As far as I know, the first auxiliary is normally treated as the head of a verb phrase (VP) in the X-Bar theory.
He was writing a letter.
In this sentence, for example, the first auxiliary, was, is normally treated as the head of the VP, was writing a letter, in the X-Bar theory.
In some respect, however, the first auxiliary, was, is merely indicating the tense and aspect of the following verb writing.
Therefore, I wonder if the first auxiliary was can be treated as the specifier of the VP and the following lexical verb writing as the head of the VP in some variant of the X-Bar theory, or if this kind of treatment is a no go in any variant thereof.