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Let's say I perform two 1st-stage regressions using

$y = x \, \alpha + \epsilon$

and

$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.

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I think I figured out the answer -- use Deming regression, which can handle standard errors in the predictor (!) and in the response variables.

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