E. Frank Stephenson has written this article for Public Choice. Here is the abstract:

Open primaries create the possibility of strategic crossover voting. On his March 3, 2008 program and subsequent broadcasts, radio personality Rush Limbaugh called on his listeners to extend the Democratic presidential contest by crossing over to vote for Sen. Hillary Clinton. Using voter registration data from North Carolina and election return data from Indiana, North Carolina, and Pennsylvania (states with open, semi-closed, and closed primaries, respectively), I find no evidence of a Limbaugh-motivated switch in political party registration or of a large or statistically significant Limbaugh-motivated increase in voting for Sen. Clinton.