There's a fight brewing in academia over which political party is better for the stock market

Does the stock market perform better under a Democrat or Republican president? The answer might not be as simple as you'd think.

For years, researchers have found that a Democrat in the White House is correlated with strong excess market returns over the past century. A new paper questions that belief, and suggests that previous results in the literature were the result of reporting bias, data mining and selection bias. Those are serious allegations in the impartial world of academic research.

The most recent evidence showing a correlation between a Democratic president and strong market returns was from earlier this year, when a working paper from two University of Chicago researchers was put out by the National Bureau of Economic Research. Lubos Pastor and Pietro Veronesi's research — which expanded upon a 2003 paper — found that in the period 1927 through 2015, the stock market saw average excess returns of 10.7 percent under Democratic presidents and negative 0.2 percent under Republicans.

Pastor and Veronesi explain that nearly 11 percent difference — the so-called "presidential puzzle" — with a model for time-varying risk aversion through occupational choices and presidential elections. When risk aversion is high, like during economic crises, voters flock to Democratic candidates, they argue. When risk aversion is low, during boom times, voters select the Republican candidate.

Not so fast, say researchers at Research Affiliates, a quantitative and "smart beta" fund manager run by Rob Arnott. The new paper argues that previous studies — including Pastor and Veronesi's — on stock markets under different parties overstate returns under Democratic presidents because of two unique occasions that throw off the average.

Specifically, they point to the economic crashes in 1929 and 2008, both of which occurred during a Republican administration (Herbert Hoover and George W. Bush.) On both occasions, a Democrat was elected following the crash (Franklin Roosevelt and Barack Obama) and was in the White House during much of the economic recovery.

"In statistical studies, it is often easy to overlook the details when examining the broad statistics," Arnott et al wrote. "Had the order of incumbencies been reversed [in those two occasions], the effect would be reversed, suggesting the finding may be serendipitous."

But Arnott's argument misses the point, Pastor said in an email to CNBC. "Removing observations from the sample would be OK if these were random outcomes, but our point is that it is not a coincidence that Democrats got elected in 2008 and 1932," he wrote. "It is not what presidents do that matters, but rather when they get elected."

Like any statistical test, removing observations will change the findings to some degree. Even so, Pastor and Veronesi tested their risk-aversion model on various sub-periods within the greater timeframe of their study. "Returns were higher under Democrats during all three subperiods," Pastor wrote.

The academics' findings were not that Democratic presidents are better for the economy, but that voters elect them when the economy has tumbled and is therefore likely to rebound.

A representative for Arnott and his co-authors said they were not available for comment.