... Another potential problem with applying 2SLS and other IV procedures is that the 2SLS standard errors have a tendency to be ‘‘large.’’ What is typically meant by this statement is either that 2SLS coefficients are statistically insignificant or that the 2SLS standard errors are much larger than the OLS standard errors. Not surprisingly, the magnitudes of the 2SLS standard errors depend, among other things, on the quality of the instrument(s) used in estimation.
This quote is from Wooldridge's "Econometric analysis of cross-sectional and panel data". I wonder why this happens? I would prefer a mathematical explanation.
Assuming homoskedasticity for simplicity the (estimated) asymptotic variance of OLS estimator is given by $$\widehat{Avar}(\hat{\beta}_{OLS}) = n\sigma^2(X'X)^{-1}$$ while for the 2SLS estimator $$\widehat{Avar}(\hat{\beta}_{2SLS}) = n\sigma^2(\hat{X}'\hat{X})^{-1}$$ where $$\hat{X} = P_zX = Z(Z'Z)^{-1}Z'X.$$
$X$ is the matrix of regressors, including the endogenous ones, and $Z$ is the matrix of instrumental variables.
So rewriting the variance for 2SLS gives $$\widehat{Avar}(\hat{\beta}_{2SLS}) = n\sigma^2\left(X'Z(Z'Z)^{-1}Z'X\right)^{-1}.$$
However, I can't conclude from above formulas that $\widehat{Avar}(\hat{\beta}_{2SLS}) \geq \widehat{Avar}(\hat{\beta}_{OLS})$.