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Question: do sandwich estimators of the standard errors equal the typical estimation of the standard error IF the data was generated with constant variance in the residuals and as the sample size approaches infinity?

Background: A sandwich estimator estimates standard error properly even if the residuals have non-constant variance. The typical standard error procedure only estimates standard error properly when we assume constant variance in the residuals. By typical standard error calculation, I mean calculating standard errors on the coefficient in an OLS:

\begin{equation} \widehat{\mbox{var}}_{OLS}\left(\hat{\beta}\right) = \left(X^TX\right)^{-1}\left(r^2X^TX\right)\left(X^TX\right)^{-1} \end{equation}

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