New home construction is soaring, with the annualized rate of housing starts jumping 20.2 in April from March to the highest level in more than seven years.

But that isn't enough to make a believer out of former Federal Reserve Chairman Alan Greenspan. He notes that the pace for new construction of both homes and commercial properties hasn't returned to the lofty levels that prevailed before the 2008 financial crisis.

"We haven't come out of the bottom," Greenspan told CNNMoney. "We are in the position now of secular stagnation" for real estate.

Before the 2007-09 Great Recession construction of homes and buildings expected to last for 20-plus years accounted for 8 percent of GDP. Now it's only 4 percent, Greenspan notes.

And that's bad news for the overall economy, because real estate construction played a key role in each of the 10 rebounds from recession since World War II, Greenspan explained.

The S&P/Case Shiller 20-city home price index rose 5 percent in the 12 months through March, and new home sales climbed 6.8 percent in the 12 months through April.

But the housing market looks bleak for the poor. "For many lower-income Americans, the housing bust of 2008 was just a prelude to a new crisis," Elyse Cherry, CEO of Boston Community Capital, which works to help the underprivileged, writes in The New York Times.

"In many areas, housing prices are stuck below their inflated pre-bubble levels. Until we deal with this fact, entire communities will continue to struggle with high foreclosure rates and a lack of economic mobility."

Ironically, an improvement after the 2008 crisis has helped create the problem, Cherry says. With the "return to sane mortgage underwriting practices, loan officers are no longer handing out mortgages left and right, but instead are tying them to borrowers' income."

But income for the poor is stagnant, she notes. Average hourly wages rose only 2.3 percent in the 12 months through May.

"Absent an extraordinary increase in income for low-income families, home prices in low-income areas aren’t going anywhere," Cherry states.

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