Test for exogenity I have a few questions about this alternative test for exogeneity.
The first question is about the auxiliary regression? I have tried to google it and still don't understand the meaning of it. What is it's  relation to the model?
The second question is that I cannot see that $\mathbb{E}(X^+_iu_i)=\mathbb{E}(v_iu_i)$.
The last question is why the equation for $\gamma$ is what it is.

 A: First question: I do not think it is related to the model (6). Rather, it is an additional assumption imposed. It says that the endogenous variable depends linearly on the exogenous variable and the instrument.
Second question: Using (7) and linearity of expectation we have $$\begin{align*}\mathbb{E}(X_i^+u_i)&=\mathbb{E}([\alpha_0+\alpha_1X_i^*+\alpha_2Z_i+v_i]u_i)\\ &=\alpha_0\mathbb{E}(u_i)+\alpha_1\mathbb{E}(X_i^*u_i)+\alpha_2\mathbb{E}(Z_iu_i)+\mathbb{E}(v_iu_i).\end{align*}$$ Now, $\mathbb{E}(u_i)=0$ since $u_i$ is a statistical error term; $\mathbb{E}(X_i^*u_i)=0$ since $X_i^*$ is exogenous; and $\mathbb{E}(Z_iu_i)=0$ since $Z_i$ is orthogonal to $u_i$ as it is an instrument for $X_i^*$ in (6). This gives $$\mathbb{E}(X_i^+u_i)=\mathbb{E}(v_iu_i).$$
Third question: If (8) is assumed then $$\mathbb{E}(u_iv_i)=\mathbb{E}([\gamma v_i+e_i]v_i)=\gamma\mathbb{E}(v_i^2)+\mathbb{E}(v_ie_i).$$ Assuming $\mathbb{E}(v_ie_i)=0$ and dividing both sides by $\mathbb{E}(v_i^2)$ gives $\gamma=\mathbb{E}(u_iv_i)/\mathbb{E}(v_i^2)$.
