In an interview with SmartMoney, former Federal Reserve Chairman Paul Volcker offered his thoughts on the financial regulatory bill, before weighing in on what isn’t in the bill: the revamp of mortgage-finance giants Fannie Mae and Freddie Mac.

Mr. Volcker doesn’t specify how he would go about remaking the mortgage market, but suggests that the hybrid shareholder-owned, government-sponsored structure of the old Fannie and Freddie needs to be discarded. At the same time, he says the current arrangement of government control for the market shouldn’t be a permanent fix.

[SmartMoney]: What’s missing [from the financial-regulatory overhaul]?

Mr. Volcker: People talk about Fannie Mae and Freddie Mac. That’s a challenge for next year and year following. We are going to have to reconstruct the whole mortgage market and you can’t do that overnight. The mortgage market now is almost a wholly owned subsidiary of the United States government. Almost all the mortgages made now are insured by the government, bought by the government, and the guys at Fannie Mae and Freddie Mac are the market.

Not much exists without the government running it. I don’t think that’s what we want. A lot of problems surround the whole mortgage market. It’s clear Fannie Mae and Freddie Mac need to go. We don’t need these hybrid institutions. You don’t know whether they should be responsible to the government or to stockholders. It’s an unfortunate invention.

On Thursday, House Democrats held the first of a series of hearings that they said would lead to legislation this fall concerning the future of Fannie, Freddie, and the broader mortgage market.

Rep. Paul Kanjorski (D., Pa.) said in a statement that those hearings would also look at “innovative ideas” to help pay back the government for the $145 billion it has injected to keep the firms afloat.