Confidence intervals in the LRM

Summary

Setup:

• The LRM with random sampling and GM

\[y_i=β_1+β_2x_i+ε_i i=1,…,n\]

• \(ε_i\) follows a normal distribution (conditionally on \(x_i\) ), \(ε_i|x_i∼N\left( 0,σ^2 \right) i=1,…,n\)

Confidence interval

• We define a \(\left( 1-α \right)∙100\) % confidence interval for \(β_2\) as

\[b_2±t_{α/2,n-2}SE(b_2)\]

• The probability that the \(β_2\) is inside the \(\left( 1-α \right)∙100\) % confidence interval is precisely \(\left( 1-α \right)∙100\) %.
• If the GM assumptions are satisfied but the errors \(ε_i\) do not follow a normal distribution then, if \(n\) is large, the confidence interval will be approximately correct (by the CLT).