0

Robust standard errors (White standard errors) are given by: $$\hat{V}(b)=(\sum_{i=1}^N x_ix_i')^{-1}(\sum_{i=1}^N e_i^2x_ix_i')(\sum_{i=1}^N x_ix_i')^{-1}$$

This helps us to estimate a asymptotic covariance matrix under heteroscedasticity. Now assume a homoscedastic model, thus $e_i\sim N(0, \sigma^2)$ and $e_i$ is $iid$. Can the estimator from above be used under these homoscedastic assumptions? How can the equivalance between this estimator and the 'classic' estimator $\hat{V}(b)=\sigma^2(\sum_{i=1}^N x_ix_i')^{-1}$ be shown?

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
Felix
  • 1
  • What I know this far is that $E(e_i^2)=\sigma^2$. Thus we can rewrite the White standard errors as: $$\hat{V}(b)= \sum_{i=1}^N e_i^2 (\sum_{i=1}^N x_ix_i')^{-1}$$ But I haven't been able to show asymptotic equivalance to the 'classic' estimator. My only try to go on was multipling this by $N/N$ which asymptotically (!) yields: $$\hat{V}(b)=E(e_i^2) (\frac{1}{N}\sum_{i=1}^N x_ix_i')^{-1}=\sigma^2 (\frac{1}{N}\sum_{i=1}^N x_ix_i')^{-1}$$ But this doesnt help me either. – Felix May 24 '21 at 13:26
  • Another comment which might be helpful. I was able to show that: $$S=\frac{1}{N}\sum_{i=1}^N e_i^2 x_ix_i'$$ is a consistent estimator for $$\Sigma=\frac{1}{N}\sum_{i=1}^N \sigma_i^2 x_ix_i'$$. If (??) we can assume that from this also follows $$S_{new}=\sum_{i=1}^N e_i^2 x_ix_i'$$ is a consistent estimator for $$\Sigma_{new}=\sum_{i=1}^N \sigma_i^2 x_ix_i'$$. We might be able to show equivalence. – Felix May 24 '21 at 14:44
  • Considering the above comment we could take the probability limit (plim) of $$\hat{V}(b)=(\sum_{i=1}^N x_ix_i')^{-1}(\sum_{i=1}^N e_i^2x_ix_i')(\sum_{i=1}^N x_ix_i')^{-1}$$ and we'd get $$plim \hat{V}(b)=(\sum_{i=1}^N x_ix_i')^{-1}(\sum_{i=1}^N \sigma_i^2x_ix_i')(\sum_{i=1}^N x_ix_i')^{-1}$$ and because of homoscedasticity assumption also $$plim \hat{V}(b)= \sigma^2(\sum_{i=1}^N x_ix_i')^{-1}$$ – Felix May 24 '21 at 14:49

0 Answers0