Multivariate linear regression model 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?).
 A: Rewrite the model in the following way:
$x_{2} =\gamma _{0}+\gamma _{1}x_{1}+\gamma _{2}y+v$
$=\left( \frac{\beta _{0}}{-\beta _{2}}\right) +\left( \frac{1}{\beta _{2}}\right) y+\left( \frac{\beta _{1}}{-\beta _{2}}\right) x_{1}+\left( \frac{u}{-\beta _{2}}\right) $
$ =\left( \frac{\beta _{0}}{-\beta _{2}}\right) +\left( \frac{1}{\beta _{2}}\right) \left( \beta _{0}+\beta _{1}x_{1}+\beta _{2}x_{2}+u\right)+\left( \frac{\beta _{1}}{-\beta _{2}}\right) x_{1}+\left( \frac{u}{-\beta_{2}}\right).$ 
You can see that y regressor is correlated with the error term since they both include u. This causes $\gamma_{2}$ to be downward-biased, which means your estimate of $\beta_{2}$ will be upward-biased. This is because the E[v|y]!=0.
Another general approach when you get stuck with the math is to simulate the problem. This will give you intuition for the proof. Here's some Stata code:
#delimit;
clear;
matrix C = (1,0,0\0,1,0\0,0,1);
drawnorm x1 x2 u, n(100000000) corr(C);
gen y=2+x1+5*x2+u;
reg y x1 x2;
reg x2 y x1;
di 1/_b[y];

