Marian Minar kind of beat me to it -- I don't know how much research is done specifically on extra credit but there is a huge body of literature on motivation, and I think that's at the heart of the question (Which I think of as, "Is extra credit valuable in the sense of getting students to engage with their work?").
According to self-determination theory motivation can be thought of in terms of intrinsic versus extrinsic motivation, that is, motivation based on a self-determined value inherent in the task itself versus motivation based on a value outside the task; and in terms of autonomous versus controlled motivation, that is, motivation that is determined by the person versus that which is determined by another.
Autonomous motivation contains all of what we classify as intrinsic motivation, as well as extrinsic motivation that has "buy-in" from the person. For example, a student who studies hard for a calculus test because she finds calculus really fascinating or because she simply enjoys the process of practicing math problems is intrinsically (therefore autonomously) motivated. A student who studies hard for a calculus test, not because he enjoys calculus but because he has a dream of being an engineer and realizes that doing well in calculus is important for becoming a successful engineer, is extrinsically but autonomously motivated.
By contrast, a student who studies for a calculus test only for a grade (in which he places no intrinsic value) or because failing a test would cause feelings of shame or uselessness, is experiencing controlled motivation -- any motivation that occurs is there on someone else's terms.
As far as extra credit goes -- does extra credit motivate students? If so is this autonomous motivation or controlled? This, I don't know, and I don't have any research references handy -- but it would be interesting to find out, and this is what you'd search for.
Speaking of which, a quick ERIC search for peer-reviewed articles with "extra credit" in either the title or abstract brings up 95 results. Just skimming these, the theme for many of them seems to be that students were given some interesting activity, often involving active learning, to do as extra credit and it seemed to improve learning. If those results hold up, I think the learning gains would be more attributable to the active learning than they would be to the fact that they are extra credit. In other words, maybe what people are doing for extra credit ought to be the normal assignments and activities in the class!
Ryan, R., & Deci, E. (2000). Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. The American Psychologist, 55(1), 68–78. http://doi.org/10.1037/0003-066X.55.1.68