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

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