Minimum sample size for panel data analysis

Sample size is determined by your willingness to be wrong and the robustness of your model. Depending on the conclusion to be drawn from the data (is someone going to die if your analysis result is wrong?) it doesn't appear that your data set is too small.

Thanks for your answer.
I am writing a master thesis, and this is actually all the data I have, since I am analyzing banks' performance and the number of banks in my country is very small (varying from 7 to 13 in different years). My supervisor is asking me to provide reference that such a small sample size is allowed to perform a statistically significant analysis, otherwise I can not proceed with analysis.

Do a power analysis on your model with an alpha set at 0.5 and you'll be able to state exactly how much power you have. Standard statistics usually look for a beta of 0.8, or better, but you can make a claim that you have sufficient power in your circumstances (no one is going to die) with a beta of 0.6, or better. If you're in a university setting you should be able to find someone in the statistics department that can help you perform the power analysis.