There's new evidence that the huge number of Americans counted as long-term unemployed has been dropping in recent months because those people have been finding jobs, not because they've given up looking.

The number of people who tell the Census Bureau they have been out of work for longer than 27 weeks has fallen by roughly 1.2 million over the last year, and by 500,000 so far in 2014.

Whether those people mostly found jobs or simply stopped telling the Census they were looking for work is an important question, one with significant implications for the health of the U.S. economy, for at least two reasons:

• Federal unemployment benefits for the long-term unemployed expired in December. What happened to those workers afterward is an important question for understanding the effects those benefits had.

• Some economists, including former Obama adviser Alan Krueger, have suggested that the long-term unemployed have little influence in the dynamics of the labor market. When it comes to the market setting wages, the short-term unemployment rate is what matters. With the short-term unemployment rate near healthy levels, that thesis suggests that there is relatively little "slack" left in the labor market that the Federal Reserve can address with monetary stimulus without generating above-target inflation.

Some analyses have suggested that so far, those out of work for long periods of time are not finding jobs but dropping out. FiveThirtyEight's Ben Casselman, for example, reviewed monthly Census data to arrive at that conclusion.

Last week, however, Fed Chairwoman Janet Yellen suggested a different answer to the question, saying in congressional testimony that "the evidence that I've seen, although perhaps not utterly definitive, suggests that the long-term -- the decline in long-term unemployment does, on balance, reflect those who have experienced long spells getting jobs and moving into employment and not simply becoming so discouraged that they move out of the labor force."

The evidence Yellen likely had in mind was revealed on Monday by economists at the Fed Board of Governors. In their own analysis of Census data, they found that on a yearly, rather than monthly, basis, "the long-term unemployed are currently more likely to transition to employment than to nonparticipation."

The reason the monthly data do not capture this improvement, the Fed economists speculate, is because in general the long-term unemployed often tell Bureau of Labor Statistics they're looking for a job one month and then that they've given up the next, "blurring" their status in any month. Over a longer period of time, however, it's easier to pick out trends.

That's not the only evidence Fed economists have presented showing that the falling headline unemployment rate has led to the long-term unemployed finding jobs in greater numbers.

In June, Fed researchers looked at what happens to long-term unemployment when the short-term rate falls, using the fact that the 50 states all have different unemployment rates, ranging from ultra-low, such as in the oil-boom state of North Dakota (2.7 percent), to terrible, such as in Rhode Island (7.9 percent).

They found that low short-term unemployment rates also lead to faster job-finding rates for the long-term unemployed.

The long-term unemployed find jobs at higher rates in states with low short-term unemployment

Not only that, but falling short-term unemployment rates also decrease the number of people who have had their hours cut or could only find part-time work.

In other words, a rising tide really does raise all boats in the labor market, despite the historically high levels of long-term unemployment the U.S. has suffered in recent years.