Journalists have searched news archives for any evidence of such footage and turned up empty handed. Public officials and law enforcement in New Jersey say Trump's account is utterly false.
But Trump's assertion might not be a bald-faced lie. Psychologists suggest that people unconsciously fabricate memories all the time, and that Trump might have done the same.
"I can't say that he's not lying," said Deryn Strange, a psychologist at the John Jay College of Criminal Justice in New York. "But my research, and the research of my colleagues, certainly supports a more charitable interpretation: that this is a false memory."
There is a well-circulated video clip of Palestinians celebrating the attacks in the West Bank, as The Washington Post's Glenn Kessler has reported. Newspapers, including The Post, stated that police had detained some people in northern New Jersey who were allegedly partying on rooftops and watching the mayhem in New York. The Newark Star-Ledger later reported that “rumors of rooftop celebrations of the attack by Muslims here proved unfounded.”
Trump might have conflated the two in his mind. "I can easily conceive how he could have developed this memory," Strange said. "All you need is a suggestion."
In the laboratory, psychologists have been able to induce more fantastic falsehoods in people. In one entertaining study, researchers doctored people's childhood photographs to place them in hot air balloons. Then they showed the manipulated images to the adults whose photographs they used, along with real photographs from birthday parties and family vacations. The researchers were able to persuade half the adults in the study that they really had taken a ride in a hot air balloon as children.
Such false memories have been shown to persist for as long as a year and a half. And they aren't always fuzzy, vague memories, either. The mind has a tendency to embellish them with lifelike details, as Elizabeth Loftus of the University of California, Irvine, has written. People will remember embarrassing themselves at a wedding or nearly drowning and being rescued by a lifeguard.
Loftus, a psychologist and an expert on the ways the mind concocts memories, said Trump could be misremembering rather than deliberately lying.
"Just because someone tells you something with a lot of confidence, detail and emotion, it doesn't mean it really happened," she said.
When it comes to the news, she noted, people are more likely to invent memories that support their political beliefs. She and her colleague showed participants in an online survey a doctored photograph of a false event involving either President George W. Bush or President Obama. They found that liberals were more likely to say they remembered President Bush vacationing with baseball ace Roger Clemens at his ranch during Hurricane Katrina, while conservatives were more likely to say they remembered President Obama meeting with former Iranian President Mahmoud Ahmadinejad. Neither meeting took place.
Trump, who has made his opposition to immigration well known, may have similarly invented a memory of seeing foreign-born residents of New Jersey on television, Loftus suggested.
"He's got this belief, and it's going to help him construct a false memory that supports that belief," she said.
Trump's story recalls that of another front-runner in a contested presidential primary -- Hillary Clinton, who said in 2008 that she had come under sniper fire in Bosnia. It wasn't a lie but a mistake, she later said.
Unlike Trump, though, Clinton apologized when journalists disproved her story. "I made a mistake. That happens," she said. "It proves I'm human, which, you know, for some people, is a revelation."
"It was on television. I saw it," Trump told Stephanopolous. "They were cheering as the World Trade Center came down."
The headline of this story has been changed.