President Trump types a tweet or makes an off-the-cuff remark to reporters about the Federal Reserve or China trade talks, and the Big Board starts moving.

For Wall Street, that is the all-too-familiar story of the Trump presidency.

“I’ve been in this business for 31 years, and I can’t recall a time when news from Washington, D.C., moved markets with such frequency,” said Randy Frederick, vice president of trading and derivatives at the Schwab Center for Financial Research. Unexpected or breaking news stories always have had the potential to create market volatility, but those events happen on an almost daily basis in the Trump era, he said.

Unlike his White House predecessors who saw danger in tying their presidency to the unpredictable stock market, Mr. Trump has made it the measure of his success. He closely monitors the market’s ups and downs, weighing in on Twitter with the same fervor in which he tweets criticism of the news media or major policy announcements.

Whether it was the president’s tweets adding fuel to the Nov. 28 rally last year that sent the Dow Jones Industrial Average up 618 points or piling on to the recent Christmas Eve selloff that shaved 653 points off the blue-chip index, Mr. Trump’s running commentary is driving the market.

“The only problem our economy has is the Fed. They don’t have a feel for the Market, they don’t understand necessary Trade Wars or Strong Dollars or even Democrat Shutdowns over Borders. The Fed is like a powerful golfer who can’t score because he has no touch — he can’t putt!” Mr. Trump fumed on Twitter that Christmas Eve morning.

A day after haranguing by Mr. Trump, Federal Reserve Chairman Jerome Powell on Nov. 28 backed down from his hawkish stance on interest rates, telling the Economic Club of New York that the central bank has “no preset policy path. We will be paying very close attention to what incoming economic and financial data are telling us.”

Investors rallied, egged on by Mr. Trump’s tweet about Steel Dynamics creating “600 good-paying U.S. JOBS” through an expansion the company announced.

An analysis by Bloomberg found 163 tweets by Mr. Trump about the economy or stock market that correlated with swings in the Dow Jones Industrial Average during the two years since his inauguration.

The tweets coincided with 95 gains and 68 declines for the day on the Dow, according to the analysis.

Mr. Frederick warned that investors, traders and Americans, in general, should get used to presidential tweets setting the mood on Wall Street.

“For better or for worse (depending upon your perspective) President Trump is the first president to embrace the use of Twitter, or any social media for that matter, to communicate directly with the American people,” he said. “It seems highly likely to me that a new precedent has been set for all future U.S. presidents.”

Mr. Trump has weathered criticism of his Twitter habit since his 2016 campaign, including that it is not presidential, lacks the careful vetting expected for official statements, and attempts to run the government in 280-character blurbs.

Making major policy moves on Twitter, Mr. Trump sometimes catches the White House and Cabinet secretaries by surprise, such as announcing the U.S. military pullout from Syria or restoring a ban on transgender people in the armed forces.

While Mr. Trump’s tweets have been effective in delivering his message to his base and keeping it energized, the results of governing with Twitter have been more mixed, said James P. Pfiffner, a professor of public policy at George Mason University.

“They show what the president is thinking at a particular time, but that is not necessarily a guide to what the policies of his administration are,” he said. “Off-the-cuff remarks are similar to tweets in their effect. They may signal what the president may do more formally in setting administration policy.”

For an example, Mr. Pfiffner pointed to the president’s tweets and off-the-cuff comments about tariffs.

“He may say something about tariffs, which might affect markets, but the tariffs will only take effect after formal actions by departments and agencies,” he said.

John P. Wihbey, a scholar of presidential communication at Northeastern University School of Journalism, is a fan of Mr. Trump’s tweets but sees other shortfalls in White House operations.

“I can’t defend the way in which he has sometimes surprised his own staff with policy announcements, but that seems a separate issue about internal management competence and coordination. I also think President Trump would be well-served to tweet a bit less frequently, as I think he dilutes his message to some extent and people tune it out,” he said.

He added that “hearing directly from leaders, in their words — and without layers upon layers of pollsters and communications staff micromanaging — is altogether a good thing.”

Mr. Wihbey noted that other politicians such as Democrats Beto O’Rourke of Texas and Rep. Alexandria Ocasio-Cortez of New York use social media in a similar way.

“Authenticity, transparency and genuine conviction are values that, because of peer-to-peer technologies, will be increasingly successful with citizens. This is not to say that we should steep all public policy in some kind of celebrity culture. Rather, I think it is increasingly important for leaders to humanize their decision-making and the personal way in which they arrive at positions. This is true even if one dislikes all of President Trump’s policy preferences,” he said.