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GuyM Colin: more excellent work. I look forward to future installment(s).
One thought/suggestion: it seems to me that a $0 free agent team is not really a replacement team. A team with an average investment in player development will have a number of above-replacement players in their system at all times who are either "wage slaves" or in their arbitration years. Even if you're including arbitration salaries in your model, the pre-arb players will contribute non-trivial value.
So I think you should estimate the average wins of a zero FA team. I would think it's at least a 60 win team, maybe a bit more. i would think this will give you a higher estimate of revenue per win, one exceeding the cost of those wins (which makes sense).
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mitchiapet I second the "excellent work" comment from Guy above.
By wins, which appears in both of your models, are you using prior season or current season?
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Millsy I'm not so sure that removing the intercept is the best approach. While I agree a 0 win team would probably not see $100 million in revenue, forcing the slope through the origin may cause some problems here. Is there any way you could report the regression with both and intercept and not with one? The R-squared values seem extremely high as well. I think a very very low win team will likely have significant revenue, as has been shown in the Forbes data even with the 2003 Tigers. The Florida Marlins were the lowest revenue team in 2008 and had a payroll uner $20 million.
I'm curious about the coefficients and scale of the variables. Are revenues and population both in millions? That's just a clarification.
Also, the use of the dummy variable for RSN and NEWPARK essentially shift the intercept. Isn't this the same as including a constant? Having no population and no wins, but including the RSN dummy as 1 and having negative revenue is literally impossible (you can have costs, but not negative revenue). The intercepts/constants usually aren't interpretable, but provide a baseline. Considering each team needs one another to exist, and many teams will be very bad...it seems like this model would undershoot the low win teams. I don't have the data, just a possible problem.
Finally, did you attempt any interaction terms with your dummy variables? I would imagine those would be fairly important. Wouldn't wins while having an RSN increase viewership as well? This could be the reason for a negative coefficient on RSN. There is significant investment on the RSN that may not be seen as returned without an interaction term.
Recent history has shown that at least the team valuation numbers have been low-balled in Forbes. It could be that the revenue does the same. Good point about acconting tricks as well. When you do the valuations, remember that owners may depriciate payroll. Given they get a tax break for spending more on a player, they could very well be willing to spend an extra buck or two to shelter their other personal income (this of course would be very difficult to incorporate but may account for some salaries looking high compared to your model).
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Millsy Addendum to above, Florida Marlins had a revenue of $128 million last year.
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Millsy Just meddling through some more stuff. Interesting point about the RSNs. However, the negative coefficient could very well be enhanced given the highly correlated nature of market size and likelihood of having an RSN. This bias could be one reason the p-value isn't significant. You do discuss this early in the article, but it could definitely be investigated since there is the negative coefficient issue that is the exact opposite of what we would expect.
I would suggest looking at a plot of the errors as well. If some of your estimations are high at the low end or low at the high end (or vise versa), this could indicate a problem and cause a coefficient to not be significant. Or perhaps there are correlated errros. Either way, a residual plot would likely give some insight into any problems there.
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Millsy Ah...and one more thing (sorry to take up the board). Are the revenues, populations, and # of households normally distributed? This could have an effect on significance of some of your predictors as well.
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MGL Guy, depends on whether you are including the cost of player develoment in the model. If you are, then certainly you are well-above replacement level. If not, then you are at replacement level, by definition. IOW, I can purchase a team, eventually trim all my payroll, invest zero money in the draft and player development, and simply hire replacement level players at or near min salary.
Now, one of the interesting things to consider is whether it is better to do that (invest no money in the draft and player development) and start with a team of replacement players and work from there OR whether it is better to invest in the draft and player development and see how much that costs me in marginal wins, even if I never invest a dime in FA.
IOW, if I run a team, first I figure out the value in revenues for a marginal win (which is probably a moving target). That is of course the number above which I cannot purchase marginal wins, no matter where they come from. Then I figure out how much it cost me in player development to come up with my own marginal wins. If that number is less than the first one (which it probably is), then I invest in player development, of course. Since I probably get diminsihing returns from that kind of investment, I need to know what limits to set on that. In addition to determining whether and how much I spend on player development, I always need to determine whether also to go into the FA market, based on how much I can purchase a FA marginal win for, compared to the value in revenue of a marginal win for my team only. If I can sometimes or occasionally (or often) buy a FA marginal win for less than the value of that win in marginal revenues, then I may do so.
