the cost of a concussion

In case you have not turned on a National Football League (NFL) broadcast in the past five years, concussions are becoming an ever-increasing hot button topic.  From 2011 to 2015, there were more than 1,200 formally diagnosed concussions during games and practices.  Yet, with all the talk about the epidemic of head injuries, there is still questions to be answered: the most pertinent to the economist being the effects of the injury on the wage of players.

My goal is to predict the effect of a concussion on future salary.  Before explaining the details of the estimation, here are some pieces of background information that are important to note:

- All players must be under contract before they play in the NFL.
- After four years of NFL service, players are free to negotiate with any of the 32 NFL teams for a contract (known as free agency).
- Anything else?

I compiled data from several sources to get a dataset of salaries from the 2013 to 2016 NFL free agency periods and then performance and concussion events from the 2012 to 2015 NFL seasons. There were 109 quarterback contracts signed in the 2013 to 2016 NFL free agency periods. Since concussion information was available only for the 2012 to 2015 NFL seasons, I considered only the concussion events that occur in a quarterback's final year of his contract. There were nine such occurrences.

I considered only quarterbacks for two reasons. First, there was a relative balance between the number of quarterback hires and the number of teams of the NFL in the sample period: with few substitutes, neither the buyer nor the seller of can have substantial bargaining power over the other. Second, there is a clear measure of productivity for the quarterback position: other positions have a wide menu of duties and/or their contribution is not easily quantifiable.

Consider the individual agent bargaining for his salary with a general manager. These negotiations are bounded by the individual's reservation wage (the minimum value he is willing to take in order to show up for work) and his marginal revenue product (MRP - the player's contribution to the employer's revenue). Denote this boundary as follows:

(1) r_i,t ≤ w_i,t ≤ MRP_i,t

Since the NFL requires players to be under contract before they are eligible to play, the player and the team must negotiate a wage for period t+1 in period t, therefore the agreed upon wage is:

(2) w_i,t+1 = MRP_i,t+1 + rent_i,t+1

where the objective function of both parties is to maximize their respective rents. Note that at the time of the negotiation, MRP and rent are both unknown and therefore some sort of estimated value of future MRP is required, such as:

(3) E(MRP_i,t+1) ≅ E(Performance_i,t+1) = f(Performance_i,t,Age_i,t+1)

where future MRP is approximated by some performance measure(s). Predicting future performance is commonly referred to as estimating the aging curve.

To determine the wage of a player in the next period, I use information on the player at the time of contract negotiations. Combining the concepts of (2) and (3), I run the follow specification using OLS:

(4) ln(wage_i,t+1) = f(Concussion_i,t,ln(wage_i,t),Performance_i,t,Age_i,t+1)

where ln(wage_i,t+1) is the natural logarithm of the average annual salary of the player: the agreed upon wage. The quarterback's win percentage and the passer rating are used as measures of performance in period t. Age is the player's age at the time of signing the new contract and ln(wage_i,t) is the natural logarithm of his current salary. I also consider the year in which the player signed the new contract.

A graphical depiction of the estimation is below. Here I have not yet separately controlled for the concussions in the regression. I plot the fitted values against the actual values. The hollow black circles represent a given quarterback contract. The red circles represent contracts signed following a season in which the quarterback had a concussion. Points above the black 45 degree line indicate the player signed a contract that was below expectations given his observable productivity. It becomes clear that the red dots appear only on or above the black line.

I then estimated the model described in (4). The results of this regression are shown below:

Variable	Coefficient
Concussion	-0.5385***
	(-2.18)
Win%	1.7096***
	(4.25)
Passer Rating	0.0053***
	(2.71)
Previous Salary	0.5182***
	(5.10)
Age	-0.0447*
	(-1.85)
Year FE	Yes
Observations	109
Adjusted R2	0.685
Estimated Concussion Effect	-39.8%
95% Confidence Interval	(-69.8%,-9.8%)

The results are rather shocking - the implied effect of a concussion is found to have a -40% impact on the next salary of the player. However, recall that the above equation (3), and therefore equation (4), would require that the negotiated salary is always at least as great as the player's reservation wage and no greater than his MRP: alternatively stated as equation (1) always holds.

Now consider that there is no feasible bargaining range to satisfy (1). There may be a reason that some players were not offered contracts in the next period and we should be wary of a selection bias. In our sample period, there were 109 quarterback contracts signed but 152 player-contracts had expired. These 152-109 = 43 censored observations were used in addition to the 109 non-censored observations in a two-step Heckman model to remove the potential selection bias. In the first step, I estimate a probit model which predicts if a player will get a new contract using only performance and age. The results of this probit model are used in the second step OLS regression to correct for any potential selection bias. You can read up on this estimation technique here for more information.

Variable	Coefficient
Concussion	-0.5344***
	(-2.15)
Win%	1.7075***
	(4.25)
Passer Rating	0.0060*
	(1.73)
Previous Salary	0.4034
	(5.10)
Age	-0.0263
	(-0.38)
Year FE	Yes
Observations	152
Adjusted R2	0.689
Estimated Concussion Effect	-39.5%
95% Confidence Interval	(-69.5%,-9.6%)

The results of the Heckman model do not change the output therefore the OLS regression likely did not suffer from selection bias. Again, a concussion is found to correlate to a 40% lower salary than would have otherwise been observed but-for the head injury. This is a huge impact but it leads me to several conclusions.

First, although the effect of a concussion is found to be negative and statistically significant, the range of the 95% confidence interval is massive. It is hard to understand what the true effect may be if the range is 60 percentage points. This issue should be corrected with more concussion data becoming available.

Secondly, this may be the first insight that the NFL managers are cautious of concussions. There may be some sort of stigma around a diagnosed concussion wherein a player will miss increasingly more time with each subsequent concussion. This is despite the evidence that individual performance is no different post-concussion compared to pre-concussion.

Lastly, money lost from concussions and injuries may be part of a larger zero-sum game. The teams of the NFL are subject to both salary floors and ceilings - there is a minimum and a maximum amount of money each team must allocate to it's players each year. If one player is paid below their estimated MRP, another player may be given more.

Thoughts? Comments? Thanks for reading.