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Proving an identity involving $E(e_i^2)$ in simple OLS

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Once expressed the simple OLS residual $e_i$ as a weighted sum of the noise terms:

begin{equation}e_{i}=sum_{j}left(delta_{i j}-frac{1}{n}-left(x_{i}-overline{x}right) frac{x_{j}-overline{x}}{n s_{X}^{2}}right) varepsilon_{j},end{equation}

where $delta_{i j}$ is 1 when $i=j$ and 0 otherwise, I need to prove that:

$$ E(e_{i}^{2})=sigma^{2}left(1-frac{1}{n}-frac{left(x_{i}-overline{x}right)^{2}}{n s_{X}^{2}}right).$$

However, I squared the first equation and then took the expected value and exploiting the fact that $E(e_ie_j) = 0$ for $ine j$, I arrive to:

$$ E(e_{i}^{2})=sigma^{2}sum_{j}left(1-frac{1}{n}-left(x_{i}-overline{x}right) frac{x_{j}-overline{x}}{n s_{X}^{2}}right) ^2.$$

What I'm doing wrong?

summation proof-explanation linear-regression

share|cite|improve this question

asked 3 hours ago

pdbpdb

205

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Cross-post: stats.stackexchange.com/questions/395975/….
  $endgroup$
  – StubbornAtom
  2 hours ago
  

  
  
  
  

add a comment | 

0

$begingroup$

Once expressed the simple OLS residual $e_i$ as a weighted sum of the noise terms:

begin{equation}e_{i}=sum_{j}left(delta_{i j}-frac{1}{n}-left(x_{i}-overline{x}right) frac{x_{j}-overline{x}}{n s_{X}^{2}}right) varepsilon_{j},end{equation}

where $delta_{i j}$ is 1 when $i=j$ and 0 otherwise, I need to prove that:

$$ E(e_{i}^{2})=sigma^{2}left(1-frac{1}{n}-frac{left(x_{i}-overline{x}right)^{2}}{n s_{X}^{2}}right).$$

However, I squared the first equation and then took the expected value and exploiting the fact that $E(e_ie_j) = 0$ for $ine j$, I arrive to:

$$ E(e_{i}^{2})=sigma^{2}sum_{j}left(1-frac{1}{n}-left(x_{i}-overline{x}right) frac{x_{j}-overline{x}}{n s_{X}^{2}}right) ^2.$$

What I'm doing wrong?

summation proof-explanation linear-regression

share|cite|improve this question

asked 3 hours ago

pdbpdb

205

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Cross-post: stats.stackexchange.com/questions/395975/….
  $endgroup$
  – StubbornAtom
  2 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

Once expressed the simple OLS residual $e_i$ as a weighted sum of the noise terms:

begin{equation}e_{i}=sum_{j}left(delta_{i j}-frac{1}{n}-left(x_{i}-overline{x}right) frac{x_{j}-overline{x}}{n s_{X}^{2}}right) varepsilon_{j},end{equation}

where $delta_{i j}$ is 1 when $i=j$ and 0 otherwise, I need to prove that:

$$ E(e_{i}^{2})=sigma^{2}left(1-frac{1}{n}-frac{left(x_{i}-overline{x}right)^{2}}{n s_{X}^{2}}right).$$

However, I squared the first equation and then took the expected value and exploiting the fact that $E(e_ie_j) = 0$ for $ine j$, I arrive to:

$$ E(e_{i}^{2})=sigma^{2}sum_{j}left(1-frac{1}{n}-left(x_{i}-overline{x}right) frac{x_{j}-overline{x}}{n s_{X}^{2}}right) ^2.$$

What I'm doing wrong?

summation proof-explanation linear-regression

share|cite|improve this question

asked 3 hours ago

pdbpdb

205

$endgroup$

share|cite|improve this question

asked 3 hours ago

pdbpdb

205

share|cite|improve this question

asked 3 hours ago

pdbpdb

205

asked 3 hours ago

pdbpdb

205

asked 3 hours ago

pdbpdb

205