I have looked at this analysis again, substituting the DOL's Compensation per Hour index. This doesn't have the negative drift over time that AHETPI does, and it produces a much stronger linear relationship between relative minimum wage and the proportion of the labor force working at or under the minimum wage.
This relationship is surprisingly linear, but it doesn't necessarily suggest job losses without a job level to use as a comparison. Luckily, there are available measures of the number of workers in a non-minimum wage context to compare this with.
One comes from The Economic Policy Institute. They issued a report on the minimum wage that included a count of the current number of workers earning $10.10 or less - about 21.3 million, or 13.5% of the labor force. I can compare this number to the number of actual minimum wage workers measured the last time the minimum wage was at that relative level. If the actual number of workers was significantly less than the current number, that would suggest that the high minimum wage caused some unemployment.
Here is the relationship, using Compensation per Hour.
The relationship is very tight, and the number of workers was much lower than 13.5%. It is highly unrealistic to think that the level of employment at minimum wage will deviate from the long term trend to anywhere near what it would need to in order to assume that there will be no job loss. This relationship can be used to estimate the number of job losses that a minimum wage increase will cause.
(Note that the relationship levels out below about 25% of the average hourly compensation, which suggests that this is essentially the natural minimum wage level. It also suggests that the initial MW hike in 2007 probably had negligible disemployment effects, but that the 2008 and 2009 hikes had more typical effects.)
One reply to this problem would be that a minimum wage hike would lead employers to boost the wages of some workers to levels slightly above the new minimum wage, and that would cause the number of measured workers at or below a minimum wage of $10.10 to be less than the current number of measured workers at $10.10. But (1) it is implausible to imagine that two thirds of workers currently earning between $10.10 and $7.25 would see wages increased to above the new MW level. And (2) this would create a sort of asymmetrical reaction to MW hikes, where we would expect to see a deviation from the long term trend as employers made relative adjustments to workers' wages after a MW wage shock, but then the proportion of workers would revert back toward the trend as the new minimum wage level aged and declined in real terms.
The relationship we do see is very linear. As the real MW level decreases over time, the number of MW workers follows the trend closely, and when MW was raised in 1990-91, 1996-97, and 2007-09, the number of workers moved right back up the trendline, with very little deviation. This suggests a surprisingly efficient labor market.
As one last exercise, I have used a previous regression of the employment-population ratio against RGDP and minimum wage levels to estimate the total potential MW labor force. I had found a correlation of the loss of .168% in EPR for each 1% rise in the ratio of the minimum wage to the average wage. I added this estimate of the number of job losers to the historical measured number of MW workers. (In all of these graphs, I am using the total number of workers at or below minimum wage.)
My regression would estimate a total decrease in employment of 4.5 million if the minimum wage were raised from $7.25 to $10.10 today. That would leave 7 million workers unaccounted for, who could be workers who start out below the new minimum but receive raises to above the new minimum upon implementation, or could be unemployed workers that weren't counted by my regression analysis. My regression was based on 2-year periods of serial MW-hikes, so the changes it was measuring would have included some period of time following the last hikes of a series where employment would have been recovering. This would cause it to understate disemployment. I think it might be realistic to expect that 7 million workers to be split between mostly workers who are bumped to above the new MW and some additional workers who would lose employment.
One could trifle with the precise estimates I am using, but I don't see how anyone could look at this very linear and long-term relationship, compare it to the current number of workers making $10.10 or less, and dismiss the expectation of significant job losses.
One note concerning the 21.3 million workers at $10.10 or less, estimated by EPI: This is based on their assumption of no job losses from MW. If one does accept my analysis here that the MW does cause job losses, then one would add the additional 1.3 million workers who are currently (as of 2012 data) unemployed because of MW to the total number of potential workers at $10.10 or less. That would increase the EPI estimate of current sub-$10.10 workers to 22.6 million workers, and it would leave 8.3 million workers unaccounted for instead of 7 million workers.
Another follow up point: My regression measured unemployment across the economy. The graph above is measuring only MW workers. This difference could be interpreted in several ways. One could wonder if my regression is undercounting the job losses by even more than it seems. One could also suggest that the MW hike would cause 12.8 million minimum wage jobs to disappear, but that substitutions across the economy would lead to the creation of 12.8 higher paying jobs to replace them.
Minor tweaks in interpretations can get us from point A to point Z, but this relationship looks regular enough and compelling enough that those interpretations have a pretty big gap to fill.