WASHINGTON—A controversial Trump administration nominee for the Federal Reserve’s policy making board cleared a major hurdle in her path to confirmation Tuesday after a Senate panel voted to support the choice.

The candidacy of Judy Shelton, an economic adviser to Mr. Trump’s 2016 presidential campaign, for the Fed’s Board of Governors was approved by the Senate banking committee in a party-line vote despite objections from Democrats. Her next and final stop will be a confirmation vote on the Senate floor.

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