<p>Consider a model <span class="math-container">$Y=X\beta+e$</span> and define the OLS error variance estimator as <span class="math-container">$\hat{\sigma^2}=1/n\sum_{i=1}^n \hat{e}_i^2$</span>, assuming that <span class="math-container">$ E[e_i^2]=\sigma^2 $</span> More precisely I want to derive the asymptotic distribution of <span class="math-container">$\sqrt{n}(\hat{\sigma^2}-\sigma^2)$</span>.</p>
<p>I can show that <span class="math-container">$E[\hat{\sigma^2}]=(n-k/n) \sigma^2$</span>, and also <span class="math-container">$\hat{\sigma^2}\xrightarrow{p}\sigma^2$</span> . Maybe I missed something but it seems quite weird to derive the variance of <span class="math-container">$\sigma^2)$</span>, as I did not see any one do so. Please help me if there is any other way to derive the asymptotic distribution.</p>

CC BY-SA 4.0
2023-12-03T07:28:26.573Z
What you define as OLS variance is not what textbooks and academic paper define as the OLS variance estimator. The OLS estimator is unbiased, while yours is biased. Yours is the maximum likelihood variance under a Gaussian likelihood. I suggest you edit your post to avoid conflict with the standard terminology. Also, in the OLS setting, $E[e_i^2]=\sigma^2$ holds by definition, so you do not need to assume that. Also, if I remember correctly, the quantity you are after will depend on the distributional assumption for $e$. What distribution do you assume $e$ to follow?
Richard Hardy
CC BY-SA 4.0
2023-12-03T07:46:07.983Z
Finally, [Cross Validated](https://stats.stackexchange.com/) Stack Exchange might be a better fit for your question. It has actually been asked there: see [**this**](https://stats.stackexchange.com/questions/468642) and [**this**](https://stats.stackexchange.com/questions/495328). And [here](https://math.stackexchange.com/questions/72975) is a somewhat related thread.
Richard Hardy
<econometrics><regression><variance>
Variance of OLS Error Variance Estimator
Paul Huang
# Variance of OLS Error Variance Estimator

(Asked by: username_0 on 2023-12-03)

Consider a model $Y=X\beta+e$ and define the OLS error variance estimator as $\hat{\sigma^2}=1/n\sum\_{i=1}^n \hat{e}\_i^2$, assuming that $ E[e\_i^2]=\sigma^2 $ More precisely I want to derive the asymptotic distribution of $\sqrt{n}(\hat{\sigma^2}-\sigma^2)$.

I can show that $E[\hat{\sigma^2}]=(n-k/n) \sigma^2$, and also $\hat{\sigma^2}\xrightarrow{p}\sigma^2$ . Maybe I missed something but it seems quite weird to derive the variance of $\sigma^2)$, as I did not see any one do so. Please help me if there is any other way to derive the asymptotic distribution.

**Tags:** econometrics, regression, variance
> **Comments:**
> **username_1**: What you define as OLS variance is not what textbooks and academic paper define as the OLS variance estimator. The OLS estimator is unbiased, while yours is biased. Yours is the maximum likelihood variance under a Gaussian likelihood. I suggest you edit your post to avoid conflict with the standard terminology. Also, in the OLS setting, $E[e_i^2]=\sigma^2$ holds by definition, so you do not need to assume that. Also, if I remember correctly, the quantity you are after will depend on the distributional assumption for $e$. What distribution do you assume $e$ to follow?  
> **username_1**: Finally, [Cross Validated](https://stats.stackexchange.com/) Stack Exchange might be a better fit for your question. It has actually been asked there: see [**this**](https://stats.stackexchange.com/questions/468642) and [**this**](https://stats.stackexchange.com/questions/495328). And [here](https://math.stackexchange.com/questions/72975) is a somewhat related thread.  

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## 0 answers

