It's not quite Morning in America, but, at least for the long-term unemployed, it's no longer the middle of the night.

Here's the good news: in the past six months, unemployment has fallen much faster than expected, from 6.7 to 6.1 percent. And as you can see above, 88 percent of that has been due to declining long-term unemployment.

But is it really good? Long-term unemployment might be falling for the bad reason that people are giving up, rather than for the good one that they're getting jobs. (Remember, you have to have looked for work in the past month to count as "unemployed"). Now, this is always a concern, but it's even more so since the start of the year when long-term unemployment benefits expired. That's because those benefits required people to try to find a job, so without them the long-term unemployed—who have trouble getting companies to even read their resumés—might just be calling it quits.

So what does the evidence say? Well, it's not clear. Ben Casselman points out that, based on the monthly data, the long-term unemployed still aren't finding work much better than they were at depths of the crisis.

But Tomaz Cajner and David Ratner of the Fed counter that these numbers can be misleading, because:

1) the jobs they are finding might be longer-lasting now, and

2) they might be finding jobs when they aren't officially "unemployed."

See, Cajner and Ratner argue that the long-term unemployed's "lower search intensity" means that they don't "always qualify as unemployed from the viewpoint of the BLS." So, because they sometimes go more than four weeks without looking work, it's possible for them to move from "out of the labor force" to "employment"—which makes it all the harder to keep track of how many of them are ultimately getting jobs.

We can get a better picture of the long-term unemployed if we look at the yearly, rather than the monthly, numbers. That eliminates most of the noise from short-term jobs, and short-term moves between unemployment and non-participation. It simply tells us how many of the long-term unemployed have a job a year later, and how many have stopped looking at that time.

By this measure, the recovery is finally starting to reach the long-term unemployed a little bit more. As you can see in the chart below from Cajner and Ranter, 38 percent of the long-term unemployed from a year ago have a job today. That's almost as many did in 2007, right before the recession hit. And even better, it's more than the number who exited the labor force in the past year.

In other words, long-term unemployment is falling more because people are getting jobs than giving up.

This doesn't mean, though, that we should stop worrying about the long-term unemployed. As former CEA Chair Alan Krueger has shown, even the long-term unemployed who do find jobs very rarely find permanent ones. Instead, they tend to bounce from one temporary gig to another.

But still, a modicum of good news is better than none. After five years, we'll take it.