Rather than trying to dig out lessons from changes in the gradually declining real minimum wage over time, lots of research in the last few decades has instead tried to look at US states or cities where the minimum wage increased over time. Then the study either does a before-and-after comparison of trends, or looks for a comparison location where the minimum wage didn't rise. 

But this kind of analysis is subject to the basic problem that the states or cities that choose to raise their minimum wages are not randomly selected. They are usually places where average wages and wages for low-skill workers are already higher. As an extreme example, the minimum wage in various cities near the heart of Silicon Valley (Palo Alto, San Francisco, Berkeley, Santa Clara, Mountain View, Sunnyvale, Los Altos) is already above $15/hour. But in general, wages are also much higher in those areas. Asking the question of whether these higher minimum wages reduced low-skill or low-wage employment in these cities is an interesting research topic, but no sensible person would extrapolate the answers from Silicon Valley to how a $15/hour minimum wage would affect employment in, say, Mississippi, where half of all workers in the state earn less that $15/hour. 

Many additional complexities arise. Clemens goes through many of the possibilities in his paper. Here are some of them. 

1) Economists commonly divided workers into those in the "tradeable" or the "nontradeable" sector. A "nontradeable" good would be working at a coffee shop, where you compete against other coffee shops in the same immediate area, but not against coffee shops in other states or countries. A "tradeable" good might be a manufacturing job where your output is shipped to other locations, and so you do compete directly against producers from other locations. 

If you work in a tradeable-sector job and the state-level or local-level minimum wage rises, it may cause real problems for the firm, which is competing against outsiders. But many low-skilled jobs are in the "nontradeable" sector: food, hotels, and others. In those situations, a rise in the minimum wage means means higher costs for all the local competing firms--in which case it will be easier to past those costs along to consumers in the form of higher prices. Of course, if an employer can pass along the  higher minimum wage to consumers, any employment effects may be muted. 

2) An employer faced with a higher minimum wage might try to offset this change by paying lower benefits (vacation, overtime, health insurance, life insurance, and so on.)  The employer might also try to get more output from workers by, for example, offering less job flexibility or pushing them harder in the workplace. 

3) A higher minimum wage means an increased incentive for employers and worker to break the law and to evade the minimum wage. Clemens cites one "analysis of recent minimum wage changes [which] estimates that noncompliance has averaged roughly 14 to 21 cents per $1 of realized wage gain."

4) An employer faced with a higher minimum wage might, for a time, not have many immediate options for adjustment. But over a timeframe of a year or two, the employer might start figuring out ways to substitute high-paid labor for the now pricier minimum wage labor, or to look for ways of automating or outsourcing minimum wage jobs. Any study that focuses on effects of a minimum wage during a relatively small time window will miss these effects. But any study that tries to look at long-run effects of minimum wage changes will find that many other factors are also changing in the long run, so sorting out just the effect of the minimum wage will be tough. 

5) A higher minimum wage doesn't just affect employers, but also affects workers. A higher wage means that workers are likely to work harder and less likely to quit. Thus, a firm that is required to pay a higher minimum wage might recoup a portion of that money from lower costs of worker turnover and training. 

There is ongoing research on all of these points. There is some evidence backing them, and some evidence not, but the evidence again often varies by place, time, occupation, and which comparison group is used. The markets for supply and demand of labor are complicated places. 

I don't mean to be a whiner about it, but figuring out the effects of a higher minimum wage from the existing evidence is a difficult question.  But of course, no one weeps for the analytical problems of economists. Most people just want a bottom line on whether a $15/hour minimum wage is good or bad, so that they know whether to treat you as friend or foe--depending on whether you agree with their own predetermined beliefs. I'm not a fan of playing that game, but here are a few thoughts on the overall controversy. 

• It's worth remember the old adage that "absence of evidence is not evidence of absence." That is, just because it's hard to provide ironclad statistical proof that a minimum wage reduces employment doesn't prove that the effect is zero--it just means that getting strong evidence is hard. 
  
