<p>They are certainly not the same, since the first contains an expected value while the second does not contain an expected value but a sample mean.  </p>

<p>Of course, they are very closely connected, and one can see this if one writes down the definition of the variance of a random variable.</p>

CC BY-SA 3.0
2018-01-05T10:28:52.517Z
In every case, i've been told that first formula ($\hat{\sigma }^2_\hat{\beta _0}=\frac{E({X_{i}}^{2})}{n\sigma _{X}^{2}}\sigma ^{2}$) is derived from this formula: $\sigma_{\hat{\beta_0}}^2=\frac{1}{n}\frac{var(H_iu_i)}{[E(H_{i}^{2})]^2}$, where $H_i=1-[\frac{\mu_X}{E(X_i^{2})}]X_i$, that is for heteroskedastic case. Could you help me to understand how derivate it?
Francesco Totti
CC BY-SA 3.0
2018-01-03T18:59:35.967Z
2018-04-03T23:18:59.153Z
2018-04-03T23:18:59.153Z
Alecos Papadopoulos
<p>Stock and Watson express the variance of $\hat{\beta _0}$ like 
$\hat{\sigma }^2_\hat{\beta _0}=\frac{E({X_{i}}^{2})}{n\sigma _{X}^{2}}\sigma ^{2}$, but starting from variance of $\hat{\beta _1}=\frac{\sigma^{2}}{n\sigma_{X}^{2}}$ i proved only that $\hat{\sigma }^2_\hat{\beta _0}=\frac{1}{n}\sigma^{2}(1+\frac{\bar{X}^2}{\sigma_{X}^{2}})$, that is the same that is showed <a href="http://www.utstat.utoronto.ca/alisong/Teaching/0708/Sta302/formulae.pdf" rel="nofollow noreferrer">here</a>. </p>

<p>How can i prove that are similar forms? </p>

<econometrics><mathematical-economics><applied-econometrics>
Variance of $\hat{\beta _0}$ in case of homoskedasticity
Francesco Totti
# Variance of $\hat{\beta _0}$ in case of homoskedasticity

(Asked by: username_0 on 2018-01-03)

Stock and Watson express the variance of $\hat{\beta \_0}$ like
$\hat{\sigma }^2\_\hat{\beta \_0}=\frac{E({X\_{i}}^{2})}{n\sigma \_{X}^{2}}\sigma ^{2}$, but starting from variance of $\hat{\beta \_1}=\frac{\sigma^{2}}{n\sigma\_{X}^{2}}$ i proved only that $\hat{\sigma }^2\_\hat{\beta \_0}=\frac{1}{n}\sigma^{2}(1+\frac{\bar{X}^2}{\sigma\_{X}^{2}})$, that is the same that is showed [here](http://www.utstat.utoronto.ca/alisong/Teaching/0708/Sta302/formulae.pdf).

How can i prove that are similar forms?

**Tags:** econometrics, mathematical-economics, applied-econometrics


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## 1 answers

### Answer by username_1 on 2018-01-03. Score: 1

They are certainly not the same, since the first contains an expected value while the second does not contain an expected value but a sample mean.

Of course, they are very closely connected, and one can see this if one writes down the definition of the variance of a random variable.

> **Comments:**
> **username_0**: In every case, i've been told that first formula ($\hat{\sigma }^2_\hat{\beta _0}=\frac{E({X_{i}}^{2})}{n\sigma _{X}^{2}}\sigma ^{2}$) is derived from this formula: $\sigma_{\hat{\beta_0}}^2=\frac{1}{n}\frac{var(H_iu_i)}{[E(H_{i}^{2})]^2}$, where $H_i=1-[\frac{\mu_X}{E(X_i^{2})}]X_i$, that is for heteroskedastic case. Could you help me to understand how derivate it?  

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