Why the joint hypothesis (F-test) cannot be substituted by multiple individual hypothesis (T-test)

Question

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!

Answer 1

When the $t$-tests are performed, they assume that the other variables are already in the model.

For example, suppose you were building a model where the dependent variable was the weight of a book, and the independent variables were $x_2$ (the number of pages in the book) and $x_3$ (the thickness of the book).

If you fit a model with both of these variables, and did t-tests for their coefficients, it's possible that you would get high p-values for both, because there is collinearity. The number of pages in a book is highly correlated with the thickness of a book.

So when you do a t-test to see if $x_2$ is needed in the model (in other words, if $B_2 = 0$), you may fail to reject, which makes sense because $x_3$ is already providing the information that $x_2$ would provide. And when you do a t-test to see if $x_3$ is needed in the model (in other words, if $B_3 = 0$), you may fail to reject as well, because $x_3$ is already providing the information that $x_2$ would provide.

However, that does not mean that not mean that it's likely that you don't need either variable in the model. You should probably have one of those variables in the model, since how thick the book is (whether measured in inches or in the number of pages) will definitely affect how much the book weighs. So even though you may fail to reject $B_2 = 0$ and $B_3 = 0$ independently, it's unlikely you would fail to reject the test $B_2 = B_3 = 0$, since one of those variables should be in your model.