Card and Krueger's paper (AER 1994, "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania") uses a difference-in-difference identification strategy to identify the causal effect of a minimum wage increase on unemployment. The main finding was that the increase in the minimum wage had a negligible or even non-existent effect of employment. There have been several criticisms to the paper---including criticizing the quality of the data or the fact that employers might have anticipated the change.
My question is what are the main economic explanations for why employment did not significantly fall? What other evidence could be used to test these explanations?