Following some musings from Tyler Cowen, Arindrajit Dube summarizes his empirical research on minimum wages. This literature is not my area, and I therefore have no opinion on the relative merits of competing empirical studies.
In my view the more relevant question is whether data on past minimum wage hikes--typically at the state level--tell us a lot about future hikes in the federal minimum wage. Even if we are persuaded that Dube's reading of the evidence is appropriate, the minimum wage is one of those issues where all future policies involve action outside of the data samples used for empirical studies. Note that this is not the case for all policy questions; I am not suggesting that all empirical research lacks policy relevance, and if your immediate reaction to my skepticism is to ask me if I find all econometric research to be useless then you're ignoring key characteristics of minimum wage policy. Different policies differ in the degree to which relevant empirical studies are externally valid.
To get to the point: Do we expect the effect of a minimum wage hike to be independent of (a) the prior level of the minimum wage and (b) the geographic region (and industrial composition) to which it applies? Is raising the wage from $6 to $7 in one state likely to have the same effect as raising the wage from $7 to $8 nationwide? It would be nice to see a model in which the employment and poverty effects of the minimum wage are linear everywhere; it seems that people relying on empirical studies to justify changes to the federal wage are assuming roughly that. But this is not the case, e.g., in the monopsonist employer model, where the effect on labor demand depends on the level of the wage. If the effects depend on levels (and other things, like industrial composition), then empirical studies are insufficient for telling us about the consequences of policies that have not yet occurred. Don't think of changes in the minimum wage only in terms of the difference between the new wage and the old; think about initial conditions as well. Theory matters here; Dube mentions the "credibility revolution," but I think credibility demands that we know why his results are the best lens through which to view potential policies in the future.
In short, the old cliche is useful here: If you support raising the minimum wage, do you support raising it to $50/hour? If not, then you admit that the effects are nonlinear. Now you need to provide evidence in favor of your preferred wage level that doesn't rely entirely on marginal effects from past data. There may be good reasons for the fact that economists have a hard time agreeing about this issue.