Constituent structure (advanced)
I. Constituent tests show that:
(a) Modal auxiliaries and periphrastic do are not part of the VP whose head is the main verb.
(b) Auxiliary have and be are not part of the VP whose head is the main verb, or of the projection containing the modal auxiliaries or periphrastic do:
(a) Modal auxiliaries and periphrastic do (which both appear only in finite clauses) are base-generated in a position, namely INFL, which is a different one from that of:
(b) Auxiliary have and be are base-generated in a V position outside of the VP whose head is the main verb.
The parallels in syntactic behaviour between modal auxiliaries/periphrastic do and auxiliary have and be can be captured syntactically (see below).
II. Modal auxiliaries, auxiliary have and be, and periphrastic do are heads, and thus, according to X-Bar theory:
(a) must project to the level of the phrase.
(b) may (and in fact must) have complements: