Suppose, I estimate a simple instrumental variable regression with 2SLS. The reduced form is
$Y = \alpha_0 + \alpha_1 Z+u,$
where $Z$ is a dummy (for simplicity). The first stage is
$X = \beta_0 + \beta_1 Z+v,$
and the second stage becomes
$Y = \gamma_0 + \gamma_1 \hat{X}+w$,
where $\hat{X}$ are fitted values. We have $\hat{\beta}_1=1$ so that the IV estimate becomes:
$\hat{\gamma_1}=\frac{\hat{\alpha_1}}{\hat{\beta_1}} = \hat{\alpha_1}$. Is this evidence that the exclusion restriction holds?