We have the standard linear multivariate regression model $y=\beta_0+\beta_1x_1+\beta_2x_2+u$ under the Gauss-Markov assumptions. Suppose we estimate $\gamma_0, \gamma_1, \gamma_2$ from $x_2=\gamma_0+\gamma_1x_1+\gamma_2y+v$ and get $\hat{\gamma_0}, \hat{\gamma_1}, \hat{\gamma_2}$. Is $\frac{1}{\hat{\gamma_2}}$ an unbiased estimate of $\beta_2$? I don't know how to answer the question.
First of all it seems that to run the regression of $x_2$ on $y$ and $x_1$ we need $\beta_2$ be nonzero. And is the expression $x_2=\frac{-\beta_0}{\beta_2}+\frac{-\beta_1}{\beta_2}x_1+\frac{1}{\beta_2}y+\frac{-1}{\beta_2}u$ somehow related to the regression model $x_2=\gamma_0+\gamma_1x_1+\gamma_2y+v$?
Another question is whether the Gauss-Markov assumptions hold in the case of this new regression, seems like some of them can fail, for instance $E(v|x_1,y)=0$ may not be true(is it important here?).