Estimating σ2

Summary

Setup

The LRM with random sampling

\[y_i=β_1+β_2x_i+ε_i \quad i=1,…,n\]

\(s^2\)

We define

\[s^2= \frac{RSS}{n-2}= \frac{1}{n-2}\sum_{i=1}^{n}{ e_i^2 }\]

About \(s^2\)

• Under GM, \(s^2\) is an unbiased estimator of \(σ^2\) . In this case, \(s^2\) is called the OLS estimator of \(σ^2\) .
• Under GM, and additional weak assumptions, \(s^2\) is a consistent estimator of \(σ^2\) .
• \(s\) , the square root of \(s^2\) , is called standard error of the regression .
• When \(s^2\) is a consistent estimator of \(σ^2\) then \(s\) is a consistent estimator of \(σ\) .