the dataset is composed of 274 male horses that are at least six years old and have made at 
 least 45 starts .
these two requirements yield horses with a sufficiently extensive career 
 for an analysis of peak performance .
a potential sample selection problem exists because the 
 top male horses often are retired to be breeding stallions well before they satisfy these 
 cutoffs .
therefore , very 
 few male horses are retired to stallion duty , and most have no breeding value and are 
 eventually castrated ( called geldings ) . the horses that meet the criteria for the dataset 
 have no residual breeding value and the majority of them are geldings .
likewise , many female 
 horses have breeding value and therefore are more likely to have shorter racing careers . for 
 this reason we focused exclusively on older male horses .
many of the most prestigious races are restricted to younger horses . the triple 
 crown , consisting of the kentucky derby , preakness , and belmont , is run in the spring and 
 restricted to three - year - olds .
of the 14 races of the breeders cup , four are for 
 two - year - olds .
a precocious horse is rewarded by the high purses in these age - restricted 
 races and often ends up retiring early to stud .
therefore , a concern is whether horses that 
 meet the criteria stipulated for the sample are an adequate representation of the male 
 thoroughbred population .
if so , then the peak age and rate of incline and decline for the 
 sample should be reasonable estimates for the entire population .
summary statistics for the 274 horses in the sample are in table 3table 3.summary statistics for horsesvariablemeanminmaxstarts65.145121age at last start8.04.712.4wins9123earnings$178,914$28,699$1,174,781avg speed figure59.131.088.7st dev speed figure15.38.625.4best speed figure84.550108avg distance ( furlongs)6.915.229.24 .
the age referred to throughout this paper is the horse s actual age rather 
 than the age according to the uniform january 1st birthday .
cool n collective , born march 
 22 , 1997 , is the oldest horse in the dataset at 12.4 years old when he ran at suffolk downs 
 on september 5 , 2009 .
ten horses have started at least 100 times with ten - year - old ww 
 conquistador running the most at 121 .
more than 40% of the horses in the sample have won at 
 least ten times .
ww conquistador also has the most wins with a record of 23 wins in all . 

multiple stakes winner fort prado is the only millionaire in the sample , and his 88.7 speed 
 figure average is the highest for the dataset .
average earnings are $ 178,914 , and the 
 overall mean speed figure is 59.1 .
lower earning horses are typically retired from racing 
 because the owner is nt able to cover variable costs ( training expenses and veterinary 
 bills ) with purse money .
fly by moonlight , an eight - year - old gelding with earnings of 
 $ 170,354 and an average speed figure of 57.8 , both very close to mean values for the 
 dataset , ties for the highest speed figure in the sample .
he earned a figure of 108 when he 
 won by 11 lengths in november 2004 .
orphan brigade also earned a 108 in october 2005 but 
 boasts an average speed figure of 83.2 , well above average for the dataset .
average distance 
 is measured in furlongs , and there are eight furlongs in a mile .
horses compete on three 
 different surfaces : dirt , grass , and a synthetic all - weather surface .
average race distance 
 is 6.91 furlongs with 30% of races at 6 furlongs and fewer than 1% at the classic distance 
 of 1 1/4 mile or farther .
the first is to 
 group horses races by age at the time of the race and compare the speed figure earned in 
 that race at that age to the mean speed figure for that horse s entire career .
a second 
 method is to estimate a system of equations and solve for the age of peak performance .
fair   develops a model for aging effects in 
 baseball that is used here for race horses .
the model assumes that horses have quadratic 
 improvement up to the age of peak performance and experience a quadratic decline in racing 
 ability past that age . the model for horse
i is : where yit is the speed figure at time t , 
 x is the age of the horse at time t , and  is the age of 
 peak performance for the sample .
the two equations have the following constraints to ensure 
 that they come together at  with zero slope : to control for fixed effects , letpjit be 
 a horse specific dummy variable equal to 1 if j = i and 0 
 otherwise . imposing the constraints on equation ( 1 )
reduces the two equations to yit 
 = j=1jpjit 
 + 1[(2xit + 
 xit)dit ] 
 + 2 [ dit + 
 ( xit 
 2xit)(1dit ) ] + 
 it ( 2 ) where dit = 1 if xit   and 0 
 otherwise . to find the correct standard errors ,
table 4table 4.racing performance and distance by age groups horses into 25 categories based on age and shows the average difference 
 between the speed figure a horse earned in a particular race and that horse s career mean 
 speed figure .
