Would an increased Federal Minimum Wage be a good idea? Would it cause unemployment?

The Federal Minimum Wage is far lower in real terms now than it was when it was raised in 1969. Many states already have passed laws to raise their Minimum Wages to $15 over several years. Would an increased Federal Minimum Wage be a good idea? Would it cause unemployment?  Should it be indexed to cost of living increases?  (analyzing the causes and broader economic implications of income inequality in the US, evaluate possible remedies to the problems caused by inequality)