  
• Since the federal minimum wage was enacted in the 1930s, it has always been a situation where a number of states have set a higher minimum wage. The recent shift is toward cities setting a higher minimum wage than the state. Thus, the effects of raising the federal minimum wage to $15/hour will not (mostly) be felt in places where the minimum wage is already at or near that level: instead, it will be felt in all the other locations. 
  
  
• Many minimum wage workers are also part-time workers. Thus, it's easy to imagine an example, where, say, the minimum wage rises 20% but for a certain person their hours worked are cut by 10%. This is a situation where the minimum wage led to fewer hours worked, but the worker still has higher annual income.  
  
  
• To the extent that a higher minimum wage does affect the demand for low-skilled labor, such effects will be less perceptible in a strong or growing economy when employment is generally expanding for other reasons, and more perceptible in a weak or recessionary economy, when fewer firms are looking to hire. 
  
  
• Everyone agrees that a smaller rise in the minimum wage will have smaller effects, and a larger rise in the minimum wage will have larger effects. I know a number of liberal-leaning, Democratic-voting economists who are just fine with the tradeoffs of raising the federal minimum wage to some extent, but who also think that a rise to $15/hour for the national minimum wage (as opposed to the minimum wage in high-wage cities and states) is too much. 

True gluttons for punishment who have read this far may want some recent minimum wage studies to look at. In this case at least, your wish is my command: 

"Wages, Minimum Wages, and Price Pass-Through: The Case of McDonald’s Restaurants," by Orley Ashenfelter and Štěpán Jurajda (Princeton University Industrial Relations Section, Working Paper #646, January 2021). "We find no association between the adoption of labor-saving touch screen ordering technology and minimum wage hikes. Our data imply that McDonald’s restaurants pass through the higher costs of minimum wage increases in the form of higher prices of the Big Mac sandwich."

"Myth or Measurement: What Does the New Minimum Wage Research Say about Minimum Wages and Job Loss in the United States?" by David Neumark and Peter Shirley (National Bureau of Economic Research,  Working Paper 28388,  January 2021). "We explore the question of what conclusions can be drawn from the literature, focusing on the evidence using subnational minimum wage variation within the United States that has dominated the research landscape since the early 1990s. To accomplish this, we assembled the entire set of published studies in this literature and identified the core estimates that support the conclusions from each study, in most cases relying on responses from the researchers who wrote these papers.Our key conclusions are: (i) there is a clear preponderance of negative estimates in the literature; (ii) this evidence is stronger for teens and young adults as well as the less-educated; (iii) the evidence from studies of directly-affected workers points even more strongly to negative employment effects; and (iv) the evidence from studies of low-wage industries is less one-sided."

"Seeing Beyond the Trees: Using Machine Learning to Estimate the Impact of Minimum Wages on Labor Market Outcomes," by Doruk Cengiz, Arindrajit Dube, Attila S. Lindner and David Zentler-Munro (National Bureau of Economic Research Working Paper 28399, January 2021). "We apply modern machine learning tools to construct demographically-based treatment groups capturing around 75% of all minimum wage workers—a major improvement over the literature which has focused on fairly narrow subgroups where the policy has a large bite (e.g., teens). By exploiting 172 prominent minimum wages between 1979 and 2019 we find that there is a very clear increase in average wages of workers in these groups following a minimum wage increase, while there is little evidence of employment loss. Furthermore, we find no indication that minimum wage has a negative effect on the unemployment rate, on the labor force participation, or on the labor market transitions.

"The Budgetary Effects of the Raise the Wage Act of 2021," Congressional Budget Office (February 2021). "CBO projects that, on net, the Raise the Wage Act of 2021 would reduce employment by increasing amounts over the 2021–2025 period. In 2025, when the minimum wage reached $15 per hour, employment would be reduced by 1.4 million workers (or 0.9 percent), according to CBO’s average estimate. In 2021, most workers who would not have a job because of the higher minimum wage would still be looking for work and hence be categorized as unemployed; by 2025, however, half of the 1.4 million people who would be jobless because of the bill would have dropped out of the labor force, CBO estimates. Young, less educated people would account for a disproportionate share of those reductions in employment."