April 21 (Reuters) - In testimony Tuesday at the Senate Banking Committee, Federal Reserve Chair nominee Kevin Warsh never said if he agrees with President Donald Trump's view that the Fed's policy rate is too high and should be cut. No lawmaker asked him that directly.

But Warsh's two and a half hours of often combative exchanges with Democratic senators and friendly ones with Republicans yielded plenty of other insights about both his approach to the job and the heated politics of the moment.

Here are five: 

POLITICS AND THE PRESIDENT

Elizabeth Warren, the panel's top Democrat, asked Warsh if Trump lost the 2020 presidential election, a question that she has asked other Trump nominees to showcase what she feels is excessive deference to a president who asserts he won it. To be clear: he did not. 

Warsh, though, sidestepped. "Umm, we try to keep politics, if I'm confirmed, out of the Fed ... I believe this body certified that election many years ago." As Warren repeated her question, Warsh spoke over her, saying it was critical to "keep the politics out of monetary policy, keep monetary policy out of politics."

He was repeatedly pressed on whether Trump had tried to get him to promise to cut rates as Fed chair. "The president never asked me to predetermine, commit, fix, decide on any interest rate decision, in any of our discussions, nor would I ever agree," Warsh told Republican Senator John Kennedy, thanking him for the question.

ON COOK AND POWELL

Asked about Fed Governor Lisa Cook, whom Trump has tried to fire for alleged misstatements on her mortgage application, Warsh said he could not comment.

"If I stand for anything, it's the Fed should stay in its lane ... it's inappropriate for me to weigh in on that." Cook has sued to keep her job. Fed Chair Jerome Powell called it "the most important legal case" in Fed history and attended the oral arguments at the Supreme Court earlier this year.

Warsh said he expects a ruling soon. "If confirmed by this body, I will follow the Constitution, the Supreme Court law, and the best of the Fed's traditions," he said. 

Warsh also sidestepped a question about whether Powell can stay on as temporary Fed chair after May 15 - as he says the law requires - if Warsh is not yet confirmed by then.

Republican Senator Thom Tillis on Tuesday repeated that he would block his confirmation as long as the Department of Justice keeps open its investigation of Powell for cost overruns on building renovations.

"I'm certainly not capable of defining whether the Vacancy Act applies to the Federal Reserve or not," Warsh said, referring to the law that allows the president to appoint temporary agency heads in some cases. 

FEWER FED MEETINGS, FED CHAIR PRESS CONFERENCES?

Warsh promised broad reform at the Fed so as to trim its balance sheet, measure inflation better, and cooperate more with the Treasury and other parts of the administration on issues not directly related to monetary policy. He also wants to make changes to how the Fed communicates to the public, and with each other. 

Warsh appeared to open the door to a change in the number of Fed policy-setting meetings, which currently take place eight times a year. Warsh noted that the law requires only four, adding, "four is not enough, so having more meetings than that is appropriate. But I've not even begun to look at the meeting schedules for 2027."

Asked to commit to doing press conferences after each Fed meeting, as Powell has done since 2018, Warsh notably did not.

"Right now, Fed chairs and other central bankers around the FOMC, they speak quite frequently. There was no lack of transparency. But I would say this, I think truth seeking is more important than repetition," Warsh said. Would he take reporters questions, he was then asked. "If a press conference were held, I think it would be incumbent to hear what the reporters of the day had in mind."

NEW APPROACH TO INFLATION? 

Warsh said he wants a better way to measure the "underlying" inflation trend, and said one of his first projects at the Fed would be a "data project" with public and private sector actors to identify such a gauge.

"The measures I prefer are looking at things that are called trimmed averages where we take out all of the tail risks," he said, an apparent reference to measures like the Dallas Fed trimmed mean that lop off data at the extremes to give policymakers a better picture of inflation's center of gravity. Many current Fed policymakers say they consult such measures regularly.

ARTIFICIAL INTELLIGENCE, INFLATION AND THE LABOR MARKET

Trump administration officials say that the productivity gains brought by the rapid adoption of artificial intelligence will push down on inflation quickly, allowing the Fed to cut interest rates.

Warsh agreed that is possible, but stopped short of signing on to the notion completely.

"It's important to revisit the Fed's models and see whether this innovation cycle - while it could have over time improvements in the price level and make the Fed's job on inflation easier - there's a question about what that means for employment, which is another part of the Fed's mandate," he said. "I am more confident that there will be improved output than I am certain about when the effects of that would be on the labor market ... the lag between the improvement on output and the effect on the labor markets, that's got to be central to the Fed's thinking."