Long story short, I'm seeing in the literature that linear instrumental variables models are identifiable, even in the presence of unobserved confounders. The unobserved confounding aspect befuddles me, since it is not clear where this insight came from.

Briefly, given a linear instrumental variables setup where

$X = Z\beta + U\theta + \epsilon, \; \epsilon \sim N(0, \sigma_{\epsilon})$

$Y = X\alpha + U\phi + \delta, \; \delta \sim N(0, \sigma_{\delta})$

where $Z\in \mathbb{R}^z$ are instruments $X \in \mathbb{R}^x$ are the exogenous variables, $Y \in \mathbb{R}^y$ are the endogenous variables and $U \in \mathbb{R}^u$ are the unobserved confounders. In the instrumental variables setup, the average causal effect $\mathbb{E}[Y|do(x)]$ is of interest.

If $U$ is observed, I can see how this works out. But it isn't clear to me how an unobserved $U$ gets dropped / marginalized out in the linear setting, and I have not been successful finding the original proof, whether it be in Bowden + Turkington 1984, Pearl 2008 or Rubin + Imbens 2015. Any references or pointers would be appreciated.

P.S. This question is similar to what was asked here, but the collider insight is only part of the story, since we know that ACE is not identifiable in a non-parametric setting.

P.S.S. This has been originally posted on mathoverflow, before I realized that this forum was better suited for causal inference questions

Long story short, I'm seeing in the literature that linear instrumental variables models are identifiable, even in the presence of unobserved confounders. The unobserved confounding aspect befuddles me, since it is not clear where this insight came from.

Briefly, given a linear instrumental variables setup where

$X = Z\beta + U\theta + \epsilon, \; \epsilon \sim N(0, \sigma_{\epsilon})$

$Y = X\alpha + U\phi + \delta, \; \delta \sim N(0, \sigma_{\delta})$

where $Z\in \mathbb{R}^z$ are instruments $X \in \mathbb{R}^x$ are the exogenous variables, $Y \in \mathbb{R}^y$ are the endogenous variables and $U \in \mathbb{R}^u$ are the unobserved confounders. In the instrumental variables setup, the average causal effect $\mathbb{E}[Y|do(x)]$ is of interest.

If $U$ is observed, I can see how this works out. But it isn't clear to me how an unobserved $U$ gets dropped / marginalized out in the linear setting, and I have not been successful finding the original proof, even though this was hinted in Bowden + Turkington 1984, Pearl 2008 and Rubin + Imbens 2015. Any references or pointers would be appreciated.

P.S. This question is similar to what was asked here, but the collider insight is only part of the story, since we know that ACE is not identifiable in a non-parametric setting.

P.S.S. This has been originally posted on mathoverflow, before I realized that this forum was better suited for causal inference questions