Fix a natural number, $n \geq 1$. Consider the class, M, of all sets hereditarily ordinal-definable using some $\Sigma_n$ formula. Since there is a universal $\Sigma_n$ formula, M is definable. Is M necessarily a model of ZF? It seems to me that it is closed under Godel operations and almost universal for the same reasons that HOD is, and therefore a model of ZF. But I feel like I'm missing something, since I've never heard anything about this model.
If it is a model of ZF, where can I learn more about it? Has anybody done any research about it? How does it relate to HOD?
Note: I require $n \geq 1$ because the formula witnessing that HOD is almost universal is $\Sigma_1$.