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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.

user6703592
  • 535
  • $y=a(x+\epsilon)+b$ iff $x=y/a-b/a-\epsilon$. Expanding $f(\hat{a})=1/\hat{a}$ around $a$ and taking expectation gives $\require{cancel}\mathbb{E}\left(\frac1{\hat{a}}\right)=\frac1a-\frac{\cancelto0{\mathbb{E}(\hat{a}-a)}}{a^2}+\frac{\mathbb{E}(\hat{a}-a)^2}{a^3}+\dots$ so the bias has leading term $\operatorname{Var}(\hat{a})/a^3$. Similar for multiple regression. – user10354138 Sep 27 '21 at 16:56
  • @user10354138 it still cannot compute $E[\hat{a}],$ could you give more detail. – user6703592 Sep 28 '21 at 02:09

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