The variance of the OLS estimators

Summary

Setup

The LRM with random sampling

\[y_i=β_1+β_2x_i+ε_i i=1,…,n\]

and the GM assumptions ( \(E\left(\varepsilon_i | x_i \right)=0\) and \(Var\left(\varepsilon_i | x_i \right)=σ^2\) ) .

The variance of the OLS estimators

\[Var\left(b_1 | x \right)=σ^2\left( \frac{1}{n}+ \frac{{\bar{x}}^2}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }} \right)\]

\[Var\left(b_2 | x \right)= \frac{σ^2}{\sum_{i=1}^{n}{ {\left( x_i-\bar{x} \right)}^2 }}\]

About the variance of the OLS estimators

• These formulas are called the OLS GM variance formulas .
• The notation \(Var\left(b_2 | x \right)\) is short for \(Var\left(b_2 | x_1,…,x_n \right)\)
• \(Var\left(b_1 | x \right)\) and \(Var\left( b_2 | x \right)\) are generally unknown as \(σ^2\) is unknown.
• The higher the \(σ^2\) the higher the variance in \(b_2\) .
• The higher the sample variance in the x -data, the lower the variance in \(b_2\) .