"We recently mentioned that x should be a strict subset of z. This has two implications.
First, any element that appears as an explanatory variable in (17.46) should also be an
explanatory variable in the selection equation. Although in rare cases it makes sense to
exclude elements from the selection equation, including all elements of x in z is not very
costly; excluding them can lead to inconsistency if they are incorrectly excluded."

First, it is NOT necessary. But excluding them can lead to inconsistency if they are incorrectly excluded. That is true and always true.

The reason is that X measurements in a selected group may be(very likely) also GOOD indicaters to tell why he selects himself into that group. Put it differently, the X in selected group has higher/lower measurement values than that in NON-selected group.

Thank you!

假洋鬼子

同意，就是卖弄一下，什么也没说、、、

参看李子奈2012新书《高级计量经济学》，probit是所有样本，ols是一个选择性样本。