Why use the partial F test instead of testing variables in an isolated model?

Question

Assume that I have $p$ independent variables:

$$X_{1}, X_{2}, \ldots, X_{p}$$

I wish to test the hypothesis

$$H_{0}\!: b_{1}=b_{2}=0$$

According to the partial $F$-test (Wald test), I need to run a full model and a reduced model with variables $3, 4, \ldots, p$, and to compare them in the $F$-statistic.

My question is why? Why can't I just run the model with these two variables and look at the "full $F$-test" for this particular model? What is the difference between running a model with these two variables and using the $F$-test, and running the partial $F$-test with two models, involving the other variables, which are not of any interest for this hypothesis?

Answer 1

The $F$-test for the model taken as a whole can be understood as a multiple partial $F$-test, but I can see what you mean.

The distinction is that one test (the nested version) tests the variables in the context of your other covariates, $X_3, \ldots, X_p$, whereas the $F$-test of the single two variable model tests them without the covariates. Put another way, the former controls for the other variables, whereas the latter ignores them¹. If the other variables are at all relevant, the former test should have greater statistical power. If your data are observational, the tests are of the variables projected on a different margin, and the true answer might differ irrespective of whether the tests yield correct decisions or not.

_{1. It may help you to read my answer here: Is there a difference between 'controlling for' and 'ignoring' other variables in multiple regression?}