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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
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