At least for gender, the evidence is mixed [1].
It’s a complicated question. For example, the evidence is compelling for French companies [2].
[1] https://hbr.org/2019/03/when-and-why-diversity-improves-your...
[2] https://www.sciencedirect.com/science/article/abs/pii/S03784...
0: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1364470
1: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3265783
http://iranarze.ir/wp-content/uploads/2018/08/E8806-IranArze...
There are quite a few problems with this sort of social science study that make it hard to draw any conclusions. You can't simply see mixed answers and say there's mixed evidence, because such answers aren't evidence to begin with.
A first problem is the standard correlation/causation fallacy. If a company that appoints women to the board is doing better is that because of the benign influence of the woman, or is it the inverse, that companies doing well feel they can appoint a token woman without running much risk? Or alternatively that women are only attracted to the sorts of low-risk companies that already do well, and thus success breeds women rather than vice-versa? Or much more likely still that it's irrelevant and there's something else at work?
A second problem is confounding factors. Many studies don't even acknowledge them, let alone control for them. This paper does better than most: it recognises that some studies are confounded because they compare companies across countries that vary in mandatory-women-on-boards-laws, which is a variant of the above problem. It selects a single country and a timeframe when (they assert) there was no social pressure to appoint women to boards. But this paper is nonetheless still ultimately a pure correlation = causation argument.
A third problem is small sample sizes. This paper claims to use a "large" sample but it's only a little under 400 firms. This is far too small to draw meaningful conclusions from, partly because (again) it's too small a set to control for confounding factors. Yet they claim it's statistically significant.
A fourth problem is social scientists are very bad at stopping when they seem to be measuring noise. The paper says:
For the various attributes, we find female directors who are foreigners, have business training, and with longer tenure significantly negatively correlate with all performance measures. However, female directors’ reputation measured by their media coverage is positively perceived by market investors and is associated with an improvement of ROA and ROE.
(ROA is return on assets, ROE is return on equity).
These sentences make a highly implausible claim, namely, the study authors are able to explain movements of the stock market by measuring "reputation" via "media coverage". Moreover that women with more experience and business training are seen as less capable by investors.
But in any other context I think most people would agree that reliably explaining the movement of stocks is extremely difficult, especially for something as abstract and long term as a change of board members. Has a major stock ever had a large movement because a non-chairperson role on a corporate board changed? I'm struggling to think of one. If these authors were really capable of analysing corporate performance in such detailed ways, surely they'd choose to become very successful investors instead of writing gender studies papers?
This sort of study is pretty common and these days I just write it off. It's not so much a problem with this specific subject but more that there is far too much research being done on a dodgy basis. Vast numbers of pseudo-scientists aren't able to step back and say "we aren't able to model this, we don't know the answer to that question". You get studies that sound clever but are just reporting random results.