I would like to explain to one of my professors why the regular methods for variance estimation are biased for complex survey data, but I am having some trouble articulating my point.
With weighted least squares in general, the variance for the regression coefficient is $var(\hat{\beta})=\sigma^2(X'WX)^{-1}$
This assumes each observation is independent, and doesn't take into account the probability of selection.
In practice, variances would be calculated using either Taylor Series linearizaton so $var(\hat{\beta})=(X'WX)^{-1}G(X'WX)^{-1}$ where $G$ is essentially the sum of the covariance matrices within stratas, or using resampling via bootstrap or jackknife.
What I am having trouble demonstrating is why exactly $\sigma^2(X'WX)^{-1}$ is biased.
Variance Estimation by Wolter derives a lot of variance estimators for complex survey data, but doesn't really demonstrate why regular variance estimating techniques are biased, which is what I would like to explain to my professor.
Any help would be appreciated.
Regards