Asymptotic variance vs. conditional variance in calculating regression estimator standard errors I am given regression model $y_i=x_i^T\beta+\varepsilon_i$ with heteroscedasticity.
Let $Avar(b)=E[x_ix_i^T]^{-1}E[\varepsilon_i^2x_ix_i^T]E[x_ix_i^T]^{-1}$
Hayashi's book states that regression coefficients are asymptotically normal:
$\sqrt{n}(b-\beta) \rightarrow^{D} N(0, Avar(b))$ therefore $b\sim N(\beta, Avar(b)/n)$ asymptotically for sufficiently large n.
Further, Hayashi states that the standard error of $b_k$ is precisely $\sqrt{Avar(b)_{kk}/n}$ where  $Avar(b)_{kk}$ implies the kkth element of the diagonal.
Question:
Why do we use $\sqrt{Avar(b)_{kk}/n}$ as a measure of standard error rather than $\sqrt{V(b|X)_{kk}}$ where
$V(b|X)$ is the conditional variance of b?
I am confused because earlier in the book he mentioned that the conditional variance should be used. For example, he specified that in the finite sample case under conditional homoscedasticity the standard error of $b_k$ was precisely:
$\sqrt{V(b|X)_{kk}}=\sqrt{\sigma^2(X^TX)^{-1}}$ with $\sigma^2=E[\varepsilon_i^2|X]$
So the asymptotic variance in the first expression that you give is the variance under heteroskedasticity as you note. The definition you gave of $V(b|X)$ is under homoskedasticity. As one of the comments states, we could use $V(b|X)$ (as you define it) under this assumption. It seems that Hayashi uses this notation because he wants to consider heteroskedastic errors.