reverse causality and endogeneity problems I am trying to analyse the effect of the gender of a CEO (women vs. men) on a performance measure of two types of firms (Conglomerate: i.e diversified firms, and Stand alone: i.e not diversified firms) for a period of a 22 years. I have panel data from 11200 firms and approximately 245,000 observations. I control for some observable firm characteristics (size , profitability, leverage) and I get positive coefficients for the gender 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 potential endogeneity problems, and try to see if there is a reverse causality, meaning do female CEOs really improve the performance of the firm, OR do they select themselves into better performing firms or better performing firms have more flexibility to nominate a woman as 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 endogeneity problems and reverse causality other than IV instruments?
I am open to any suggestions.
 A: To me this question is outside the realm of any standard econometric, textbook answer or solution. I can see several approaches to addressing it but no one "correct" or "best" solution. 
Personally, I like the panel data model with OLS estimation structure. It makes sense especially wrt pooling the relatively sparse information available for female CEOs. Not to mention that this approach has a long and venerable history in econometric modeling of corporate performance. Just give consideration to transformations to the dependent variable(s) to ensure that it's scale invariant, as appropriate. A key question is whether you use an ANOVA or mixed model (hierarchical) functional form. The latter approach is motivated by the fact that firms can be nested within SIC codes, forming a hierarchical structure. It's been demonstrated that this class of models reduces the variance considerably vs non-hierarchical ANOVA.
You haven't stated what your performance metrics are. This seems like a useful piece of information to know. Lots of metrics suggest themselves such as stock price (or its natural log), year-over-year % return,  market cap, total return to shareholders (aka EVA or economic value added), etc. I would have a hard time picking just one. In fact, my view is why would you want to pick just one when MANOVA-type frameworks are available and are potentially quite insightful? I would want to leverage a combination of dependent variables including relative (% return, TRS, etc.) with absolute (stock price, market cap, etc.) measures to assess how the drivers of performance evolve as a function of the type of measure under analysis.
A Bayesian such as Andrew Gelman, in his book with Jennifer Hill Data Analysis Using Regression and Multilevel/Hierarchical Models (chap. 13), would contend that a sample size of 1 remains analyzable, at least in terms of the posterior information produced by Bayesian models. Adopting this framework might make the sparsity of female CEO information more tractable. 
In terms of one of the tests you want to perform:

In the second part of my analysis, I am trying to correct for
  potential endogeneity problems, and try to see if there is a reverse
  causality, meaning do female CEOs really improve the performance of
  the firm, OR do they select themselves into better performing firms or
  better performing firms have more flexibility to nominate a woman as
  CEO.

