<p>So the asymptotic variance in the first expression that you give is the variance under heteroskedasticity as you note. The definition you gave of <span class="math-container">$V(b|X)$</span> is under homoskedasticity. As one of the comments states, we could use <span class="math-container">$V(b|X)$</span> (as you define it) under this assumption. It seems that Hayashi uses this notation because he wants to consider heteroskedastic errors.</p>

CC BY-SA 4.0
2021-05-18T19:28:47.687Z
2021-05-18T19:28:47.687Z
Ariel
<p>I am given regression model <span class="math-container">$y_i=x_i^T\beta+\varepsilon_i$</span> with heteroscedasticity.</p>
<p>Let <span class="math-container">$Avar(b)=E[x_ix_i^T]^{-1}E[\varepsilon_i^2x_ix_i^T]E[x_ix_i^T]^{-1}$</span></p>
<p>Hayashi's book states that regression coefficients are asymptotically normal:</p>
<p><span class="math-container">$\sqrt{n}(b-\beta) \rightarrow^{D} N(0, Avar(b))$</span> therefore <span class="math-container">$b\sim N(\beta, Avar(b)/n)$</span> asymptotically for sufficiently large n.</p>
<p>Further, Hayashi states that the standard error of <span class="math-container">$b_k$</span> is precisely <span class="math-container">$\sqrt{Avar(b)_{kk}/n}$</span> where  <span class="math-container">$Avar(b)_{kk}$</span> implies the kkth element of the diagonal.</p>
<p><strong>Question:</strong></p>
<p>Why do we use <span class="math-container">$\sqrt{Avar(b)_{kk}/n}$</span> as a measure of standard error rather than <span class="math-container">$\sqrt{V(b|X)_{kk}}$</span> where
<span class="math-container">$V(b|X)$</span> is the conditional variance of b?</p>
<p>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 <span class="math-container">$b_k$</span> was precisely:</p>
<p><span class="math-container">$\sqrt{V(b|X)_{kk}}=\sqrt{\sigma^2(X^TX)^{-1}}$</span> with <span class="math-container">$\sigma^2=E[\varepsilon_i^2|X]$</span></p>

CC BY-SA 4.0
2021-05-17T16:27:01.910Z
Why do you write $V(b|X)$ and say 'unconditional' variance when clearly you are using notation for conditioning on $X$. Makes question a little confusing.
bomadsen
CC BY-SA 4.0
2021-05-18T13:18:56.163Z
If $\sigma^2$ is known, then use $V(b|X)$.
Bertrand
<econometrics>
Asymptotic variance vs. conditional variance in calculating regression estimator standard errors
HHQ
# Asymptotic variance vs. conditional variance in calculating regression estimator standard errors

(Asked by: username_0 on 2021-05-17)

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 $\sigma^2=E[\varepsilon\_i^2|X]$

**Tags:** econometrics
> **Comments:**
> **username_1**: Why do you write $V(b|X)$ and say 'unconditional' variance when clearly you are using notation for conditioning on $X$. Makes question a little confusing.  
> **username_2**: If $\sigma^2$ is known, then use $V(b|X)$.  

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## 1 answers

### Answer by username_3 on 2021-05-18. Score: 1

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.


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