Suppose you have a single cross-section of data where individuals are located within groups (e.g. students within schools) and you wish to estimate a model of the form Y_i = a + B*X_i where X is a vector of individual level characteristics and a a constant.
In this case, suppose unobserved between-group heterogeneity biases your point estimates of B and their SEs since it is correlated with your independent variable of interest.
One option is to cluster your SEs by groups (schools). Another is to include group FEs. Another is to use both. What should one consider when choosing between these options? It is particularly unclear why one might cluster SEs by group AND use group FE. In my specific case, I have 35 groups and 5,000 individuals nested within each group. I have followed the discussion in this pdf, but it is not very clear on why and when one might use both clustered SEs and fixed effects.
(Please discuss the pros and cons of clustered SEs vs. FEs instead of suggesting I just fit a multilevel model.)
Best Answer
Both approaches, using group fixed effects and/or cluster-adjusted standard error take into account different issues related to clustered (or panel) data and I would clearly view them as distinct approaches. Often you want to use both of them:
First of all, cluster-adjusted standard error account for within-cluster correlation or heteroscedasticity which the fixed-effects estimator does not take into account unless you are willing to make further assumptions, see the Imbens and Wooldridge lecture slides for an good discussion of short and long panels and various issues related to this problem. There is also a novel paper about this topic by Cameron and Miller: A Practitioner's Guide to Cluster-Robust Inference which might be interesting for you. If you do not want to model the variance-covariance matrix and you suspect that within-cluster correlation is present, I advise to use cluster robust standard error because the bias in your SE may be severe (much more problematic than for heteroscedasticity, see Angrist & Pischke Chapter III.8 for a discussion of this topic. But you need enough cluster (Angrist and Pischke say 40-50 as a role of thumb). Cluster-adjusted standard error take into account standard error but leave your point estimates unchanged (standard error will usually go up)!
Fixed-effects estimation takes into account unobserved time-invariant heterogeneity (as you mentioned). This can be good or bad: On the hand, you need less assumptions to get consistent estimations. On the other hand, you throw away a lot of variance which might be useful. Some people like Andrew Gelman prefer hierarchical modeling to fixed effects but here opinions differ. Fixed-effects estimation will change both, point and interval estimates (also here standard error will usually be higher).
So to sum up: Cluster-robust standard error are an easy way to account for possible issues related to clustered data if you do not want to bother with modeling inter- and intra-cluster correlation (and there are enough clusters available). Fixed-effects estimation will take use only certain variation, so it depends on your model whether you want to make estimates based on less variation or not. But without further assumptions fixed-effects estimation will not take care of the problems related to intra-cluster correlation for the variance matrix. Neither will cluster-robust standard error take into account problems related to the use of fixed-effects estimation.