One immediate problem with this hypothesis as stated is that you haven't indicated how you are measuring firm "flexibility." In other words, it's a great question but how is "flexibility" operationalized? Comparative firm performance is a more readily testable hypothesis by focusing on performance before and after a female CEO is introduced. 
Endogeneity is a particularly tough issue to address and even the most hardcore econometric modelers will admit that using IVs introduces as many problems as it solves, i.e., the cure can be worse than the disease. Among the recommended "solutions" is 2SLS. Wooldridge, in his book Econometric Analysis of Cross Section and Panel Data devotes several sections to this discussion. My problem with his book is that it's from a theoretician's perspective, i.e., his examples are based on "toy" data where the input matrices are neat, balanced and completely observed, i.e., he doesn't have to deal with missing values, unbalanced matrices wrt time and sample size, censoring or any other irregularities. This is rarely the reality in applied work and is certainly not the case with your information. 
Given that endogeneity is mostly a theoretical problem, the analyst is almost forced to use theory to drive a solution. Empirically, this makes it an almost insoluble, indeterminate problem. Not having read anything in the corporate modeling literature that discusses this specific challenge, I would draw an analogy to "age-period-cohort" models where the aspiration is to disentangle separate effects for each factor independent of the other two. These are issues in fields such as geriatric aging, sociology, marketing or political science, requiring quite long time frames to even initialize the models. (Note age refers to the chronological age of a unit, period refers to a era or set of chronological years, e.g., the 60s or 90s and cohort refers to a a group typically based on birth year, e.g., millennials or baby boomers). Indeterminancy is inherent to the model's structure. For instance, statistician and political scientist, Andrew Gelman has written in an unpublished paper, Thoughts on new statistical procedures for age-period-cohort analyses, (here ... http://www.stat.columbia.edu/~gelman/research/unpublished/apc.pdf), "There can be no general solution to the age-period-cohort problem." He then goes on to say that, "Some methods of constraining the possible space of solutions seem more reasonable than others," but offers little beyond intuition and common sense as selection criteria. 
In your case, the relative recency of female CEOs further complicates the problem. This is a kind of censoring which potentially argues for addressing the issue as a survival model. Another approach might be to treat it as a regression discontinuity model where companies with at least one female CEO are your treatment group and companies never having a female CEO are the control group. The effect modeled is performance before and after the introduction, by collapsing or averaging the model-predicted time series, after controlling for the other factors of course, creating an eight cell design (pre vs post, treatment vs control, male vs female).
Some interactions suggest themselves. Most stat packages offer a variety of sum-of-squares metrics in ANOVA-type models, such as hierarchical or independent effects, which can be used in determining the relative importance of the variables in the model. Ulrike Groemping has papers and an R module (RELAIMPO) which review the heuristics suggested in the literature for relative variable importance. Her approach in her R module is to leverage the various ANOVA-type effects in a computationally intensive way to tease out average or overall variable effects under differing model specifications. Her stuff is worth a look. 
Anyway, these are just a few thoughts. As already noted, a single answer to your questions will not suffice.
/** Later Thoughts **/
You mentioned "reverse causality." To me, that suggests an interest in determining lead-lag structures without an a priori theory about the direction of the causal structure. This paper by Sornette suggests a nonparametric approach to understanding lead-lag structures in which time's arrow is treated agnostically. Non-parametric Determination of Real-Time Lag Structure between Two Time Series: the “Optimal Thermal Causal Path” Method, Sornette and Zhou.
https://arxiv.org/pdf/cond-mat/0408166.pdf
A: One angle could be to look at it from a time perspective. Although not sufficient in itself effect should follow after cause (in time). 
So you could look and see if the appointment of a CEO of a certain gender is followed by higher performance and then flip it and look for the other way around etc
A: There's multiple things going on here.
First, it is not clear to me what your causal quantity of interest is. If it is the ATE of having a female CEO as opposed to having a male CEO, you basically need make sure that you block all back-door paths from the decision to hire a CEO of a certain gender to the firm performance outcome. This is very hard, as the decision is undertaken internally, and the board may have much more information than the researcher. This information drives them to pick a certain CEO and also influences future performance. Also, I am not sure the panel structure helps much in this case, because this unobserved information changes over time and can probably not be captured by firm or time fixed effects.
Second, you mention that you adjust for "size , profitability, leverage". Depending on the time at which you measure these, these variables may actually be influenced by the gender of the CEO. If you're interested in the ATE, you must not adjust for them (but it would make sense to adjust for lagged values of them). However, if you're interested in some kind of "direct effect" of CEO gender, you would need to adjust for them, but in general, identifying the direct effect is even harder than identifying the ATE. See the article Neil G referred to, Pearl, Judea. "Interpretation and identification of causal mediation." Psychological Methods 19.4 (2014).
Lastly, you mention reverse causality. In your context, this would mean future performance influencing past gender of the CEO. This is physically impossible, and so not an issue.
A: 
I am trying to analyse the effect of the gender of a CEO (women vs.
  men) on a performance measure…I control for some observable firm
  characteristics…

Unfortunately, this doesn't work.  Without causal assumptions, you can't support counterfactual conclusions with observational data.  In your case, these assumptions would be of the form of assuming no back doors, which is unrealistic.
When you just control for your possible mediating variables, you open back doors to unobserved confounders.  This is like cutting heads of the hydra.  Each back door that you open would need to be controlled for and doing that can open many more back door paths, and so on.
You need to freeze the value (this is an intervention since it disconnects causal parents) of your mediating variables as you change the treatment.  I think this is what you're getting at in the second paragraph.  This freezing is Pearl's counterfactual operator.  He writes:

The reason such conditioning does not work for the natural direct
  effect is that the latter is not defined in terms of a population
  experiment (i.e., control M to level M = m and change T from T = 0 to
  T = 1) but in terms of a hypothetical manipulation at the unit level,
  namely, for each individual u, freeze M at whatever level it attained
  for that individual, then change T from T =0 to T =1 and observe the
  change in Y).

(Where, $T$ is the treatment, $M$ is the mediator, $Y$ is the outcome.)
Almost your exact example is discussed by Pearl in his recent paper: Pearl, Judea. "Interpretation and identification of causal mediation." Psychological Methods 19.4 (2014): 459.
If you can't do experiments, I guess you can try to be very thorough about closing back doors, but the validity of your conclusions will depend on your assumptions.
