I am drafting my personal statement for my application to a PhD program at a university in the US. This university, like many other universities in the US, does not require PhD applicants to submit a research proposal or statement. However, I have already prepared one before I have decided to apply to this university. So, is it advisable to briefly discuss my research proposal's framework/objectives in my personal statement? I believe I can discuss it in around three hundred words and relate it somehow to the research interests of some of the university's faculty members. I also do not have to worry about my personal statement's word count, because, according to my chosen university's website, there is no word count limit. I still aim to keep my personal statement in under two pages, though.
There is a bit of a balance here. Yes, I'd suggest that you spend a few words on some topic that interests you a lot, though perhaps not so much as a "proposal". I'd also recommend that you don't commit yourself too deeply to following that precise line of interest for dissertation studies. Why...
On the one hand, saying something indicates seriousness of purpose, which is a plus in admissions. On the other hand, the place you apply to may not have anyone interested in taking you on as an advisee if your proposal is not in their main research line.
Unlike some other places a student doesn't start out with a dissertation advisor and there is an opportunity to look around and find someone compatible - more so if you enter with a BA/BS, of course. But the first task in a doctoral program in the US is almost always to pass comprehensive/qualifying exams. Depending on what you learned in a masters it may require some coursework.
If you can find an advisor during this period that is interested in your proposal, all the better, but it is unlikely that they will be involved in the admissions process directly.
Your application is directed to a committee that will have people with different specialties. They aren't there to select individual advisees, but to accept a more general pool of people they can predict will be successful. So, having a specific goal is good, as long as you are willing to accept other ideas.
And, even if you follow a different thread to obtain a doctorate you can always return to your original ideas later if you stay in academia.