Actually, contrary to my pessimistic expectation, some studies exist, e.g."Direct Democracy and Resource Allocation:
Experimental Evidence from Afghanistan":
Direct democracy is designed to better align policy outcomes with citizen preferences.
To test this proposition, we randomized whether 250 villages across Afghanistan
selected projects by secret-ballot referenda or by consultation meetings. We find that
referenda reduce the influence of local elites over both project type and location.
Consistent with previous experimental results, we also find that referenda improve
villagers’ perceptions of the local economy and of the quality of local governance.
However, we find no systematic evidence that selecting projects via referenda increases
the average impact of such projects
(Let me look at their related-work section to see what else is there.) One other similar study they mention:
Using experimental variations similar to those employed in this
study, Olken (2010) finds a strong positive effect of the use of direct democracy in Indonesia on villagers’ satisfaction, but no effect on the choice of the general project itself, other than a relocation
effect of women’s projects towards poorer areas.
The paper also cites a fairly cited (elsewere) work: Matsuska, J. (2005), “Direct Democracy Works,” The Journal of Economic Perspectives, 19(2), 185-206. but this paper turns out to be mostly a historical & observational (rather than experimental) account. It does have some theoretical & empirical observations which pertain to your (farily open) question though:
Of course, the bottom line is whether direct democracy makes government
better or worse. This question may not have an objective answer since whether a
policy is good or bad so often is in the eye of the beholder. Nevertheless, some
research has attempted to assess the quality of direct democracy policymaking in
terms of efficiency, utility and other concepts that economists traditionally use to evaluate outcomes.
There's 5 pages following that reviewing various works, which is too much to cover here.