The Federal Reserve has linked rising student debt to a drop in homeownership among young Americans and the flight of college graduates from rural areas, two big shifts that have helped reshape the U.S. economy.

The effect of student debt on the economy has been debated in recent years, as the total has soared to $1.5 trillion, surpassing Americans’ credit-card and car-loan bills. Congress and various White House administrations have pointed to federal student loans as a key way for Americans to pay for college and boost their...