panel data model

[mesamir]

I am interested in running a panel data model to explain the push factor driver of two African countries' outward FDI. I succeeded in collecting data about 8 potential explaining variables. Yet the problem is which model should I run to test the significance of my independent my variables.
According to my knowledge, I have to run both random effect and fixed effect models and choose between them by hausman test. But in my case, It's impossible to run a random effect model as the number of regressors is higher than the cross sections.
Therefore, I would appreciate if you could advise on what is the exist from this problem? is it acceptable to run only the fixed effect? if yes, how can I defend this approach? if not what is the alternative approach.