The Details of the General Linear F-test
This video provides two examples using the auto dataset with Stata to show how the general linear F-test is constructed and calculated by hand. At the end of the examples, the testparm command is used to show how to do these tests in Stata.
ERRATUM
At the 6:39 minute mark, there is an error in the flyout formula that uses an incorrect sum of squares. The first part of the formula should be (2,443-851.469256) AND NOT (2443-1591.9).
If you skip ahead a little bit until the formula goes off the screen and look at line 21 of the Stata program, you can see that the F-statistic is calculated correctly. Line 21: di "F=" ((2443.45946-851.469256) / ((74-2)-(74-1))) / (851.469256/(74-2)).
Thank you to MISC Vid for finding this error (see the comments below).