OLS Estimator, derivation sigmahat(beta0hat)

[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:

[itex] {\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 }} [/itex]

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

[itex] {\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) }} [/itex]

I tried to get it to this form:

[itex] {\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 [/itex]

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

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

 

[mmwave]

Hello,

Thank you for reaching out with your question. I understand that you are struggling with understanding the derivation of the formula for the heteroskedasticity-robust standard errors in linear regression with one regressor. This can be a complex topic, so let me try to break it down for you.

First, let's review some basic concepts. In linear regression, we are trying to estimate the relationship between a dependent variable (Y) and an independent variable (X). The estimated equation is usually written as Y = β0 + β1X + u, where β0 and β1 are the intercept and slope coefficients, respectively, and u is the error term. The goal is to minimize the sum of squared residuals (SSR) in order to find the best fitting line.

Now, let's focus on the formula you provided:

{\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 }}

This formula is the calculation for the variance of the intercept coefficient, β0. It is derived from the general formula for the variance of an estimated coefficient, which is:

{\hat{\sigma }_{\hat{\beta }_{j }}}^{2 }=\frac{\text{var }{\left( \hat{\beta }_{j }\right) }}{1 }=\frac{\text{var }{\left( \bar{Y }-\hat{\beta }_{1 }\cdot \bar{X }\right) }}{1 }=\frac{\text{var }{\left( \bar{Y }\right) }+\text{var }{\left( \hat{\beta }_{1 }\cdot \bar{X }\right) }-2\cdot \text{cov }{\left( \bar{Y