With all the hub-bub about the 2 Red Cards for the U.S. team and the overall number of cards that are being presented in this year's tournament, I gathered data from the 2002 and 2006 World Cups and estimated a little instrumental variables model.

The main dependent variable is the number of Red Cards (total) each team received during the tournament finals. There are more matches in the data for the 2002 Cup, so I control for that with a dummy variable. I model the number of red cards as being a linear combination of the number of team yellow cards, the number of matches played, a 2006 dummy varaible and dummy variable for whether the team is from the host country.

Yellow Cards and Red Cards are related to each other by some underlying physicality of play - on the part of the particular team of focus and its opponents. While it is possible for a direct Red Card to be issued, often the Red Cards stem from previous Yellow Cards.

I therefore consider Yellow Cards as an endogenous regressor and model Yellow Cards as being a function of the number of fouls a team commits and the number of fouls committed against a team. It is expected that the more fouls committed, the more Yellow Cards issued. I contend that the impact of Fouls Suffered is ambiguous (although colleague Dennis Wilson suggests the impact should be positive).

Here are STATA results of the two-stage regression. The first stage suggests that the more fouls committed, the more yellow cards issued. In fact, on average over the two tournaments there was one Yellow Card issued for every 5.5 fouls committed, ceteris paribus. A couple of interesting results from the first stage regression: the host team receives 4 fewer Yellow Cards over the course of their participation in the tournament and the more fouls committed against a team the fewer Yellow Cards issued to the victim team, ceteris paribus. This suggests (to me) that referees might let the victim exact some revenge on the pitch without punishing the victim for their redress.

. ivreg2 reds (yellows =foulscomit foulssuff match06) matches yr06 hc, r first

First-stage regressions
-----------------------

First-stage regression of yellows:

OLS regression with robust standard errors
------------------------------------------

                                                      Number of obs =       64
                                                      F(  6,    57) =    12.35
                                                      Prob > F      =   0.0000
Total (centered) SS     =   846.359375                Centered R2   =   0.5672
Total (uncentered) SS   =         5799                Uncentered R2 =   0.9368
Residual SS             =  366.2885346                Root MSE      =    2.535

------------------------------------------------------------------------------
             |               Robust
     yellows |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     matches |   .0851108   1.004385     0.08   0.933    -1.926136    2.096357
        yr06 |   2.726882   4.007503     0.68   0.499    -5.298004    10.75177
          hc |   -4.63538   1.161479    -3.99   0.000    -6.961201   -2.309558
  foulscomit |   .1837463   .0298394     6.16   0.000     .1239939    .2434988
 foulssuffer |  -.0605243   .0319657    -1.89   0.063    -.1245345     .003486
     match06 |  -.1744405   1.222848    -0.14   0.887    -2.623151     2.27427
       _cons |  -.5548464    1.78105    -0.31   0.757    -4.121337    3.011644
------------------------------------------------------------------------------
Partial R-squared of excluded instruments:   0.3870
Test of excluded instruments:
  F(  3,    57) =    13.76
  Prob > F      =   0.0000

The next set of output is basically a battery of tests to determine a) whether there is an endogeneity problem and b) whether the instruments selected, namely the fouls committed and fouls suffered, are good instruments (i.e., they are independent of the shocks to Red Cards). On the surface, one wonders if the number of fouls committed or suffered would be independent of Red Cards. However, the statistical tests suggest they are - there is more to be done here.

Then comes the good stuff. The instrumental variable regression suggests the following. Based on the 2002 and 2006 World Cups, there is one Red Card issued for approximately every 12-13 Yellow Cards. There also seems to be a little home cooking for the host team above and beyond what consideration the team gets from the Yellow Cards issued.

IV (2SLS) regression with robust standard errors
------------------------------------------------

                                                      Number of obs =       64
                                                      F(  4,    59) =     5.86
                                                      Prob > F      =   0.0005
Total (centered) SS     =           33                Centered R2   =   0.2031
Total (uncentered) SS   =           58                Uncentered R2 =   0.5466
Residual SS             =  26.29657278                Root MSE      =     .641

------------------------------------------------------------------------------
             |               Robust
        reds |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     yellows |    .079954   .0378841     2.11   0.035     .0057027    .1542054
     matches |  -.0317426   .1084575    -0.29   0.770    -.2443155    .1808302
        yr06 |   .0989152   .1865216     0.53   0.596    -.2666604    .4644908
          hc |  -.5537637   .2174461    -2.55   0.011    -.9799503   -.1275771
       _cons |   .0132214   .3341736     0.04   0.968    -.6417467    .6681896
------------------------------------------------------------------------------
Anderson canon. corr. LR statistic (identification/IV relevance test):  31.320
                                                   Chi-sq(3) P-val =    0.0000
------------------------------------------------------------------------------
Hansen J statistic (overidentification test of all instruments):         2.849
                                                   Chi-sq(2) P-val =    0.2406
------------------------------------------------------------------------------
Instrumented:         yellows
Included instruments: matches yr06 hc
Excluded instruments: foulscomit foulssuffer match06
------------------------------------------------------------------------------

The Hansen J-statistic is a test for whether the instruments are jointly independent from the Red Cards error term. Fancy talk for whether these instruments can be considered exogneous. The test supports the applicability of the instruments (although I mention that there is more to be done here).

What I found interesting is that the U.S. team received 2 Red Cards on 5 Yellow Cards. Given the estimation results, the U.S. should have received 0.39 Red Cards (round down to zero) and yet received two. Was there bias against the U.S.? Hard to tell without more data. From these two tournaments (2002 and 2006), the 95% confidence interval of the number of Yellow Cards per Red Card is [0.89, 24.12], centered on 12.5. Therefore, this evidence suggests that the U.S. experience was within the 95% confidence interval, but right on the lower bound. My gut tells me there is still statistical bias in the estimation results, and therefore the confidence interval is a bit too high. This would suggest that the U.S. was likely not discriminated against.

Some might quibble with the linear regression model, versus a count data or some other non-linear specification. I plead guilty, but the qualitative results are likely to hold [fingers crossed!!]. With a little time, I might come back with some additional results from non-linear estimation.

[STATA data file]