Studies of democracy and growth are plagued with what economists call "identification problems"—it's hard to identify whether relationships you see between variables are due to those variables affecting each other, or some other variable. For example, across the world, democracies are all-round better countries than non-democracies: richer, freer, safer, and happier—but is this due to democracy itself? It's hard to say: these countries also have different institutions, favourable history and geography, and different cultures—all with roots deeper than democracy.
So economists try and come up with clever methods to work out whether democracy itself makes countries more prosperous (or indeed whether it makes them freer, happier, and/or safer). For example, you can try and control for other factors you think you might be missing out, like the rule of law, human capital, and free market institutions. I survey some of this work in a previous post.
Or instead of looking at a "cross section" of countries—a bunch of countries in one time period—and looking how countries vary, you might look at lots of countries over time, and check whether increased democratisation changed their growth trajectories. If everyone's growth seems to pick up just when they democratise, that's much stronger evidence in favour of democracy, because it controls for all the idiosyncratic factors particular to individual states (what economists call "country fixed effects").
You can improve the methodology yet further. When countries democratise, it often isn't the only thing they do at the same time, and it may not be a cause of or necessary feature of the other stuff they do (e.g. formalise land titles and allow markets in various areas). What's more, it may be caused by special circumstances—like an economic crisis—that would lead us to expect accelerated growth afterwards even if democracy itself didn't have any special effect.
A new paper takes these seriously, and attempts to identify the effect of democracy on growth by surveying 165 country-specific political experts, asking them whether an episode of democratisation came out of economic turmoil, or came out of an "exogenous" (i.e. unrelated) desire to increase democracy. They find that once you control for these "endogenous" (i.e. turmoil-related) increases in democracy, democracy no longer causes growth—previous positive findings came from faulty identification.
Now, democracy may still be good for other reasons—people may just like it, so even if it's only equal to the alternatives, we might prefer it—but we should be yet more sceptical that it is responsible for the things we enjoy in our society.