about multiple dimension.
We know that the assumption of OLS is
$$y = ax+b+\epsilon,\quad \epsilon\sim N(0,\sigma^2).$$
And its estimation is unbiased. Then if we change the assumption to
$$y = a(x+\epsilon)+b,\quad \epsilon\sim N(0,\sigma^2),$$
what's the bias of estimation?
It seems we should solve the estimation of $a,b$ by MLE first (replace $\sigma$ by $a\sigma$) and the solution is no longer equivalent to OLS solution, which is more complicate.

$$y = a(x+\epsilon)+b,\quad \epsilon\sim N(0,\sigma^2)$$
is the same as
$$ y= ax+b+a\epsilon,\quad \epsilon\sim N(0,\sigma^2)$$
and
$$ y= ax+b+\epsilon,\quad \epsilon\sim N(0,a^2\sigma^2)$$
So it is the same situation, except the error terms have different variance.

This only remains the same for simple linear regression. If there are more terms, e.g quadratic terms then there will be bias.

Another thing is when the measurements of $x$ have an error. So the $x$ that we use in the estimate, is not the real $x$. In that case  you have an errors in variables model and the OLS estimator will be biased.
https://en.m.wikipedia.org/wiki/Errors-in-variables_models
