In the book Basic Econometrics (Page254), the author writes:
In testing the significance of β2_hat under the null hypothesis, it was assumed tacitly that the testing was based on a different sample from the one used in testing the significance of β3_hat under the null hypothesis that β3 = 0.
And the author said that is the reason why the F-test cannot be substituted by multiple t-tests since any single hypothesis is 'affected' by the information in the other hypothesis.
However, I am still not sure about what does it mean by tests are independent of each other. Does it assume that there is no multicollinearity in the variables so that the result of one test does not affect the result of the other one?
May someone gives an example of how the test can be dependent of each other and explain why exactly the joint hypothesis cannot be substituded by multiple individual hypothesis? Thanks!