During her four-year term, the unemployment rate has fallen to 4.2 percent while inflation has remained below 2 percent. Both metrics are close to the levels the Fed regards as ideal. Indeed, it has seldom come closer. She also has presided over the gradual reduction of the Fed’s stimulus campaign, which was implemented in the wake of the 2008 financial crisis, as the economy struggled through a bruising recession with persistent unemployment that peaked at 10 percent in October 2009.

Ms. Yellen’s calm and careful leadership style has won fans, both inside and outside of the Fed. John Williams, president of the San Francisco Fed, said that Ms. Yellen had been effective in building a consensus about the direction of policy among Fed officials, and in communicating policy to the markets and the public.

“When she came in there was a lot of concern about how would we unwind our policies and would that be disruptive and difficult, especially in the context of a global economy where other central banks were going in the other direction,” he said. “Now it’s almost like, ‘That was easy!’ But it wasn’t. It took hard work.”

Some of the other candidates, notably Mr. Taylor and Mr. Warsh, have argued the Fed should be raising interest rates more quickly. But financial regulation is the area in which a new chairman likely would make the largest changes. The Fed supervises the nation’s largest banks and oversees the broader financial system.

Mr. Trump wants to loosen financial regulations, which he regards as an impediment to economic growth. Ms. Yellen, who played a key role implementing the new banking rules after the 2008 crisis, has acknowledged room for improvement. But in a high-profile speech in August, she also issued a warning against going too far.

The Treasury secretary, Steven Mnuchin, has argued that installing a new Fed chairman would help Mr. Trump achieve his goal of reducing regulation.

Charles W. Calomiris, a finance professor at Columbia University, said that Ms. Yellen’s approach to regulation was reason enough to replace her. “It’s not just that regulation is excessively costly and complex,” he said. “It’s also failing to achieve its objectives.”