Let's say I perform two 1st-stage regressions using
$y = x \, \alpha + \epsilon$
$w = v \, \beta + \ldots $
where $y$, $x$, $w$, and $v$ are vectors of length $n$. I obtain the regression coefficients $\alpha$ and $\beta$ along with their standard errors. Assume that the assumptions of ordinary least squares (OLS) are met.
If I repeat the above many times, I have enough samples of $\alpha$ and $\beta$ to perform a 2nd-stage regression to obtain the coefficient $\gamma$:
$\alpha = \beta \, \gamma + \ldots$
How do I estimate the standard errors for $\gamma$? Note that the predictor variable $\beta$ in the 2nd-stage regression is not error-free.