Testing a single linear restriction

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• \(ε_i|x_i \sim N(0,σ^2)\)
• Result: if \(r={\left( r_1, \ldots ,r_k \right)}'\) is a \(k×1\) vector of parameters then (conditionally on \(X\) )

\[r'b \sim N(r'β,r'Var\left( b \right)r)\]

• and

\[ \frac{r'b-r'β}{SE\left( r'b \right)} \sim t_{n-k}\]

• where

\[SE\left( r'b \right)=s\sqrt{r'{\left( X'X \right)}^{-1}r}\]

• Null hypothesis: \(H_0:r'β=q\) where \(q\) is a constant.
• Result: under \(H_0\)

\[ \frac{r'b-q}{SE\left( r'b \right)} \sim t_{n-k}\]

• Reject \(H_0\) if

\[\left| \frac{r'b-q}{SE\left( r'b \right)} \right|>t_{n-k,α/2}\]