Test Whether 𝛽₂ = 0

Train the following 2 models against the dataset

Full Model	$\overset{y}{^}_{f} = \overset{𝛽}{^}_{f 1} x_{1} + \overset{𝛽}{^}_{f 2} x_{2}$ $\overset{y}{^}_{f} = X_{r} \overset{𝛽}{^}_{f}$
Reduced Model	$\overset{y}{^}_{r} = \overset{𝛽}{^}_{r 1} x_{1}$ $\overset{y}{^}_{r} = X_{r} \overset{𝛽}{^}_{r}$

After training, compute the residual sum of squares:

Full Model	$\overset{e}{^}_{f} = y - \overset{y}{^}_{f}$
Reduced Model	$\overset{e}{^}_{r} = y - \overset{y}{^}_{r}$

The corresponding residual sum of squares are independent 𝒳-square random variables:

Hence, from the definition of an 𝐹 variable (note that the 𝜎 cancels):

$F = \frac{( e ^ _{r}^{T} e ^ _{r} - e ^ _{f}^{T} e ^ _{f} ) / ( 1 )}{e ^ _{f}^{T} e ^ _{f} / ( n - p )}$

／var／log marcus chiu