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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? $$ H_0': \beta_1 = \beta_4 = 0 $$

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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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    $\begingroup$ Perhaps you could expand your post a bit to summarize the main ideas. It's common practice on this site when linking to an external source. This makes the answer self-contained, and preserves it for the benefit of future readers in case the link ever goes dead. $\endgroup$ – user20160 Feb 28 '18 at 12:54
  • $\begingroup$ @user20160, you are right! I edited as suggested. $\endgroup$ – Bruno Feb 28 '18 at 18:17

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