Of course, in addition to all this, I have to determine the proper balance and magnitude of FA acquisitions and player development investments in order to maximize my net revenue. I also have to look at things in the long-term as what I do now not only effects my net revenue but in the future as well, and that is not such a clear-cut effect.
I find it interesting that monies spent on FA salaries always seems to exceed MRP by a few hundred thousand per win (8-10%). It should be the other way around of course. Why is that? Either teams pay too much for FA, which is possible for a number of reasons, the model has some flaws, or the actual revenue and thus the marginal revenue per win, is understated for a number of reasons.
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Colin Wyers Including the intercept for the estimates of other revenue, I get:
Other_Revenue=87.4759+-0.0931562*WAR+4.96328*Households+1.17828*RSN_DUMMY
Removing the RSN term doesn't remove the negative coefficent from the wins term. Using wins above average or absolute wins doesn't change it, either.
I'm open to other ways to address the issue, but so far the only workable solution I've found was to remove the intercepts.
Revenue data and wins both appear to be normal, or at least close to. Population/households do not.
The residual plots look okay, no extreme outliers. I don't like the spread on the one for Other_Revenues, but the spread for Gate_Receipts looks fine to me. Again, I'm not especially trained in this - if you want to shoot me an e-mail (pontifexexmachina at hotmail.com) I can send you the full results of the regressions with p-values, along with the residual graphs.
I've got a lot of things to mull over here - Guy's point is well-taken as well. I just didn't want to leave things too long without a response here.
What I would be really interested to hear from you, Millsy, is your thoughs on the use of rep-level in the model.
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Millsy Understandable that you can't stick everything in the article. I was just curious if those things could be at the heart of the problem. You do a fine job in explaining the possible problems at the beginning of the article.
As to the replacement level. The economics experience in me tells me that it does not go along with things I know. I have, however, seen well-structured examples in which both having replacement level, and not having replacement level in the real world would seem to make sense. It's interesting that the coefficient on wins does not change using absolute wins. I plan to fiddle with the idea of replacement level with some labor market stuff, but have been insanely bogged down of late. (of course, that doens't stop me from reading baseball blogs online...and making less than completely thought out comments at times). I do believe that what Guy and Studes discuss have to do with replacement level...by signing a FA, you do not need to spend money on development for another player. This would bring the costs of each move closer together. The problem here, though, is assuming that you balanced your risk with development to begin with, you should have taken that into account, rather than have to sign a free agent to save money on a player that was developed (paying Jeff Francouer $4 million bucks or whatnot). But I guess a sunk cost is a sunk cost. It's a fact of baseball development. My guess is that assuming Quad-A player moves are costless is the first problem.
Could it be that population is an overriding factor with these variables? We do know that wins/WAR and population/market size are very collinear. As well as well as whether or not they have an RSN. Like I said before, this could account for the strange signs on the coefficients. These types of regressions are tough given how intermingled all the independent-or not so independent-variables are. Perhaps there are good proxies or instrumental variables that can be used for market size that are uncorrelated with wins (personally, I don't like IV's, especially since they can become outdated, but they may be necessary in a problem such as this). This may be extremely hard, given that what we know about revenues is they're tied to wins, with wins and revenues both tied to market size.
Just more input. Buying wins is not a linear process. The 100th win is going to be much more expensive even at the marginal level than is the 90th win. Why is this the case? I have no clue, but it seems to follow how teams spend their money. I know that you know this as well. That makes it really really difficult to capture it with a linear regression. Have you tried any polynomial on the Win variables?
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studes I agree with Guy, and this is something I've been ranting about for several years (now I sound like Chris Dial!). Clearly, all baseball teams develop players in their farm systems, and to ignore that fact is to model something that has little basis in reality. At one point, I calculated that all the players making the league minimum in a given year played .450 baseball, or something around that, using WSAB.
