What you are describing is a multilevel version of what Arnold Zellner called the 'seemingly unrelated regressions' model.  In the single-level version of this, two (or more) outcomes are measured on the same set of units, each following an OLS model.  Joint estimation of the two models is equivalent to separate estimation under the univariate OLS models if either of the following conditions holds (but not in general otherwise):  (1) the two outcomes are measured on the same cases with the same predictors (identical design matrices), or (2) the residuals for the two models are known to be uncorrelated.  If these conditions do not hold, you get a different (and more efficient) estimate of the coefficients by estimating jointly.  In the multilevel setting, think of the units of analysis as schools; it seems highly unlikely that the design matrices for boys and girls would be identical.  (I haven't worked it out but I suspect you would need at least proportional numbers of boys & girls, and identical distributions of hours homework for boys & girls, in each school.)  And you clearly do not want to assume independence of boy & girl random effects (school level residuals).

It is that nonzero correlation parameter that is the one additional model feature of the joint versus separate model, allowing the results to be different.  To understand this effect, consider the extreme case in which the correlation of random effects is close to 1 and there are many times as many girls as boys in each school.  With your large girl samples, the girl random effects are estimated quite precisely.  In separate models, residual variance for boy random effects might be large, but bringing in the highly correlated girl random effects allows you to predict the boy random effects pretty well, leaving small residual variance and slightly different but more precise estimates of the coefficient for boys.

Think of it this way -- To simplify suppose there were only 2 random effects in your model, boy and girl intercepts.  Then at level 2 (schools) there are 2 variables -- boy intercept, girl intercept.  The fact that they are based on the same measurement doesn't change the way they relate in the model.  It could have been an algebra test for the boys and a trigonometry test for the girls, or a biology test in 9th grade and a physics test in 11th grade.  (could have even been 2 tests on same group of kids but then you have to deal with correlations at both levels.)

In fact if you make the samples of boys and girls in each school very large, then you could treat the boy and girl means in each schools as directly observed rather than estimates and drop out level 1, treating it as just 2 observations per school in a single level model.  (in your case 4 because you have a slope and intercept for each sex).  Perhaps this makes the relationship to SUR clearer.







Zaslavsky, Alan M. Harvard University