Michael L. Anderson (UC-Berkeley), The Benefits of College Athletic Success: An Application of the Propensity Score Design with Instrumental Variables:

[U]nexpected regular season football victories by NCAA Division I-A
schools increase alumni athletic donations by $134,000. These victories
also increase applications by 1%, and they improve a college's
25th percentile SAT score by 1.8 points.

Anderson uses data on bookmaker spreads to estimate the probability
of winning each football game, and thus to identify unexpected success.
He then estimates the effect of unexpected success on donations and
applications. He suggests that his observed effects likely operate
through one of two channels. First, a team that plays well may be more
enjoyable to watch, and if alumni and prospective students spend more
time watching a college's team, they may feel more connected to the
school. Second, fans and alumni may enjoy winning itself.

Anderson notes that a simultaneous investment of $1 million in every
one of these teams probably would generate smaller effects on donations
and applications than the surprise victories he studies, because team
won/loss records are a zero sum game and improving the level of overall
play would not create any more wins for a given team.

About 8% of the teams in Anderson's sample improve their
season wins by five games over a one-year period. Improvements of that
magnitude increase alumni athletic donations by $682,000 (28%),
applications by 677 (5%), and 25th percentile SAT scores by 9
points (1%).

Here is the abstract:

Spending on big-time college athletics is often justified on the grounds
that athletic success attracts students and raises donations. Testing
this claim has proven difficult because success is not randomly
assigned. We exploit data on bookmaker spreads to estimate the
probability of winning each game for college football teams. We then
condition on these probabilities using a propensity score design to
estimate the effects of winning on donations, applications, and
enrollment. The resulting estimates represent causal effects under the
assumption that, conditional on bookmaker spreads, winning is
uncorrelated with potential outcomes. Two complications arise in our
design. First, team wins evolve dynamically throughout the season.
Second, winning a game early in the season reveals that a team is better
than anticipated and thus increases expected season wins by more than
one-for-one. We address these complications by combining an instrumental
variables-type estimator with the propensity score design. We find that
winning reduces acceptance rates and increases donations, applications,
academic reputation, in-state enrollment, and incoming SAT scores.