You ideally should specify all the stages, but there's a common approximation of specifying the full weights but only strata and clusters at the first stage, and then pretending the clusters were sampled with replacement. This 'with-replacement' approximation is surprisingly accurate when all the sampling probabilities for individuals are either 100% or fairly small, which tends to be the case in national surveys.
The reason for using the approximation is twofold
- Specifying the full sampling design can be problematic for confidentiality reasons
- Code to handle the full sampling design is more complicated. In the past, it might have been hard to find software that did it correctly.
Public-use datasets from large national surveys are often only published with stage-one information, so external users have to use the approximation.
There are some national survey datasets where the approximation isn't good. For example, Stats Canada have some post-census follow-up surveys of First Nations people with high sampling rates in some communities. And non-national surveys might well have high sampling fractions in some groups and need more detailed variance calculations