A lot of economists seem to make this aggregation of data mistake and of glorifying results of studies without reading about the underlying assumptions. Levitt and also apparently Prof. Mankiw are apparently applauding the Patriots' for their 'gutsy' fourth down move to go for it on their own 30 yard-line at 4th and 2. They cite a study by Prof. Romer which states that, using historical data, contrary to supposed maximizing behavior, teams are often too conservative on fourth down situations (they punt too much).
First, Romer analyzed 1st quarter attempts, NOT 4th quarter attempts with 2 minutes remaining. Behavior and optimal choices are and should be different in toward the end of a finite game, than early on when one failed risk taking measure might hurt you but not doom you. The risk is substantially greater at the end of the game.
From Romer himself:
"I am implicitly assuming that a team that wants to maximize
its chances of winning should be risk-neutral over points scored. Although
this is clearly not a good assumption late in a game, I show in
Section IV that it is an excellent approximation for the early part.
...direct evidence about the impact of points on the probability
of winning suggests that risk neutrality is an excellent approximation
for the early part of the game. Because teams adjust their play late in
the game on the basis of the score, one cannot just look at the distribution
of actual winning margins."
Second, Romer actually used third downs to estimate what 'would have' happened on fourth downs since fourth down attempts are so rarely attempted. The assumption of course is that third down plays and results would be similar to fourth down plays and results, which I don't necessarily agree with.
Finally, the study is analysis which uses simulation estimation, which means it is wholly dependent on the author's underlying assumptions and judgments of behavior - ie. it is inherently subject to bias. It can model behavior of a 'typical' NFL team based on a host of assumptions, but it can not take into account the specific variables in a specific football game (like Colts V. Patiots).
Make no mistake, Bill Belicek made a dumb move.