I'm still working on an update of the minimum wage and employment. One problem with analysis of the national minimum wage is that there are really only 7 episodes of isolated minimum wage increases, so even if it has very poor employment effects, it would be hard to find statistically significant results. Five of the seven episodes just happen to coincide with significant downturns in the labor market. Here is a graph of part time employment since 1987:
It seems like there are correlations within the broad national data that point to a large disemployment effect, yet, as can be seen in this graph, interpretation can be difficult. Here we have 2 episodes where a drop in age 16-19 part time employment drops and age 20-24 part time employment climbs, which could result from a substitution effect, and these both coincide with broad recessionary labor markets. Then, a third episode seems to have no effect on employment at all. And, a fourth notable event is another recession that has the same signature of the recessions that coincided with MW hikes.
Political Calculations points out that the 2001 episode coincides with a large MW hike above the federal level in California, but I'm not convinced that the timing and scale of the labor market declines fit that story.
And, the 1994 episode happened to come during what was possibly the strongest labor market in the last 80 years, where the labor demand was strong enough to create a bulge in the greater-than-full-time labor force unprecedented for this period.
I think I've got some ways to get some indications through the fog of data, but it's a funny situation, where MW hikes have an unlikely and uncanny coincidence with poor labor markets, yet there is enough noise to cast doubts on using the broad national data to confirm anything definitive. Maybe it's not worth my time, as this has been a frequently studied topic, but there are interesting things to learn from the data along the way.
PS. One other thing this graph makes clear, again, is that the reports of a labor market that is only strong among part time workers because of Obamacare seem to be based on specious measurements. There is a surge in part time employment and a plateau in full time work over the past few months, but these are noisy indicators, and the same thing could have been said at some point each year since the recession ended, so significant confirmation would be required to be able to say that. The lack of any trend in the year over year part time employment data in the chart here suggests that a confirmation is unlikely.