At first make clear your question.
Your model has several equations and the data set is panel data. You want to know how to estimate this model.
This is SEM (Simultaneous-Equations Model). You can use 2sls or 3sls to estimate it whatever the data set is panel or not.
But the problem is if the data set is panel, the heteroscedasticity problems are always exist. To increase efficiency, you can use 2SIS-GLS or 3SLS-GlS to estimate this model, here the structure of error terms of GLS estimates follows the property of panel data-- ei`*ei/T. i is the number of groups and T is the number of periods. (chapter 13 Greene)
If you want to calculate the fixed effect in this model, you only need to add dummy variables for each group.(remember fixed effect model is consistant but not efficient if Heteroscedasticity exists. actually it is an OLS with dummy variables)
I do not know which software can do this directly, But SAS/IML can do it by programming.
You can refer the greene's book-- econometric analysis( fifth version chapter 15).
If you have any comments or opinions, please let me know.