Solved – Test for subset of coefficients being zero

Given a linear model
y = beta_0 + beta_1x_1 + beta_2x_2+beta_3x_3 + beta_4x_4
we can perform an $F$-test for the null hypothesis
H_0: beta_1 = beta_2 = beta_3 = beta_4 = 0
However, what test is appropriate for a subset of these coefficients being zero in the same model?
H_0': beta_1 = beta_4 = 0

You can use still an F-test. The test statistic is: $$F_0 = dfrac{SS_R/2}{MS_E},$$ where $SS_R$ is the increasing in the residual sum of squares of the reduced model (setting $beta_1=beta_4=0)$ with respect the full model with all the parameters. $MS_E$ is the residual sum of squares of the full model divided by $n-5$, where $n$ is the number of the observations.

Under $H_0$, $F_0$ is distributed as a $F_0$ of Fisher-Snedecor with $d_1=2$ (i.e. number of restrictions) and $d_2=n-5$.

Remark: clearly the result is valid under the usual assumptions of the Linear regression (e.g. Gaussian residuals).

See here the details of what they call partial F Test.

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