Note that there is no need for every individual to have both responses and so long as we can consider 'missing' responses as random, the IGLS algorithm will supply maximum likelihood estimates. We can add further covariates to the model in a straightforward manner by forming interactions between them and the dummy variables defining the separate response intercepts.
The ability to fit a multivariate linear model with randomly missing responses finds a number of applications, for example where matrix or rotation designs are involved (reference , Chapter 4), each unit being allocated, at random, a subset of responses. The possibility of having additionally level-1 responses allows this to be used as a very general model for meta analysis where there are several studies (level-2 units) for some of which responses are available only in summary form at level 2 and for others detailed level-1 responses are available. Goldstein et al.  provide a detailed example.
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