The question I want to ask is close to but not exactly what stated in the title:


Fix a language $L$, it is known that a statement $\sigma$ is universal in the language if whenever $M$ satisfies $\sigma $ and $N$ is a substructure of $M$ then $N$ also satisfy $\sigma$. It is also known that a statement $\sigma$ is existential if whenever $M$ satisfies $\sigma $ and $N$ is an extension of $M$ then $N$ also satisfy $\sigma$. 


I can not find generalization of these criteria for formulas with more quantifiers. I wonder why this is the case?



Is there a model theoretic realization of the concept of Arithmetical Hierachy?
There is one more well-known equivalence for $\forall \exists$ sentences. 


**Theorem** (Chang-Los-Suszko). A theory $T$ is preserved under taking unions of increasing chains of structures if and only if $T$ is equivalent to a set of $\forall \exists$ sentences.


For a proof, see Keisler, "Fundamentals of model theory", *Handbook of Mathematical Logic*, p. 63.


I found a related paper, which is older and doesn't quite answer your question but may be of interest. R. C. Lyndon, "Properties preserved under algebraic constructions", Bull. Amer. Math. Soc. 65 n. 5 (1959), 287-299, [Project Euclid](http://projecteuclid.org/euclid.bams/1183523309)


According to that paper, and MathSciNet, a general solution to your question should be contained in H. J. Keisler, "Theory of models with generalized atomic formulas", J. Symbolic Logic v. 25 (1960) 1-26, 
[MathSciNet](http://www.ams.org/mathscinet-getitem?mr=130169), [JStor](http://www.jstor.org/stable/2964333)


