Can a Turing machine or Push Down Automaton construct languages of type 3?

I am not quite sure, whether automata can construct languages over their types. For example, a Push down automaton can construct a language of type 2 - does that mean that a PDA also can construct a language of type 3? The same for turing machine?

The set of languages of type $$3$$ is a proper subset of the of languages of type $$2$$. In other words, a language of type $$3$$ is also a language of type $$2$$.
If some automaton can recognize languages of type $$2$$, then it can also recognize languages of type $$3$$.