GATE 2021 Question 14: Endogeneity, instruments and simultaneous equations

Consider a regression model $y=\beta_0+\beta x+u$ where the continuous variable $y$ is regressed on a dummy variable $x$, which takes the value either 1 or 0. However, the model was estimated using the instrumental variable (IV) estimation method, wherein the indicator variable $z$ is used as an instrument of $x$.

Let:

$\bar y_1$ and $\bar y_0$ be the sample averages of $y$ when $z$ takes the value 1 and 0, respectively

$\bar x_1$ and $\bar x_0$ be the sample averages of $x$ when $z$ takes the value 1 and 0, respectively

$\bar y^1$ and $\bar y^0$ be the sample averages of $y$ when $x$ takes the value 1 and 0, respectively

$\bar z^1$ and $\bar z^0$ be the sample averages of $z$ when $x$ takes the value 1 and 0, respectively

Then the estimated coefficient of $\beta_{IV}$ is