“The economy, stupid.” Few quotes are so memorable that they move beyond the world of political pundits and even make it into the popular consciousness. This one isn’t just well known because it’s catchy; it’s also because it’s largely true – at least when it comes to winning elections. These three words were written on a lowly whiteboard at the Clinton campaign headquarters in Little Rock, Arkansas. It was 1992, and Bill Clinton was on course to become America’s forty-second president. But his campaign needed to stay ruthlessly focused. Penned by Clinton’s indefatigable campaign manager, James Carville, these words offered a daily point of reference to everyone on the campaign. If they wanted to see their man reach the White House, they couldn’t afford to meander off point. And it paid off.

I’m guessing that virtually every politician, every pundit, and every political activist in the English-speaking world (and beyond) is familiar with those three words. That’s why it is all the more remarkable that the Vote Leave campaign is taking a different approach to try and win the upcoming EU referendum: “It’s immigration, stupid.” It’s a strategically bizarre move, but it’s because the Brexiteers realize that if it really is all about the economy, they know they can’t win. Instead of trying to win over new supporters, they’re falling back on mobilising the core vote.

It’s too early to tell for sure, but I get the sense that last week fundamentally changed the dynamic of this referendum campaign. It began with a stark warning from the Treasury: if we vote to leave the EU, each family in the UK is forecast to be an average of £4,300 worse off each year. The report was quickly dismissed by the Leave campaign (like most things) as “scaremongering.” They said that the Treasury’s forecasts relied on too many assumptions, and confidently declared that Britain would be able to negotiate very favourable trade deals with other countries upon leaving the EU.

There were two problems, however. The first big problem was exposed by Kate Hoey’s performance in a revealing interview with Andrew Neil. “Can you name any reputable, independent study that shows us better off if we leave?” she was asked. Her shaky response: “I can’t actually produce a study that says that.” Regardless of whether or not people actually believed the Treasury’s projections in the first place, Kate Hoey’s interview response will at the very least remind voters that the Brexit side of the argument isn’t supported by any serious economic projections.

On the other hand, suppose for a second that voters couldn’t care less about abstract economic projections. Perhaps many will come to the conclusion that it’s simply impossible to be sure either way. Even if an authoritative report does emerge that says Britain would be economically better off leaving the EU, they may be just as skeptical of its numbers as the Treasury’s. They’re not interested in numbers, they’re interested in the credibility of who is making the argument.

This is where the second, and arguably much bigger, problem emerged for the Leave campaign. When Air Force One touched down at Stansted Airport on Thursday night, turbulence was in the air for Boris Johnson, Nigel Farage and co. If they had been able to dismiss the gloomy assumptions behind the Treasury report by saying the UK would negotiate fantastic trade deals with other powerful countries, not least the United States, they were dealt a hammer blow.

This is also why the president’s intervention was perfectly reasonable. “They are voicing an opinion about what the United States is going to do. I figured you might want to hear from the president of the United States what I think the United States is going to do.” His message was clear: “I think it’s fair to say that maybe some point down the line there might be a UK-US trade agreement, but it’s not going to happen any time soon because our focus is in negotiating with a big bloc, the European Union, to get a trade agreement done”. Until then, however, “the UK is going to be in the back of the queue.”

At first sight, the president’s intervention might have backfired. British people, of course, don’t like to be told what to do. Indeed, that’s one of the main arguments behind leaving the EU. A YouGov poll found that 53% of respondents felt it was inappropriate for the president to express a view; only 35% thought it was appropriate. Whether people like it or not, however, the US president has intervened. As the dust settles, the people who are wavering between voting to remain or leave the EU are unlikely to weigh up various statistical analyses. Instead, they will remember that time Obama came to London, stood beside David Cameron, and warned that Brexit will come at a cost.

The Brexiteers know this. That’s why Boris Johnson wrote his pre-emptive in the Sun, and it’s why Nigel Farage has been keen to portray the president as a ‘lame duck’ with only a matter of months left in office. On the other hand, it’s precisely because Obama isn’t facing re-election that he is being so candid. His warning simply won’t be forgotten.

So, the Brexiteers have switched tack. Nigel Farage admitted as much on BBC News yesterday. Instead of simply keeping the ball in their half, they need to score a goal. And how does he think they can do that? By changing the rules of the game. Out goes the economy. In comes immigration. Last week Michael Gove set out his vision of post-EU trade. Today he has warned that staying in the EU risks an immigration “free for all.”

There is no doubt that many people in Britain care a great deal about immigration. Many of them will vote to leave the EU because of it. There are also plenty of people who care about sovereignty, the money we spend on EU membership, and the apparently excessive number of laws that emanate from Brussels instead of London. Many of them, too, will vote to leave the EU.

On top of these voters, however, the committed Brexiteers need to persuade millions of floating voters that their argument is the winning argument. If Britain does vote to remain a member of the European Union on 23 June, it will have been the events over the last week that will have made the difference. That’s for the simple reason that this referendum won’t be won on immigration, debates over sovereignty or the nature of decision-making. It will still be about the economy, stupid.