Testing several linear restrictions jointly

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• \(ε_i|x_i \sim N(0,σ^2)\)
• Result: if \(R\) is an \(J×k\) matrix of parameters then

\[Rb \sim N(Rβ,RVar\left( b \right)R')\]

• and

\[{\left( RVar\left( b \right)R' \right)}^{-1/2}(Rb-Rβ) \sim N(0,I_J)\]

• and

\[{\left( Rb-Rβ \right)}'{\left( RVar\left( b \right)R' \right)}^{-1}\left( Rb-Rβ \right) \sim χ_J^2\]

• and

\[ \frac{{\left( Rb-Rβ \right)}'{\left( R{\left( X'X \right)}^{-1}R' \right)}^{-1}\left( Rb-Rβ \right)}{Js^2} \sim F_{J,n-k}\]

• Null hypothesis: \(H_0:Rβ=q\) where \(q\) is a \(J×1\) vector of constants.
• Result: under \(H_0\)

\[ \frac{{\left( Rb-q \right)}'{\left( R{\left( X'X \right)}^{-1}R' \right)}^{-1}\left( Rb-q \right)}{Js^2} \sim F_{J,n-k}\]

• Reject \(H_0\) if

\[ \frac{{\left( Rb-q \right)}'{\left( R{\left( X'X \right)}^{-1}R' \right)}^{-1}\left( Rb-q \right)}{Js^2}>F_{J,n-k,α}\]

• or reject \(H_0\) if

\[ \frac{{\left( Rb-q \right)}'{\left( R{\left( X'X \right)}^{-1}R' \right)}^{-1}\left( Rb-q \right)}{s^2}>χ_{J,α}^2\]