With so many heavyweights, from Barack Obama to Mark Carney, saying that we will be worse off with Brexit, why are the polls still neck and neck? There seem to me two reasonable explanations: that the tabloid media have a strong influence, and that immigration is a big issue among voters. But perhaps the two are connected, for reasons that will become clear.
It is a well known result that worries about immigration tend to be greatest in areas where there is little immigration. In areas where there are a high proportion of migrants, like London, UKIP do rather poorly. For most, immigration is not a problem that is facing them directly, but rather an issue they feel is facing the country.
For some this concern about immigration is cultural, but for others it is economic. But if it is economic, on what is this concern based? All too often I come across arguments that make simple economic errors. Like more migrants put greater pressure on public services. Study after study suggest exactly the opposite: because migrants tend to be young adults who work, they pay more in taxes and take less advantage of public services or benefits than the average non-migrant. To his great credit, when Jonathan Portes (from the non-aligned research institute NIESR) was confronted in this Newsnight debate by nonsense from someone from the Centre for Policy Studies (right wing think tank, rated D for funding openness), he did not attempt to win the argument by quoting statistics or academic studies, but by trying to show why what he was saying made common sense once you explain it.
Sometimes it is simply false correlations: austerity has put pressure on public services and the recession and productivity slowdown has held back real wages, but both have happened at a time of high immigration. For a very good and simple explanation of the facts about recent EU immigration, see this LSE analysis. (For those that can access it, here is a similar take from Gemma Tetlow at the FT.) The only area where there might be some negative effect from migration is on the wages of unskilled labour, but even where a negative impact is found it seems to be small as the chart in this post shows. As Portes suggests, this negative impact could be wiped out by positive effects from higher growth and better public finances.
In some sense what we have is very similar to the austerity problem, with the combination of simplistic ideas and non-causal association. It feels right that governments should tighten their belts when households are doing the same, and the ‘clearing up the mess’ idea is reinforced because the deficit went up when Labour was in power. With both austerity and immigration we have a visual media that normally makes no attempt to ‘educate and inform’, and a tabloid media that actively reinforces these mistakes. (If I open the MailOnline as I write this, here is the top story.) We have a governing party that does the same, and in the past an opposition that was reluctant to say that immigration benefits the economy as a whole. 
So the referendum debate amounts to economics versus immigration. But here is a revealing bit of information from the YouGov analysis cited earlier.
“We recently conducted an experiment in which we asked people to imagine how they would vote if they knew Brexit would make them just £100 worse off per year. This instantly changed a neck-and-neck result to a 12 point victory for ‘Remain’. The effect is even stronger among undecided voters, who flip 18 points from veering towards ‘Leave’ to veering strongly towards ‘Remain’ in this scenario.”
It is of course a classic technique economists use to quantify how strongly people feel about an issue, and it suggests the immigration concern is not worth that much to many people. Given that the economic assessments of the costs of Brexit are of the order of at least 10 times £100 a year, the economic argument is key. Which is why it is worrying that the BBC seem to ignore the consensus among academic economists, as expressed in the letter from the 196 (who cannot be accused of being part of the establishment) but instead find time to publish a 'fact check' that is, to put it politely, misleading.
 Labour’s line should be (and occasionally is) to note that recent immigration from the EU has benefited the economy, but not every part of the economy, and government needs to be active in spreading the benefits.