OLS Estimator, derivation sigmahat(beta0hat)

1. Oct 21, 2013

chrisoutwrigh

Good day,
in the lectures of emperical economic research of my uni, we got to the topic of Linear Regression with one regressor. There I encountered upon:

${\hat{\sigma }_{\hat{\beta }_{0 }}}^{2 }=\frac{1 }{n }\cdot \frac{\text{var }{\left( {\left[ 1 -{\left( \frac{\mu _{x }}{E {\left( {X _{i }}^{2 }\right) }}\right) }\cdot X _{i }\right] }u _{i }\right) }}{{{\left[ E {\left( {{\left[ 1 -{\left( \frac{\mu _{x }}{E {\left( {X _{i }}^{2 }\right) }}\right) }\cdot X _{i }\right] }}^{2 }\right) }\right] }}^{2 }}$

When this obeys [heteroskedasticity-robust standard errors], the formula becomes:

${\hat{\sigma }_{\hat{\beta }_{0 }}}^{2 }=\frac{1 }{n }\cdot \frac{\frac{1 }{n -2 }\cdot \sum _{i =1 }^{n }{{\left[ 1 -\frac{\bar{X }}{\frac{1 }{n }\cdot \sum _{i =1 }^{n }{\left( {X _{i }}^{2 }\right) }\; }\cdot X _{i }\right] }}^{2 }\cdot {\hat{u }_{i }}^{2 }\; }{{\left( \frac{1 }{n }\cdot {{\left[ 1 -\frac{\bar{X }}{\frac{1 }{n }\cdot \sum _{i =1 }^{n }{\left( {X _{i }}^{2 }\right) }\; }\cdot X _{i }\right] }}^{2 }\right) }}$

I tried to get it to this form:

${\hat{\sigma }_{\hat{\beta }_{0 }}}^{2 }=\frac{\text{var }{\left( \hat{\beta }_{0 }\right) }}{1 }=\text{var }{\left( \bar{Y }-\hat{\beta }_{1 }\cdot \bar{X }\right) }=\text{var }{\left( \bar{Y }\right) }+\text{var }{\left( \hat{\beta }_{1 }\cdot \bar{X }\right) }-\text{cov }{\left( \bar{Y },\hat{\beta }_{1 }\cdot \bar{X }\right) } \\ \qquad{\hat{\sigma }_{\hat{\beta }_{0 }}}^{2 }=0 +{\bar{X }}^{2 }\cdot {\left( \frac{1 }{n }\frac{\frac{1 }{n -2 }\cdot \sum _{i =1 }^{n }{{\left( {X _{i }}^{}-\bar{X }\right) }}^{2 }\cdot {\hat{u }_{i }}^{2 }\; }{{{\left[ \frac{1 }{n }\cdot \sum _{i =1 }^{n }{{\left( {X _{i }}^{}-\bar{X }\right) }}^{2 }\; \right] }}^{2 }}\right) }-\text{cov }{\left( \hat{\beta }_{0 }+\bar{X }\cdot \hat{\beta }_{1 },\hat{\beta }_{1 }\cdot \bar{X }\right) }\\{\hat{\sigma }_{\hat{\beta }_{0 }}}^{2 }={\bar{X }}^{2 }\cdot {\left( \frac{1 }{n }\frac{\frac{1 }{n -2 }\cdot \sum _{i =1 }^{n }{{\left( {X _{i }}^{}-\bar{X }\right) }}^{2 }\cdot {\hat{u }_{i }}^{2 }\; }{{{\left[ \frac{1 }{n }\cdot \sum _{i =1 }^{n }{{\left( {X _{i }}^{}-\bar{X }\right) }}^{2 }\; \right] }}^{2 }}\right) }+0 \qquad$

so far no luck... [cov comprises again ${\hat{\beta }_{0 }}$ and ${\hat{\beta }_{1 }}$ so how to resolve? is it null?]

and where does the paraphrased term $\qquad{\hat{H }_{i }}^{}=1 -\frac{\bar{X }}{\frac{1 }{n }\cdot \sum _{i =1 }^{n }{\left( {X _{i }}^{2 }\right) }\; }\cdot X _{i }$ in the first equation come from?
it would be glad to get the complete derivation ;-)

Last edited: Oct 21, 2013