Donald Trump with Fox News anchor Megyn Kelly, who interviewed the presumptive Republican presidential nominee for her Fox network special “Megyn Kelly Presents.” (Eric Liebowitz/Fox)

One of the unexpected features of the 2016 primary election campaign was Donald Trump’s feuding with Fox News — unexpected because in recent years Fox has been a reliable booster of the Republican party. As with many of Trump’s actions, this fight seemed at first to represent political suicide in the context of contesting a Republican nomination, yet it seemed to have paid off.

Now that the primary season has ended, it will make sense for Fox to move to a general-election footing and give full support to the Republican candidate. Assuming this candidate is Trump, and that there is no serious Republican alternative in the general election, we can expect Fox to supplement its usual anti-Hillary programming with a positive attitude toward Trump. It’s natural to ask how much this will help in the general election. The effect of Fox News is, presumably, something more than zero.

Two smart and thoughtful economists, Ray Fisman and Andrea Prat, write in Slate of two new studies which, in their view, show that Fox News “could easily prove pivotal in deciding the 2016 presidential election.”

Here’s what Fisman and Prat report:

1. A study by economists Stefano DellaVigna and Ethan Kaplan published in 2007 “found a modest impact: Republican vote share was 0.4 to 0.7 percentage points higher in locales with Fox News” during 1996-2000, a period during which Fox was expanding throughout the country so that it was possible to directly compare votes in areas that did or did not receive the cable channel.

2. A newly posted study by Gregory Martin and Ali Yurukoglu “use the differences in audience size that come from Fox’s random positioning in the cable lineup to calculate how much impact the channel had on voting in the 2000, 2004, and 2008 presidential elections. For 2000, they generated an estimated impact pretty close to what DellaVigna and Kaplan calculated a decade before.

But they also found that Fox’s impact took off during the decade that followed: It increased Republican vote share by more than 3.5 percentage points in the 2004 election and more than 6 percentage points by 2008. And then, “one more vote for the Republican nominee is typically one vote less for the Democrats. So an extra 6 percent share for the Republicans means 6 percent less for the Democrats, suggesting Fox News’ support could erase a 12-percentage point Democratic lead in the popular vote.”

According to Pollster’s moving average, Clinton is currently leading Trump by less than 6 percent in the polls, so a swing of 12 percent could indeed decide the election.

I read this all with interest — and skepticism. Where did that 6-percentage-point estimate come from? I clicked through to Martin and Yurukoglu’s article, which begins as follows:

“We measure the persuasive effects of slanted news and tastes for like-minded news, exploiting cable channel positions as exogenous shifters of cable news viewership. Channel positions do not correlate with demographics that predict viewership and voting, nor with local satellite viewership. We estimate that Fox News increases Republican vote shares by 0.3 points among viewers induced into watching 2.5 additional minutes per week by variation in position. We then estimate a model of voters who select into watching slanted news, and whose ideologies evolve as a result. We quantitatively assess media-driven polarization, and simulate alternative ideological slanting of news channels.”

That 0.3 percentage points is a lot less than 6 percentage points. You can, however, get pretty close if you scale up from 2.5 minutes per week to 1 hour per week: .003*60/2.5 = 0.072, and maybe the difference between that and .006 can be explained by rounding. (For example, if the reported 0.3 is a rounded 0.026, then you get .026*60/2.5 = 0.0624, which rounds to 0.06.)

I then searched the Martin and Yurukoglu paper for the 6-percentage-point estimate. I found it in table 14, which reports the estimate that were Fox News to have disappeared in 2008, the Republican candidate would’ve lost 6.3 percentage points of the vote.

President Obama beat John McCain in 2008 by the margin 53 percent to 46 percent, so according to this model, had Fox News that year disappeared (or, I suppose, switched to a politically neutral format), Obama’s electoral margin would’ve been a Reaganesque 59 percent to 40 percent.

I don’t believe it.

What, then, went wrong in this analysis? Lots of little things, many of which are indeed mentioned by Martin and Yurukoglu. First is the extrapolation from 2.5 minutes per week to an hour per week, which assumes a linear effect (no diminishing returns) and also which takes the estimate far from what can be seen directly from the data.

Second, there’s the assumption that a change in Fox News would happen in a vacuum.

Third, there’s uncertainty. I’ll take the authors’ word that these estimates are statistically significant — that is, that one would not see such a pattern from chance alone — but there’s still going to be a lot of variation in these numbers. And in such settings, effect sizes tend to be overestimated. Estimates near zero are discarded and high estimates are reported. We call this the “statistical significance filter.”

There’s an additional problem in the reporting by Fisman and Prat, because they extrapolate the (already exaggerated, I believe) estimate of what would have happened in 2012 if Fox had moved to neutrality, to make a claim about what would happen in 2016 if Fox were to fully throw its support to Trump.

This would be an appropriate comparison if Fox were currently neutral — but that is not the case. Fox has sent both positive and negative messages regarding Trump while being consistently anti-Clinton and generally pro-Republican and anti-Democrat. So Fox has a lot less room to move to the right than would be implied by Fisman and Prat.

I think Martin and Yurukoglu’s paper is interesting and I think they’re admirably careful both in their presentation and their summary. They very appropriately gave their empirical estimate of 0.03 percentage points in the abstract, putting the larger claims deep in the paper with lots of qualifiers. Fisman and Prat were also careful to report that their conclusions were based on a model.

Still, something went wrong in the presentation, because the Slate article reads like this out-of-control 6-percentage-point extrapolation is real.

What do I think about Fox, Trump, and the 2016 election? I agree with Fisman and Prat that the slant of a major news outlet can make a difference. If Fox decides to go full on against Trump (perhaps supporting a third-party candidate or even switching to Clinton; as Fisman and Prat write, who knows what negotiations are going in the background between the various candidates and the Murdoch-affliated network?), this could hurt his chances in the general election. Conversely, were Fox to move to fully support Trump, I’d expect that would help him some small amount — but nothing like 6 percentage points.

It’s possible that Fox could “get Trump elected” in the sense that it’s possible the election could end up superclose, in which case anything could be considered the deciding factor. But I don’t buy the claimed large effect.

For further background, see this 2014 paper by political scientists Dan Hopkins and Jonathan Ladd, who analyze data from a 2000 pre-election poll and find a positive effect of Fox News on support for George W. Bush, but “only on the vote intentions of Republicans and pure independents.” In summarizing this study, Hopkins writes that media influence “fosters political polarization. For Republicans and pure independents, Fox News access in 2000 reinforced GOP loyalties.”