First, this one (pdf) estimates, from a sample of 63 countries, that immigration actually raises GDP per head; a 10% rise in the stock of migrants raises it by 2.2%. This controls, as far as possible for the reverse causality - the tendency for rich countries to attract migrants - and for the correlation between openness to migrants and openness to trade.
There are two possible mechanisms generating this. One is simply that an increased supply of labour reduces inflationary pressures, and so allows real interest rates to stay lower for longer, which encourages economic activity*. The other is that migrant workers might be complements for native ones. If Polish builders cause more houses to get built, British plumbers and electricians can get more work done and earn more. If Latvians cause more fruit to be picked, British market stall holders can sell more. Or if you can see a doctor quicker because of migrant GPs, you can get back to work sooner.
This finding is entirely consistent with the notion that some unskilled native workers lose out from immigration. What it means, though, is that migration is a potential Pareto improvement, in the sense that winners from it can in theory compensate the losers with everyone becoming better off. This provides a justification for the Left’s response to fears about migration - that the way to protect unskilled native workers is not to restrict migration, but to have better redistributive mechanisms.
Our second paper (pdf), however, is slightly more pessimistic. It estimates, from 14 OECD countries, that immigration simply increases raw GDP; native per capita GDP is unaffected by migrants.
This effect, though, is the average of good times and bad times. In normal economic times (when GDP is within 1% of its trend level) immigration actually increases native employment. In recessions, however, it depresses it slightly; for every 10 immigrants, employment expands by only 9, implying that either one immigrant is unemployed, or that he finds work at the expense of a native.
Whether this is an argument for restricting immigration in recessions, or for better counter-cyclical policies, is, however, questionable even in narrow economic terms.
In one important sense, though, both papers agree - that, on average, immigration does no harm to the native economy, although you could easily argue that 2.2% is a small gain.
The problem with the BNP, then, is not merely that they are vicious thugs. It’s that they are plain, factually, wrong.
* I suppose you could use this mechanism to argue that immigrants caused the banking crisis - but only in the same way that rope-makers cause people to hang themselves.