The New York Fed did a study on exactly this question. They found that there was a correlation: https://www.newyorkfed.org/medialibrary/media/research/staff...

"We find a pass-through effect on tuition of changes in subsidized loan maximums of about 60 cents on the dollar, and smaller but positive effects for unsubsidized federal loans. The subsidized loan effect is most pronounced for more expensive degrees, those offered by private institutions, and for two-year or vocational programs."