reverse causality and endogeneity problems

#1
Hi all,

I am trying to analyse the effect of the sex of a CEO (female vs male) on a performance measure of two types of firms (Conglomerate and Stand alone) for a period of a 22 years. I have panel datas of 11200 firms and approximately 245 000 observations. I control for some observable firm characteristics (size , profitability, leverage) and i get positive coefficients for the sex variable in conglomerate firms using OLS, also when i control for the year effect (year dummy variables), and when i control for the ONE digit SIC Code (dummy or fixed effect).

In the second part of my analysis, I am trying to correct for potentiel endogeneity problems and try to see if there is a reverse causality, meaning do the female CEO really improve the performance of the firm OR do they select themselves in better performing firms or better performing firms have more flexibility to nominate a Women CEO.
The problem is I do not have any instrument variables, so I am looking for another approach. I was wondering if there was another way for correcting for any endogeny problems and reverse causality other than IV instruments?
I am open to any suggestions.
Thank you very much!

Amira