for example , 130 horses made 357 total starts when they were less than 2 1/2 
 years old .
speed figures for the horses running in these 357 races average 17.2 points or 
 1.14 standard deviations below their career mean speed figure ( diff and st diff ) .
the speed figure difference 
 is negative but smaller in size through age 3 1/4 years .
every category between 3 1/4 years 
 and 6 1/2 years has a positive difference , indicating that horses typically run faster than 
 their career mean betw een these ages .
the peak difference occurs between ages 4 1/4 and 4 
 1/2 when horses run 5 points faster than their career mean ( two lengths faster for a 6 
 furlong sprint and three lengths faster for a 9 furlong route ) .
1.deviation from mean speed figure by age . provides a graphical representation of speed figure differences across age 
 categories . beyond age 6 3/4
the speed figure differences are once again negative , because a 
 typical horse is past his peak age and is consequently slowing down .
horses older than 9 
 years run an average 7.1 points slower than their career mean and 12.1 points slower than 
 the peak age category . according to the table ,
horses improve by 22.2 points as young 
 two - year - olds to the middle part of their four - year - old year and then decline by 12.1 points 
 over the next five years . in sprint races
this translates to a 9 length improvement to the 
 peak and a 5 length decline . in route races
it is a 13 1/2 length improvement to peak and a 
 7 1/2 length decline .
pct high sf is the percentage of horses that earned their career 
 highest speed figure when they were the age of the corresponding category .
just over 3% of 
 the sample earned their best figure as a two - year - old .
this increases to 23.1% as a 
 three - year - old , 32.9% as a four - year - old , and 21.2% as a five - year - old .
80% of all horses in 
 the sample earned their top speed figure below the age of six .
three horses earned their top 
 speed figure as nine - year - olds : sunshine bear ran a 91 , winning by 14 lengths at 9.3 years 
 of age , messenger springs earned a 73 to win by a length at 9.46 years of age , and my friend 
 jim ran a 63 at 9.94 years of age .
the effect of age on racing distance is studied by looking at the difference and standard 
 difference between the distance of a specific race and the mean distance of all races that 
 the horse competed in .
there is a large increase in racing distance over the course of a 
 horse s two- year - old year . in the spring of the two - year - old year
, horses compete in short 
 sprint races between 4 1/2 furlongs and 6 furlongs .
typically a two - year - old will not run a 
 route race ( 8 furlongs or longer ) until the fall .
therefore , horses less than 2 1/2 years 
 old have the smallest average distance run by a significant margin .
horses in this category 
 run 0.75 furlongs less in distance than their career average .
rk is the difference between the predicted speed figure at age 
 k and the predicted speed figure at the peak performance age . 
 dk is the approximate percentage change in speed figure for 
 a horse at age k. peak age is estimated to be 4.45 years . at age 2 a horse is predicted to be 26 points 
 slower than at peak age .
the improvement over the first two years is dramatic and the 
 subsequent decline is more gradual .
predicted speed figure by age for the mean ability horse 
 is depicted in fig .
 using a dataset of 274 male thoroughbred racehorses in the united states , we study the 
 effect of age on racing performance . 
 beyer speed figures , which are uniform measures of 
 racing performance across distance and racing surface , are utilized in this study . 
 a 
 system of equations is estimated to determine quadratic improvement and decline in racing 
 performance . 
 we find that a typical horse s peak racing age is 4.45 years . 
 the rate of 
 improvement from age 2 to 4 1/2 is greater than the rate of decline after age 4 1/2 . 
 a 
 typical horse will improve by 10 ( horse ) lengths in sprints ( less than 1 mile ) and 15 
 lengths in routes ( one mile or greater ) from age 2 to 4 1/2 . over the next five years 
 the 
 typical decline is 6 lengths for sprints and 9 1/2 lengths for routes . 
Materials and Methods
Results and Discussion