I noticed there are a lot of discussions about pro and con of clustered standard errors v.s. multilevel modeling. But how about doing both of them in a model? Can someone tell me whether it is a good idea to do multilevel models and clustering standard errors at the same time? I am working on a nested data with three levels: school, class, and individual. I am only interested in the school and individual level. Is it appropriate to use HLM of two levels--school and individual--while clustering standard errors at class level?
One way to handle clustered errors in a linear model is to write down a model for the within-cluster error correlation, consistently estimate the parameters of this correlation model, and then estimate the original model by feasible generalized least squares. Hierarchical/multilevel models are one example of this approach, which provides valid statistical inference, as well as more efficient estimates of the parameters compared to OLS. However, your model of within-cluster error correlation needs to be correctly specified to get the benefit.
Cluster-robust standard errors do not require specification of a model for within-cluster error correlation, but do require the additional assumption that the number of clusters, rather than just the number of observations, shoots off to infinity. There's no free lunch.
You can combine the two approaches, which usually entails clustering at the highest level.