##
Wins and Revenue
*March 31, 2014*

*Posted by tomflesher in Baseball, Economics.*

Tags: linear model, Marginal Revenue Product of a win, Revenue, Wins

add a comment

Tags: linear model, Marginal Revenue Product of a win, Revenue, Wins

add a comment

Forbes has released its annual list of baseball team valuations. This is interesting because it accounts for all of the revenue that each team makes, ignoring a lot of the broader factors that play into what causes a team’s value to rise or fall. It also includes a bunch of extra data, including which teams’ values are rising, which are falling, and what each team’s operating income is for the year.

Without getting too in-depth, there are a lot of interesting relationships we can observe by crunching some of the numbers. First, the relationship between wins and revenue is often taken for granted, but the correlation is really very small – only about .26. That means that there’s a great deal more in play determining revenue than just whether a team wins or loses. (This, of course, assumes a linear relationship – one win is worth a fixed dollar amount, and that fixed dollar amount is the same for every team. Correcting this for local income – allowing a win to be worth more in New York than in Pittsburgh – would be an easy extension.)

Under the same assumptions, we can also run a quick linear regression to determine what an average team’s revenue would be at 0 wins and then determine what each marginal win’s revenue product is. Those numbers tell us that, roughly, a 0-win team would make about $129.68 million dollars, gaining around $1.31 million for each win. Again, though, there are a lot of problems with this – obviously, a 0-win team doesn’t exist and would probably have significantly lower revenue than we’d estimate. Even the worst team last year came in at 51 wins. Also, the p-values don’t exactly inspire confidence – the $130 million figure is significant at the 10% level, but the Wins factor comes in around 16%. That’s a pretty chancy number.

Extending it out to include a squared value for wins, we come up with numbers that are astonishingly nonpredictive – the intercept drops to -$34.9 million for a 0-win team (much more reasonable!) with the expected positive marginal value for wins ($5.6 million) and a negative coefficient for squared wins (-$.027), indicating that wins have a decreasing marginal effect as would be predicted. (Once you have 97 wins, the 98th doesn’t usually provide much value.) However, those numbers are basically no better than chance, with respective p-values of .936, .619, and .701. Although the signs look nice, the magnitudes are up in the air.

The sanest model that I can come up with is a log-log regression – that is, starting off with the natural log of revenue and regressing it on the natural log of the number of wins. This gives you an elasticity – a value that explains a percentage change in revenue for a 1% change in the number of wins. This isn’t the most realistic value, of course, since baseball teams play a fixed number of games, but the values look much better – the model looks like:

log(Revenue) = 3.6608 + .4058*log(Wins)

The 3.6608 value is highly significant (p = .00299) and the .4058 coefficient on the number of wins is the strongest we’ve seen yet (p = .1253). It still gives us an unfortunate $38 million operating budget for a zero-win team, but says that doubling a team’s wins should give a 40% increase in revenue. That seems a bit more reasonable.

There are a couple of other, nicer functional forms we could use, but for now, that’s the best we can do with purely linear models.