# Exclusion Restriction in instrumental variable regression

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?

You are right in saying that the IV coefficient is proportional to the reduced form and, in the case of $\widehat{\beta}_1$ it is the same. However, this is not a test for whether the exclusion restriction holds. In fact, there is no formal way of testing this assumption which makes the use of instrumental variable techniques tricky. This is the reason as to why most papers in economics usually devote a great deal of space to making an argument for why their instrument is valid.
The point about the exclusion restriction is to make an argument for the assumption that $\text{Cov}(Z,w)=0$ such that the instrument affects the outcome $Y$ only through the channel $X$ but not directly. Given that you never observe $w$ there is no direct way of testing this assumption.