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I estimated an unconditional cross-classified model with 3 levels. The intercept is quite a bit off from the sample mean. Any ideas for why that is? Thanks in advance.

I checked this with George and the behaviour is expected and is covered in Snijders and Bosker.

George gives the following explanation for the simplest multilevel case (two-level variance components model):

The intercept is interpreted as the population average of the cluster-specific population means, however the problem is that we only observe the cluster-specific sample means and some cluster-specific samples are larger than others. We therefore need to give more weight to the larger samples as those sample means will be more reliable estimates of their unknown population means.

We could frequency weight the cluster-specific sample means but in the presence of clustering we can do something better which is to precision weight the cluster-specific sample means, essentially giving more weight to smaller clusters than in the frequency weighted case.

The explanation is that the more clustering there is the fewer observations we need on a cluster to obtain a reliable estimate. In the absence of clustering the precision weighted average is just the frequency weighted average and with complete clustering the precision weighted average is the unweighted average. In balanced data the precision weighted average equals the frequency weighted average equals the unweighted average whatever the clustering.

Many thanks, Chris!