The fact that, as the Lichter study shows, "A majority of Democratic statements (54 percent) were rated as mostly or entirely true, compared to only 18 percent of Republican statements," probably has more to do with how the statements were picked and the subjective bias of the fact checker involved than anything remotely empirical. Likewise, the fact that "a majority of Republican statements (52 percent) were rated as mostly or entirely false, compared to only 24 percent of Democratic statements" probably has more to do with spinning stories than it does with evaluating statements.
https://www.usnews.com/opinion/blogs/peter-roff/2013/05/28/study-finds-fact-checkers-biased-against-republicans