If we're going to talk about 'shy Trump' voters, it's worth being clear about what we're really talking about. The old version of the theory just doesn't make sense, but there are others that can't be discounted.
The old version went like this: Trump voters won't tell pollsters the truth because supporting Trump is socially undesirable.
Honestly, it just doesn't add up. Trump doesn't do worse online than live polls. And Biden's >= 50; that's not bc of Trump voters saying they're pro Biden
There was no serious evidence to back this up in 2016, and this version of the tale also just doesn't comport with common sense. There's nothing quiet about the MAGA boat parade vote. And there's no evidence of it in the tabs. He's >90 with Republicans, after all
TLDR: if the race was Biden 45, Trump 42 online, and Biden 45, Trump 37 on phone, and there were a lot of undecided GOPers in one but not the other, you'd have the makings of a real case! I don't see anything like that.
A better version of the argument is that Trump voters just refuse to participate in surveys, all together. There are a variety of forms of this argument: suspicion of the media/polling/establishment; an 'FU' attitude toward volunteerism; higher rates of manual, non-office work
There's at least *something* to this argument: after all, we know that voters without a degree are far less likely to participate in surveys. Who says it's not a particular kind of voter without a degree that's particularly receptive to Trump, even controlling for education?
This argument just can't be dismissed. It is quite obvious that response rates have degraded to the point where polls depend on robust controls; once that's true you just can't guarantee that you've controlled for everything that's material, even if you take a robust approach.
This, to me, is simply not a 'shy' Trump vote. It is a Trump vote that's spitting in the face of the pollster, not one afraid to show off the confederate flag hat that they're wearing. But it would be more dangerous for polling and it would be harder to identify
Now for that same reason, there's just not a ton of evidence that this is a real problem, either. Education + undecided voters + turnout covers a whole lot of what went wrong in '16 and 2018 was good enough for most purposes. But I don't see how you can disprove it, either
One thing that's clear: an apples-to-apples comparison, today, unmistakably shows Biden doing better than Clinton, including among the kind of white, working class voters where we might expect problems. Any issues would almost certainly need to have gotten much worse since '16
It is pretty tough, IMO, to come up with an explanation for how a panel like the Pew ATP or USC (weighting issues aside) would be pretty good in '16, retain many of the same panelists, and be badly biased in '20.
How do you get enough Trump vote in '16, but too much Trump-Biden vote in 20? It would need to be an awfully targeted kind of bias, though I could make up a story where committed Trump voters become disillusioned with polling and drop out while weaker Trump support remains.
This seems fairly testable by the folks at Pew or USC, though even then panel attrition is a real and expected thing and I'm not sure whether we could read something more generalizable into anything they saw. It would still be an interesting data point imo
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