I was wondering why standard errors are (severely) downward biased when you are using the (general) instrumental variable - estimator or the generalized method of moments (gmm) estimator.
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See slides 8 and 9 of these notes. In a simplified setting,
$$y=\beta_{0}+\beta_{1}x + u$$
where $\text{Cov}(x,u) \ne 0$ but $z$ is a valid instrument for $x$, the estimated variance for $\hat{\beta_{1}}$ is the OLS estimated variance divided by the $R^2$ from the first-stage regression of $x$ on $z$. As long as $R^2 < 1$, the IV estimated variance will always be larger than the OLS estimated variance, hence the IV standard errors will also always be larger. If $z$ is a weak instrument measured by a low $R^2$, the IV standard errors will be much larger than the OLS standard errors, all else equal.
