They spend the first part of the paper presenting a formal conditional multifactor model. They then immediately note that "however, the true model in equation (1) is infeasible to examine because the true set of factors is unknown and the true conditional factor loadings are unobservable." Well, it was fun to go over anyway I guess.
Then they develop a factor mimicking portfolio by finding stocks that mimic the changes in the VIX index, and then sorting stocks by the prior correlations with the VIX changes and creating a portfolio of stocks correlated with future VIX changes. They then show, in true Fama-French fashion, that cross-tabbing portfolios formed on sensitivities to the changes in the VIX, are sensitive to the changes in the VIX! But the effect is small, as they estimate the price of volatility risk as -0.08% per month, factor loadings on the pre-sorted portfolios fall by a factor of 100, and adding or removing one August 1998 or October 1987 reverses the sign of the result. So, statistically significant, but not economically significant.
And totally unnecessary because it was motivated by the following, which basically insured the above '25-portfolio-Fama-French' test would work statistically. When they group stocks by low to high total volatility the stocks with the highest volatility had the lowest returns, so the 'price' of volatility is negative. They check robustness by presorting the stocks first by size (market cap), and the effect still shows, They do this for 8 or 9 other things. That's all we needed to know. The early part was less convincing, less intuitive, than the latter part. Good statistics is mainly about knowing what to control for, and these tests make it pretty clear that the obvious alternative hypotheses are not going to explain this. If you know X is correlated with returns in the presence of other variables, the Fama-French test is redundant, and suggests a spurious rigor that is not there.
Then, they get to the explanation, which is solid gold. Remember, higher volatility is correlated, cross sectionally, with lower returns empirically. What rational story might explain this?
our estimate of a negative price of risk of aggregate volatility is consistent with a [unspecified] multifactor model or Intertemproral CAPM. In these settings, aggregate volatility risk is priced with a negative sign because risk-averse agents reduce current consumption to increase precautionary savings in the presence of higher uncertainty about future market returns.
You see, in the Stone Ages we were taught higher risk implies higher return. Modern asset pricing just adds the nuance that higher volatility assets have lower returns, because of the offsetting demand for more savings (huh?). The specific model that generates that is an exercise for the reader, but I guess it could work, especially if we factor in the many worlds interpretation of reality, because I'm told there are universes where the Statue of Liberty waves at tourists when they sail by. But even there, the fanciful interpretation only works if you emphasize the factor loading stuff that was not economically significant, and not very compelling (note, it was not duplicated in their follow-up paper that documented this internationally). It's kind of like how Global Warming causes Global Cooling when you read the above rationalization. The bottom line is that stocks with higher volatility, that perform worst when times are really bad, have lower relative returns.
You can only get away with this kind of enlightened blather if you appear appear 100% serious and rigorous, even though the rigor only occluded the real result of this paper. If you just said, the standard model can't explain this, well, I know what happens to those papers.