Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

OLS Estimator, derivation sigmahat(beta0hat)

  1. Oct 21, 2013 #1
    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 ;-)
     
    Last edited: Oct 21, 2013
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: OLS Estimator, derivation sigmahat(beta0hat)
Loading...