The GSEs are quasi-public, federally chartered corporations with publicly traded stock and a government mandate to provide a specific public service.
In one aspect they are public corporations with shareholders owning publicly traded company stock and in another aspect they sponsored by the US government to perform a public purpose, making them quasi-public versus being wholly private or wholly public.
The GSEs are not private state corporations as usually defined in corporate law and they are not a government agency.
Examples: 1. Ginnie Mae - a wholly owned government corporation
2. Sallie Mae, once a quasi-public corporation (GSE) and now a public corporation with publicly traded stock. Took a decade to end Sallie Mae's status as a quasi-public corporation or GSE and to become a private sector corporation.