Ohio Sen. JD Vance, the GOP vice presidential nominee, argued during the debate on Tuesday night that the massive influx of illegal immigrants is contributing to rising housing costs and when challenged by one of the CBS News moderators to identify his source, Vance pledged to post it to social media.

After the debate, Vance followed through on his promise and posted remarks from Michelle Bowman of the Federal Reserve Board of Governors as well as two studies.

"Given the current low inventory of affordable housing, the inflow of new immigrants to some geographic areas could result in upward pressure on rents," Bowman said.

Bowman's comments are supported by reports of illegal immigrants benefitting from affordable housing programs in the U.S.

According to a study from the Journal of Housing Economics, "immigration inflows into a particular Metropolitan statistical area (MSA) is associated with increases in rents and with house prices in that MSA while also seeming to drive up rents and prices in neighboring MSAs." 

The Congressional Budget Office analysis he posted "generally found that increases in immigration raise state and local governments’ spending—particularly on education, health care, and housing—more than their revenues."