Of course, that's just a starting place, but I think the level of "no free agent" wins that would reflect reality would be somewhere between .296 (Colin's number) and .450.
Plus, I have to say that I'm a bit confused by the article (I know I'm slow at this stuff). I see that you "created a model for the value of a win above replacement to the average team" and I see a reference to specific variables in the footnote section, but what, exactly did your model do and why should we be surprised (or not) that it fitted Dave's data?
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Colin Wyers I haven't studied this in-depth, and probably should do more, but I didn't find a significant difference in hitting performance between players who make the league minimum, whether or not they're controlled. This isn't much of a study either, and I should probably do one at some point: looking at the VORP page for rookie position players, I added up total VORP and prorated it out per 650 PAs, and came up with 4.22 VORP.
Doing the same with pitcher VORP and you get a bit more of a difference, 11.5 VORP per 180 IP.
So yes, we can probably tweak the rep-level to better fit these circumstances - I chose the one I did because it matched up well with available rep-level metrics. But I don't think the difference is as drastic as .450.
As for the colliniarity issue between market size and wins, I have no idea of what I could do about that. Given that there are only 30 markets, I think it'd be very difficult to find an IV that doesn't correlate with wins.
Quad-A players aren't costless - they aren't free, even when they are freely available - but for the most part, you can't opt out of developing them. Top prospects are different, but teams have to have a minor-league system. Given the minimal investment in player development and the ability to sign free agent substitutes - players who will play for the league minimum even though they aren't controlled - still guarantees a certain performance baseline. I suppose you could apply a similar MRP analysis to a team's spending on player development, but you run into a lot of problems there simply getting the data to use.
Dollars per win
THE BOOK--Playing The Percentages In Baseball —
Colin does more great work. He gives a dollar per win of 3.8 for 2007 and 4.2 for 2008, compared to the 4.0 and 4.4, respectively, I’ve been using. Close enough!
Some Kudos for Colin
GoatRiders of the Apocalypse —
... In case you are not aware, our very own Colin whips out his big rig (his computer) and performs exemplary analyses from time to time on ...
How good is Oakland's 2009 Rotation, With a Cameo By Pythagoras and More Thoughts on Pitcher Value
Driveline Mechanics —
... [NB: All dollar values are millions. I've taken the dollar value of each WAR in 2009 to be $4.62 million on the basis of Colin Wyers' finding that the cost was $4.2 million in 2008 and assuming the general 10% annual inflation of that salary + $0.4 M replacement player salary. It seems too early to say for sure how to adjust for the new economic climate, and it's better to adjust too little than too much...] ...
The Numbers Are In: What Do the Projections Say About the Royals' Rotation in 2009?
Royals Review —
... [NB: All dollar values are millions. I've taken the dollar value of each WAR in 2009 to be $4.62 million on the basis of Colin Wyers' finding that the cost was $4.2 million in 2008 and assuming the general 10% annual inflation of that salary + $0.4 M replacement player salary. It seems too early to say for sure how to adjust for the new economic climate, and it's better to adjust too little than too much... Note also that this just notes the value of the contracts as if they were free agent contracts. to correctly get a read on how "good" they ...
Could Zack Greinke Be Worth His Entire Contract in One Season?
Driveline Mechanics —
... current market price of a marginal win is more difficult to determine. Prior to the 2008-2009 offseason beginning, most people had the dollars per WAR predicted to keep inflating to around $5M given the rate of inflation in free agents salaries in previous years... and then the economy and the free agent market crashed. Long story short, we'll keep things at about the previous season's level, and say that the cost of a marginal win in the 2008-2009 offseason stuck at around $4.5M, as Colin Wyers calculates. ...
Marginal dollar values per run and first-order win, Pt. 1
Marlin Maniac | A Florida Marlins Blog —
... and payroll for the 2008 season. Cameron’s work was just a quick study done near the end of 2008, and more has been done to try and break down revenue based on hitters’, pitchers’, and fielders’ performance. The Baseball Economist by J.C. Bradbury goes in far more detail thatn I’d ever be able to or have the time for, so if you’re really interested in this, you can take a look. Also, check out the explanations, issues, and other things involved (it’s a lot of reading, I haven’t gone through it all, but ...
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