WASHINGTON – Tuba player Andrew Schwartz quit the Manhattan School of Music in 2011 when he saw opportunities shrinking and orchestras struggling. After a series of low-paid jobs such as selling stocks by phone, he left the workforce in August to pursue a master’s degree in business administration.

“The three letters M-B-A are going to be incredibly valuable to me,” said Schwartz, 26, whose undergraduate degree is in music. “I’m sure 99.9 percent of HR people throw my applications in the garbage and the other 0.1 percent laugh at me having a music degree.”

Schwartz is among millions of Americans whose departure from the labor force since the start of the last recession adds a layer of complexity to the Federal Reserve’s effort to attack unemployment by linking monetary policy to the jobless rate.

When people like Schwartz stop looking for work, the government no longer counts them as unemployed, which lowers the official jobless rate.

That means the Fed will need to look at other gauges of labor-market strength because the unemployment rate may send misleading signals of improvement long before payrolls start showing the substantial gains policymakers are seeking. Fed officials have pledged to keep the main interest rate near zero as long as unemployment remains above 6.5 percent.

“It complicates policy,” said Michael Feroli, chief U.S. economist at JPMorgan Chase in New York and a former Fed researcher in Washington. “The unemployment rate, as they’ve pointed out, may be the best single number for judging the health of the labor market, but it’s still imperfect, and those imperfections have grown because of the participation rate.”

The unemployment rate dropped to a four-year low of 7.5 percent in April, with the labor force participation rate at 63.3 percent, the lowest level since May 1979, Labor Department data show. The participation rate measures those with jobs, or looking for work, as a percentage of the civilian U.S. population age 16 and older.

The number of so-called discouraged workers who have given up on finding work stood at 835,000 in April, near the 861,000 average over the past year, Labor Department data show. Participation may remain depressed for years, San Francisco Fed researchers said last week.

Fed officials also say they must see substantial employment gains before they curb bond buying. Vice Chairman Janet Yellen has said the unemployment rate has limits as a policy guide and should be supplemented by gauges such as payroll growth.

San Francisco Fed President John Williams said last week the speed and magnitude of the participation rate’s drop over the past couple of years has been “striking.” Researchers are trying to decipher how much is because of a weak economy, which Fed policy can influence, and how much is the work of structural forces such as the retirement of millions in the post-World War II baby boom generation, he told reporters this month in Portland, Ore.

In December, Fed policymakers for the first time linked the outlook for the main interest rate to unemployment and inflation. Abandoning their previous practice of projecting how long current stimulus levels would continue, policymakers instead said borrowing costs will stay low “at least as long” as the jobless rate remains above 6.5 percent and inflation is projected to be no more than 2.5 percent. Policymakers still may hold rates low even if unemployment declines below 6.5 percent.

“It’s a big puzzle for the Fed,” said Yelena Shulyatyeva, a U.S. economist at BNP Paribas SA in New York. “What do they do if we reach 6.5 percent without significant job growth and just due to people leaving the labor force? Are they going to tighten? I don’t think so. They need to tighten the message.”

Even as unemployment declines, other gauges show U.S. workers remain distressed. The U.S. has only regained 6.2 million of the 8.7 million jobs lost after the recession began in December 2007. More than 4.35 million Americans have been out of work for six months or more, accounting for 37.4 percent of the jobless. Since the Labor Department began collecting monthly data in 1948, that share never exceeded 30 percent until after the 18-month recession ended in June 2009.