"The statute that created the FHFA states that DeMarco can only be removed when a new director is either confirmed or installed using a recess appointment"
The Obama administration was prodded to fire Edward DeMarco in 2014. However, at that time the Republicans had retaken control of the Senate. Also, it was speculation the Obama Administration kept DeMarco in office because he was taking the blame for not engaging in principal reductions for underwater homeowners..a move that would have made tax payers pay for peoples home loans.
Basically, the Democrats labeled DeMarco as someone against helping lower income homeowners. However, this kind of program was probably not necessary since the delinquent loan rate has dropped substantially since 2014 and home prices have rebounded. This information was probably something the insiders already knew and cited as "executive privilege" in the attempt to cover up what was really happening at the FHFA/Treasury and FnF.